Topological quantum matter with cold atoms
Dan-Wei Zhang, Yan-Qing Zhu, Y. X. Zhao, Hui Yan, Shi-Liang Zhu
AApril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401
To appear in
Advances in Physics
Vol. 00, No. 00, Month 20XX, 1–153
REVIEW ARTICLETopological quantum matter with cold atoms
Dan-Wei Zhang , , Yan-Qing Zhu , , Y. X. Zhao , , Hui Yan , and Shi-Liang Zhu , ∗ Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials,SPTE, South China Normal University, Guangzhou 510006, China ; National Laboratory of Solid State Microstructures and School of Physics, Nanjing University,Nanjing 210093, China ; Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China ; ( April 3, 2019 )This is an introductory review of the physics of topological quantum matter with cold atoms. Topolog-ical quantum phases, originally discovered and investigated in condensed matter physics, have recentlybeen explored in a range of different systems, which produced both fascinating physics findings andexciting opportunities for applications. Among the physical systems that have been considered torealize and probe these intriguing phases, ultracold atoms become promising platforms due to theirhigh flexibility and controllability. Quantum simulation of topological phases with cold atomic gasesis a rapidly evolving field, and recent theoretical and experimental developments reveal that some toymodels originally proposed in condensed matter physics have been realized with this artificial quan-tum system. The purpose of this article is to introduce these developments. The article begins with atutorial review of topological invariants and the methods to control parameters in the Hamiltoniansof neutral atoms. Next, topological quantum phases in optical lattices are introduced in some detail,especially several celebrated models, such as the Su-Schrieffer-Heeger model, the Hofstadter-Harpermodel, the Haldane model and the Kane-Mele model. The theoretical proposals and experimentalimplementations of these models are discussed. Notably, many of these models cannot be directly re-alized in conventional solid-state experiments. The newly developed methods for probing the intrinsicproperties of the topological phases in cold atom systems are also reviewed. Finally, some topologicalphases with cold atoms in the continuum and in the presence of interactions are discussed, and anoutlook on future work is given.
PACS:
Keywords:
Topological matter, Cold atoms, Chern number and topological invariants, Opticallattices, Artificial gauge fields
Contents
1. Introduction 32. Topological invariants in momentum space 62.1. Gauge fields in momentum space 62.2. Quantized Zak phase 72.3. Chern numbers 82.4. Spin Chern number and Z topological invariants 9 ∗ Corresponding author. Email: [email protected] a r X i v : . [ c ond - m a t . qu a n t - g a s ] A p r pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 topological insulators 674.3.5. 3D Chiral topological insulators 694.3.6. Hopf topological insulators 724.3.7. Integer quantum Hall effect in 3D 754.4. Higher and synthetic dimensions 784.5. Higher-spin topological quasiparticles 825. Probing methods 865.1. Detection of Dirac points and topological transition 865.2. Interferometer in momentum space 875.3. Hall drift of accelerated wave packets 905.4. Streda formula and density profiles 925.5. Tomography of Bloch states 935.6. Spin polarization at high symmetry momenta 965.7. Topological pumping approach 985.8. Detection of topological edge states 996. Topological quantum matter in continuous form 996.1. Jackiw-Rebbi model with topological solitons 996.2. Topological defects in Bose-Einstein condensates 1016.3. Spin Hall effect in atomic gases 1057. Topological quantum matter with interactions 1067.1. Spin chains 1067.1.1. Spin-1/2 chain 1062 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401
1. Introduction
Topology is an important mathematical discipline, starting its prosperity in the early partof the twentieth century. It is concerned with the properties of space that are preservedunder continuous deformations, such as stretching, crumpling, and bending, but nottearing or gluing. Topological methods have recently played increasingly important rolesin physics, and it is now difficult to think of an area of physics where topology does notapply. In early development in this field, Paul Dirac used topological concepts to showthat there are magnetic monopole solutions to Maxwell’s equations [1], and Sir RogerPenrose also used topological methods to show that singularities are a generic feature ofgravitational collapse [2]. However, it was not until the 1970’s that topology really cameto prominence in physics, and that was thanks to its introduction into gauge theoriesand condensed matter physics.What we now know as “topological quantum states” of condensed matter may go backto the Su-Schrieffer-Heeger model for conducting polymers with topological solitons inthe 1970’s [3–5] and were encountered around 1980 [6], with the experimental discoveryof the integer [7] and fractional [8] quantum Hall effects (QHE) in the two-dimensional(2D) electron systems, as well as the theoretical discovery of the entangled gapped spin-liquid states in quantum integer-spin chains [9]. Until then, phases of matter have beenlargely classified based on symmetries and symmetries breaking known as the Landauparadigm. The discovery of the “quantum topological matter” made it clear that theparadigm based on symmetries is insufficient, as the quantum Hall phases do not breakany symmetry and would seem “trivial” from the symmetry standpoint.A topological phase is an exotic form of matter characterized by non-local propertiesrather than local order parameters. An early milestone was the discovery by David Thou-less and collaborators in 1982 of a remarkable formula [Thouless-Kohmoto-Nightingale-den Nijs (TKNN) formula] for QHE [10], which was soon recognized by Barry Simon asthe first Chern invariant for the mathematically termed U (1) fiber bundles in topology[11] with an essential connection to the geometric phase discovered by Michael Berry [12].3 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 The identification of the TKNN formula as a topological invariant marked the beginningof the recognition that topology would play an important role in classifying quantumstates. The TKNN result was originally obtained for the band structure of electrons inuniform magnetic fields. In 1988, F. D. M. Haldane realized that the necessary condi-tion for a QHE was not a magnetic field, but broken time-reversal invariance [13]. Heinvestigated a graphene-like tight-binding toy model (now called the Haldane model)with next-nearest-neighbor hopping and averaged zero magnetic field, constructing thefirst model for the QHE without Landau levels. The QHE without Landau levels is nowknown as the quantum anomalous Hall effect or Chern insulator, and is the first topo-logical insulator discovered, although it is one with a broken time reversal symmetry(TRS). D. J. Thouless, J. M. Kostrlitz, and F. D. M. Haldane were awarded the 2016Nobel Prize in physics “for theoretical discoveries of topological phase transitions andtopological phases of matter”.Another major development in this field is the discovery of topological insulators withTRS in 2-4 dimensions [14, 15]. S.-C. Zhang and J. Hu predicated a kind of four-dimensional QHE, which is characterized by the second Chern number [16]. It is thefirst topological insulator with TRS predicted and only recently was experimentally real-ized with ultracold atoms [17]. C. Kane and E. Mele [18, 19] theoretically combined twoconjugate copies of the Haldane model, one for spin-up electrons for which the valenceband has Chern number ± ∓
1. Since the total Chern number of the band vanishes, thereis no QHE. However, they discovered that so long as the TRS is unbroken, the systemhas a previously unexpected Z topological invariant related to Kramers degeneracy. In-dependently, B. A. Bernevig, T. L. Hughes, and S.-C. Zhang [20] predicted the quantumspin Hall effect [21] in quantum well structures of HgCdTe, which is known as a stateof 2D topological insulators, paving the way to its experimental discovery [22]. The 3Dgeneralization of this Z invariant was independently and simultaneously predicted in2007 by three groups [6], which led to the experimental discovery of the 3D time rever-sal invariant topological insulators. The discovery of topological insulators signaled thestart of a wider search for topological phases of matter, and this continues to be fertileground. Since topological quantum numbers are fairly insensitive to local imperfectionsand perturbations, topological protection offers fascinating possibilities for applicationsin quantum technology.Besides topological insulators, topological phases are generalized to topological(semi)metals, such as Weyl and Dirac semimetals in 3D solids [23, 24], and new topolog-ical materials are being discovered and developed at an impressive rate, the possibilitiesfor creating and probing exotic topological phases would be greatly enhanced if thesephases could be realized in systems that are easily tuned. Ultracold atoms with theirflexibility could provide such a platform. In particular, some idealized model Hamiltoni-ans for topological quantum matter, which are unrealistic in other quantum systems, canbe realized with ultracold atoms in optical lattices (OLs). Below, we briefly summarizethe toolbox that has been developed to create and probe topological quantum matterwith cold atoms.i) The lattice structure of a single-particle energy band in a solid is fundamental forsome topological quantum phases. For instance, both the topological insulators proposedby Handane and Kane and Mele exist in a honeycomb lattice, while spin liquid states favora Kagome lattice. Ultracold atoms can be trapped in the potential minima formed by thelaser beams. By changing the angles, wavelengths and polizations of the laser beams, onecan create different lattice geometries. OLs with various geometric structures, such assquare/cubic, triangular, honeycomb, and Kagome lattice, and superlattice structures,have been experimentally realized (see the review on engineering novel OLs [25]). Inaddition, OLs provide convenient ways to control various factors in cold atoms such as4 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 the strength of interatomic interactions, the band structures, the spin composition, andthe levels of disorder more easily than in real crystals.ii) A necessary condition for the QHE or topological insulators with broken TRS is amagnetic field (flux). Although atomic gases are neutral particles, artificial gauge fieldscan be realized for them [26, 27]. Therefore, one can use atomic gases to simulate chargedquantum particles, such as electrons in external electromagnetic fields. Artificial magneticfields for atomic gases have been implemented through several ways: rotating an isotropic2D harmonic trap, generating a space-dependent geometric phase by dressing the atom-light interaction, and suitably shaking an OL. Importantly, the methods based on theatom-light interaction and shaking lattices are well-suited for implementing an artificialgauge field in an OL. Artificial gauge fields, combined with OLs, lead to the realization ofseveral celebrated toy models proposed but unrealistic in condensed matter physics. Forinstance, the Haldane model [28] and the Hofstadter model [29, 30] have been directlyrealized for the first time with ultracold gases.iii) Spin-orbit coupling (SOC) is a basic ingredient for a Z topological insulator withTRS. It can also be realized by a non-Abelian geometric phase due to the atom-laserinteraction. To simulate an SOC of spin-1/2 particles, one can use a configuration wheretwo atomic dressed states form a degenerate manifold at every point in the laser field.When an atom prepared in a state in the manifold slowly moves along a closed trajectory,a non-Abelian geometric phase is accumulated in the wave function, and an SOC is gen-erated if the non-Abelian geometric phase is space dependent. Recently, one-dimensional(1D) and 2D SOCs for bosonic and fermionic atoms have been experimentally createdin the continuum or OLs [31–36], which are the first step towards the simulation of atopological insulator with TRS.iv) The concept of synthetic dimensions offers an additional advantage for the exper-imental exploration of topological states in cold gases. One kind of synthetic dimensionconsists of interpreting a set of addressable internal states of an atom, e.g. Zeeman sub-levels of a hyperfine state as fictitious lattice sites; this defines an extra spatial dimen-sion coined synthetic dimension. Therefore, driving transitions between different internalstates corresponds to inducing hopping processes along the synthetic dimension. In turn,loading atoms into a real N-dimensional spatial OL potentially allows one to simulatesystems of N + 1 spatial dimensions. Synthetic dimensions were recently realized in 1DOLs for investigating the chiral edge states in the 2D QHE [37–39]. Notably a dynamicalversion of the 4D QHE [16, 40, 41] has been experimentally achieved with cold atoms ina 2D optical superlattice with two synthetic dimensions [17].v) Besides the possibility of engineering single particle Hamiltonians, there are sev-eral methods to flexibly tune complex many-body interactions in cold atoms. Strongcorrelation plays important roles for some typical topological quantum matter, such asfractional quantum Hall states and spin liquids. More recently, there has been intenseinterest in the possibility of realizing fractional quantum Hall states in lattice systems:the fractional Chern insulators. The tunability of atomic on-site interactions [42] or long-range dipole-dipole interactions in ultracold dipolar gases [43, 44] opens up the possibilityof realizing various new topological states with strong correlations, including fractionalanyonic statistics, an unambiguous signature of topological phases.vi) Compared with condensed-matter systems, ultracold atoms allow detailed studiesof the relation between dynamics and topology as the timescales are experimentally easierto access. For example, time-dependent OLs constitute a powerful tool for engineeringatomic gases with topological properties. Recently, the identification of non-equilibriumsignatures of topology in the dynamics of such systems has been reported by usingtime- and momentum-resolved full state tomography for spin-polarized fermionic atomsin driven OLs [45–47]. These results pave the way for a deeper understanding of theconnection between topological phases and non-equilibrium dynamics.5 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 vii) Another remarkable advantage for studying topological phases with cold atomsis that the topological invariants can be directly detected in this system. For example,the Chern number has been directly detected by measuring the quantized center-of-mass response [48]. It can also be observed through the Berry-curvature-reconstructionscheme [45, 46] or by measuring the spin polarization of an atomic cloud at highly-symmetric points of the Brillouin zone (BZ) [36]. Furthermore, nontrivial edge statescan be visualized in real space since the high-resolution addressing techniques offer thepossibility of directly loading atoms into the edge states and cold atoms can be visualizedby imaging the atomic cloud in-situ. Momentum distributions and band populations canalso be obtained through time-of-flight imaging and band-mapping, respectively.In this review, we take a closer look at the merger of two fields: topological quantummatter as discussed in condensed matter physics and ultracold atoms. Both are activefields of research with a large amount of literature. For readers interested in more special-ized reviews of quantum simulation with ultracold atoms, we recommend review articles[26, 27, 49–59]. For readers interested in more dedicated reviews on topological phasesin condensed matter, we recommend Refs. [14, 15, 23, 24, 60, 61]. The aim of this re-view is to satisfy the needs of both newcomers and experts in this interdisciplinary field.To cater to the needs of newcomers, we devote Sec. 2 to a tutorial-style introduction totopological invariants commonly used in condensed matter physics, and the more generalintroductions are put in the Appendix A. In Sec. 3, we describe how the Hamiltonianscan be fully engineered in cold atom systems. A reader new to condensed matter physicsor ultracold atomic physics would find these two sections beneficial. In Sec. 4, our em-phasis is on recent theoretical and experimental developments on how to realize varioustopological states (models) or phenomena in different OL systems. In Sec. 5, we introducethe developed methods for probing topological invariants and other intrinsic propertiesof the topological phases in cold atom systems. In Sec. 6 and Sec. 7, we move beyondsingle-particle physics of Bloch bands in lattice systems to describe some quantum matterin the continuum and interacting many-body phases that have topologically nontrivialproperties. Finally, an outlook on future work and a brief conclusion are given.
2. Topological invariants in momentum space
The purpose of this section is to briefly introduce various topological invariants referencedin the following sections. The more general introduction of topological invariants withthe derivations of many formulas in this section are put in the Appendix A.
Gauge fields in momentum space
We denote the momentum-space Hamiltonian of an insulator as H ( k ) with k in the firstBrillouin zone (BZ), and assume finite number of bands, namely H ( k ) is a ( M + N )-dimensional matrix at each k , where M and N are numbers of conduction and valencebands, respectively. At each k , H ( k ) can be diagonalized and the conduction and valenceeigenpairs are ( E + ,a , | + , k , a (cid:105) ) and ( E − ,b , |− , k , b (cid:105) ), respectively, with a = 1 , · · · , M and b = 1 , · · · , N . At each k , valence states |− , k , b (cid:105) span an N dimensional vector spaceand these vector space spread smoothly over the whole BZ. We can define the Berryconnection (gauge potential) as A µb,b (cid:48) ( k ) = (cid:104)− , k , b | ∂∂k µ |− , k , b (cid:48) (cid:105) (1)6 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 with µ = 1 , , · · · , d labeling momentum coordinates. Accordingly, the Berry curvature(gauge field strength) is given by F µν = ∂ µ A ν − ∂ ν A µ + [ A µ , A ν ] . (2)To have a basic idea of the Berry connection and curvature in momentum space, wetake a general two-band model as an example. The Hamiltonian reads H b ( k ) = d ( k ) · σ, (3)where σ i with i = 1 , , (cid:15) ( k )1 should also be added into Eq. (3). But it is ignored here because it is irrelevant tothe topology of the band structure, noticing that it only shifts the energy spectrumand does not affect eigenstates. As the spectrum is given by E ± ( k ) = ±| d ( k ) | , forinsulator | d ( k ) | is not equal to zero for all k . The valence eigenstates can be representedby |− , k (cid:105) = e − iσ φ ( k ) / e − iσ θ ( k ) / | ↓(cid:105) , where θ ( k ) and φ ( k ) are the standard sphericalcoordinates of ˆ d ( k ) ≡ d ( k ) / | d ( k ) | , and | ↓(cid:105) is the negative eigenstate of σ . The Berryconnection can be straightforwardly derived as A µ ( k ) = i θ ( k ) ∂ µ φ ( k ) . (4)Under the U (1) gauge transformation |− , k (cid:105) → e iϕ ( k ) |− , k (cid:105) , the Berry connection A µ ( k )is transformed to be A µ ( k ) + i∂ k µ ϕ ( k ). But the Berry curvature is invariant under gaugetransformations, and is given from Eq. (2) by F µν ( k ) = − i sin θ ( k )[ ∂ µ θ ( k ) ∂ ν φ ( k ) − ∂ ν θ ( k ) ∂ µ φ ( k )], which can be recast in terms of ˆ d ( k ) as F µν ( k ) = 12 i ˆ d · ( ∂ µ ˆ d × ∂ ν ˆ d ) . (5) Quantized Zak phase
The simplest example of topological invariant in momentum space is the so-called quan-tized Zak phase. The Zak phase is a Berry’s phase picked up by a particle moving acrossa 1D BZ [62]. For a given Bloch wave ψ k ( x ) with quasimomentum k , the Zak phase canbe conveniently expressed through the cell-periodic Bloch function u k ( x ) = e − ikx ψ k ( x ): γ = i (cid:90) G/ − G/ A k dk, (6)where the gauge potential in Eq. (1) is given by A k = (cid:104) u k | ∂ k | u k (cid:105) and G = 2 π/a is thereciprocal lattice vector and a is the lattice period. As i∂ k in Eq. (6) is the positionoperator, physically γa/ (2 π ) is just the center of the Wannier function corresponding to u k ( x ). Accordingly, it is noticed that the Zak phase γ , Eq. (6), is well defined module2 π , because a shift of the lattice origin by d , which corresponds to u k ( x ) → e ikd u k ( x ),changes Eq. (6) by 2 πd/a . So the Zak phase γ can be any real number mod 2 π , andtherefore is not a topological invariant. However, certain symmetries can quantize it intointegers in units of π . The quantization of Eq. (6) was first discussed in 1D band theoryby Zak taking into account the inversion symmetry [62]. In order to preserve the inversionsymmetry, the wanner-function center has to be either concentrated at lattice sites or atthe midpoints of lattice sites. 7 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 A paradigmatic 1D model with the topological invariant being the Zak phase is pro-vided by the Su-Schrieffer-Heeger model of polyacetylene [3], which exhibits two topolog-ically distinct phases. A unit cell in this model has two sites with sublattice symmetry,which quantizes the Zak phase. Accordingly the cell-periodic wave function u k can beviewed as a two-component spinor u k = ( α k , β k ), and the Zak phase, Eq. (6), in units of π takes an simple form ν Z = ( i/π ) (cid:82) G/ − G/ ( α ∗ k ∂ k α k + β ∗ k ∂ k β k ) dk . Chern numbers
The second example of the topological invariants in momentum space is the famousChern number, which can be formulated for any even-dimensional spaces. For 2 n di-mensions, the corresponding Chern number is called the n th Chern number, and thecorresponding integrand is called the n th Chern character. We first introduce the firstChern number (conventionally called Chern number), which appears in 2D momentumspace k = ( k x , k y ). The BZ forms a torus T and the Chern number for a 2D insulatoris given as C = i π (cid:90) T d k tr F xy . (7)Noticing that the trace over the commutator in Eq. (2) vanishes, we find that the Chernnumber essentially comes from the Abelian connection a µ = tr A µ , which is just the sum ofthe Abelian Berry connection of all valence bands, namely, that a j = (cid:80) α (cid:104) k , α | ∂/∂k j | k , α (cid:105) with α labeling the valence bands. Accordingly the Chern number of Eq. (7) can berewritten in terms of the Abelian connection as C = i π (cid:90) (cid:90) BZ f xy ( k ) dk x dk y , f xy ( k ) = ∂a y ( k ) ∂k x − ∂a x ( k ) ∂k y , (8)The Chern number of Eq. (7) is also called the Thouless-Kohmoto-Nightingale-den Nijs(TKNN) invariant, which was shown to be the transverse conductance in units of e /h using the Kubo formula, and therefore is the topological invariant to characterize theinteger quantum Hall effect [10]. A nonvanishing transverse conductance requires theTRS breaking, which is consistent with Eq. (7), since i F is odd under TRS. For thetwo-band model of Eq. (3), the Chern number can be expressed explicitly by C = 14 π (cid:90) (cid:90) BZ ˆ d · ( ∂ k x ˆ d × ∂ k y ˆ d ) dk x dk y , (9)which can be derived by directly substituting Eq. (5) into Eq. (8).If n = 2, the second Chern number for a 4D insulator is given by C = − π (cid:90) T d k (cid:15) µνλσ tr F µν F λσ . (10)For more than one valence bands, the second Chern number, Eq. (10), cannot be ex-pressed in terms of the Abelian Berry connection a µ ( k ), which is in contrast to the firstChern number, and therefore is essentially non-Abelian. It was predicted that the secondChern number C can be used to characterize a quantum Hall effect in 4D space [16],which was realized in a recent experiment with ultracold atoms loaded in an optical latticewith synthetic dimensions [17]. Furthermore, in contrast to that all systems with TRShave vanishing first Chern number, the second Chern number of Eq. (10) can preserve8 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 TRS, namely, that there exist nontrivial time-reversal-invariant 4D Chern insulators. Inaddition, the meaning of the second Chern number for electromagnetic response can befound in Refs. [63, 64].Let us consider isolated gap-closing points in a (2 n + 1)D BZ, where the Berry con-nection is not well-defined. Although the Berry connection is singular at any gap-closingpoint, a (2 n )D sphere S n can be chosen to enclose it, restricted on which the spectrumis gapped with the well-defined Berry connection. Accordingly the Chern number can becalculated on the S n , and is referred to as the monopole charge of the singular point.For monopoles in 3D space, the monopole charge can be calculated by the Abelian Berryconnection a µ = tr A µ , and therefore are termed as Abelian monopoles. For instance theWeyl points described by the Hamiltonian H W ( k ) = ± k · σ can be interpreted as unitAbelian monopoles in momentum space for the respective Abelian gauge field of valenceband restricted on S surrounding the origin. The monopole charges defined in higherdimensions are introduced in the Appendix. Spin Chern number and Z topological invariants We further consider particles with spin-1/2 (or pseudo-spin-1/2) in 2D momentum space.If the U (1) spin-rotation symmetry to any specific direction (denoted as z -direction here)is preserved, the corresponding spin polarization s = ↑ , ↓ is a good quantum number, andtherefore the notation | k , α (cid:105) of valence bands used above should be refined as | k , α, s (cid:105) .Then each spin s can be individually assigned a Chern number C s as that of Eq. (8),which is the sum of the Chern numbers of all valence bands with the correspondingspin s and naturally integer valued. As a topological insulator it is now characterized bytwo topological indices, the usual Chern number C and the spin Chern number C s [65],respectively given by C = C ↑ + C ↓ , C s = ( C ↑ − C ↓ ) / . (11)Provided TRS is preserved (thus C = 0) as well as the U (1) spin-rotation symmetry, thespin Chern number C s is also integer valued. In this case C s can be used to characterizedthe quantum spin Hall effect [65].Notably, the U (1) spin-rotation symmetry can usually be broken by generic spin-orbitalcouplings and therefore is not a good symmetry, but TRS is still preserved in the absenceof magnetic field. In the general situation with only TRS, the spin Chern number C s isno longer well-defined and should be replaced by a Z topological invariant for character-izing the 2D topological insulators with TRS [19–21], which was first proposed by Kaneand Mele in Ref. [18]. The Z topological invariants proposed there can be generalizedto characterize 3D time-reversal-invariant topological insulators. These Z topologicalinvariants are briefly introduced in the Appendix A. The Hopf invariant
There is a kind of topological insulator restricted in both two bands and three dimensions.For a two-band insulator, the Hamiltonian (3) at each k can be topologically regardedas a point ˆ d ( k ) on a unit sphere S , and thereby it gives a mapping from the 3D BZto S . Because of the homotopy group π (S ) ∼ = Z, there exist (strong) 3D two-bandtopological insulators with Z classification, which is termed the Hopf insulators [66]. The9 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 corresponding topological invariant is called the Hopf invariant [66, 67], and is given by ν H = − π (cid:90) T d k (cid:15) µνλ A µ ∂ ν A λ , (12)where A µ = (cid:104)− , k | ∂ k µ |− , k (cid:105) is the Berry connection of the valence band defined in Eq. (1).
3. Engineering the Hamiltonian of atoms
For particles of mass m and index i , charge q and magnetic moment µ B , in an electro-magnetic field described by the vector potential A = ( A x , A y , A z ) and scalar potential V ( r ), the Hamiltonian is given by H = (cid:88) i (cid:20) m ( p i − q A ) + V ( r i ) − µ B · B ( r i ) (cid:21) + U int , (13)where p i = − i (cid:126) ∇ i is the momentum operator, B is the magnetic field, and U int is theHamiltonian caused by the interaction between particles. One of the great advantagesof ultracold atomic systems is that, all terms in the Hamiltonian (13) are tunable inexperiments, and thus many exotic quantum phases, including various topological phasesaddressed latter in this review, can be realized. In this section, we first briefly review themethods to modify the mean kinetic energy (cid:104) p i (cid:105) / m related to the temperature and theinteraction U int , and then address more detailed the approaches to engineer the so-calledartificial gauge fields for neutral atoms (the vector potential A , the scalar potential V ( r i ),and the effective Zeeman field B ( r i )), which are fundamentally important in creatingvarious exotic topological phases. Laser cooling
The mean kinetic energy of the atoms (cid:104) p i (cid:105) / m is mainly determined by the temperatureof the atomic cloud and can be controlled by laser cooling, which refers to a number oftechniques in which atomic samples are cooled down to near absolute zero. Laser coolingtechniques rely on the fact that when an atom absorbs and re-emits a photon its mo-mentum changes. For an ensemble of particles, their temperature is proportional to thevariance in their velocities. That is, more homogeneous velocities among particles cor-responds to a lower temperature. Laser cooling techniques combine atomic spectroscopywith the mechanical effect of light to compress the velocity distribution of an ensemble ofparticles, thereby cooling the particles. A Nobel prize was awarded to three physicists, S.Chu, C. N. Cohen-Tannoudji, and W. D. Phillips, for their achievements of laser coolingof atoms in 1997.The first proposal of laser cooling by H¨ansch and Schawlow in 1974 [68] was based uponDoppler cooling in a two-level atom. It was suggested that the Doppler effect due to thethermal motion of atoms could be exploited to make them absorb laser light at a differentrate depending on whether they moved away from or toward the laser. Consider an atomirradiated by counterpropagating laser beams that are tuned to the low frequency side ofatomic resonance. The beam counterpropagating with the atom will be Doppler shiftedtowards resonance, thus increasing the probability of photon absorption. The beam co-propagating with the atom will be frequency-shifted away from resonance, so there willbe a net absorption of photons opposing the motion of the atom. The net momentum kickfelt by the atom could then be used to slow itself down. By surrounding the atom with10 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 three pairs of counter-propagating beams along the x , y and z axes, one can generate adrag force opposing the velocity of the atom. The term ”optical molasses” was coined todescribe this situation.When this simple principle was finally applied in the early 1980s, it immediately ledto low temperatures only a few hundreds of micro-Kelvins above absolute zero. As anexample, the mean velocity of a Rb atomic gas in temperature of 100 mK is about0 .
17 m/s, which is much slower than the velocity of several hundred meters per secondat room temperature. Ultracold atoms also turned out to be an ideal raw material forthe realization of magnetic traps for neutral atoms. Held in place by magnetic dipoleforces, such atomic gases can then be evaporatively cooled by successively lowering thetrap depth, thus letting the most energetic atoms escape and allowing the remainingones to rethermalize. In this way, the fundamental limitations of laser cooling due tophoton scattering can be overcome and the temperature as low as a few nano-Kelvinscan be reached. The mean velocity of a Rb atomic gas in temperature of 1 nK is about5 . × − m/s. Effective interactions
The term U int in Eq. (13) is induced by interatomic interactions and can be manipulatedwith a powerful method called the Feshbach resonance (for a review, see Ref. [42]). Thefundamental result for the atom-atom scattering is that under appropriate conditions,the effective interaction potential U int ( r = r i − r j ) of two atoms (particle indices i and j )of reduced mass m r can be replaced by a delta function of strength 2 π (cid:126) a s /m r , where a s is the low-energy s -wave scattering length. As for two similar particles with mass m ,the commonly quoted form of the effective interaction is U int ( r ) = 4 πa s (cid:126) m δ ( r ) . (14)Alternative, it can be understood in the following way: the mean interaction energy ofthe many-body system is given by the expression (cid:104) E int (cid:105) = 12 4 πa s (cid:126) m (cid:88) ij | Ψ( r ij → | , (15)where Ψ is the many-body wave function and the notation r ij → r ij of the two atoms, while large compared to a s , is small compared to anyother characteristic length (e.g., thermal de Broglie wavelength, interparticle spacing,etc). The conditions necessary for the validity of Eq. (15) in the time-independent case arethe following: First, the orbital angular momentum l (cid:54) = 0 scattering must be negligible.Second, the existence of the limit r ij → k c a s (cid:28)
1, where k c is thecharacteristic wave-vector scale of the many-body wave function Ψ (for a very generalargument, see Ref. [69]).The scattering length a s can be manipulated by a Feshbach resonance [42]. It oc-curs when the bound molecular state in the closed channel energetically approaches thescattering state in the open channel. Then even weak coupling can lead to strong mix-ing between the two channels. The energy difference can be controlled via a magneticfield when the corresponding magnetic moments are different. This leads to a magnet-ically tuned Feshbach resonance. The magnetic tuning method is the common way toachieve resonant coupling and it has found numerous applications. A magnetically tunedFeshbach resonance without inelastic two-body channels can be described by a simple11 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 expression, introduced by Moerdijk et al. [70], for the s-wave scattering length a s as afunction of the magnetic field strength B , a s ( B ) = a (cid:18) − ∆ rw B − B (cid:19) . (16)The background scattering length a represents the off resonant value. The parameter B denotes the resonance position, where the scattering length diverges ( a s → ±∞ ),and the parameter ∆ rw is the resonance width. Note that both a and ∆ rw can bepositive or negative, thus the interaction energy U int can be positive or negative andeven infinity by just controlling the magnetic field strength B . Alternatively, resonantcoupling can be achieved by optical methods, leading to optical Feshbach resonanceswith many conceptual similarities to the magnetically tuned case. Such resonances arepromising for cases where magnetically tunable resonances are absent. Dipole potentials and optical lattices
The dipole potentials.
The potentials V ( r ) in Eq. (13) can be manipulated with the laserbeams. As for the topological band structures reviewed in this paper, we are particularlyinterested in OLs formed by the light-atom interactions. OLs and other optical trapswork on the principle of the ac Stark shift. In order to understand the origin of light-induced atomic forces and their applications in laser cooling and trapping it is instructiveto consider an atom oscillating in an electric field. When an atom is subjected to a laserfield, the electric field E induces a dipole moment p d in the atom as the protons and sur-rounding electrons are pulled in opposite directions. The dipole moment is proportionalto the applied field, p d = α ( ω ) E , where the complex polarizability of the atoms α ( ω )is a function of the laser light’s angular frequency ω . The potential felt by the atoms isequivalent to the ac Stark shift and is defined as V ( r ) = − (cid:104) p d · E (cid:105) = − α ( ω ) (cid:104) E ( t ) (cid:105) , (17)where the angular brackets (cid:104)·(cid:105) indicate a time average in one cycle.For a two-level atomic system, away from resonance and with negligible excited statesaturation, the dipole potential can be derived semiclassically. To perform such a calcu-lation, the polarizability is obtained by using Lorentzs model of an electron bound to anatom with an oscillation frequency equal to the optical transition angular frequency ω .The natural line width has a Lorentzian profile as the Fourier transform of an exponen-tial decay is a Lorentzian. Then the dipole potential calculated by the two-level modelis given as V ( r ) = − πc Γ2 ω (cid:18) ω − ω + 1 ω + ω (cid:19) I ( r ) , (18)where Γ is the natural line width of the excited state and has a Lorentzian profile, and I ( r ) = (cid:15) c | E ( r ) | / r . For small detuning ∆ d = ω − ω and ω/ω ≈
1, the rotating wave approximation can be made and the 1 / ( ω + ω ) termin Eq. (18) can be ignored. Under such an assumption, the scale of the dipole potential V ( r ) ∝ I ( r ) / ∆ d . Therefore, a blue-detuned laser (i.e., the frequency of the light field islarger than the atomic transition frequency (∆ d > pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 on the atom, points in the direction of decreasing field. On the other hand, an atom willbe attracted to the red-detuned (∆ d <
0) regions of high intensity.
Optical lattices.
A stable optical trap can be realized by simply focusing a laser beamalong the z direction to a waist of size w under the red-detuned condition. If the crosssection of the laser beam is a Gaussian form, with w and z R = w π/λ being the spot(waist) and Rayleigh lengths, respectively, the resulting dipole potential is given as V ( r, z ) = V exp (cid:32) − r w (cid:112) z/z R ) (cid:33) , (19)where the trap depth V = I p / ∆ d with I p being the peak intensity of the beam. Expandingthis expression at the waist z = 0 around r = 0, we obtain that in the harmonic ap-proximation the radial trap frequency in such a potential is given by ω ⊥ = (cid:112) V /m/w .Besides this radial trapping force, there is also a longitudinal force acting on the atoms.However, this force is much less than the radial one owing to the much larger length scalegiven by the Rayleith length z R . To confine the atoms tightly in all spatial directions,one can use several crossed dipole traps or superpose an additional magnetic trap.The possibility to create dipole potentials proportional to the laser intensity allowsfor the creation of OL potentials from standing light waves [71], as artificial crystals oflight to trap ultracold atoms. As an example, we first address how to realize a 1D latticecreated by two counterpropagating laser beams with wave vectors k L and − k L . Weconsider two identical laser beams of peak intensity I p and make them counterpropagatein such a way that their cross sections completely overlap. In addition, we also arrangetheir polarizations to be parallel. In this case, the two beams can create an interferencepattern, with a distance λ L / λ L = 2 π/k L and k L = | k L | ) between two maxima orminima of the resulting light intensity. Therefore, the potential seen by the atoms issimply given by V lat ( x ) = V cos ( πx/d ) , (20)where the lattice spacing d = λ L / V is the lattice depth.Note that mimicking solid-state crystals with an OL has the great advantage that,in general, the two obvious parameters in Eq.(20), the lattice depth V and the latticespacing d can be easily controlled by changing the laser fields. Rather than directly cal-culating the lattice depth V from the atomic polarizability in Eq. (17), one typicallyuses the saturation intensity I of the transition and obtains V = η (cid:126) Γ Γ∆ I p I , where theprefactor η of the order unit depends on the level structure of the atom in questionthrough the Clebsh-Gordan coefficients of the various possible transitions between sub-levels. Thus, the lattice depth V is proportional to the laser intensity I p , which can beeasily controlled by using an acousto-optic modulator. This device allows for a preciseand fast (less than a microsecond) control of the lattice beam intensity and introduces afrequency shift of the laser light of tens of MHz. Typically, the lattice depth is measuredin units of the recoil energy E R = π (cid:126) / (2 md ), and often the dimensionless parameter s = V /E R is used. It corresponds to the kinetic energy required to localize a particle onthe length of a lattice constant d . Recoil energies are of the order of several kilohertz,roughly corresponding to microkelvin or several picoelectron volts. The lattice depth cantake values of up to hundreds of recoil energies. On the other hand, the lattice spacing d = λ L / θ < π . Assuming that the polarizationsof the two beams are perpendicular to the plane spanned by them, this will give rise toa periodic potential with lattice constant d ( θ ) = d/ cos( θ/ ≥ d .13 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 In experiments, a 1D OL can be created in several ways. The simplest way is to take alinearly polarized laser beam and retro-reflect it with a high-quality mirror. If the retro-reflected beam is replaced by a second phase-coherent laser beam, which can be obtainedby dividing a laser beam in two with a polarized beam splitter, we can introduce afrequency shift δν L between the two lattice beams. The periodic lattice potential will nowno longer be stationary but move at a velocity v lat = δν L d . If the frequency difference isvaried at a rate δ ˙ ν L , the lattice potential will be accelerated with a lat = δ ˙ ν L d . Therefore,there will be a force F = ma lat = mδ ˙ ν L d , acting on the atoms in the rest frame of thelattice. We shall see latter that this gives a powerful tool for manipulating the atoms inan OL.A superlattice or disordered lattice can be realized with two pairs of counterpropa-gating beams. We consider two counterpropagating beams, where the polarizations areperpendicular and the wave vectors are k L and k L , respectively. In this case, each paircan form a lattice which is similar to that of Eq. (20), and the resulting total potentialis then given by V lat ( x ) = s E R cos ( πx/d ) + s E R cos ( πx/d ) , (21)where d j = π/k Lj ( j = 1 , s and s measure the height of the lattices in units ofthe recoil energies. A superlattice with the period pq is created when the ratio d /d = p/q (with p, q being integers) is a rational number. For instance, a dimerized latticewith two sites per unit cell is realized when d /d = 1 /
2, which is the famous Su-Schrieffer-Heeger model with a topological band structure (see Sec. 4.1.1). On the otherhand, a disordered lattice can be formed when the ratio d /d is an irrational number.Especially, when s (cid:28) s the disordering lattice has the only effect to scramble theenergies, which are nonperiodically modulated at the length scale of the beating betweenthe two lattices (2 /λ L − /λ L ) − with λ Lj = 2 π/k Lj . Theoretical and experimental workshave demonstrated that in finite-sized systems this quasi-periodic potential can mimic atruly random potential and allow the observation of a band gap [72, 73]. Alternatively,for a system of ultracold atoms in a lattice one can introduce controllable disorders byusing laser speckles [74].By combining standing waves in different directions or by creating more complex in-terference patterns, one can create various 2D and 3D lattice structures. To create a 2Dlattice potential for example, one can use two orthogonal sets of counter propagatinglaser beams. In this case the lattice potential has the form V lat ( x, y ) = V [cos ( k L x ) + cos ( k L y ) + 2 (cid:15) · (cid:15) cos φ cos( k L x )cos( k L y )] , (22)where (cid:15) and (cid:15) are polarization vectors of the counter propagating set and φ is therelative phase between them. In derivation of this equation, we have assumed that the twopairs of laser beams have the same wave vector magnitude k L and the same laser density I p . A simple square lattice can be created by choosing orthogonal polarizations betweenthe standing waves. In this case the interference term vanishes and the resulting potentialis just the sum of two superimposed 1D lattice potentials. Even if the polarization of thetwo pair of beams is the same, they can be made independent by detuning the commonfrequency of one pair of beams from that the other. A more general class of 2D latticescan be created from the interference of three laser beams [25, 75–77], which in generalyield non-separable lattices. Such lattices can provide better control over the number ofnearest-neighbor sites and allow for the exploration of richer topological physics, suchas the honeycomb lattices for the Haldane model or Kane-Mele model. Moreover, 3D14 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 lattices can be created with more laser beams. For example, a simple cubic lattice V lat ( x, y, z ) = V [cos ( k L x ) + cos ( k L y ) + cos ( k L z )]can be formed with three orthogonal sets of counter propagating laser beams when theyhave the same wave vector magnitude k L and the same laser density I p , but have orthog-onal polarizations. The tight-binding Hamiltonian.
A useful tool to describe the particles in OLs is thetight-binding approximation. It deals with cases in which the overlap between localizedWannier functions at different sites is enough to require corrections to the picture ofisolated particles but not too much as to render the picture of localized wave functionscompletely irrelevant. In this regime, one can only take into account overlap betweenWannier functions in nearest neighbor sites as a very good approximation. Wannierfunctions are a set of orthonormalized wave functions that fully describes particles ina band that are maximally localized at the lattice sites. They can form a useful basisto describe the dynamics of interacting atoms in a lattice. Furthermore, if initially theatoms are prepared in the lowest band, the dynamics can be restricted to remain in thisband. In the absence of the gauge potential and Zeeman field, the Hamiltonian in Eq.(13) for the interacting particles in OLs is given by H = − (cid:126) m (cid:88) i ∇ i + V lat ( x ) + V ( x ) + U int , (23)where V lat ( x ) is the periodic lattice potential, and V ( x ) denotes any additional slowly-varying external potential that might be present (such as a harmonic confinement usedto trap the atoms). In the grand canonical ensemble, the second-quantized Hamiltonianreads H = (cid:90) Ψ † ( x ) (cid:20) (cid:126) m ∇ + V lat ( x ) + V ( x ) + U int − µ (cid:21) Ψ( x ) d x , (24)where Ψ † ( x ) is the bosonic or fermionic field operator that creates an atom at the position x , and µ is the chemical potential and acts as a Lagrange multiplier to the mean numberof atoms in the grand canonical ensemble.We first consider the noninteracting situation. For sufficiently deep lattice potentials,the atomic field operators can be expanded in terms of localized Wannier functions.Assuming that the vibrational energy splitting between bands is the largest energy scaleof the system, atoms can be loaded only in the lowest band, where they will reside undercontrolled conditions. Then one can restrict the basis to include only lowest band Wannierfunctions w ( x ), i.e., Ψ( x ) = (cid:80) j a j w ( x − x j ), where a j is the annihilation operator atsite j which obeys bosonic or fermonic canonical commutation relations. The sum is takenover the total number of lattice sites. If Ψ( x ) in this form is inserted in Eq. (24), and onlythe tunneling processes between nearest neighbor sites are kept (Next-nearest-neighbortunneling amplitudes are typically two orders of magnitude smaller than nearest-neighborones and they can be neglected.), one obtains the single-particle Hamiltonian H = − (cid:88) (cid:104) i,j (cid:105) J ij a † i a j + (cid:88) j ( V j − µ ) a † j a j , (25)where V j = V ( x j ) and the notation (cid:104) i, j (cid:105) restricts the sum to nearest-neighbor sites. J ij pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 is the tunneling matrix element between the nearest neighboring lattice sites i and jJ ij = − (cid:90) dxw ∗ ( x − x i ) (cid:20) − (cid:126) m ∇ + V lat ( x ) (cid:21) w ( x − x i +1 ) , (26)Equation (25) is a general noninteracting tight-binding Hamiltonian for atoms in OLs. The Hubbard models.
For interacting atoms in an OL, the Hubbard model can be con-sidered an improvement on the single-particle tight-binding model [50, 78]. The Hubbardmodel was originally proposed in 1963 to describe electrons in solids and has since beenthe focus of particular interest as a model for high-temperature superconductivity. Theparticles can either be fermions, as in Hubbard’s original work and named the (Fermi-)Hubbard model, or bosons, which is referred to as the Bose-Hubbard model. For stronginteractions, it can give behaviors qualitatively different from those of the single-particlemodel and correctly predict the existence of the so-called Mott insulators, which areprevented from becoming conductive by the strong repulsion between the particles. TheHubbard model is a good approximation for particles in a periodic potential at suffi-ciently low temperatures where all the particles are in the lowest Bloch band, as long asany long-range interactions between the particles can be ignored. If interactions betweenparticles on different sites of the lattice are included, the model is often referred to asthe “extended Hubbard model”.The simplest nontrivial model that describes interacting bosons in a periodic potentialis the Bose-Hubbard Hamiltonian. It can be derived from Eq. (25) with the additionalinteracting term U int . In the grand canonical ensemble and assuming the interactions aredominated by s-wave interactions, i.e., U int = πa s (cid:126) m | Ψ( x ) | , the Bose-Hubbard Hamil-tonian is given by [79], H BH = − J (cid:88) (cid:104) i,j (cid:105) b † i b j + (cid:88) j ( V j − µ ) b † j b j + U (cid:88) j b † j b † j b j b j , (27)where U = (4 πa s (cid:126) /m ) (cid:82) | w ( x ) | d x accounts for interatomic interactions and measuresthe strength of the repulsion of two atoms on the same lattice site. To express thatthe atoms are bosons, the notation of the annihilation operator in Eq. (27) is explicitlydenoted as b j . While the parameter J decreases exponentially with lattice depth V , U increases as a power law of V D/ , where D is the dimensionality of the lattice. The Bose-Hubbard model has been used to describe many different systems in solid-state physics,such as short correlation length superconductors, Josephson arrays, critical behaviors of He and, recently, cold atoms in OLs. The Bose-Hubbard Hamiltonian exhibits a quantumphase transition from a superfluid to a Mott insulator state [80]. Its phase diagram hasbeen intensively studied via analytical and numerical approaches with many differenttechniques and experimentally confirmed using ultracold atomic systems in 1D, 2D, and3D lattice geometries [50, 78].The ultracold atomic system also provides an almost ideal experimental realization ofthe originally proposed Fermi-Hubbard model with highly tunable parameters [81]. Tosimulate the spin-1/2 electrons in condensed matter physics, we may need two-componentFermi gas trapped in OLs. The Fermi-Hubbard Hamiltonian then takes the form H FH = − J (cid:88) (cid:104) i,j (cid:105) ,σ ( c † i,σ c j,σ + h.c. ) + U N (cid:88) i =1 n i, ↑ n i, ↓ + (cid:88) j (cid:15) j n j . (28)Here the annihilation operator for spin σ on j -th site is denoted as c j,σ , and n j,σ = c † j,σ c j,σ pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 is the spin-density operator, with the total density operator n j = n j, ↑ + n j, ↓ . The lastterm takes account of the additional confinement V ( x ) of the atom trap, which is usuallyharmonic, with (cid:15) j the corresponding energy offset on the j -th lattice site.Experimentally, the tunnel amplitude in the Hubbard models is controlled by theintensity of the standing laser waves. This allows for a variation of the dimensionality ofthe system and enables tuning of the kinetic energy. The energy width of the lowest bandis W = 4 J D . Due to the low kinetic energy of the atoms, two atoms of different spinsusually interact via s-wave scattering and the coupling constant is given by g = 4 πa s /m .With this, the Hubbard interaction U can be tuned to negative or positive values byexploiting Feshbach resonances. However, a single component Fermi gas is effectivelynoninteracting because Pauli’s principle does not allow s -wave collisions of even parity. Artificial magnetic fields and spin-orbit couplings
A magnetic field plays a crucial role in topological quantum matter with broken TRS,whereas an SOC is a basic ingredient for those having TRS. Atoms are, however, electri-cally neutral; therefore, it is highly desirable to make them behave as charged particles inan electromagnetic field. This capability has been explored and demonstrated in a seriesof publications, including several nice review papers [26, 27, 51, 54–56]. In this sectionwe describe three typical methods (geometric gauge potentials, laser-assisted tunnelingand periodically driven OLs) to generate artificial magnetic fields and SOCs for ultracoldneutral atoms.
When a quantum particle with internal structure moves adiabatically in a closed path,Mead [82] and Berry [12] discovered that a geometric phase, in addition to the usualdynamic phase, is accumulated on the wave function of the particle. This geometricphase is a generalization of Aharonov-Bohm phase [83] that a charged particle movingin a magnetic field acquires. Therefore, an artificial magnetic field can emerge in coldatom systems when the atomic center-of-mass motion is coupled to its internal degrees offreedom through laser-atom interaction. Based on this geometric phase approach, Refs.[84–88] proposed setups for systematically engineering vector potentials associated witha non-zero artificial magnetic field for quantum degenerate gases, and they have beenexperimentally realized for both bosonic [89, 90] and fermionic atoms [91]. When the localatomic internal states dressed by the laser fields have degeneracies, effective non-Abeliangauge potentials can be formed [92–96], manifesting as artificial SOCs in Bose-Einsteincondensations [97–101] or degenerate Fermi gases [102, 103]. The artificial SOCs havebeen experimentally realized by several groups [31–35, 104–107], and they lead to anatomic spin Hall effect [84, 108], which has been experimentally demonstrated [104].To understand these artificial gauge fields, we consider the adiabatic motion of neutralatoms with N internal levels in stationary laser fields. The full Hamiltonian of the atomsreads H = p m + V ( r ) + H AL (29)where H AL represents the laser-atom interaction. H AL depends on the position of theatoms and is a N × N matrix in the representation of the internal energy levels | j (cid:105) . Inaddition, the potential V ( r ) is assumed to be diagonal in the internal states | j (cid:105) withthe form V ( r ) = (cid:80) Nj =1 V j ( r ) | j (cid:105)(cid:104) j | . In this case, the full quantum state of the atoms(including both the internal and the motional degrees of freedom) can then be expanded17 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 to | Φ( r ) (cid:105) = (cid:80) Nj =1 φ j ( r ) | j (cid:105) .We may discuss the problem in the representation of the dressed states | χ n (cid:105) that areeigenvectors of the Hamiltonian H AL , that is, H AL | χ n (cid:105) = ε n | χ n (cid:105) . Then the dressedstates | χ (cid:105) = ( | χ (cid:105) , | χ (cid:105) , · · · , | χ N (cid:105) ) (cid:62) (with (cid:62) denoting the transposition) are related tothe original internal states | j (cid:105) with the relation | χ (cid:105) = U ( | (cid:105) , | (cid:105) , · · · , | N (cid:105) ) (cid:62) , where thetransform matrix U is a unitary operator. In the new basis | χ (cid:105) , the full quantum stateof the atom | Φ( r ) (cid:105) is written as | Φ( r ) (cid:105) = (cid:80) j ψ j ( r ) | χ j ( r ) (cid:105) , where the wave functions | Ψ (cid:105) = ( | ψ (cid:105) , | ψ (cid:105) , · · · , | ψ N (cid:105) ) (cid:62) obey the Schr¨odinger equation i (cid:126) ∂∂t | Ψ (cid:105) = H eff | Ψ (cid:105) , with theeffective Hamiltonian H eff = U HU † taking the following form: H eff = 12 m ( − i (cid:126) ∇ − A ) + εI N + ˜ V ( r ) . (30)Here A = i (cid:126) U ∇ U † , ˜ V ( r ) = U V ( r ) U † , ε = ( ε , ε , · · · , ε N ) (cid:62) , and I N is the N × N unitmatrix [84, 92, 109]. In the derivation we have used the operator identity U P U † = (cid:0) − i (cid:126) ∇ − i (cid:126) U ∇ U † (cid:1) because of ∇ ( U † U ) = 0. From Eq. (30), one can see that in thedressed basis the atoms can be considered as moving in an induced (artificial) vectorpotential A and a scalar potential ˜ V ( r ), where the potential A is usually called theMead-Berry vector potential [12, 82]. They come from the spatial dependence of theatomic dressed states with the elements A mn = i (cid:126) (cid:104) χ m ( r ) |∇ χ n ( r ) (cid:105) , ˜ V mn = (cid:104) χ m ( r ) | V ( r ) | χ n ( r ) (cid:105) . (31) Abelian gauge potential.
An Abelian U (1) gauge potential is induced for each dressedstates provided that the off-diagonal elements of the matrices A and ˜ V are much smallerthan the energy difference between any pair of the dressed states, which implies thatthe eigenstates must be non-degenerate. In this case an adiabatic approximation can beapplied which is equivalent to neglecting the transitions between the specific dressed state | χ n (cid:105) and the remaining | χ l (cid:105) with n (cid:54) = l . Therefore, atoms in the dressed state | χ n (cid:105) evolveaccording to a separately effective Hamiltonian H n . We project the full Hamiltonian inEq. (30) to the specific state | χ n (cid:105) and obtain an effective Hamiltonian given by H n = 12 m ( − i (cid:126) ∇ − A n ) + ε n + ˜ V n + ˜ V (cid:48) n , (32)where A n = A mn δ nn , ˜ V n = ˜ V mn δ nn and ˜ V (cid:48) n = m (cid:80) Nl (cid:54) = n A n,l · A l,n . So an Abelian gaugepotential U (1) is induced for the neutral atoms. Non-Abelian gauge potential.
A non-Abelian gauge potential introduced by Wilczek andZee [109] can also be induced in this way if there are degenerate (or nearly degenerate)dressed states [92]. In this case the adiabatic approximation fails and then the off-diagonalcouplings between the degenerate dressed states can no longer be ignored. Assume thatthe first q atomic dressed states among the total N states are degenerate, and these levelsare well separated from the remaining N − q states, we neglect the transitions from thefirst q atomic dressed states to the remaining states. In this way, we can project the fullHamiltonian onto this subspace. Under this condition, the wave function in the subspace˜Ψ = ( ψ , . . . , ψ q ) (cid:62) is again governed by the Schr¨odinger equation i (cid:126) ∂∂t ˜Ψ = ˜ H eff ˜Ψ, wherethe effective Hamiltonian reads˜ H eff = 12 m ( − i (cid:126) ∇ − A ) + εI q + ˜ V + ˜ V (cid:48) . (33)Here the matrices A , εI q , and ˜ V are the truncated q × q matrices in Eq. (30). The18 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 projection of the term A in Eq. (30) to the q dimensional subspace cannot entirely beexpressed in terms of the truncated matrix A . This gives rise to an additional scalarpotential ˜ V (cid:48) which is also a q × q matrix,˜ V (cid:48) n,j = 12 m N (cid:88) l = q +1 A n,l · A l,j = (cid:126) m (cid:32) (cid:104)∇ χ n |∇ χ j (cid:105) + q (cid:88) k =1 (cid:104) χ n |∇ χ k (cid:105)(cid:104) χ k |∇ χ j (cid:105) (cid:33) (34)with n, j ∈ (1 , . . . , q ). Since the adiabatic states | χ (cid:105) . . . | χ q (cid:105) are degenerate, any basisgenerated by a local unitary transformation U ( r ) within the subspace is equivalent. Thecorresponding local basis change as ˜Ψ → U ( r ) ˜Ψ , which leads to a transformation of thepotentials according to A → U ( r ) A U † ( r ) − i (cid:126) [ ∇ U ( r )] U † ( r ) , ˜ V → U ( r ) ˜ V U † ( r ) . (35)These transformation rules show the gauge character of the potentials A and ˜ V . Thevector potential A is related to a curvature (an effective “magnetic” field) B as: B i = 12 (cid:15) ikl F kl , F kl = ∂ k A l − ∂ l A k − i (cid:126) [ A k , A l ] . (36)Note that the term ε ikl [ A k , A l ] = ( A × A ) i does not vanish in general, since the com-ponents of A do not necessarily commute. This term reflects the non-Abelian characterof the gauge potentials. The generalized “magnetic” field transforms under local rota-tions of the degenerate dressed basis as B → U ( r ) B U † ( r ) . Thus, as expected, B is a truegauge field. In the following, we employ this general scheme to create laser-induced gaugepotentials for ultracold atoms using two typical laser-atom interacting configurations. Spin-dependent gauge potentials in three-level Λ -type atoms . We first take an atomic gaswith each atom having a Λ-type level configuration as an example to illustrate the aboveidea [84–86]. As shown in Fig. 1(a), the ground states | (cid:105) and | (cid:105) are coupled to an excitedstate | (cid:105) through spatially varying laser fields, with the corresponding Rabi frequencies Ω and Ω , respectively. We assume off-resonant couplings for the single-photon transitionswith the same large detuning ∆ d . In this case the atom-laser interaction Hamiltonian H AL in the basis {| (cid:105) , | (cid:105) , | (cid:105)} is given by H AL = Ω ∗ Ω ∗ ∆ d . (37)We may parameterize the Rabi frequencies through Ω = Ωsin θe iϕ and Ω = Ωcos θ ,with Ω = (cid:112) | Ω | + | Ω | ( θ and ϕ are in general spatially varying). We are interested inthe subspace spanned by the two lowest dressed states {| χ (cid:105) , | χ (cid:105)} (called respectivelythe dark and the bright states). This gives an effective spin-1/2 system, and in the spinlanguage we also denote | χ ↑ (cid:105) ≡ | χ (cid:105) and | χ ↓ (cid:105) ≡ | χ (cid:105) . In the case of a large detuning(∆ d (cid:29) Ω), both states | χ ↑ (cid:105) and | χ ↓ (cid:105) have negligible contribution from the initial excitedstate | (cid:105) , so they are stable under atomic spontaneous emission. Furthermore, we assumethe adiabatic condition, which requires that the off-diagonal elements of the matrices ˜A and ˜ V are much smaller than the eigenenergy differences | λ i − λ j | ( i, j = 1 , , | χ i (cid:105) . This gives the quantitative condition ∆ D (cid:28) Ω / ∆ d , where ∆ D =cos θ | v · ∇ (tan θe iϕ ) | ( v is the typical velocity of the atom) represents the two-photonDoppler detuning [85]. Under this adiabatic condition, the effective Hamiltonian for the19 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 d (a) x y g(b) xy (d) (c) Figure 1. (Color online) Schematic of atom-laser interactions for artificial gauge potentials. (a) Three-level Λ-typeatoms interacting with laser beams characterized by the Rabi frequencies Ω and Ω through the Raman-typecoupling with a large single-photon detuning ∆ d . (b) The configuration of the Raman laser beams for a spin Halleffect. (c) and (d) Atoms with tripod-level configuration interacting with three laser beams characterized by theRabi frequencies Ω , Ω , and Ω . wave function Ψ in the subspace spanned by {| χ ↑ (cid:105) , | χ ↓ (cid:105)} is [84] H eff = (cid:18) H ↑ H ↓ (cid:19) , (38)where H σ = m ( − i (cid:126) ∇− A σ ) + V σ ( r ) ( σ = ↑ , ↓ ). The gauge potentials A σ can be obtainedas A ↑ = − A ↓ = − (cid:126) sin θ ∇ ϕ, and the related gauge field B σ = ∇ × A σ = − η σ (cid:126) sin(2 θ ) ∇ θ × ∇ ϕ, (39)where η ↑ = − η ↓ = 1. We obtain precisely a spin-dependent gauge field that is critical forthe spin Hall effect. A typical scheme to generate atomic spin Hall effect is shown in Fig1(b), which was demonstrated experimentally in Ref. [104]. Spin-orbit couplings in a tripod configuration . The second example we address is an SU (2) non-Abelian gauge field created in a tripod-level configuration [93–96]. Considerthe adiabatic motion of atoms in x - y plane with each having a tripod-level structure ina laser field as shown in Fig. 1(c) and (d). The atoms in three lower levels | (cid:105) , | (cid:105) and | (cid:105) are coupled with an excited level | (cid:105) through three laser beams characterized by theRabi frequencies Ω = Ωsin θ e − iκx / √
2, Ω = Ωsin θ e iκx / √
2, and Ω = Ωcos θ e − iκy , re-spectively, where Ω = (cid:112) | Ω | + | Ω | + | Ω | is the total Rabi frequency and the mixingangle θ defines the relative intensity. The atom-laser interaction Hamiltonian H AL in theinteraction representation reads H AL = − (cid:126) (cid:88) j =1 Ω j | (cid:105)(cid:104) j | + H . c .. (40)Diagonalizing this Hamiltonian yields two degenerate dark states with zero energy as wellas two bright states separated from the dark states by the energies ± (cid:126) Ω. If Ω is sufficientlylarge compared to the two-photon detuning due to the laser mismatch and/or Dopplershift, the adiabatic approximation is justified and one can safely study only the internalstates of an atom evolving within the dark state manifold. In this case, the non-Abeliangauge potential A in the present configuration of the light field can be obtained as A = (cid:126) κ (cid:18) e y − e x cos θ − e x cos θ e y cos θ (cid:19) . (41)20 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 Furthermore, let the mixing angle θ = θ with cos θ = √ −
1, such that sin θ = 2cos θ .Thus, the vector potential takes a symmetric form A = (cid:126) κ (cid:48) ( − e x σ x + e y σ z ) + (cid:126) κ e y I , where κ (cid:48) = κ cos θ and κ = κ (1 − cos θ ). Using a unitary transformation ˜ H (cid:48) = U † ˜ HU with U = exp( − iκ y ) exp (cid:0) − i π σ x (cid:1) , one obtains the Hamiltonian for the atomic motion H = 12 m [( p x + (cid:126) κ (cid:48) σ x ) + ( p y + (cid:126) κ (cid:48) σ y ) ] + V. (42)This Hamiltonian provides a coupling between the atomic center-of-mass motion and theinternal pseudospin degrees of freedom, thus giving rise to an effective SOC.However, the two degenerate dark states in this tripod configuration are not the lowest-energy states, so the atoms may quickly decay out of the dark states due to collisions andother relaxation processes. This problem may be solved by using the blue-detuned lasers[110] or a closed loop Raman coupling configuration [111]. Furthermore, the combinationof an SOC and an effective perpendicular Zeeman field is required for the emergence oftopological superfluid. To this end, five or two additional laser beams superposing into theabove tripod configuration were proposed [103, 112]. However, locking the phases of theselaser beams is challenging in experiments. It was thus proposed and then experimentallydemonstrated that controlling polarizations of the Raman lasers is sufficient to generatesimultaneously an effective SOC and a perpendicular Zeeman field for atoms [35, 113]. (a) (b) y/ λ m ix eJ x/ λ m m m n n n n n y J y J m ix eJ x/ λ V/V n n n n n g g e Figure 2. (Color online) Scheme for realizing artificial magnetic fields based on laser-assisted tunneling [114].Open (closed) circles denote atoms in state | g (cid:105) ( | e (cid:105) ). (a) Hopping in the y -direction J y is the same for particles instates | e (cid:105) and | g (cid:105) , while the x -direction hopping is laser-assisted. (b) Laser-assisted tunneling along the x -direction.Adjacent sites are set off by an energy ∆. The laser Ω is resonant for transitions | g (cid:105) and | e (cid:105) while Ω is resonantfor transitions | e (cid:105) and | g (cid:105) due to the offset of the lattice sites. The atoms hopping around one plaquette get phaseshifts of 2 πα due to the created artificial magnetic fields. Laser-induced tunneling was the first method proposed to generate artificial magneticfields in OLs [114], and it was used to experimentally realize the Hofstadter-Harper model[48]. Furthermore, this method was further proposed to simulate artificial SOCs in OLs[115–117], and was implemented very recently in an experiment realizing artificial 2DSOC in a Raman OL [36]. In this section we illustrate the basic idea of laser-inducedtunneling, following the original proposal in Ref. [114]. We then introduce recent devel-opments in Sec. 4.Consider a gas of atoms trapped in a 3D OL created by standing wave laser fields, whichgenerates a potential for the atoms V ( r ) = V x sin ( kx ) + V y sin ( ky ) + V z sin ( kz ) with k = 2 π/λ being the wave vector of the light. We assume the lattice to trap atoms intwo different internal hyperfine states | e (cid:105) and | g (cid:105) and the depth of the lattice in the x − and z − directions to be so large that hopping in these directions due to kinetic energyis prohibited. Furthermore, we assume that adjusting the polarization of the lasers that21 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 confine the particles in the x -direction allows us to place the potential well trapping atomsin the different internal states at distances λ/ x -direction of a x = λ/ y -direction of a y = λ/
2. Wefocus on one layer of the OL in the xy -plane since in the following, there will neither behopping nor interactions between different layers. The dynamics of atoms occupying thelowest Bloch band of this OL can be described by the Hamiltonian H l = (cid:88) n,m J y ( a † n,m a n,m − + h.c. ) + (cid:88) n ∈ even,m ω eg a † n,m a n,m , where J y is the hopping strength for particles to tunnel between adjacent sites alongthe y -direction. The energy difference between the two hyperfine states is ω eg > a n,m ( a † n,m ) are destruction (creation) operators for atoms in the lowestmotional band located at the site x n,m = ( x n , y m ), where x n = nλ/ y m = mλ/ x -direction,as shown in Fig. 2(b). This can be achieved by accelerating the OL along the x -axiswith a constant acceleration a acc , which induces an additional potential energy term H acc = M a acc x with M being the mass of the atoms. Alternatively, if both of the internalatomic states | e (cid:105) and | g (cid:105) have the same static polarizability µ an inhomogeneous staticelectric field of the form E ( x ) = δEx , where δE is the slope of the electric field in the x -direction, can be applied to the OL, which leads to a potential energy term H acc = µ p δEx .We keep this additional potential energy small compared to the OL potential and treat H acc as a perturbation. In second quantization this yields H acc = ∆ (cid:80) n,m na † n,m a n,m ,where ∆ = µ p δEλ/ M a acc λ/ δ << ν x with ν x = 4 √ E R V x being the trapping frequency of the OL in the x -direction.Here E R = k / M is the recoil energy.Finally, the laser-induced tunneling can be activated along the x -direction by couplingtwo internal states | g (cid:105) and | e (cid:105) with two additional lasers forming Raman transitions.The Raman beams consist of two running plane waves chosen to give space-dependentRabi frequencies of the form Ω , = Ω e ± iqy , where Ω denotes the magnitude of the Rabifrequencies, and ± ∆ is the detuning. We assume the lasers not to excite any transitionsto higher-lying Bloch bands with detuning of the order of ∆, i.e. Ω (cid:28) ∆ (cid:28) ν x . Then thelasers Ω will only drive transitions n − ↔ n if n is even (odd) and we can neglect anyinfluence of the nonresonant transitions. Then one can find the following Hamiltoniandescribing the effect of the Raman lasers H AL = (cid:88) n,m ( γ n,m a † n,m a n − ,m + h.c. ) − (cid:88) n,m ∆ a † n,m a n,m . Here the matrix elements γ nm can be written as γ n,m = 12 e iπαm ΩΓ y ( α )Γ x , where α = qλ/ π , and the matrix elementsΓ x = (cid:90) dxw (cid:63) ( x ) w ( x − λ/ , Γ y ( α ) = (cid:90) dyw (cid:63) ( y )cos(4 παm ) w ( y ) , with w ( r ) = w ( x ) w ( y ) w ( z ) being the Wannier function. To achieve a symmetric Hamil-22 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 tonian, we assume hopping amplitudes J x = ΩΓ x Γ y / J y = J , and thus the totalHamiltonian describing the configuration is given by H α = J (cid:88) n,m (cid:16) e iπαm a † n,m a n +1 ,m + a † n,m a n,m +1 + h.c. (cid:17) . This Hamiltonian H α is equivalent to the Hamiltonian for electrons with charge e movingon a lattice in an external magnetic field B = 2 πα/A cell e , where A cell is the area of oneelementary cell. Driving cold-atom systems periodically in time is a powerful method to engineer effectivemagnetic fields or SOCs, and thus can trigger topological quantum phases. For instance,the OL shaking method has been used to experimentally realize the Hofstadter model[29, 30, 118]. Modulating a honeycomb OL also led to the experimental realization of theHaldane model [28].We first describe two simple examples to illustrate the basic concept of creating arti-ficial gauge fields with the periodically driven method. In the first example we considerultracold atoms trapped in a 1D shaken bichromatic OL [119]. This lattice is generatedby the superposition of two shaken OLs. The single-particle Hamiltonian of an atom inthis 1D shaken lattice system reads H s = p x m + V sin [ k ( x − x ( t ))] + V sin [ k ( x − x ( t ) + φ )] , (43)where V i , k i = 2 π/λ i , and λ i ( i = 1 ,
2) are the lattice depth, laser wave vector andwavelength, respectively; and φ is the phase of the second laser, x i ( t ) = b sin( ωt ) is theperiodic time-dependent lattice shaking. Here we assume that the two lattices experiencethe same shaking amplitude b and frequency ω . Experimentally, a shaking sinusoidallattice can be realized through a modulation of the driving frequency and by changingthe relative phase of the acousto-optic modulators. The tunneling between neighboringsites decreases exponentially with the intensity of the lasers creating the lattice, whereasthe shape of the wavepacket (the Wannier functions) has a much weaker dependence.Therefore, by varying the laser intensity, one can rapidly vary the tunneling. With aunitary rotation, the Hamiltonian is transferred to a new frame x → x + b sin( ωt ) H r = ( p x − A x ) m + V sin ( k x ) + V sin ( k x + φ ) , (44)with a shaking-induced vector potential A x = mω cos( ωt ) [119].The second example is the topological phases of a 2D honeycomb lattice proposed byHaldane [13], which can also be realized with the method of shaking lattices [28, 120].We consider the following time-dependent lattice potential V ( x, y, t ) = − V X cos [ k r ( x + b cos ωt ) + θ/ − V X cos [ k r ( x + b cos ωt )] (45) − V Y cos [ k r ( y + b sin ωt )] − α (cid:112) V X V Y cos[ k r ( x + b cos ωt )]cos[ k r ( y + b sin ωt )] , which leads to a honeycomb lattice realized by the ETH group when b = 0 [77]. Here θ controls the energy offset between two sublattices A and B in the honeycomb lattice. The b (cid:54) = 0 case describes a shaking lattice in both x and y directions with a phase difference23 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 π/
2. Similar to the 1D case, transferring into the moving frame x → x + b cos( ωt ) and y → y + b sin( ωt ), one obtains a Hamiltonian with time-dependent vector potential term H ( t ) = 12 m [ p − A ( t )] + V ( x, y ) , (46)where A x ( t ) = mωb sin( ωt ) and A y ( t ) = − mωb cos( ωt ) [120]. It is equivalent to aHamiltonian that describes a particle in an ac electrical field in the 2D plane E ( t ) = mω b (cos( ωt ) , sin( ωt )). The phase diagram in this Hamiltonian has been calculated inRef. [120], and it shows a similar phase diagram with that of the Haldane model [13]; i.e.,it contains topological trivial and nontrivial phases characterized by a Chern number.This shaking lattice method has been experimentally used to realize the Haldane model[28], as addressed in detail in Sec. 4.2.3.After addressing the basic ideas, we now turn to some general frameworks that de-scribe periodically driven quantum systems. A general theoretical treatment of periodi-cally driven quantum systems is based on the Floquet theory. For a periodically drivenHamiltonian H ( t ) with period τ , its Floquet operator ˆ F o is defined asˆ F o ≡ U ( τ i + τ, τ i ) = T exp (cid:20) − i (cid:90) τ + τ i τ i H ( t ) dt (cid:21) , (47)where τ i is the initial time, and T denotes the required time-ordered integral as theHamiltonian at different times do not necessarily commute. The eigenvalue and eigen-states of the Floquet ˆ F o are given byˆ F o | ϕ n (cid:105) = e − i(cid:15) n τ | ϕ n (cid:105) , (48)where (cid:15) n ∈ ( − π/τ, π/τ ) is the quasi-energy. A general method to explore the topologicalphases, which is free from any further approximation, is to numerically evaluate Floquetoperator ˆ F o according to Eq. (47) and determine its eigenvalues and eigenfunctions fromEq. (48). If a periodically driven system exhibits nontrivial topology, there must be in-gapquasi-energies (cid:15) n and their corresponding wave functions ϕ are spatially well localized atthe edge of the system [120].A physically more transparent method is introducing a time-independent effectiveHamiltonian H eff via the Floquet operator [120, 121] U ( τ f , τ i ) = e − iK ( τ f ) e − iτH eff e iK ( τ i ) , (49)where we impose that (1) H eff is a time-independent operator, (2) K ( t ) is a time-periodicoperator K ( t + τ ) = K ( t ) with zero average over one period, and (3) H eff does notdependent on the starting time τ i , which can be realized by transferring all undesiredterms into the kick operator K ( τ i ). Similarly, H eff does not depend on the final time τ f .Equation (49) shows that the initial (final) phase of the Hamiltonian at time τ i ( τ f ) mayhave an important impact on the dynamics. However, the topological phenomena in theperiodically driven systems can be connected to those in equilibrium systems described bythe effective Hamiltonian H eff . Consider a static system described by a Hamiltonian H that is driven by a time-periodic modulation V ( t ), whose period τ = 2 π/ω is assumedto be much smaller compared to any characteristic time scale in the problem. In thishigh-frequency regime, one can obtain the effective Hamiltonian by using a perturbationexpansion. 24 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 We consider a time-periodic Hamiltonian H ( t ) = H + V ( t ) with V ( t ) = ∞ (cid:88) j =1 [ V ( j ) e ijωt + V ( − j ) e − ijωt ]between times τ i and τ f , with the period of the driving τ = 2 π/ω . The time-dependentpotential has been explicitly expanded with its Fourier’s form. By using a perturbationexpansion in powers of 1 /ω , one can obtain [121] H eff = H + 1 ω ∞ (cid:88) j =1 { j [ V ( j ) , V ( − j ) ]+ 12 ω ([[ V ( j ) , H ] , V ( − j ) ] + [[ V ( − j ) , H ] , V ( j ) ] } + O ( τ ) , (50) K ( t ) = (cid:90) t V ( t (cid:48) ) dt (cid:48) + ( τ ) . (51)In Eq. (50), the second-order terms that mix different harmonics have been omitted.To understand the emergence of topological nontrivial phases, we usually write theHamiltonian into momentum space. As for two-band systems, the general Hamiltonian inmomentum space k can be rewritten as H eff = B ( k ) · σ . By using this kind of perturbationexpansion, one can obtain the explicit expressions of B ( k ) for the models described inEqs. (43) and (45) [119, 120], which shows that the nontrivial topological phases can beinduced in the periodically driven OLs.The perturbation expression in Eq. (50) can also be used in the derivation of theeffective Hamiltonian for the general situation where a pulse sequence is characterizedby the repeated N -step sequence γ N = { H + V , H + V , · · · , H + V N } , (52)where the V m ’s are arbitrary operators [121]. For simplicity, we assume that the durationof each step is τ /N , and we further impose that (cid:80) Nm =1 V m = 0. The Hamiltonian γ N canbe expanded in terms of the harmonics H ( t ) = H + (cid:80) j (cid:54) =0 V j e ijωt , where V j = 12 πi N (cid:88) m =1 j e − i πjm/N ( e ij (2 π/N ) − V m . By applying Eqs. (50) and (51), one can derive the effective Hamiltonian and the initial-kick operator as H eff = H + 2 πiN ω N (cid:88) m 4. Topological quantum matter in optical lattices In the previous section, we introduced the techniques for engineering the Hamiltonian ofcold atoms, specially the techniques of creating artificial magnetic fields and SOCs. Theuse of these techniques in OLs has led to the realization and characterization of sometopological states for cold atoms. Compared with conventional solid-state systems, coldatoms offer an ideal platform with great controllability to study topological models. Forinstance, the laser fields that couple hyperfine states of atoms can be used to synthesizeeffective physical fields, such as gauge fields, SOCs, and Zeeman fields. The forms andstrengths of those synthetic fields are tunable as they are determined by the atom-lasercoupling configurations. The structure of an OL can be designed via several counter-propagating lasers to realize various unconventional lattice potentials, which include thedouble-well superlattices, honeycomb lattices, spin-dependent lattices, and so on.This section systematically discusses some important lattice models with topologicalquantities originally introduced in condensed matter theories and describes their pro-posed schemes as well as current implementation methods. The topological bands andphenomena in these models can be created and detected with cold atoms in OLs. Theselattice models range from 1D to 3D and even higher-dimension geometries, which can beimplemented with OLs of various geometric structures. These systems mainly focus onenergy bands in the absence of interactions, and hence the topological phenomena ad-dressed here correspond to the single-particle physics. Some advances in their extensionto the interacting regime will also be briefly discussed.This section is divided into five parts. In the first, we describe some basic topologicalmodels with nontrivial bands realized in 1D OLs, which include the famous Su-Schrieffer-Heeger (SSH) model and its implementation for topological pumping. In the second partwe discuss the physics of Dirac fermions, the topological properties of the Hofstadtermodel, Haldane model and Kane-Mele model, and their experimental realization anddetection in 2D OLs. Some typical 3D topological insulating states of Z or Z types andtopological gapless (semimetal) states with emergent Dirac or Weyl fermions in 3D OLsare presented in the third part. The last two parts are respectively devoted to topologicalstates in higher dimensions with the newly developed synthetic dimension technique andunconventional topological quasi-particles with higher pseudospins for cold atoms in OLs,both of which are currently absent or extremely challenging to realize in condensed mattersystems. One-dimension The SSH model [3, 4] for polyacetylene is the simplest 1D model of band topology incondensed matter physics. Such a model describes the polyacetylene with free fermionsmoving in a 1D chain with dimerized tunneling amplitudes. The essence of the SSH modelis manifested by two topological characters. The first character is the nontrivial Zakphase that describes distinct topological phases in 1D lattice systems with zero-energyedge modes in a finite chain with open boundaries. The second one is the topologicalsolitons with fractional particle numbers, which emerge on the domain walls in the latticepotential to separate two dimerization structures. The physics of such a dimerized latticewith two sites per unit cell is captured by the SSH Hamiltonian H SSH = − (cid:88) n ( J a † n b n + J (cid:48) a † n b n − + h.c. ) , (59)27 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 where J and J (cid:48) denote the modulated hopping amplitudes, and a † n ( b † n ) are the creationoperators for a particle on the sublattice site a n ( b n ) in the n th lattice cell, as shown inFig. 3(a). Written in momentum space, the Hamiltonian (59) takes the form H SSH = (cid:80) k Ψ † k H SSH ( k )Ψ k with Ψ † k = (cid:16) a † k , b † k (cid:17) and H SSH ( k ) = − [ J + J (cid:48) cos( ka )] σ x − J (cid:48) sin( ka ) σ y , (60)where a denotes the lattice spacing. Consequently, there are two bands with the energydispersion E ± = ± (cid:112) [ J + J (cid:48) cos( ka )] + [ J (cid:48) sin( ka )] .It can be found that H SSH ( k ) possesses the chiral symmetry σ z H SSH ( k ) σ z = −H SSH ( k )and the TRS ˆ T H SSH ( k ) ˆ T − = H SSH ( − k ), where ˆ T = ˆ K with ˆ K being the complexconjugate operator. Note that the chiral symmetry here is a sublattice symmetry andrequires that hoppings only exist between two sublattices. The chiral symmetry gives riseto an additional particle-hole (charge-conjugation) symmetry because for any eigenstate | u E (cid:105) with energy E there exists a corresponding eigenstate | u − E (cid:105) = σ z | u E (cid:105) with energy − E . Thus, the SSH model is classified in the BDI class of topological insulators [123].It is known that the SSH model has two topologically distinct phases with differentdimerization configurations, D J > J (cid:48) and D J < J (cid:48) , separated by a topologicalphase transition point at J = J (cid:48) . The topological features can be characterized by theZak phase [62] ϕ Zak = i (cid:90) G/ − G/ (cid:104) u ± ( k ) | ∂ k | u ± ( k ) (cid:105) dk, (61)where G = 2 π/d is the reciprocal lattice vector with d = 2 a and | u ± ( k ) (cid:105) denote theBloch wave functions of the higher (+) and lower ( − ) bands. The Zak phase in eachlattice configuration is a gauge dependent quantity depending on the choice of originof the unit cell. For our choice, the Zak phase ϕ Zak = 0 for D D ϕ Zak = π and the system hosts twodegenerate zero-energy edge states, yielding a gapped topologically nontrivial phase. Thedifference between the Zak phases for the two dimerization configurations is well definedas δϕ Zak = π , which is gauge invariant and thus can be used to identify the differenttopological characters of the Bloch bands. In the topologically nontrivial phase, thereare two degenerate zero-energy modes respectively localized at two edges of the systemunder the open boundary condition.Another topological feature in the SSH model is that a kink (anti-kink) domain be-tween the two dimerization configurations gives rise to an undegenerate, isolated soli-ton (anti-soliton) state on the domain, which is a zero-energy mid-gap state. Due tothe particle-hole ambiguity of the energy spectrum, the zero mode takes the fractionalfermion number N = ( N = − ) when this mode is occupied (unoccupied). The solitonstate is topologically protected in the sense that it is impossible to remove it withoutclosing the bulk energy gap, which is due to the fact that it is on an interface betweentopologically distinct phases. Historically, such topological solitons with fractional parti-cle numbers were first found in a 1D modified Dirac equation in the context of the fieldtheory by Jackiw and Rebbi [124] (see Sec. 6.1), and the SSH model provides the firstphysical demonstration of this remarkable phenomenon in lattice systems.The SSH model was generalized to describe the linearly conjugated diatomic polymerswith the Rice-Mele Hamiltonian [126] H RM = − (cid:88) n ( J a † n b n + J (cid:48) a † n b n − + h.c ) + ∆ (cid:88) n ( a † n a n − b † n b n ) , (62)28 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 (a) Δ 𝑑𝑑𝐽𝐽 𝐽𝐽 𝐽𝐽𝐽𝐽′ 𝐽𝐽′𝑎𝑎 𝑛𝑛 𝑏𝑏 𝑛𝑛 𝑎𝑎 𝑛𝑛−1 𝑏𝑏 𝑛𝑛−1 𝑎𝑎 𝑛𝑛+1 𝑏𝑏 𝑛𝑛+1 Optical superlattice 𝑥𝑥𝑉𝑉 ( 𝑥𝑥 ) | > | > 𝐷𝐷 : 𝐽𝐽 > 𝐽𝐽′𝐷𝐷 : 𝐽𝐽 < 𝐽𝐽′ , B.O. 𝐷𝐷 𝐷𝐷 , spin flip, B.O. (b) Figure 3. (Color online) (a) Schematic illustration of the 1D optical superlattice for realizing the SSH model(∆ = 0) and the Rice-Mele model (∆ (cid:54) = 0) in the experiment [125]. The yellow box denotes the unit cell containingtwo sites with staggered hopping strengths ( J and J (cid:48) ) and tunable energy offset (2∆). (b) Schematic illustrationof experimental three-step sequence of measuring the Zak phase difference δϕ Zak , based on a combination ofspin-dependent Bloch oscillation (B.O.) and Ramsey interferometry [125]. The preparation and state evolution ofthe atomic gas in a superposition of two spin-states with opposite magnetic moment (brown and green balls) aredescribed in the text. where ∆ is the energy offset between neighboring lattice sites. For a heteropolar dimerconfiguration with ∆ (cid:54) = 0, the particle-hole symmetry (and chiral symmetry) in theoriginal SSH model is broken. Consequently, the Zak phase is fractional in units of π anddepends on the energy offset ∆. Strictly, the Rice-Mele model is not topological in thetheory of topological classification with symmetry [61]. However, one can investigate theexistence of edge modes to determine which configuration has nontrivial properties in thiscase, noting that the two edge modes in the topological phase are no longer degeneratewhen ∆ (cid:54) = 0. If there is a domain wall in the Rice-Mele model, where a Dirac Hamiltonianemerges in the continuum limit as the generalized Jackiw-Rebbi model (see Sec. 6.1), theunpaired soliton state in general has non-zero energy and carries an irrational particlenumber [126–128]. The Rice-Mele/SSH model with band topology (geometry) provides aparadigmatic system for studying topological quantum pumps [129–131] (see Sec. 4.1.2).Compared with conventional solid-state materials, cold atom systems offer a perfectlyclean platform with high controllability to study topological states of matter. The firstscheme to simulate the SSH/Rice-Mele model along this direction was proposed to en-gineer the spatial profiles of the hopping amplitudes for cold atoms in a 1D opticalsuperlattice in such a way that an optically induced defect as the domain wall carriesfractional particle numbers in the lattice [132–134]. In the proposed system, a two-speciesgas of fermionic atoms is trapped in a state-dependent OL, where the internal states aredenoted by | ↑(cid:105) and | ↓(cid:105) . The two atomic species experience different optical potentialsthat are shifted relative to each other by λ/ 4, where λ is the wavelength of light ofthe confining OL. Such a state-dependent OL is achieved when the laser beam is bluedetuned from the internal transition of the atoms in | ↑(cid:105) and red detuned by the sameamount from the internal transition of the atoms in | ↓(cid:105) . When the lattice is sufficientlydeep, each site is assumed to support one mode function that is weakly coupled to twonearest-neighbor sites, such that the hopping of the atoms between adjacent lattice sitesonly occurs as a result of driving by coherent electromagnetic fields. The coupling couldbe a far-off-resonant optical Raman transition via an intermediate atomic level. In thisway, the required dimerized OL with the alternating hopping for atoms can be realized byusing coupling lasers of proper two-photon Rabi frequency. Since the laser phase directlymodulates the hopping configuration, when there is a jump in one particular lattice site inthe laser phase, it will generate a domain wall to separate the D D pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 (a) (b) Figure 4. (Color online) Determination of the Zak phase. (a) The atom number in the two spin states N ↑ , ↓ ismeasured following the sequence and the fraction of atoms in the |↑(cid:105) spin state n ↑ = N ↑ / ( N ↑ + N ↓ ) is plotted asa function of the phase of the final microwave π/ δϕ Zak . Blue (black) circles correspond to the fringe in which the dimerization was(not) swapped. (b) Measured relative phases for 14 identical experimental runs (left), which give an average valueof δϕ Zak = 0 . π . The corresponding histogram is shown on the right with a binning of 0 . π . Reprinted bypermission from Macmillan Publishers Ltd: Atala et al. [125], copyright c (cid:13) (2013). OLs [133, 134]. By measuring the intensity of the scattered light, one can detect thefractional expectation value of the atom number and its fluctuations. It was recentlydemonstrated that the fractional particle number in the SSH/Rice-Mele model can besimulated in the momentum-time parameter space in terms of Berry curvature without aspatial domain wall [136]. In the simulation, a hopping modulation is adiabatically tunedto form a kink-type configuration, and the induced current plays the role of an analogoussoliton distributing in the time domain. Thus the mimicked fractional particle number isexpressed by the particle transport and can be detected from the center-of-mass motionof an atomic cloud. Two feasible experimental setups of OLs for realizing the requiredSSH Hamiltonian with tunable parameters and time-varying hopping modulation werepresented in Ref. [136].Other schemes for creating 1D topological bands and soliton/edge states have recentlybeen proposed by using cold atoms trapped in double-well OLs. The authors in Ref.[137] considered a single species of fermionic atoms occupying an sp orbital ladder ofthe two wells, where the staggered hopping pattern for realizing the topological phasenaturally arises. In the noninteracting limit, the sp orbital ladder naturally reproducesthe SSH model with a quantized Zak phase and fractionalized zero-energy edge states.The stability of the topological phase against atomic interactions and the emergence oftopological flat bands of edge modes in the presence of inter-ladder coupling were alsodiscussed [137]. It was shown that an atomic gas of attractively interacting fermions in a1D periodically shaken OL can give rise to the emergent Rice-Mele model with controlleddomain walls, which comes from the density-wave ground state [138]. By using cold atomsin a spin-dependent optical double-well lattice, one may realize a two-leg generalizedSSH model with glide reflection symmetry [139], which is topologically characterized byWilson lines and automatical fractionalization without producing domains in the latticedue to the interplay between the glide symmetry and atomic repulsive interactions.The SSH/Rice-Mele model described by the Hamiltonian (62) with the tunable en-ergy offset ∆ has already been experimentally realized with a Bose-Einstein condensate(BEC) of Rb trapped in a 1D optical superlattice [125]. The superlattice potentialshown in Fig. 3(a) is created by superimposing two standing optical waves of short- andlong-wavelengths differing by a factor of two ( λ l = 2 λ s = 1534nm), which leads to thetotal lattice potential V ( x ) = V l sin ( k L x + φ/ 2) + V s sin (2 k L x + π/ k L is thewave vector of the short wavelength trapping lasers, and V and V s are the correspondingstrengths of the two standing waves. The lattice potential can be controlled by varyingthe laser intensity of long-wavelength standing-wave lasers to make the system well de-30 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 scribed by the SSH/Rice-Mele Hamiltonian in the tight-binding regime. Phase controlbetween the two standing wave fields enables one to fully control φ for tuning the atomichopping. This makes the OL into the D D φ = 0 and φ = π was used to rapidly access the two differentdimerized configurations with ∆ = 0, whereas a tunable energy offset ∆ (cid:54) = 0 was alsointroduced by tuning φ slightly away from these symmetry points.Moreover, the Zak phase δϕ Zak characterizing the topological Bloch bands was de-tected, even though the atoms used in the experiment are bosons. A three-step sequenceshown in Fig. 3(b) was employed, which is based on a combination of spin-dependentBloch oscillations and Ramsey interferometry [125]. The first step is to start with anatomic condensate in the state | ↓ , k = 0 (cid:105) and bring it into a coherent superpositionstate 1 / √ | ↑ , k = 0 (cid:105) + | ↓ , k = 0 (cid:105) ) using a microwave π/ σ = ↑ , ↓ denotestwo spin states of the atoms with opposite magnetic moment and k is the central mo-mentum of the condensate. Then a magnetic field gradient is applied to create a constantforce in opposite directions for the two spin components, leading to spin-dependent Blochoscillations. In this process, the atomic wavepacket evolves into the coherent superpo-sition state 1 / √ | ↑ , k (cid:105) + e iδϕ | ↓ , − k (cid:105) ). When the two states reach the band edge, thedifferential phase between them is given by δϕ = ϕ Zak + δϕ Zeeman , where δϕ Zeeman denotesthe Zeeman phase difference induced by the magnetic field. For the TRS Hamiltonianhere, the dynamical phase acquired during the adiabatic evolution is equal for the twospin states and thus cancels in the total phase difference. The second step is to eliminatethe Zeeman phase difference by applying a spin-echo π -pulse and switching dimerizationconfigurations following the first step. For atoms located at the band edge k = ± G/ k = 0. At this point,a final π/ ϕ MW is applied in order to make the two spin componentsinterfere and read out their relative phase δϕ Zak through the resulting Ramsey fringe.Experimental results for the two Ramsey fringes obtained with and without dimeriza-tion swapping during the state evolution are shown in Fig. 4(a), and the obtained phasedifferences are shown in Fig. 4(b), together with the corresponding histogram. Thus, theZak phase difference between the two dimerized configurations was determined to be δϕ Zak = 0 . π , which agrees well with theoretical prediction of the topological Blochbands in the SSH model. This method was used to further study the dependence of theZak phase on the offset energy δϕ Zak (∆), which corresponds to the Rice-Mele model withthe fractional Zak phase. This work establishes a general approach for probing the topo-logical invariant in topological Bloch bands in OLs. The measurement technique can beextended to more complicated topological models, such as detecting the Chern numbersof the Hofstadter model and the Haldane model [28, 48], and the π Berry flux associatedwith a Dirac point in 2D OL systems [140].After measuring the bulk topological index in the SSH model, cold atom experimentshave begun to probe topological boundary states at the intersection of two differenttopological phases. Recently, two different approaches for synthesizing and observingtopological soliton states in the SSH model using BECs in 1D OLs were experimentallydeveloped [141, 142]. In the first experiment [141], the authors used a 1D OL for coldrubidium atoms with a spatially chirped amplitude to create a domain wall and thendirectly observed the soliton state by optical real-space imaging of atoms confined atthe interface. The lattice potential is realized using a rubidium atomic three level con-figuration with two ground states of different spin projections and one excited state. Toachieve a topological interface, two four-photon potentials V ( x ) and V ( x ) with oppositespatial variation of the Raman coupling are superimposed, as shown in Fig. 5(a). Thisgives rise to a 1D lattice potential V ( x ) = V ( x ) + V ( x ) ≈ a · x cos(4 k L x ), creating a31 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 (b) x xxx (a) (c)(d) xx Figure 5. (Color online) Realizing the interface and probing the topological bound state from temporal evolutionof atomic clouds. (a) The 1D OL with spatially varying lattice depth from the amplitude-chirped lattice potentials V , ( x ), which increases (decreases) along x for positive (negative) values of the Raman detuning. (b) Measuredspatial variation of band ordering. Relative atomic population transferred into the upper (green circles) andlower (blue squares) band on loading from an initial state with an atomic cloud centered at position x . Seriesof absorption images for a relative phase of the initially prepared atomic wavepacket of (c) ϕ = + π/ ϕ = − π/ ϕ = + π/ 2, trapping of atoms in the topological edge statewas observed, while for ϕ = − π/ 2, the cloud splits up. Reprinted with permission from Leder et al. [141]. zero crossing at x = 0, where k L = 2 π/λ with λ being the wavelength of the laser beams.For x > x < 0) the maxima (minima) of the potential are located at integer multiplesof λ/ 4, This phase change is reflected in the inversion of ordering bands, which cannotbe transformed into each other by continuous deformation without closing the gap. Forsuch a situation a non-degenerate topologically protected bound state localized around x = 0, where the bands intersect, is expected in the SSH model. The dynamics of atomsin such a structure near the band crossing is described by the Dirac Hamiltonian with aspatially dependent effective mass (the Jackiw-Rebbi model) in the continuum limit withgood accuracy, as the width of the topological bound state is two orders of magnitudelarger than the lattice spacing [141]. To verify the band inversion on sign change of x , theadiabatically expanded atomic cloud centered at different lateral positions x along thelattice beam axis was transferred to the state φ i ( x ) = √ φ x ( x − x )( e ikx + e − ikx ) viatwo simultaneously performed Bragg pulses. The band populations following activationof the lattice were determined. For x < 0, loading is enhanced in the lower band, whilefor x > x = 0 the curvescross, as shown in Fig. 5(b). This experimentally verifies the spatial variation of the bandstructure and the exhibition of a topological interface at x = 0. Next in the experiments,the atoms were loaded into the interface and the topological bound state was observedvia an initial state φ i ( x ) = √ φ x ( x )( e ikx + e − iϕ e − ikx ) with ϕ = π/ 2. The atomicwavepacket was centered at x = 0, after which the lattice beams were activated and aseries of atomic absorption images were recorded after a variable holding time in Fig.5(c). This shows that the atomic cloud remains trapped at the expected position of thetopological bound state. On the other hand, for a phase of ϕ = − π/ 2, no such trappingin the bound state was observed, as shown in Fig. 5(d). As expected, there is no overlapwith the topological bound state when the initially prepared atomic wavepacket is π outof phase, such that the wavepacket is split into two spatially diverging paths. Furtherevidence for the successful population of the topological bound state was obtained fromthe phase ( ϕ ) dependence and the dependence of the initial atomic momentum width on32 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 𝛿𝛿 𝛿𝛿𝛿𝛿𝛿𝛿𝛿𝛿𝛿𝛿 LL Figure 6. (Color online) Adiabatic preparation of the topological bound state in the momentum-space lattice.(a) Time sequence of the smooth ramp of the weak tunneling links (blue), holding the strong (red) links fixed. (b)Simulated (top) and averaged experimental (bottom) absorption images for an adiabatically loaded edge-defectlattice. (c) Same as (b), but for an adiabatically loaded central-defect lattice. (d) Decay length of the atomicdistribution on even sites of the edge-defect lattice vs. δJ final /J . The dashed line represents the results of anumerical simulation of the experimental sequence. Reprinted with permission from Meier et al. [142]. loading efficiency.In another experiment [142], a momentum-space lattice was used to simulate the hard-wall boundaries in the SSH model and the simulated topological bound state was thenobserved by site-resolved detection of the populations in the lattice. The physics of theSSH lattice model was emulated by using the controlled evolution of momentum-spacedistributions of cold atomic gases [143, 144]. Controlled coupling between discrete free-particle momentum states is achieved through stimulated two-photon Bragg transitions,driven by counterpropagating laser fields detuned far from atomic resonance. The laserscoherently coupled 21 discrete atomic momentum states in the experiment [142], creatinga ”momentum-space lattice” of states in which atomic population may reside [see, e.g.,Figs. 6(b,c)]. The momentum states are characterized by site indices n and momenta p n = 2 n (cid:126) k L . The coupling between these states is fully controlled through 20 distincttwo-photon Bragg diffraction processes, allowing simulation of 1D tight-binding modelswith local control of all site energies and tunneling terms. This enables one to create thehard-wall boundaries and lattice defects in the SSH model. The authors then directlyprobed the topological bound states in the SSH model simulated in such an OL throughquench dynamics, phase-sensitive injection, and adiabatic preparation. The first detectionmethod is to abruptly expose the condensate atoms initially localized at only a singlelattice site and observe the ensuing quench dynamics with single-site resolution. Whenpopulation was injected onto a defect site, a large overlap with the topological boundstate was found, resulting in a relative lack of dynamics as compared to injection at anyother lattice site. The second method is to probe the inherent sensitivity of the boundstate to a controlled relative phase of initialized states, following a Hamiltonian quench.It was observed that the dynamics is nearly absent when the phase matches that of thebound state, while defect-site population is immediately reduced when the phase doesnot match. Lastly, the topological mid-gap bound state was directly probed through aquantum annealing procedure. The bound eigenstate was initially prepared in the fullydimerized limit of the time-dependent SSH Hamiltonian [142] H SSH ( t ) = − (cid:88) n ∈ odd ( J + δJ final )( c † n +1 c n + h.c.) − (cid:88) n ∈ even J even ( t )( c † n +1 c n + h.c.) , (63)with only the odd tunneling links present at a strength J odd = J + δJ final . Atomicpopulation was injected at the decoupled zeroth site, identically overlapping with themid-gap state. Next, the tunneling on the even links was slowly increased from zero to J − δJ final , as depicted by the smooth ramp in Fig. 6(a). For adiabatic ramping, the atomic33 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 wavefunctions should follow the eigenstate of H SSH ( t ). This adiabatic preparation methodwas performed for a lattice with the defect on the left edge and at its center, as shown inFig. 6(b) and 6(c), respectively. As known in the SSH model, the amplitude of the mid-gap state wavefunction is largest at the defect site and decays exponentially into the bulk.The decay length ξ (in units of the spacing between lattice sites) should scale roughlyas the inverse of the energy gap. Thus, highly localized mid-gap states for δJ final /J ∼ δJ final /J (cid:28) δJ final /J , as shown in Fig. 6(d). The technique of creating momentum-space latticeswith direct and full control of tunneling configuration demonstrated in this work hasbeen extended to create 2D artificial flux lattices for cold atoms [145]. Quantum pumping as originally proposed by Thouless [129] entails the transport ofcharge in a 1D periodic potential through an adiabatic cyclic evolution of the underlyingHamiltonian. In contrast to classical pumping, the transported charge is quantized andpurely determined by the topology of the pump cycle, which is related to the Chernnumber. Topological pumping can be generalized to interacting systems [130], and isrobust against moderate perturbations and finite-size effects.Topological Thouless pumping is closely related to the modern theory of polarization.Consider a lattice site at x = R in a 1D periodic lattice, with the Bloch function of thelowest band | ψ k (cid:105) = e ikx | u k (cid:105) , the corresponding Wannier function is given by | R (cid:105) = 1 √ N π/d (cid:88) k = − π/d e − ikR | ψ k (cid:105) = (cid:114) dL π/d (cid:88) k = − π/d e ik ( x − R ) | u k (cid:105) , (64)where N = L/d is the number of unit cells in the system with L being the system lengthand d being the lattice constant. The expected shift of the Wannier center from thelattice site R at time t is denoted by the polarization P ( t ) = (cid:104) R ( t ) | x − R | R ( t ) (cid:105) = dL π/d (cid:88) − π/d (cid:104) u k ( t ) | i∂ k | u k ( t ) (cid:105) = d (cid:90) π/d − π/d dk π A k ( k, t ) . (65)Here A k ( k, t ) = i (cid:104) u k ( t ) | ∂ k | u k ( t ) (cid:105) is the Berry connection, and its integration over thefirst BZ is the Zak phase in Eq. (61). The spatial shift of the Wannier function afterone pumping cycle at t = T can be represented by the change of the polarization ∆ P = (cid:82) T dt ∂P∂t , which can be obtained by using Stokes’s formula as∆ P = d π (cid:90) π/d − π/d dk (cid:90) T dt [ ∂ t A k ( k, t ) − ∂ k A t ( k, t )] . (66)Such an integral over two parameters corresponds to a topological invariant, i.e., the firstChern number C in a k - t BZ: C = 12 π (cid:90) π/d − π/d dk (cid:90) T dt F ( k, t ) , (67)with F ( k, t ) = ∂ t A k ( k, t ) − ∂ k A t ( k, t ) as the Berry curvature. Thus the shift of the34 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 (b) (c) (d)(f)(a) 𝛥𝛥𝛿𝛿 J 𝑡𝑡 = 0𝑡𝑡 = ⁄𝑇𝑇 4𝑡𝑡 = ⁄𝑇𝑇 2 𝑡𝑡 = ⁄3𝑇𝑇 4 𝑡𝑡 = 𝑇𝑇 cfde (e) 𝑪𝑪𝑪𝑪𝑪𝑪 J JJJ W ν = − W +1 ν = W ν = W +1 ν = Figure 7. (Color online) Topological pumping in the Rice-Mele OL. (a) A pump cycle sketched in δJ -∆ spacewith schematic of the pumping sequence. The pink shaded packet indicates the wave function of a particular atominitially localized at the unit cell. The wave function shifts to right as the pumping proceeds and moves the atomto the next unit cell after one pump cycle. (b) Results of four typical topological/trivial pumping (characterizedby the Chern number C = ± , 0) with schematic pumping sequences in the δJ -∆ plane shown in (c-f) and thewinding number ν w of each trajectory around the origin. (c) Charge pumped during a simple Rice-Mele pumping;(d) topologically nontrivial pumping; (e) topologically trivial pumping; and (f) negative sweep pumping. Reprintedby permission from Macmillan Publishers Ltd: Nakajima et al. [146], copyright c (cid:13) (2016). Wannier center after one pumping cycle is quantized in units of the lattice constant of aunit cell: ∆ P = Cd . This indicates that the transported particle in the adiabatic cyclicevolution is quantized and related to topological Bloch bands. The SSH/RM model canserve as a pictorial model for implementing the topological pumping [131].Although topological charge pumping was first proposed more than thirty years ago, ithas not yet been directly realized in condensed matter experiments. With ultracold atomsin tunable optical superlattices, the implementation of topological pumping has been ex-tensively discussed [119, 136, 147–155]. In the context of the SSH/Rice-Mele model, itwas theoretically demonstrated that the quantized particle pumping characterized by thenon-zero Chern number can be realized in the cold atom systems [119, 136, 148], which isrobust with respect to some perturbations in realistic experiments, such as nonzero tem-perature and the effects of finite sizes, non-adiabatic evolutions and trapping potentials.Moreover, as an extension of Thouless pumping, the topological pumping of interact-ing bosoinc and fermionic atoms trapped in OLs for specific models was also studied[147, 149, 151–153, 156]. For example, in the strongly interacting region, the bosonicatoms share the same transport properties as non-interacting fermions with quantizedtransport [147, 156]. Due to the degeneracy of the many-body ground states of the inter-acting bosons or fermions, the so-called topological fractional pumping (fractional valuesof the pumped particle) related to the many-body Chern number can be realized atcertain fractional fillings [149, 151–153].Recently, topological pumping has been realized in two experiments [146, 157] withultracold fermionic and bosonic atoms in 1D optical superlattices, respectively. In theexperiment in Ref. [146], an ultracold Fermi gas of ytterbium atoms Yb was loadedinto a dynamically controlled optical superlattice, occupying the lowest energy band.The superlattice is formed by superimposing a long lattice V L and a short lattice V S with periodicity difference by a factor of two, and its time-dependent potential takes theform V ( x, t ) = − V S ( t )cos (cid:18) πxd (cid:19) − V L ( t )cos (cid:16) πxd − φ ( t ) (cid:17) , (68)where φ is the phase difference between the two lattices. By slowly sweeping φ over time,the lattice potential returns to its initial configuration whenever φ changes by π , thuscompleting a pumping cycle. The ability to tune all parameters of the lattice potential35 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 independently in a dynamic way offers the opportunity to realize various pumping pro-tocols. In the absence of a static short lattice, V ( x, t ) describes the simple sliding latticethat Thouless originally proposed [129]. In addition to this term, the double well lat-tice shown in Fig. 7 is realized, which can be described by the tight-binding Rice-MeleHamiltonian in Eq. (62) for the alternating pumping. Figure 7 shows the schematics ofthe continuous pumping protocol as a closed trajectory in the δJ -∆ parameter plane,where δJ ≡ ( J − J (cid:48) ) / φ ( t ) = πt/T , one can periodically modulate the hoppingamplitudes and on-site energies. In the experiment, topological pumping was detectedas a shift of the center-of-mass of the atomic cloud measured with in situ imaging andthe first Chern number C of the pumping procedure was extracted from the averageshift of the center-of-mass per pumping cycle, which is consistent with the ideal value C = 0 , ± 1. The topological nature of the pumping was revealed by the pumping trajec-tories’ clear dependence on the topology in parameter space (denoted by the windingnumber ν w ) as to whether or not a trajectory encloses the degenerate point. Pumpingin the sliding lattice was demonstrated to be topologically equivalent to the continuousRice-Mele pumping because of the same Chern numbers of the first band, which can beconnected by a smooth crossover without closing the gap to the second band. It was alsoverified that the topological pumping indeed works in the quantum regime by varyingthe pumping speed and the temperature in the experiment [146].In another experiment [157], the authors realized the topological pumping with ultra-cold bosonic atoms Rb forming a Mott insulator in a similar dynamically controlledoptical superlattice. Due to the large on-site interaction, each atom is localized on anindividual double well, resulting in homogeneous delocalization over the entire first BZ.By taking in situ images of the atomic cloud in the lattice, they also observed a quantizeddeflection per pump cycle. The genuine quantum nature of the pumps was revealed by acounterintuitive reversed deflection of particles in the first excited band and a controlledtopological transition in higher bands when tuning the superlattice parameters.Also with bosonic atoms, a quantum geometric pump for a BEC in the lowest Blochband of a tunable bipartite magnetic lattice was realized [158]. In contrast to the topolog-ical pumping yielding quantized pumping set by the global topological properties of thefilled bands, the geometric pumping for a BEC occupying just a single crystal momen-tum state exhibits non-quantized particle pumping set by local geometrical propertiesof the band structure. For each pump cycle, a non-quantized overall displacement anda temporal modulation of the atomic wave packet’s position in each unit cell (i.e., thepolarization) were observed [158].These cold atom systems can be extended to implement more complex quantum pump-ing schemes, including spin degrees of freedom and in higher dimensions [159]. Analogousto Thouless pumping, topological pumping for spins without a net transport of chargemay be constructed by imposing the TRS [160]. A spin pump with spin conservation canbe composed of two independent pumps, where up and down spins have inverted Berrycurvature and are transported in opposite directions. Spin pumps could serve as spincurrent sources for spintronic applications.In a recent experiment [159], quantum spin pumping was implemented with ultracoldbosonic atoms in two hyperfine states in a spin-dependent dynamically controlled opticalsuperlattice, where each spin component is localized to a Mott insulator with negligibleinterspin interaction. In addition, the two spin components are coupled via spin-isotropicon-site interactions. For strong interactions U (cid:29) J (tunneling J is suppressed) and unit36 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 (b)(a) Figure 8. (Color online) Spin pumping in a spin chain of cold atoms. (a) A spin pump cycle in parameter space(green) of spin-dependent tilt ∆ and exchange coupling dimerization δJ ex . The path can be parametrized by theangle φ , acting as the pump parameter. The insets in the quadrants show the local mapping of globally tilteddouble wells to the corresponding local superlattice tilts with the black rectangles indicating the decoupled doublewells. Between φ = 0 and π , |↑(cid:105) and |↓(cid:105) spins exchange their position, which can be observed by site-resolved bandmapping images detecting the spin occupation on the left (L) and right (R) sites. (b) Center-of-mass position ofup (red) and down (blue) spins as a function of φ . Different absorption images of both sequences for |↑(cid:105) and |↓(cid:105) spins are shown on the right side. The solid lines depict the calculated motion of a localized spin for the idealcase (light gray) and for a reduced ground state occupation and a pump efficiency per half pump cycle that wasdetermined independently through a band mapping sequence (gray). Reprinted with permission from Schweizer et al. [159]. Copyright c (cid:13) (2016) by the American Physical Society. filling, the system can be described by a 1D spin chain [159]: H SP = − (cid:88) n [ J ex + ( − n δJ ex ] (cid:0) S + n S − n +1 + h.c. (cid:1) + ∆2 (cid:88) n ( − n S zn (69)with spin-dependent tilt ∆ and alternating exchange coupling ( J ex ± δJ ex ). For largetilts ∆ (cid:29) ( J ex + δJ ex ) the many-body ground state forms an antiferromagnetic orderedspins, while for strong exchange coupling ( J ex + δJ ex ) (cid:29) ∆ dimerized entangled pairsare favored. Varying δJ ex and ∆ during the pump cycle modulates ( δJ ex , ∆) in theinteracting 1D spin chain and encircles the degeneracy point, as shown in Fig. 8(a). Aftera full cycle, the two spin components move by a lattice site in opposite directions. Thisleads to a quantized spin transport described by the Z invariant as in the topologicallyequivalent case of independent spins [160]. Such a spin pump can be regarded as adynamical version of the quantum spin Hall effect, where the parameter φ (the phase ofthe superlattice) is an additional dimension in a generalized 2D momentum space. Theatomic spin current was measured and the net spin transport was further verified through in situ measurements of the spin-dependent center-of-mass displacement, as shown in Fig.8(b). The topological or trivial character of a 1D insulator is completely determined by thepresence or absence of chiral symmetry [61, 123, 161]. There are two distinct classes of1D topological insulators, as chiral symmetry is the composition of time-reversal andcharge-conjugation (particle-hole) symmetries. The first class is invariant under both thetwo symmetries and is called the BDI symmetry class represented by the SSH model.The second one is the AIII class with broken time-reversal and charge-conjugation sym-metries, which still lacks experimental realization in condensed matter materials. Thetopology of the AIII class phase is quantified by an integer winding number. Recently,several works have been presented to study the 1D AIII class topological insulator usingcold fermionic atoms in OLs [162–165].A simple dimerized lattice model for realizing the AIII class topological insulator was37 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401Figure 9. (Color online) Proposed optical Raman lattice for realizing the Hamiltonian (71). (a) Cold fermionstrapped in 1D optical lattice with internal three-level Λ-type configuration coupled to radiation. (b) Energy spectrawith open boundary condition in the topological (∆ = 0) and trivial (∆ = 3 J s ) phases. The SO coupled hopping J so = 0 . J s . Reprinted with permission from Liu et al. [162]. Copyright c (cid:13) (2013) by the American Physical Society. proposed in Ref. [163]. The Hamiltonian is given by H AIII = − (cid:88) n (cid:16) J a † n b n + J (cid:48) e iθ a † n b n − + h.c. (cid:17) , (70)which is a generalization of the SSH model by introducing an acquired complex phasefactor e iθ ( θ is a phase difference between the inter-cell and extra-cell hoppings) whenparticles tunnel from one unit cell to the next. It is worth noting that for open boundaryconditions, the Hamiltonian H AIII in Eq. 70 is equivalent to H SSH in Eq. 59 through thegauge transformations, ( a † n , b † n ) → e − inθ ( a † n , b † n ). However, under the periodic boundaryconditions, it is equivalent to the SSH model in a ring of N lattice sites threaded by amagnetic flux φ = N θ . Therefore, under the periodic boundary conditions, this Hamilto-nian still corresponds to the SSH model for θ = 0, while in the presence of this extra phase θ (cid:54) = 0 , π (and N θ (cid:54) = 2 mπ with m an integer), the model enters the AIII symmetry class.This model can be realized with spinless cold atoms by combining a 1D optical super-lattice with the Raman assisted tunneling [163]. The Bloch Hamiltonian in this model isgiven by H AIII ( k ) = − [ J + J (cid:48) cos( k − θ )] σ x − J (cid:48) sin( k − θ ) σ y , which exhibits chiral symme-try for any value of θ since σ z H AIII ( k ) σ z = −H AIII ( k ). For θ (cid:54) = 0 , π , it is not time reversalsymmetric and thereby not charge-conjugation symmetric because H ∗ AIII ( − k ) (cid:54) = H AIII ( k )and no 2 × U such that U H ∗ AIII ( − k ) U † = H AIII ( k ) in this case.Another proposed model of 1D AIII class topological insulator is using spin-orbit-coupled fermionic atoms in an optical Raman lattice [162]. The atoms with the inter-nal three-level Λ-type configuration are coupled through the transitions | g ↑ (cid:105) , | g ↓ (cid:105) → | e (cid:105) driven by the laser fields with Rabi-frequencies Ω ( x ) = Ω sin( k x/ 2) and Ω ( x ) =Ω cos( k x/ | ∆ d | (cid:29) Ω and a small two-photon detuning 2 | ∆ | (cid:28) Ω for the transitions, the systemHamiltonian reads H = H + H , with H = (cid:80) σ = ↑ , ↓ (cid:2) p x m + V σ ( x ) (cid:3) | g σ (cid:105)(cid:104) g σ | + 2 (cid:126) ∆ | g ↓ (cid:105)(cid:104) g ↓ | ,and H = (cid:126) ∆ d | e (cid:105)(cid:104) e | − (cid:126) (cid:0) Ω | e (cid:105)(cid:104) g ↑ | + Ω | e (cid:105)(cid:104) g ↓ | + H . c . (cid:1) . Here the potentials V ↑ , ↓ ( x ) = − V sin ( k x ) form a 1D spin-independent OL. For | ∆ d | (cid:29) Ω , the lasers Ω , induce atwo-photon Raman transition between | g ↑ (cid:105) and | g ↓ (cid:105) . The effect of the small two-photondetuning is equivalent to a tunable Zeeman field along z axis Γ z = (cid:126) ∆. Eliminating theexcited state by | e (cid:105) ≈ (Ω ∗ | g ↑ (cid:105) + Ω ∗ | g ↓ (cid:105) ) yields the effective Hamiltonian H eff = p x m + (cid:88) σ = ↑ , ↓ (cid:2) V σ ( x ) + Γ z σ z (cid:3) | g σ (cid:105)(cid:104) g σ |− (cid:2) M ( x ) | g ↑ (cid:105)(cid:104) g ↓ | + H . c . (cid:3) , (71)where M ( x ) = M sin( k x ) with M = (cid:126) Ω / d represents a transverse Zeeman field38 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 induced by the Raman process. In the tight-binding regime, the system Hamiltonian canbe recast into [162]˜ H AIII = − J s (cid:88) ( c † i ↑ c j ↑ − c † i ↓ c j ↓ ) + (cid:88) j Γ z ( n j ↑ − n j ↓ ) + (cid:88) j J so ( c † j ↑ c j +1 ↓ − c † j ↑ c j − ↓ ) + H . c .. (72)Here time reversal and charge conjugation operators are respectively defined by ˆ T = i ˆ Kσ y with ˆ K being the complex conjugation, and ˆ C : ( c σ , c † σ ) (cid:55)−→ ( σ z ) σσ (cid:48) ( c † σ (cid:48) , c σ (cid:48) ). The topo-logical phase in this free-fermion system belongs to the chiral AIII class because whileboth ˆ T and ˆ C are broken in ˜ H AIII , the chiral symmetry, defined as their product, is re-served since ( ˆ C ˆ T ) ˜ H AIII ( ˆ C ˆ T ) − = ˜ H AIII , with ( ˆ C ˆ T ) = 1. In particular, this Hamiltoniandescribes a 1D topological insulator for | Γ z | < J s with two mid-gap zero edge modes andotherwise a trivial insulator, with the bulk gap E g = min {| J s − | Γ z || , | J so |} , as shownin Fig. 9(b). It was shown that the zero edge modes in this 1D AIII topological insulatorare spin polarized, with left and right edge spins polarized to opposite directions andforming a topological spin qubit of cold atoms [162]. A similar Raman lattice schemewas proposed to simulate symmetry-protected topological states using alkaline-earth-likeatoms [164]. The interaction-driven topological phase transition for interacting fermionicatoms and a Z reduction of the 1D AIII class was also studied [162, 164].A recent experiment [165] was reported to realize the 1D symmetry-protected topolog-ical state with cold fermionic atoms of Yb in the optical Raman lattice. The exper-imental setup is similar as that proposed in Ref. [162], except that the potential formsa spin-dependent lattice with spin-dependent hopping strengths that explicitly breakthe locally defined chiral symmetry. In this case, the topological phase is protected bya magnetic group symmetry (defined as the product of time-reversal and mirror sym-metries) and a nonlocal chiral symmetry. The topology of the cold atom system wasmeasured via Bloch states at symmetric momenta and the spin dynamics after a quenchbetween trivial and topological phases [165]. This work may open the way to explore thesymmetry-protected topological states with ultracold atoms, including the chiral AIIIclass by considering spin-independent rather than spin-dependent OLs. Further general-ization of this study to higher dimensional systems or interacting regimes also offers thesimulation of quantum phases beyond natural conditions in solid-state materials. Initially introduced in the context of lattice gauge theory, the Creutz model [166] hasgained a foothold in condensed matter physics as a versatile model to study fractional-ization, Dirac fermions, and topological phases [167]. The model describes free fermionshopping on a two-leg ladder pierced by a π magnetic flux. The Creutz ladder shown inFig. 10(a) is described by the following Hamiltonian H CL = 12 (cid:88) n (cid:2) Ω c † n σ x c n + c † n +1 ( iJ σ z − J σ x ) c n (cid:3) + H . c ., (73)where n is the site index containing up and down legs as the spin basis for the Pauli matri-ces, with the particle annihilation operators c n = ( c nu , c nd ). The fermions can jump fromone site to the nearest-neighboring site within the same leg with a complex amplitude ± iJ , i.e., gaining or losing a Peierls phase π/ 2. The fermions can hop between sites withamplitude J while flipping between the legs, mimicking an SOC, and also horizontallyalong the legs with amplitude Ω, mimicking a Zeeman field along x axis. The correspond-ing Bloch Hamiltonian is given by H CL ( k ) = J sin kσ z + ( w − J cos k ) σ x . Consequently,39 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 n +1 n +2 n -1 n Up legDown leg iJ - iJ Ω - J (a) … … J (b) k π Figure 10. (Color online) (a) The Creutz ladder with up and down legs. Each lattice site encompasses a verticalbond with two leg states, containing an on-site coupling Ω and a leg-conserving hopping ± iJ and a leg-fliphopping − J , respectively. (b) Energy dispersion of the Creutz ladder model with gapped Dirac cones at k = 0and π for Ω (cid:54) = ± J . The 1D Dirac points exhibit when Ω = ± J . there are two bands with the energy dispersion E ± = ± (cid:112) ( J sin k ) + (Ω − J cos k ) , asshown in Fig. 10(b). For periodic boundary conditions, the bands display a pair of mas-sive Dirac fermions with different Wilson masses m = J − Ω at k = 0 and m π = J + Ωat k = π . When J = Ω ( J = − Ω), the system is gapless with a Dirac cone at momentum k = 0 ( k = π ). When both J and Ω vanish, the system exhibits two Dirac cones.The Creutz ladder model is classified in the BDI class of topological insulators sincethe Hamiltonian has the particle-hole symmetry σ z H ∗ CL ( k ) σ z = −H CL ( − k ), the TRS σ x H ∗ CL ( k ) σ x = H CL ( − k ), and a chiral symmetry represented by σ y with {H CL ( k ) , σ y } =0. Therefore, the Creutz ladder is in the same symmetry class as the SSH model, ex-hibiting nontrivial band topology and edge states. In the SSH model, there are only twophases with different dimerization configurations. In the Creutz ladder model of fullygapped insulators, there are three phases distinguished by their bulk band topology ascharacterized by the Zak phase for all values of the parameters ( J , Ω): a trivial insu-lator with ϕ Zak = 0 when | J / Ω | < 1, two nontrivial insulators with ϕ Zak = π when | J / Ω | < J > 0, and with ϕ Zak = − π when | J / Ω | < J < 0. Equivalently,the band topology can be characterized by the winding number ν w defined from the com-plex phase of the Bloch vectors: ν w = (cid:2) sgn(Ω + J ) − sgn(Ω − J ) (cid:3) . In the topologicallynontrivial phase there are two zero-energy bound states at the edges of the ladder withhalf fractional particle numbers.Several recent works proposed schemes to realize and study the topological propertiesof the Creutz ladder model with cold atoms [155, 168–173]. The general scheme of usingcold atoms with artificial SOCs in a ladder-like OL under a synthetic magnetic field wassuggested for simulation of the Creutz ladder [168]. The experimental setups capable ofimplementing the tunable Creutz ladder model were proposed in Refs. [155, 170, 172].Several protocols that can be used to extract the topological properties in the Creutzmodel from atomic density and momentum distribution measurements as well as topolog-ical quantum pumping were presented [155, 169, 170]. By engineering a quantum walkwith cold atoms, one can observe the topological phases and the bound states in theCreutz model [171]. By adding an energy imbalance between the two legs of the ladder,the symmetry class of the topological insulator changes from BDI to AIII [172]. More-over, the interaction-induced topological phase transition in the presence of interatomicinteractions in the optical Creutz ladder were investigated [170, 172]. A topological insu-lating phase protected by the inversion symmetry was found in a three-leg ladder model[173]. Notably, the realization of an optical two-leg ladder for ultracold bosonic atoms ex-posed to a uniform artificial magnetic field created by laser-assisted tunnelling has beenexperimentally achieved [174]. In the experiment, the atomic current on either leg of theladder and the momentum distribution were observed for demonstrating chiral Meissner-like edge currents. Very recently, an experimental realization of a three-leg chiral ladder40 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401Figure 11. (Color online) Energy spectrum of the Aubry-Andr´e-Harper model as a function of the phase φ for(a) α = 1 / V = J = 1; and (b) α = ( √ / V = 0 . J = 1 under the open boundary condition in a latticeof site L = 98. The density distribution of two typical in-gap edge states (red dotted lines) is depicted. with ultracold fermionic atoms was reported [175], where the legs were formed by theorbital states of a 1D optical lattice and the complex inter-leg links were generated bythe orbital-changing Raman transitions. Recently, the topological properties of 1D quasiperiodic lattices have been theoreticallyand experimentally revealed in the context of cold atoms and photonic quasicrystals[176, 177]. The system exhibits topological edge states and nontrivial Chern numbers,equivalent to those of 2D quantum Hall systems on periodic lattices. The tight-bindingHamiltonian of the 1D quasiperiodic lattices takes the form: H AAH = − J (cid:88) n ( c † n c n +1 + H . c . ) + (cid:88) n V n c † n c n , (74)where V n = V cos(2 παn + φ ) is the spatially modulated on-site potential with V beingthe strength, φ being the tunable modulation phase, and α controlling the periodicityof the modulation. When α is rational (irrational), the modulation is commensurate(incommensurate). Alternatively, the system Hamiltonian can be rewritten as H AAH ψ n = − J ( ψ n +1 + ψ n − ) + V cos(2 παn + φ ) ψ n , (75)where ψ n is the wave function at site n . Historically, this model is known as the Aubry-Andr´e model [178] or Harper model [179].Figure 11 shows the energy spectrum of the Aubry-Andr´e-Harper model as a functionof the modulation phase φ for a finite lattice of the length L under the open boundarycondition. In the commensurate potential ( α = 1 / φ varies from 0 to 2 π . The position of the edge states in the gaps also varies continuouslywith the change of φ , which is localized either on the left or on the right boundary ofthe system. In the incommensurate potential ( α = ( √ / − J x ( ψ n − + ψ n +1 ) − J y cos(2 παn − k y ) ψ n = E ( k y ) ψ n ,41 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 where J x ( J y ) is the hopping amplitude along the x ( y ) direction. By substitutions of J → J x , V → − J y , and φ → − k y , the current 1D problem can be mapped to the latticeversion of the 2D integer QHE problem. Adiabatically varying φ from 0 to 2 π for eachBloch band forms an effective 2D manifold of Hamiltonian H ( k, φ ) in the k - φ parameterspace, where the first Chern number can be defined.For cold atoms, the Aubry-Andr´e-Harper model has been experimentally realized in 1Doptical superlattices for studying localization phenomena [73, 180]. In the experiments,the quasiperiodic lattices were created using a primary 1D OL V and an additionalweak lattice V , with the wave numbers k and k , respectively. For deep potentials, theatomic system is governed by the tight-binding Hamiltonian H = H + H , with H = − J (cid:80) n ( c † n +1 c n + c † n − c n ) from the primary lattice, and H = V (cid:80) n c † n c n cos(2 παn + φ )from the interference of the perturbation lattice, where V ∼ J (cid:28) V , α = k /k , and φ is the tunable relative phase. It was suggested that the energy spectrum and the Chernnumbers can be revealed by observing the density profile of trapped fermionic atoms [176],which display plateaus with their positions uniquely determined by varying the parameter α of the optical superlattices. Another method to create and study quasiperiodic OLsunderlying all quasicrystals by the abstract cut-and-project construction was recentlyproposed [181].It was shown that the commensurate off-diagonal Aubry-Andr´e-Harper model is topo-logically nontrivial in the gapless regime and supports zero-energy edge modes [182],which is attributed to the topological properties of the 1D Majorana chain of Z class.By generalizing the spinless Aubry-Andr´e-Harper model to a spinfull version, which canbe realized with spin-1/2 atoms in a spin-dependent quasiperiodic OL, one can realize theZ topological insulators and the topological spin pumping [183]. The phases of ultracoldspin-1/2 bosons with SOCs in the quasiperiodic optical lattice were also studied [184]. Inthe presence of the pairings or interactions, the topological superconducting phase withMajorana end modes [185–187], the fractional topological phases connecting to the 2Dfractional QHE [188], and the topological Bose-Mott insulators [189] in the quasiperiodiclattices have been theoretically investigated. Two-dimension The graphene material, formed with a single layer of carbon atoms arranged in a honey-comb lattice with its low-energy quasipaticles described by the massless Dirac equation[190], has recently attracted strong interest in condensed-matter physics. The crystalstructure of graphene, as shown in Fig. 12(a), consists of sublattices A (solid) and B(open). Its energy spectrum is shown in Fig. 12 (b), where two inequivalent points de-noted as K and K (cid:48) are Dirac points. At a Dirac point, two energy bands intersect linearlyand the quasiparticles in the vicinity of these points are described by the Hamiltonian H D and are frequently called “Dirac fermions”, where H D is the Dirac Hamiltonian intwo spatial dimensions given by H D = v x σ x p x + v y σ y p y + ∆ g σ z . (76)Here σ x,y,z are the three Pauli matrices. Compared with the standard energy-momentumrelation for the relativistic Dirac particles, here ∆ g and v x,y denote rest energy andthe effective velocity of light respectively. For ∆ g (cid:54) = 0, the energy spectrum with a gapdenotes the massive Dirac fermions , as shown in Fig. 12(b). In graphene, by contrast,the Dirac points appear at the corners ( K and K (cid:48) points) where the dispersion relation of42 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 the honeycomb lattice shows the conical intersection between the first and second bands.Here, the low-energy fermionic excitations are the massless Dirac fermions described bythe Dirac Hamiltonian (76) with ∆ g = 0, v x = ± v y = ± v . Moreover, the associatedBerry phases of the Dirac points are ± π [140, 190, 191] where the corresponding Berrycurvatures have the form Ω n = ± πδ (cid:0) k − K n (cid:1) . (77)Here K n denotes the Dirac points K or K (cid:48) . One can see that the Berry curvatures tendto infinity at Dirac points. Here the Dirac points with the quantization of Berry phase(to 0 and π ) require symmetry protection, which can be the composition of inversion andtime reversal symmetries. This is in contrast to Weyl points in 3D (see Sec. 4.3.2), whichdo not require any symmetry protection at all. J J J AB (a) K' K (b) (c) Figure 12. (Color online) (a) Crystal structure of a honeycomb lattice consisting of sublattices A (solid) andB (open). The nearest-neighbor hopping amplitudes denoted as J , J , J corresponding to the three differentdirections. (b) Energy bands of the low-energy excitations with a gap. The first BZ is outlined by the dashed line,and two inequivalent valleys are labelled K and K (cid:48) , respectively. Reprinted with permission from Xiao et al. [192].Copyright c (cid:13) (2007) by the American Physical Society. (c) The energy spectrum (dashed) and Berry curvature(solid) of the conduction bands of a honeycomb lattice with broken inversion symmetry. Since the relativistic Dirac fermions were found in graphene, a substantial amountof effort has been devoted to the understanding of exotic relativistic effects in solid-state systems and other artificial quantum systems [14, 23, 52]. Given these excitingresults and the state-of-the-art technologies in quantum control of atoms, one topic thatnaturally arises is how to mimic the graphene and the relativistic quasiparticles withcold atoms in a similar 2D hexagonal lattice [76, 193–196]. In cold atomic systems, it iseasy to realize the anisotropy Hamiltonian with mass term ∆ g (cid:54) = 0 and v x (cid:54) = v y due tothe highly controllable experimental parameters [76]. Furthermore, the detection of theBerry curvature can spread over a finite range, which provides a feasible way to measurethe Berry phase over the first BZ in reciprocal space [140]. In addition to the honeycomblattice, Dirac fermions can emerge in some lattices of other geometric structures [197–205]. Figure 13. (Color online) The honeycomb optical lattices. (a),(b) The contours with three potentials described inEq. (78). The minima of the potentials are denoted by the solid dots. All V j are the same in (a), and V = V =0 . V in (b). The dispersion relations are shown in (c) for β = 1 (gapless state) and (d) for β = 2 . et al. [76]. Copyright c (cid:13) (2007) by the American Physical Society. Simulating Dirac equations with cold atoms loaded in a honeycomb OL was initiallyproposed in Ref. [76]. In the proposal, single-component fermionic atoms (e.g., spin-43 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 polarized atoms K, Li, etc.) trapped in a 2D ( x - y plane) hexagonal OL are consid-ered. The hexagonal OL is constructed by three standing-wave laser beams with thecorresponding potential V ( x, y ) = (cid:88) j =1 , , V j sin (cid:104) k L ( x cos θ j + y sin θ j ) + π (cid:105) , (78)where θ = π/ θ = 2 π/ θ = 0, and k L is the optical wave vector. It is easy totune the potential barriers V j by varying the laser intensities along different directionsto form a standard hexagonal lattice for V = V = V , or a hexagonal lattice with afinite anisotropy for different V j as depicted in Fig. 13(a) and 13(b), respectively. Thetight-binding Hamiltonian of the system is then given by H = − (cid:88) (cid:104) i,j (cid:105) J ij ( a † i b j + H . c . ) , (79)where (cid:104) i, j (cid:105) represents the neighboring sites, a i and b j denote the fermionic mode oper-ators for the sublattices A and B, respectively. The tunneling amplitudes J ij depend onthe tunneling directions in an anisotropic hexagonal lattice, denoted as J , J , and J corresponding to the three different directions as illustrated in Fig. 12(a). For simplicity,assume J = J = J and J = βJ with β being the anisotropy parameter. As the atomictunneling rate in an OL is exponentially sensitive to the potential barrier, this controlprovides an effective method to control the anisotropy of the atomic tunneling by laserintensities. The first BZ of this system also has a hexagonal shape in the momentumspace with only two of the six corners in Fig. 12(b) being inequivalent, corresponding totwo different sites A and B in each cell in the real hexagonal lattice, usually denoted as K and K (cid:48) . One can choose K = (2 π/a )(1 / √ , 1) and K (cid:48) = − K , where a = 2 π/ ( √ k L )is the lattice spacing. Taking a Fourier transform a † i = (1 / √ N ) (cid:80) k exp( i k · A i ) a † k and b † j = (1 / √ N ) (cid:80) k exp( i k · B j ) b † k , where A i ( B j ) represents the position of the site insublattice A ( B ) and N is the number of sites of the sublattice, the Hamiltonian (79)can be diagonalized with the expression of energy spectrum[76] E k = ± J (cid:113) β + 2cos( k y a ) + 4 β cos( √ k x a/ k y a/ . (80)As plotted in Fig. 13(c) and 13(d), there are two branches of the dispersion relation,corresponding to the ± sign in Eq. (80). When 0 < β < 2, the two branches touchwith each other, and a Dirac cone structure appears around the touching points. It hasthe same standard Dirac cones as the graphene material with β = 1 [190, 191, 206].The Dirac cones squeeze in the x or y direction when β deviates from 1, but they stilltouch each other. When β > 2, a finite energy gap ∆ g = | J | ( β − 2) appears betweenthe two branches. So, across the point β = 2, the topology of the Fermi surface changes,corresponding to a quantum phase transition without breaking any local symmetry [207].Such a topological phase transition associated with the production or annihilation of apair of Dirac points has been investigated in Ref.[208]. The evolution of the Dirac pointsin the hexagonal lattice by varying the asymmetric hopping and the resulting phasetransition was also studied in Ref.[194]. With this phase transition, the system changesits behavior from a semimetal to an insulator at the half filling case (which means oneatom per cell; note that each cell has two sites). Around half filling, the Fermi surfaceis close to the touching points, and one can expand the momentum k around one of thetouching points K ≡ ( k x , k y ) as ( k x , k y ) = ( k x + q x , k y + q y ). Up to the second order of44 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 q x and q y , the energy spectrum (80) becomes E q = ± (cid:113) ∆ g + v x q x + v y q y , (81)where ∆ g = 0, v x = √ βJ a/ 2, and v y = J a (cid:112) − β / < β < 2; ∆ g = | J | ( β − v x = J a (cid:112) β/ 2, and v y = J a (cid:112) β/ − β > 2. This simplified energy sprectrum E q is actually a good approximation (named long wavelength approximation) as long as q x , q y (cid:46) / a . The wave function for the quasiparticles around half filling then satisfiesthe Dirac equation i (cid:126) ∂ t Ψ = H D Ψ, where the relativistic Hamiltonian H D is the DiracHamiltonian with the form H D = τ z v x σ x q x + v y σ y q y + ∆ g σ z , (82)where τ z = ± k ): it is an odd function in thepresence of TRS and even in the presence of inversion symmetry, as shown in Fig. 12(c).From Eq. (82), we can obtain the Berry curvature near the valleys for the conductionband[192] Ω( q ) = τ z v x v y ∆ g g + v x v y q ) / . (83)Through an analogy to the graphene physics, one can realize both massive and masslessDirac fermions and observe the phase transition between them by controlling the latticeanisotropy [76]. This proposal was demonstrated to be experimentally feasible in Ref.[195], where the temperature requirement and critical imperfections in the laser configu-ration were considered in detail. Even in the presence of a harmonic confining potential,the Dirac points are found to survive [209]. In the presence of atomic interactions, themany-body physics of Dirac particles in graphene-type lattices, such as novel BCS-BECcrossover [210], topological phase transition between gapless and gapped superfluid [196]and even charge and bond ordered states with the p -orbital band of lattices [193, 211],have been investigated. Notably, a honeycomb lattice has been realized and investigatedusing a BEC [212, 213], but no signatures of Dirac points were observed. Even so, theseimportant theoretical works pave the way for mimicking relativistic Dirac fermions andthe aforementioned beyond-graphene physics with controllable systems. Figure 14. (Color online) Optical lattice with adjustable geometry. (a) Three retro-reflected laser beams createthe 2D lattice potential of Eq.(84). (b) The real-space potential of the honeycomb lattice. (c) Left: sketch of thefirst and second BZs of the honeycomb lattice, indicating the positions of the Dirac points. Right: the energyspectrum of honeycomb lattice in the first BZ showing the linear intersection of the bands at the two Dirac points.The color scale illustrates lines of constant energy. W and E G denote the full bandwidth and the minimum energygap at the edges of the BZ, respectively; q B = 2 π/λ is the Bloch wavevector. Reprinted by permission fromMacmillan Publishers Ltd: Tarruell et al. [77], copyright c (cid:13) (2012). Subsequently, an experiment to realize Dirac points with adjustable properties usingsingle-component ultracold fermionic atoms in a tunable honeycomb OL was reported45 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 in Ref. [77]. Furthermore, an artificial graphene consisting of a two-component ultracoldatomic Fermi gas with tunable interactions was realized [214]. To create and manipulateDirac points, the authors studied an ultracold Fermi gas of K atoms in a 2D tunableOL [77]. In the experimental setup, three retro-reflected laser beams of wavelength λ =1 , X and Y can form a chequerboard latticeof spacing λ/ √ 2. The third beam ¯ X , collinear with X but detuned by a frequency δ ,creates an additional standing wave with a spacing of λ/ 2. The yielding potential takesthe form V ( x, y ) = − V ¯ X cos ( kx + θ/ − V X cos ( kx ) − V Y cos ( ky ) − α (cid:112) V X V Y cos( kx )cos( ky )cos( ϕ ) , (84)where V ¯ X , V X and V Y denote the single-beam lattice depths, α is the visibility of theinterference pattern and k = 2 π/λ . Varying the relative intensities of the beams canrealize various lattice structures, such as the chequerboard, triangle, square and honey-comb lattices. We focus on the honeycomb lattice with real-space potential as shown inFig. 14(b). The primitive lattice vectors are perpendicular, leading to a square BZ withtwo Dirac points inside, as shown in Fig. 14(c). This lattice is called a brick-wall lattice,which is topologically equivalent to the honeycomb lattice.The Dirac points here were characterized by probing the energy split between the twolowest-energy bands through inter-band transitions. They are topological defects in theband structure with the associated Berry phases ± π , which guarantee their stability whilea perturbation only moves the positions of Dirac points. However, breaking the inversionsymmetry of the potential by introducing an energy offset ∆ between the sublatticesopens an energy gap at the Dirac points, as shown in the insets of Fig. 15(a). The bandstructure can be measured with the Bloch-Landau-Zener-oscillation technique [77, 215,216] (see Sec. 5.1), and the results are plotted in Fig. 15(a), where the total fractionof atoms transferred to the second band ξ is plotted as a function of the detuning δ .The maximum indicates the point of inversion symmetry, where ∆ = 0 and the gap atthe Dirac point vanishes. Therefore, one can identify the points of maximum transferwith the Dirac points. To investigate how breaking the inversion symmetry of the latticeaffects the Dirac points, the authors varied the sublattice offset ∆, which is controlledby the frequency detuning δ between the lattice beams, and measured the total fractionof atoms transferred ξ . The population in the second band decreases symmetrically onboth sides of the peak as the gap increases, indicating the transition from massless tomassive Dirac fermions.The position of the Dirac points inside the BZ and the slope of the associated lineardispersion relation are determined by the relative strength of the tunnel couplings (i.e., J /J , J /J ) [76, 194, 217], which can be adjusted simply by controlling the intensityof the laser beams. Therefore, it was observed that the positions of the Dirac pointscontinuously approach the corners of the BZ when the tunnelling in the x directiongradually increases by decreasing the intensity of ¯X. When they reach the corners ofthe BZ, the two Dirac points merge, annihilating each other. Beyond this critical point,a finite band gap appears for all quasimomenta of the BZ. This situation signals thetransition between band structures of two different topologies, one containing two Diracpoints and the other containing none. This corresponds to a Lifshitz phase transition froma semimetallic phase to a band-insulating phase in 2D honeycomb lattices at half-filling[76, 194]. The topological transition line was experimentally mapped out by recording thefraction of atoms transferred to the second band, ξ , as a function of the lattice depths V ¯X and V X , while keeping the value of V Y /E R . The results are shown in Fig. 15(b). There the46 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401Figure 15. (Color online) Energy offset and topological transition. (a) The total fraction ξ of atoms transferredto the second band as a function of the detuning δ , which controls the sublattice energy offset ∆. The maximumindicates the Dirac point. Insets: away from the peak, the atoms behave as Dirac fermions with a tunable mass.Solid line is a Gaussian fit to the data.The topological transition occurs along q x (b) and q y (c) directions inthe quasi-momentum space. The dashed line is the theoretical result for the transition line, and the dotted lineindicates the transition from the triangular lattice to the dimer lattice. The bottom diagrams show cuts of theband structure along the q x axis ( q y ;(b)) and q y axis ( q x ; (c)) for the values of V X and V ¯ X indicated. Reprintedby permission from Macmillan Publishers Ltd: Tarruell et al. [77], copyright c (cid:13) (2012). onset of population transfer to the second band signals the appearance of Dirac pointsin the band structure of the lattice. For a given value of V X , the transferred fraction, ξ , decreases again for large values of V ¯X , as the Dirac points lie beyond the momentumwidth of the cloud. The ξ as a function of q y for a 1D lattice structure ( V ¯X (cid:29) V X ) wasobtained in Fig. 15(c), where the transition line is clearly demonstrated. This work opensthe way to realize and investigate other topological models with cold atoms in OLs, suchas the Haldane model [13] and Kane-Mele model [18, 19] to be addressed in Sec. 4.2.3and 4.2.4, respectively.After the realization of the Dirac points in the honeycomb OL using ultracold atoms,a further experiment to detect the π Berry flux located at each Dirac point by realizingan atomic interferometer was reported [140]. The idea of detecting the Berry flux isanalogous to using an Aharonov-Bohm interferometer to measure a magnetic flux inreal space. As we know, the Aharonov-Bohm effect describes a charged particle wavepacket being split into two parts that encircle a given area in real space [Fig. 16(a)].Any magnetic flux through the enclosed area gives rise to a measurable phase differencebetween the two components. In analogy to the magnetic field, the Berry curvature Ω n fora single Bloch band in the reciprocal space can be probed by forming an interferometeron a closed path in reciprocal space [Fig. 16(b)]. The geometric phase acquired along thepath can be calculated from the Berry connection A n ( k ), which is given by A n ( k ) = (cid:104) u n k | i ∇ k | u n k (cid:105) . Here, | u n k ( r ) (cid:105) is the cell-periodic part of the Bloch wave function | ψ n k ( r ) (cid:105) = e i k · r | u n k ( r ) (cid:105) with quasimomentum k in the n th band. Accordingly, the phase along aclosed loop in reciprocal space is ϕ Berry = (cid:73) L A n ( k ) d k = (cid:90) S Ω n ( k ) d k . (85)where S is the area enclosed by the path L = ∂ S , and Ω n ( k ) = ∇ × A n ( k ).In the experimental setup, the graphene-like hexagonal OL for ultracold Rb atomsis implemented by superimposing three linearly polarized blue-detuned running wavesat 120(1) ◦ angles (Fig. 16(c)). Fig. 16(d) shows that the resulting dispersion relationincludes two inequivalent Dirac points with opposite Berry fluxes ± π located at K and K (cid:48) , respectively. The interferometer sequence begins with an almost pure Rb BEC inthe state | ↑(cid:105) = | F = 2 , m F = 1 (cid:105) at quasimomentum k = 0 in a V = 1 E r deep lattice,47 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401Figure 16. (Color online) Aharonov-Bohm analogy and geometric properties of the hexagonal lattice. (a) In theAharonov-Bohm effect, electrons encircle a magnetic flux in real space. (b) In the interferometer, the particlesencircle the π Berry flux of a Dirac point in reciprocal space. (c) Sketch of the hexagonal lattice in real space,which is realized by interfering three laser beams (arrows) of intensity I i and frequency w L , with linear out-of-plane polarizations. A linear frequency sweep of the third lattice beam creates a uniform lattice accelerationalong the y direction. A magnetic field gradient along the x axis creates an additional spin-dependent force. (d)Dirac points are located at the corners ( K and K (cid:48) points) of the BZ (gray hexagons). Black diamond is a typicalinterferometer path. (e) Summary of phase shifts measured relative to the zero-area reference interferometer fordifferent final quasimomenta k finy (Inset shows the interferometer paths). Lines follow ab initio theory using a fullband structure calculation with no momentum spread σ k = 0 and perfectly localized Berry curvature δk Ω = 0(black) or σ k = 0 . k L and δk Ω (cid:39) − k L (blue). Reprinted from Duca et al. [140]. Reprinted with permissionfrom AAAS. where E r = h / (2 mλ L ) ≈ h × h is Planck’s constant. Infact, the method of detecting the Berry phase here is a 2D extension of the Zak phase’smeasurement addressed in Sec. 4.1.1. The first step is to create a coherent superpositionof | ↑(cid:105) and | ↓(cid:105) = | F = 1 , m F = 1 (cid:105) states by using a resonant π/ x axis is applied to create a constant force in oppositedirections for the two spin components. Meanwhile, an orthogonal, spin-independentforce from lattice acceleration is created by a linear frequency sweep of the third latticebeam and can move atoms along the y direction [Fig. 16(c,d)]. As a consequence, the twospin components move symmetrically along the interferometer path in reciprocal space.After an evolution time τ , a spin-echo π -microwave pulse is applied to swap the states | ↑(cid:105) and ↓(cid:105) . Subsequently, the two atomic wave packets experience opposite magnetic forcesin the x direction, such that both spin components arrive at the same quasimomentum k fin after an additional evolution time τ . At this point, the coherent superposition stateis given by | ψ fin (cid:105) ∝ | ↑ , k fin (cid:105) + e iϕ | ↓ , k fin (cid:105) with relative phase ϕ . Finally, a second π/ ϕ MW is applied in order to close the interferometer andconvert the phase information into spin population fractions n ↑ , ↓ ∝ ± cos( ϕ + ϕ MW ).Notably, the phase difference ϕ = ϕ B + ϕ d consists of the geometric phase ϕ B and anydifference in dynamical phases ϕ d between the two paths of the interferometer and isequal to the Berry phase of the region enclosed by the interferometer. This is becausethe dynamical contribution should vanish due to the symmetry of the paths and the use ofthe spin-echo sequence. To ascertain that the measured phase is truly of geometric origin,the authors additionally employed a zero-area reference interferometer, which comprises48 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 a V-shaped path produced by reversing the lattice acceleration after the π -microwavepulse.The experimental results for detecting the Berry phase (flux) of the varying regionenclosed by the interferometer are shown in Fig. 16(e). The results show that a phasedifference ϕ (cid:39) π when a Dirac point is enclosed in the measurement loop, which agreeswell with the theoretical prediction of the Berry phase for a single Dirac point. In contrast,the phase difference vanishes when enclosing zero or two Dirac points. The shift in thephase jump results from the momentum spread σ k , the broadening of the edges is causedby δk Ω , and the shaded area accounts for a variation in σ k = 0 . − . k L . The contrast islimited by inhomogeneous broadening of the microwave transition, the finite momentumspread of the condensate, and, for large final quasimomenta, the dynamical instability ofthe Gross-Pitaevskii equation. The celebrated Hofstadter model (also named as the Hofstadter-Harper model) describescharged particles moving in a 2D periodic lattice under a uniform magnetic flux perunit cell [218]. In the tight-binding regime, the single-particle energy spectrum dependssensitively on the number of flux quanta per unit cell and a band splits into narrowmagnetic bands. At high magnetic fields, the self-similar energy spectrum was predictedto emerge, known as the Hofstadter butterfly. Moreover, for filled bands of non-interactingfermions when the Fermi energy lies in one of the band gaps, the Hall conductance ofthe system is quantized [10]. In this case, the Hofstadter model realizes the paradigmaticexample of a topological insulator that breaks TRS and can be characterized by the firstChern numbers.Consider the non-interacting spinless particles moving in a 2D square lattice in thepresence of an artificial magnetic field, which are described by the Hofstadter Hamiltonian[218] H H = − J (cid:88) m,n (cid:0) a † m +1 ,n a m,n + e iϕ m,n a † m,n +1 a m,n + h . c . (cid:1) , (86)where a † m,n ( a m,n ) is the creation (annihilation) operator of a particle at lattice site( m, n ), and ϕ m,n denotes the spatially-varying hopping phase induced by a magneticflux 2 πφ . Taking the Landau gauge, the Hofstadter Hamiltonian can be rewritten as H H = − J (cid:88) m,n (cid:16) a † m +1 ,n a m,n + e i πmφ a † m,n +1 a m,n + h . c . (cid:17) . (87)With y = na as the periodic coordinates on the system, this Hamiltonian can be diag-onalized as a block Hamiltonian H H = (cid:76) H x ( k y ), where k y is the quasimomenta alongthe periodic directions. The decoupled block Hamiltonian takes the form H x ( k y ) = − J (cid:88) m ( a † m +1 a m + h.c.) − (cid:88) m V m a † m a m , (88)where V m = 2 J cos(2 πφm + k y a ). The single-particle wave function is written as Ψ mn = e ik y y ψ m , and then the Schr¨odinger equation H x ( k y )Ψ mn = E Ψ mn reduces to the Harperequation [179] − J ( ψ m − + ψ m +1 ) − V m ψ m = Eψ m . (89)49 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401Figure 17. (Color online) Hofstadter-butterfly energy spectrum. Dashed lines represent the Fermi energy fordifferent values of φ , namely φ = 1 / 3, 1 / 5, and 1 / 10. Regions marked by × and (cid:78) have the band Chern numbers C = ± 1; those marked by ◦ and (cid:4) have C = ± For rational fluxes φ = p/q with p and q being relatively prime integers, and under theperiodic boundary condition along x axis, the wave function ψ m satisfies ψ m = e ik x x u m ( k )with u m ( k ) = u m + q ( k ). In this case, the spectrum of this system consists of q energybands and each band has a reduced (magnetic) BZ: − π/qa ≤ k x ≤ π/qa , − π/a ≤ k y ≤ π/a . In term of the reduced Bloch wave function u m ( k ), Eq. (89) becomes − J ( e ik x u m − + e − ik x u m +1 ) − V m u m = E ( k ) u m . (90)Since u m ( k ) = u m + q ( k ), the problem of solving the equation (90) reduces to solving theeigenvalue equation, M Υ = E Υ, where Υ = ( u , ..., u q ) is the Bloch wave function forthe q bands and M is the q × q matrix. The Hofstadter energy spectrum is displayedin Fig. 17, where the band gaps form continuous regions in the φ − E plane. When theFermi energy lies in a gap, the system is an insulator, and the topological nature andthe Hall conductance of the insulator do not change as long as the Fermi level remainswithin the same gap [10]. When the Fermi energy is in the gap between two bands N and N + 1, the quantized Hall conductance is σ xy = Ce /h with the topological Chernnumber C = 12 π (cid:88) n (cid:54) N (cid:90) π/qa − π/qa dk x (cid:90) π/a − π/a dk y F ( n ) xy ( k ) , (91)where F ( n ) xy is the Berry curvature of the n -th subband. As marked in Fig. 17, the largesttwo gaps correspond to topological insulators with the Chern number C = ± 1, and thesecond largest ones have C = ± pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401Figure 18. (Color online) (a) Raman-assisted tunneling in the lowest band of a tilted lattice with an energy offset∆ between neighboring sites. (b) Experimental geometry to generate uniform magnetic fields using a pair of far-detuned laser beams and a uniform potential energy gradient. (c) A schematic depicting the position-dependentphases of the tunneling process with resulting magnetic flux quanta per unit cell α = φ y / π . Reprinted withpermission from Miyake et al. [29]. Copyright c (cid:13) (2013) by the American Physical Society. of the Raman coupling, the wave function of an atom tunneling from one lattice site toanother acquires an effective spatially-varying Berry phase. This method can also createnon-Abelian U(2) gauge potentials acting on cold atoms in the OLs, leading to general-ized Hofstadter butterfly spectra with new fractal structures and topological properties[222, 223].The laser-assisted hopping scheme was further proposed by using a long-livedmetastable excited state for alkaline-earth or ytterbium atoms in an optical superlat-tice to produce uniform magnetic fields for realizing the Hofstadter Hamiltonian [224]. Itwas also suggested to realize the uniform synthetic magnetic fields for neutral atoms byperiodically shaking square OLs, and thus provided the Floquet realization of the Hof-stadter Hamiltonian [121, 225–228]. The atomic gas of noninteracting spinless fermionsin a rotating OL was considered to study the Hofstadter butterfly and to measure thequantized Hall conductance of the Hofstadter insulator from density profiles using theStˇreda formula [229]. The evolution of the Hofstadter butterfly in a tunable OL amongthe square, checkerboard, and honeycomb structures was studied [230, 231]. A methodfor detecting topological Chern numbers in the Hofstadter bands by simply countingthe number of local maxima in the momentum distribution from time-of-flight imagesof ultracold atoms was presented [232]. The detection of the fractal energy spectrum ofthe Hofstadter model from the density distributions of ultracold fermions in an externaltrap and under finite temperatures was analyzed [233]. The chiral edge states in theHofstadter insulator may be created by using a steep confining potential in the OLs, andthen they can be detected from the atomic Bragg spectroscopy and from their dynamicsafter the potential is suddenly removed [234, 235].Experimentally, the laser-assisted technique was used to generate large staggered mag-netic fields for ultracold bosonic atoms [118], where the two internal states in the propos-als [114, 224] were replaced by doubling the unit cell of the OL using superlattices. Twoexperiments were subsequently implemented for realizing the Hosftadter Hamiltonianbased on the generation of homogeneous and tunable artificial magnetic fields with ul-tracold atoms in tilted OLs [29, 30]. In the experiments, the bosonic atoms were loadedin a square OL with a tilt potential along the y direction, as shown in Fig. 18. Theatomic tunneling in this direction was then suppressed by the linear tilt of energy perlattice site ∆ (cid:29) J , which can be created with magnetic field gradients, gravity, or an acStark shift gradient. The tunneling is resonantly restored by the laser-assisted hoppingmethod with two far-detuned Raman beams of two-photon Rabi frequency Ω, frequencydetuning δω = ω − ω = ∆ / (cid:126) , and momentum transfer δ k = k − k ≡ ( δk x , δk y ),as shown in Fig. 18(a). Here the two Raman beams couple different sites, but do notchange the internal state of the atoms, similar to the scheme proposed in Ref. [236]. Inthe dressed atom picture (for resonant tunneling along the x axis) and high-frequency51 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 (a) (b) (c) (d) C C Figure 19. (Color online) Measurements of the Chern number of Hofstadter bands with ultracold bosonic atoms.(a) The setup consists of a 2D OL with a staggered potential. The magnetic unit cell (gray shaded area) is fourtimes larger than the usual lattice unit cell. (b) The Chern number is extracted from the transverse displacement ofthe atomic cloud, in response to an external force generated by an optical gradient. Measured Chern number C exp as a function of (c) gradient strength F a ; and (d) staggered detuning δ . Reprinted by permission from MacmillanPublishers Ltd: Aidelsburger et al. [48], copyright c (cid:13) (2014). limit ( δω (cid:29) J/ (cid:126) ), the tilt disappears and time averaging over rapidly oscillating Ramanbeams yields an effective time-independent Hamiltonian for the lattice system, whichtakes the single-band form (∆ < E Gap ) of the Hofstadter Hamiltonian:˜ H H = − (cid:88) m,n (cid:0) Ke − iφ m,n a † m +1 ,n a m,n + J a † m,n +1 a m,n + h.c. (cid:1) . (92)The induced hopping strength K along the x axis and the spatially-varying phase φ m,n = δ k · R m,n = mφ x + nφ y correspond to the vector potential A = (cid:126) ( δk x x + δk y y ) /a ˆ x insteadof the simple Landau gauge. Adding up the accumulated phases around a closed pathleads to an enclosed phase φ y = δk y a per lattice unit cell of area a , thus realizing theHofstadter Hamiltonian with the magnetic flux α = φ y / π . When the frequencies of theRaman beams are similar to those used for the OL, one can tune the magnetic flux overthe full range between zero and one by adjusting the angle between the Raman beams.In the experiments, the laser-assisted tunneling processes was characterized by study-ing the expansion of the atoms in the lattice [29], and the local distribution of fluxes weredetermined through the observation of cyclotron orbits of the atoms on lattice plaque-ttes [30]. Since the laser-assisted hopping used does not require near-resonant light forconnecting hyperfine states, this method can be implemented for any atoms, includingfermionic atoms. Moreover, for two atomic spin states | ↑ , ↓(cid:105) with opposite magneticmoments and the titled potential created by a magnetic field gradient, two different spincomponents experience opposite directions of the magnetic field [30, 237], and the systemnaturally realizes the time reversal symmetric spinfull Hofstadter Hamiltonian: H ↑ , ↓ = − (cid:88) m,n (cid:0) Ke ± iφ m,n a † m +1 ,n a m,n + J a † m,n +1 a m,n + h.c. (cid:1) , (93)which gives rise to the quantum spin Hall effect, topologically characterized by a Z spinChern number. In a recent experiment [238], the weakly interacting ground state of theHofstadter Hamiltonian (92) was studied, which for bosonic atoms is a superfluid BEC.The Chern numbers of the Hofstadter bands have been measured with ultracold bosonicatoms from the transverse deflection of an atomic cloud as a Hall response [48]. The exper-imental setup consisted of an ultracold gas of Rb atoms loaded into a two-dimensionallattice created by two orthogonal standing waves with wavelength λ s = 767 nm. An ad-ditional standing wave with twice the wavelength λ L = 2 λ s was superimposed along x to create the staggered potential as shown in Fig. 19(a), with an energy offset ∆ muchlarger than the bare tunneling J x . The modulation restoring resonant tunneling was cre-52 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 ated by two additional pairs of far-detuned laser beams with wave number k L = 2 π/λ L and frequency ω = ∆ / (cid:126) . This system realized an effective time-independent HofstadterHamiltonian with the magnetic flux α = π/ 2. In contrast to previous experiments gen-erating uniform flux in tilted OLs [29, 30], this scheme does not rely on magnetic fieldgradients, providing a higher degree of experimental control. The incoherent distributionof bosonic atoms (where the population within each band is homogeneous in momentumspace) was then loaded into the lowest Hofstadter band via an experimental sequenceusing an auxiliary superlattice potential, which introduces a staggered detuning δ alongboth directions [48]. For δ > J the topology of the bands is trivial and all Chern num-bers are zero. When crossing the topological phase transition at δ = 2 J (the spectral gapsclose), the system enters the topologically non-trivial regime, where the lowest band hasa Chern number C = 1. The Chern numbers were finally extracted from the transverseHall drift by exploiting Bloch oscillations. Under a constant force F = F ˆ e y , atoms on alattice undergo Bloch oscillations along the y direction. The cloud also experiences a netperpendicular (Hall) drift shown in Fig. 19(b) when the energy bands have a nonzeroBerry curvature, which leads to an anomalous velocity that can be isolated by uniformlypopulating the bands. In the absence of inter-band transitions, the contribution of the n -th band to the center-of-mass motion along the x direction can be written in terms ofits band Chern number C n [48]: x n ( t ) = − a Fh C n t = − a C n tτ B , (94)where the factor 4 a corresponds to the extended unit cell and τ B = h/ ( F a ) is the char-acteristic time scale for Bloch oscillations. The center-of-mass evolution of the atomiccloud was measured in-situ with opposite directions of the flux α for subtracting thedifferential shift x ( t, α ) − x ( t, − α ) = 2 x ( t ), as shown in Fig. 19(c), where the deflectionis symmetric with respect to the direction of the applied force and gives an experimentalChern number C exp ≈ 1. The measured drifts for α = 0 and for a staggered-flux distribu-tion do not show any significant displacement, corresponding to zero Chern number. Thedependence of the Chern-number measurement with respect to the force was studied, asshown in Fig. 19(d).The chiral edge states of the Hofstadter lattice were experimentally observed in a ribbongeometry with an ultracold gas of neutral fermions [37] and bosons [38] subjected to anartificial gauge field and a synthetic dimension (see Sec. 4.4 for synthetic dimensions).Very recently, the following interacting Hofstadter model of bosons in the two-body limitwas realized in OLs [239]: H = − (cid:88) i,j (cid:0) Ke − iφ i,j a † i +1 ,j a i,j + J a † i,j +1 a i,j + h.c. (cid:1) + U (cid:88) i,j n i,j ( n i,j − , (95)where U is an on-site repulsive interaction energy. Through microscopic atomic controland detection [239], it was shown that the inter-particle interactions affect the populatingof chiral bands, giving rise to chiral dynamics whose multi-particle correlations indicateboth bound and free-particle characteristics. The novel form of interaction-induced chi-rality observed in these experiments provides the key piece for future investigations ofhighly entangled topological phases of many-body systems. The superfluid pairing andvortex lattices for interacting fermions in OLs under a uniform magnetic field was studied[240]. The Hofstadter-Hubbard model on a cylinder geometry with fermionic cold atomsin OLs was shown to allow one to probe the Hall response as a realization of Laughlin’scharge pump [241]. 53 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 The well-known QHE in 2D electron systems is usually associated with the presence ofa uniform externally generated magnetic field, which splits the electron energy spectruminto discrete Landau levels. In order to realize the integer QHE seen in the Landau-level problem while keeping the translational symmetry of the lattice, in 1988, Haldaneproposed a spinless fermion model for the integer QHE without Landau levels[13]. Heproposed that the QHE may result from the broken TRS without any net magnetic fluxthrough the unit cell of a 2D hexagonal lattice, as illustrated in Fig. 20(a). The Haldanemodel based on breaking both time reversal and inversion symmetries is the first exampleof a topological Chern insulator, and the Hamiltonian is as follows H = J (cid:88) (cid:104) i,j (cid:105) c † i c j + J (cid:88) (cid:104)(cid:104) i,j (cid:105)(cid:105) e − iv ij φ c † i c j + (cid:88) i (cid:15) i c † i c i . (96)Here the on-site energy (cid:15) i is ± M , depending on whether i is on the A or B sublattice, J is the nearest-neighbor-hopping energy, J is the next-nearest-neighbor energy, and v ij = sgn(ˆ d i × ˆ d j ) z = ± , (97)where ˆ d i,j are the unit vectors along the two bonds constituting the next-nearest neigh-bors the particle traverses going from site j to i . As depicted in Fig. 20(a), a periodicmagnetic flux density B ( r ) is added normal to the plane with the full symmetry of thelattice and with zero net flux through the unit cell. Thus, the flux φ a and the flux φ b inthe regions a and b respectively has the relation φ b = − φ a : since the net flux is zero andthe next-neighbor hoppings form closed loops in the hexagonal cell, the hopping terms J are not affected but the hopping terms J acquire a phase φ = 2 π (2 φ a + φ b ) /φ where φ is the flux quanta. C= -1 C= 1C= 0 (a) (b) (c) A BA AB B ij J ' ij iij J e Φ Figure 20. (Color online) (a) The Haldane honeycomb model showing nearest-neighbor bonds (solid lines) andnext-nearest-neighbor bonds (dashed lines). The white and black dots represents the two sublattice sites A and B with different on-site energy M and − M . The areas a and b are threaded by the magnetic flux φ a and φ b = − φ a ,respectively. The area c has no flux. (b) A distorted honeycomb lattice realized in the experiment [28]. (c) Phasesof the Haldane model. Under the periodic condition, we can diagonalize the Haldane Hamiltonian by using thebasis of a two-component spinor c † k = ( c † k,A , c † k,B ) of Bloch states constructed on the twosublattices. Let a , a , a be the displacements from a B site to its three nearest-neighbor A sites, as shown in Fig. 20(a), then a = ( √ a, − a ) , a = (0 , a ) , a = ( − √ a, − a ) , where a is the bond length. Taking a Fourier transform c † i = (1 / √ N ) (cid:80) k e i k · r i c † k , where r i represents the position of the site in sublattice A or B and N is the number of sitesof the sublattice, the Haldane Hamiltonian can be expressed as H ( k ) = (cid:15) ( k ) + d ( k ) · σ, (98)54 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 where (cid:15) ( k ) = 2 J cos φ (cid:88) i =1 cos( k · b i ) , d ( k ) = J (cid:88) i =1 cos( k · a i ) ,d ( k ) = J (cid:88) i =1 sin( k · a i ) , d ( k ) = M − J sin φ (cid:88) i =1 sin( k · b i ) , (99)with b = a − a , b = a − a , and b = a − a . The BZ is a hexagon rotated π with respect to the Wigner-Seitz unit cell: At its six corners ( k · a , k · a , k · a ) is apermutation of (0 , π , − π ). The two distinct corners k α , are defined so that k α · b i = α π , α = ± 1. The energy spectrum of this system can be easily obtained by diagonalizing theHamiltonian (98). There are two bands that touch only if all three Pauli matrix termshave vanishing coefficients, and only occur at the zone corner k α while M = α √ J sin φ .To guarantee that the two bands never overlap and are separated by a finite gap unlessthey touch, in the following, we consider the case for | J /J | < / 3. One can choose thecorner point K = π a (1 / √ , K · a , K · a , K · a ) = (0 , π , − π ). We expand theHaldane Hamiltonian around the point K to linear order in q = k − K : H + = v ( q x σ x − q y σ y ) + m + σ z (100)where v = J a and m + = M − √ J sin φ . Hereafter, we ignore the k -independentterm − J cos φ , which plays no role in topology. At the other point K (cid:48) = − π a (1 / √ , K (cid:48) · a , K (cid:48) · a , K (cid:48) · a ) = (0 , − π , π ), around K (cid:48) we have H − = v ( − q x σ x − q y σ y ) + m − σ z (101)where v = J a and m − = M + 3 √ J sin φ . The Chern number of the whole system isdetermined by C = 12 [sgn( m − ) − sgn( m + )] . (102)The phase diagram of the Haldane model as a function of M/J and φ is shown in Fig.20(c). For φ = 0 , π , the model (98) is under time reversal, and the two mass m + and m − are equal, the system is trivial with C = 0. Moreover, the system has the inversionsymmetry when M = 0. If M and (or) J sin φ vanish, the two bands touch with gaplessDirac fermions. The model can have the nontrivial phases with C = ± | M | < √ J sin φ and φ (cid:54) = 0 , π . Note that along the critical lines in the phase diagram whereeither m + or m − vanishes, the system experiences a topological phase transition, andhas a low-lying massless spectrum around K or K (cid:48) simulating nondegenerate relativisticchiral Dirac fermions.Although the Haldane model has been proposed for nearly 30 years, it has not beenrealized in any condensed matter systems since it is extremely hard to realize the requiredstaggered magnetic flux assumed in the model. The technology of ultracold atoms inan OL provides an approach to realize and explore this model originally proposed incondensed matter physics. The idea of realizing and detecting the QHE of the Haldanemodel in an OL was first proposed in Ref. [242]. In the proposal, three standing-wavelaser beams are used to construct a honeycomb OL where different on site energies in twosublattices required in the model can be implemented through tuning the phase of onelaser beam. The other three standing-wave laser beams are used to create the staggered55 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401Figure 21. (Color online) (a) Laser beam set-up for forming the OL. The laser ¯ X is frequency-detuned from theother beams. Piezo-electric actuators sinusoidally modulate the retro-reflecting mirrors, with a controllable phasedifference ϕ that induces the complex tunneling phase Φ. Acousto-optic modulators ensure the stability of thelattice geometry. (b) Topological phase diagram measured in the experiments. Solid lines show the theoreticallycomputed topological transitions. Dotted lines represent the uncertainity of the maximum gap originating fromthe uncertainty of the lattice parameters. Data are the points of maximum transfer for each Dirac point. Reprintedby permission from Macmillan Publishers Ltd: Jotzu et al. [28], copyright c (cid:13) (2014). magnetic field. Firstly, to generate the honeycomb lattice with different on site energiesin sublattices A and B , the three laser beams with the same wave length but differentpolarizations are applied along three different directions: e y and √ e x ± e y , respectively.Thus, the corresponding potential is given by V = V [sin ( α + + π ( yk L + π ( α − − χ , (103)where α ± = √ xk L / ± yk L / V is the potential amplitude and k L is the wave vectorof the laser. The ingenuity of the design is that the different site energies of sublattices A and B are controllable by the phase of the laser beam χ . An interesting methodto realize the staggered magnetic field in the Haldane model is to use two opposite-travelling standing-wave laser beams to induce Berry phase [84], which can create effectivestaggered magnetic fields with zero net flux per unit cell. For the two laser beams withRabi frequencies Ω = Ω sin( yk L + π ) e ixk L and Ω = Ω sin( yk L + π ) e − ixk L , the effectivegauge potential is generated as A ( r ) = (cid:126) k L sin(2 yk L ) e x . Here, k L = k L cos θ and k L = k L sin θ with k L being the wave vector of the laser and θ being the angle between the wavevector and the e x axis. The choice of wave vector k L of the laser beams must be a multipleof √ π a in order to be commensurate with the OL, such as k L = √ π a . Since the latticehas the symmetry of point group C v , the other two vector potentials can be rotatedby ± π from the vector potential A . Finally, the total accumulated phases along thenearest-neighbor directions are found to cancel each other out because of the symmetryof the honeycomb lattice. However, the total accumulated phases for the next-nearest-neighbor hopping are preserved as the hopping phase ϕ = k L a sin ak L √ . Consequently,the total Hamiltonian of this cold atomic system is described by the Haldane model.However, in the proposal, the lasers for the honeycomb lattice and artificial magneticfields are different and thus the required lasers are extremely complicated and hard torealize in practice.Another scheme to realize the Haldane-like model was proposed in Ref. [243], with anorbital analogue of the anomalous QHE arising from orbital angular momentum polariza-tion without Landau levels. This effect arises from the energy-level splitting between theon site p x − ip y and p x + ip y orbitals by rotating each OL site around its own center. Atlarge rotation angular velocities, this model naturally reduces to two copies of Haldane’squantum Hall model. An improved experimental proposal to realize the generalized Hal-56 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 dane model using OLs loaded with fermionic atoms in two internal states was proposedin Ref. [244]. In this simulation, the original phase factors in the next-nearest-neighborhopping in Haldane’s paper are replaced by that in the nearest-neighbor, whose phasedepends on the momentum imparted by the Raman lasers. An experimental scheme torealize the quantum anomalous Hall effect in an anisotropic square OL was proposed[203].Instead of using the extra laser beams, one can create effective magnetic fields in thehoneycomb OL by shaking the lattice [28, 120]. In 2014, the first experimental realiza-tion of the Haldane model and the characterization of its topological band structurewere reported [28], which used ultracold fermionic atoms in a periodically modulatedhoneycomb OL. In the experiment, the spin-polarized non-interacting ultracold Fermigas of K atoms was prepared in the OL created by several laser beams at wavelength λ = 1064 nm. The lattice potential is given by V ( x, y, z ) = − V ¯ X cos ( k lat x + θ/ − V X cos ( k lat x ) − V Y cos ( k lat y ) − α (cid:112) V X V Y cos( k lat x )cos( k lat y )cos( ϕ lat ) − V ˜ Z cos ( k lat z ) , (104)where V ¯ X,X,Y, ˜ Z are the single-beam lattice depths and k lat = 2 π/λ . The energy offset∆ AB can be controlled by varying θ around π and changing the frequency detuning δ between the ¯ X and the X (which has the same frequency as Y ) beams using anacousto-optic modulator [77], as depicted in Fig. 21(a). ϕ lat is the relative phase of thetwo orthogonal retro-reflected beams X and Y , the geometry of the lattice is activelystabilized at ϕ lat = 0, and the visibility of the interference pattern is α = 0 . ij . The complex tunneling e i Φ ij J (cid:48) ij (see Fig. 20(b)) can be induced by circularmodulation of the lattice position. The modulation applied in this experiment consists ofmoving the lattice along a periodic trajectory r lat ( t ). Here, the time-dependence of thelattice position r lat ( t ) = − A (cos[ ωt ) e x + cos( ωt − ϕ ) e y ] , (105)where A is the amplitude of the motion, and ω/ π denotes the modulation frequency.Thus, after the atoms are loaded into the honeycomb lattice, a phase-modulated hon-eycomb lattice will be realized by ramping up the sinusoidal modulation of the latticeposition r lat along the x and y directions with a final amplitude of 0 . λ , frequencyof 4 . ϕ . This gives access to linear ( ϕ = 0 ◦ or 180 ◦ ), circular( ϕ = ± ◦ ) and elliptical trajectories.At this point, the effective Hamiltonian of this phase-modulated honeycomb lattice canbe well described by the Haldane model, where the energy offset ∆ AB ≷ A and B sublattices breaks inversion symmetry and opens a gap | ∆ AB | . TRS canbe broken by changing ϕ . This controls the imaginary part of the next-nearest-neighbortunneling, whereas its real part, as well as the nearest-neighbor tunneling J ij and ∆ AB ,are mostly unaffected. To explore the topological properties of this system, the authorsmeasured the band structure and probed the Berry curvature for the lowest band withdifferent parameter ∆ AB and ϕ by applying a constant force to the atoms, and it wasfound that orthogonal drifts are analogous to a Hall current. Meanwhile, one can mapout the transition lines in the topological phase diagram of the Haldane model, as shownin Fig. 21(c), by identifying the vanishing gap at a single Dirac point.57 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 In 2005, Kane and Mele [18, 19] generalized the Haldane model into the time reversalsymmetric electron systems with spin. They introduced the spin-orbit interaction betweenelectron spin and momentum to replace the periodic magnetic flux and predicted anew quantum phenomenon – the quantum spin Hall effect. Unlike the QHE where themagnetic field breaks TRS, the spin orbit interaction preserves TRS.The Kane-Mele model takes the tight-binding Hamiltonian H KM = J (cid:88) (cid:104) i,j (cid:105) c † i c j + iλ SO (cid:88) (cid:104)(cid:104) i,j (cid:105)(cid:105) v ij c † i s z c j + iλ R (cid:88) (cid:104) i,j (cid:105) c † i ( s × ˆd ij ) z c j + λ v (cid:88) i ξ i c † i c i . (106)The first term is the usual nearest neighbor hopping term on a honeycomb lattice [Fig.20(a)], where c † i = ( c † i, ↑ , c † i, ↓ ). The second term connecting next-nearest neighbors with aspin dependent amplitude is a mirror symmetric spin-orbit interaction. Here v ij = ± s i arethe Pauli matrices describing the electron’s spin. The third term is a nearest neighborRashba term, which explicitly violates the z → − z mirror symmetry. The last termis a staggered sublattice potential with ξ i = ± i is the A or B site, which will describe the transition between the quantum spin Hall phase andthe simple insulator. If the Rashba term vanishes, the Kane-Mele model then reduces toindependent copies for each spin of a Haldane model. In this case with s z being conserved,the distinction between graphene and a simple insulator is easily understood. Each spinhas an independent Chern number C ↑ and C ↓ . The TRS gives rise to C ↑ + C ↓ = 0,but the difference C ↑ − C ↓ is nonzero and defines a quantized spin Hall conductivity.This characterization breaks down when s z non-conserving terms are present ( λ R (cid:54) = 0),which makes the system more complicated. The electrons with spin-up and spin-downare coupled, and thus the spin Hall conductance is not quantized. arise due to a perpendicular electric field or interactionwith a substrate. The fourth term is a staggered sublatticepotential ( (cid:5) i (cid:1) (cid:7) ), which we include to describe thetransition between the QSH phase and the simple insulator.This term violates the symmetry under twofold rotations inthe plane. H is diagonalized by writing (cid:6) s (cid:2) R (cid:4) (cid:7) d (cid:3) (cid:1) u (cid:7)s (cid:2) k (cid:3) e i k (cid:8) R . Here s is spin and R is a bravais lattice vectorbuilt from primitive vectors a ; (cid:1) (cid:2) a= (cid:3)(cid:2) (cid:1)(cid:1)(cid:1) p ^ y (cid:7) ^ x (cid:3) . (cid:7) (cid:1) ; is the sublattice index with d (cid:1) a ^ y = (cid:1)(cid:1)(cid:1) p . For each k theBloch wave function is a four component eigenvector j u (cid:2) k (cid:3)i of the Bloch Hamiltonian matrix H (cid:2) k (cid:3) . The 16components of H (cid:2) k (cid:3) may be written in terms of theidentity matrix, 5 Dirac matrices (cid:1) a and their 10 commu-tators (cid:1) ab (cid:1) (cid:9) (cid:1) a ; (cid:1) b (cid:10) = (cid:2) i (cid:3) [9]. We choose the followingrepresentation of the Dirac matrices: (cid:1) (cid:2) ; ; ; ; (cid:3) (cid:1)(cid:2) (cid:1) x (cid:11) I; (cid:1) z (cid:11) I; (cid:1) y (cid:11) s x ; (cid:1) y (cid:11) s y ; (cid:1) y (cid:11) s z (cid:3) , where thePauli matrices (cid:1) k and s k represent the sublattice and spinindices. This choice organizes the matrices according to T . The T operator is given by (cid:2) j u i (cid:12) i (cid:2) I (cid:11) s y (cid:3)j u i (cid:13) . Thefive Dirac matrices are even under T , (cid:2)(cid:1) a (cid:2) (cid:5) (cid:1) (cid:1) a while the 10 commutators are odd, (cid:2)(cid:1) ab (cid:2) (cid:5) (cid:1) (cid:5) (cid:1) ab .The Hamiltonian is thus H (cid:2) k (cid:3) (cid:1) X a (cid:1) d a (cid:2) k (cid:3) (cid:1) a (cid:4) X a (cid:1)(cid:1)(cid:1) p (cid:3) SO the gap is domi-nated by (cid:3) v , and the system is an insulator. (cid:1)(cid:1)(cid:1) p (cid:3) SO > (cid:3) v describes the QSH phase. Though the Rashba term violates S z conservation, for (cid:3) R < (cid:1)(cid:1)(cid:1) p (cid:3) SO there is a finite region ofthe phase diagram in Fig. 1 that is adiabatically connectedto the QSH phase at (cid:3) R (cid:1) . Figure 1 shows the energybands obtained by solving the lattice model in a zigzagstrip geometry [7] for representative points in the insulat-ing and QSH phases. Both phases have a bulk energy gapand edge states, but in the QSH phase the edge statestraverse the energy gap in pairs. At the transition betweenthe two phases, the energy gap closes, allowing the edgestates to ‘‘switch partners.’’The behavior of the edge states signals a clear differencebetween the two phases. In the QSH phase for each energy in the bulk gap there is a single time reversed pair ofeigenstates on each edge. Since T symmetry preventsthe mixing of Kramers’ doublets these edge states arerobust against small perturbations. The gapless statesthus persist even if the spatial symmetry is further reduced[for instance, by removing the C rotational symmetry in(1)]. Moreover, weak disorder will not lead to localizationof the edge states because single particle elastic backscat-tering is forbidden [7].In the insulating state the edge states do not traverse thegap. It is possible that for certain edge potentials the edgestates in Fig. 1(b) could dip below the band edge, reduc-ing—or even eliminating —the edge gap. However, this isstill distinct from the QSH phase because there will nec-essarily be an even number of Kramers’ pairs at eachenergy. This allows elastic backscattering, so that theseedge states will in general be localized by weak disorder.The QSH phase is thus distinguished from the simpleinsulator by the number of edge state pairs modulo 2.Recently two-dimensional versions [10] of the spin Hallinsulator models [11] have been introduced, which underconditions of high spatial symmetry exhibit gapless edgestates. These models, however, have an even number ofedge state pairs. We shall see below that they are topologi-cally equivalent to simple insulators.The QSH phase is not generally characterized by aquantized spin Hall conductivity. Consider the rate ofspin accumulation at the opposite edges of a cylinder ofcircumference L , which can be computed using Laughlin’sargument [12]. A weak circumferential electric field E canbe induced by adiabatically threading magnetic fluxthrough the cylinder. When the flux increases by h=e each momentum eigenstate shifts by one unit: k ! k (cid:4) (cid:2)=L . In the insulating state [Fig. 1(b)] this has no effect,since the valence band is completely full. However, in theQSH state a particle-hole excitation is produced at theFermi energy E F . Since the particle and hole states donot have the same spin, spin accumulates at the edge.The rate of spin accumulation defines a spin Hall conduc-tance d h S z i =dt (cid:1) G sxy E , where TABLE I. The nonzero coefficients in Eq. (2) with x (cid:1) k x a= and y (cid:1) (cid:1)(cid:1)(cid:1) p k y a= . d t (cid:2) (cid:4) x cos y (cid:3) d (cid:5) t cos x sin yd (cid:3) v d (cid:3) SO (cid:2) x (cid:5) x cos y (cid:3) d (cid:3) R (cid:2) (cid:5) cos x cos y (cid:3) d (cid:5) (cid:3) R cos x sin yd (cid:5) (cid:1)(cid:1)(cid:1) p (cid:3) R sin x sin y d (cid:1)(cid:1)(cid:1) p (cid:3) R sin x cos y −5 0 5−505 IQSH λ / λ R λ / λ v SOSO E / J ka ka π π (a) (b) FIG. 1 (color online). Energy bands for a one-dimensional‘‘zigzag’’ strip in the (a) QSH phase (cid:3) v (cid:1) : t and (b) theinsulating phase (cid:3) v (cid:1) : t . In both cases (cid:3) SO (cid:1) : t and (cid:3) R (cid:1) : t . The edge states on a given edge cross at ka (cid:1) (cid:2) . The insetshows the phase diagram as a function of (cid:3) v and (cid:3) R for <(cid:3) SO (cid:14) t . PRL week ending30 SEPTEMBER 2005 Figure 22. (Color online) Energy spectrum for a 1D zigzag strip in the (a) Quantum spin Hall phase λ v = 0 . J and (b) the insulating phase λ v = 0 . J . In both cases λ SO = 0 . J and λ R = 0 . J . The inset shows thephase diagram as a function of λ v and λ R for 0 < λ SO (cid:28) J . Reprinted with permission from Kane et al. [18].Copyright c (cid:13) (2005) by the American Physical Society. Following the method introduced in the Haldane model, we diagonalize the Hamilto-nian by using a basis of the four-component spinor c † k = ( c † k,A ↑ , c † k,A ↓ , c † k,B ↑ , c † k,B ↓ ) of Blochstates constructed on the two sublattices and two spins. The generic 4 × H KM ( k ) = (cid:88) a =1 d a ( k )Γ a + (cid:88) a √ λ SO , the gap is dominated by λ v , and thesystem is a normal insulator since both Chern numbers C ↑ and C ↓ are zero. In contrast,for λ v < √ λ SO , the corresponding Chern number becomes nonzero, C ↑ = − C ↓ =sgn( λ SO ). Although the total Chern number C = C ↑ + C ↓ = 0, their difference C ↑ − C ↓ = ± 2, which describes the quantum spin Hall phase with a pair of edge states crossingthe bulk gap, as depicted in Fig. 22(a). For λ R (cid:54) = 0, the s z symmetry is broken, andelectrons with spin-up and spin-down mix together. Thus, we cannot introduce the spin-dependent Chern number to describe this system. Instead, Kane and Mele introducedthe Z invariant (see Sec. A.3) to describe it.Although the Kane-Mele model has been proposed for more than a decade, physicistsstill have not realized it or found such materials in condensed matter physics. To directlyimplement the Kane-Mele model is difficult, but the quantum spin Hall effect predicted byKane and Mele was first experimentally realized in HgTe quantum wells [22]. Comparedwith conventional solid-state systems, cold atomic systems provide a perfectly cleanplatform with high controllability to construct and investigate the Kane-Mele-like model.There are several works in recent years proposing schemes to realize and study thetopological properties of the quantum spin Hall insulators with cold atoms [116, 237,245, 246]. An experimental scheme to simulate and detect the 2D quantum spin Hallinsulator in a kagome OL was proposed in Ref. [245]. In this proposal, a kagome OLwith the trimer and SOC terms can host the 2D quantum spin Hall insulator phase withonly the nearest-neighbor hopping instead of the next-nearest-neighbor hopping in theKane-Mele model. Moreover, the nearest-neighbor intrinsic SOC generated by the laser-induced-gauge-field method can be directly implemented in cold atomic experiments.Based on the investigation of the Hofstadter model on a 2D square OL [29, 30], onecan construct the Kane-Mele-like model from two time-reversal copies of the spinlessHofstadter model [116, 237, 246]. Here, we briefly introduce a recent proposal on realizing59 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 time-reversal invariant topological insulators in alkali atomic gases [116], where quantumspin Hall states will emerge. In this model, particles with spin σ = ↑ , ↓ experience auniform magnetic flux per plaquette, but opposite in sign for the two spin components.The corresponding Hamiltonian is given by[116] H = − J (cid:88) m,n c † m +1 ,n e i ˆ θ x c m,n + c † m,n +1 e i ˆ θ y c m,n + H.c. + λ s (cid:88) m,n ( − m c † m,n c m,n , (111)where c † m,n is a two-component creation operator for fermonic atoms defined on a latticesite ( ma, na ). The last term describes an on-site staggered potential with amplitude λ s ,along the x direction, which has been introduced to drive transitions between differenttopological phases. The Peierls phases ˆ θ x = 2 πγσ x and ˆ θ y = 2 πm Φ σ z resulted froman artificial gauge field, are engineered within this tight-binding model to simulate theanalog of SOCs. The effect of the SU (2) link variable ˆ U y ( m ) = e i ˆ θ y ∝ σ z is thereforeanalogous to the intrinsic SOC in Eq.(106). It corresponds to opposite magnetic fluxes ± Φ for each spin component and generates quantum spin Hall phases. The link variableˆ U x = e i ˆ θ x ∝ sin(2 πγ ) σ x plays a role similar to the Rashba SOC in Eq. (106). For γ = 0,this model corresponds to two decoupled copies of the spinless Hofstadter model. Besides,ˆ θ x mix the two spin components as they tunnel from one site to its nearest-neighbor sites.This model therefore captures the essential effects of the Kane-Mele model in a multi-band framework and offers the practical advantage of only involving nearest-neighborhopping on a square lattice. Three-dimension As introduced in the previous sections, the 2D Dirac fermions have been extensively stud-ied in graphene and honeycomb OLs. In recent years, it is of great interest to search forrelativistic quasiparticles in 3D materials or artificial systems with stable band touchingpoints, such as 3D Dirac(-like) fermions, which can exhibit transport properties differ-ent from those of 2D Dirac fermions. The Dirac equation in the Weyl representation iswritten as i (cid:126) ∂ Ψ /∂t = H D Ψ , where Ψ denotes the four-component bispinor for 3D Diracfermions and the Dirac Hamiltonian is given by H D = (cid:18) v F σ · p mm − v F σ · p (cid:19) , (112)with the linear dispersion E ± D = ± (cid:113) v F p + m . Here σ = ( σ x , σ y , σ z ) are Pauli matrices, p = ( p x , p y , p z ) is the 3D (quasi-)momentum, and the Fermi velocity v F and the massterm m represent the effective speed of light and rest energy, respectively. Notably, theoff-diagonal term m in Eq. (112) mixes the two Weyl fermions (see the following section)of opposite chirality.The Hamiltonian (112) can describe the transition between a 3D topological insulatorand a trivial insulator in the critical case m = 0. Recently, it was revealed that the Diracpoints with fourfold degeneracies can be protected by certain symmetries [247–249] suchas rotation or nonsymmorphic symmetries, which are not accidental band crossings at thetransition between topological and trivial insulators. Thus, one has a topological Diracsemimetal with four band degeneracy, which can be viewed as 3D graphene, possessing 3DDirac fermions in the bulk with linear dispersions along all momentum directions. These60 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 massless ( m = 0) 3D Dirac fermions have recently been observed in some compounds(for a comprehensive review of Dirac semimetals in 3D solids, see Ref. [24]).Before the Dirac semimetals were discovered in materials, several schemes for simu-lating massless and massive 3D Dirac fermions with cold atoms in 3D OLs have beenproposed [115, 168, 250, 251]. With cold fermions in an edge-centered cubic OL for properparameters, the linear dispersion characterizing 3D Dirac-like particles with tunable masscan exhibit [250]. The system was proposed to realize 3D massless Dirac fermions in acubic OL subjected to a synthetic frustrating magnetic field [251], and the mass termmay be induced by coupling the ultracold atoms to Bragg pulses in the system. It wassuggested that the massless and massive 3D relativistic fermions can also be simulatedwith ultracold fermionic atoms in 3D optical superlattices with Raman-assisted hopping[115, 168]. Moreover, by tuning the Raman laser intensities, the system may allow the de-coupling of fermion doublers from a single Dirac fermion through inverting their effectivemass [115], providing a quantum simulation of Wilson fermions [252]. The Dirac equation (112) for massless particles can be rewritten in a simpler form: i (cid:126) ∂ψ ± ∂t = H W ± ψ ± , H W ± = ± v F σ · p , (113)where ψ ± are effectively two-component vectors acting as two chiral modes. This is theWeyl equation and ψ ± are referred to as Weyl fermions, which propagate parallel (orantiparallel) to their spin and thus defines their chirality. There are no fundamentalparticles currently found to be massless Weyl fermions. In some 3D lattice systems, Weylfermions can emerge as low-energy excitations near band crossings, and they always arisein pairs with opposite chirality and separated momenta. These systems are the so-calledWeyl semimetals [253–255], which have been intensively investigated in the last couple ofyears. For more details, see the comprehensive review of vast theoretical and experimentalstudies of Weyl semimetals in 3D solids [24] and the references therein. z k y k x k W - W +Surface BZBulk BZ Fermi Arc kz C = − kz C = kz C = Figure 23. (Color online) A pair of Weyl points as the Berry flux monopole and anti-monopole in the bulk BZ.They are connected by surface Fermi arcs. Let us use the following minimal two-band model for discussing the properties of Weylsemimetals in a simple cubic lattice. The Bloch Hamiltonian is given by H ( k ) = d x ( k ) σ x + d y ( k ) σ y + d z ( k ) σ z (114)61 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 with the Pauli matrices acting on two (pseudo-)spin states and the Bloch vectors d x = 2 J s sin k x , d y = 2 J s sin k y , d z = m z − J (cos k x + cos k y + cos k z ) . (115)Here J s and J denote the hopping amplitudes of spin-flipping hopping in the xy planeand spin-dependent hopping along all the three dimensions, respectively, and m z repre-sents the Zeeman field. For 2 J < m z < J , the bands E ± ( k ) = ± (cid:113) d x + d y + d z havetwo crossings in the first BZ located at W ± = (0 , , ± arccos k w ) with k w = ( m − J ) / J .They are actually a pair of Weyl points (nodes) since one can obtain the effective WeylHamiltonian for low-energy excitations by approximating H W ± ( k ) ≈ H ( W ± + q ): H W ± = v x q x σ y + v y q y σ x ± v z q z σ z , (116)where v x = v y = 2 J s and v z = 2 J . When the Fermi level is at E F = 0, the Fermi surfaceconsists solely of two Weyl points and the system is the Weyl semimetal with emergentWeyl fermions. The Hamiltonian H W ± can be written as H W , ± = (cid:80) i,j q i α ij σ j , where[ α ij ] is a 3 × α xy = α yx = 2 J s , α zz = ± J and zero otherwise.Thus the chirality of the two Weyl points W ± can be defined as χ ± = sgn(det[ α ij ]) = ± H W ± and opensa band gap since all three Pauli matrices are used up in H W ± . To further realize thetopological stability of Weyl points, one can obtain the Berry flux F ± ( k ) = ± k | k | , (117)which is the source and sink of the Berry curvature, forming vector fields in momentumspace that wraps around the Weyl points W ± . As shown in Fig. 23, the pair of Weylpoints act as the monopole and anti-monopole in the bulk BZ, which are characterizedby the topological charges (Chern numbers) C W ± = 12 π (cid:73) S F ± ( k ) · d S = ± χ ± , (118)through any surface S enclosing the points. This implies that the Weyl points alwaysexhibit in pairs of opposite chirality, because the field lines of the Berry curvature mustbegin and end somewhere within the BZ. The only way to eliminate the Weyl pointsis to annihilate them pairwise by moving them at the same point in momentum space.Therefore, the stability of the Weyl points comes from their intrinsical topology. It isworth noting that unlike Dirac points, Weyl points necessitate the breaking of either (orboth) time-reversal or space-inversion symmetry in lattice systems.To further discuss the topological properties of Weyl semimetals, we consider the BlochHamiltonian (114,115) using the dimension reduction method. Treating k z as an effectiveparameter ( k z being a good quantum number), we can reduce the original system to a ( k z -modified) collection of effective 2D subsystems described by H k z ( k x , k y ). If k z (cid:54) = ± k c , the2D bulk bands of H k z ( k x , k y ) are fully gapped and thus can be topologically characterizedby the first Chern number C k z = 14 π (cid:90) π − π dk x (cid:90) π − π dk y ˆ d · (cid:16) ∂ k x ˆ d × ∂ k y ˆ d (cid:17) = (cid:26) − , − k c < k z < k c ;0 , | k z | > k c , where ˆ d ≡ (cid:126)d/ | (cid:126)d | . For any plane − k c < k z < k c , one has C k z = − pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 insulator, while elsewhere C k z = 0 signals a trivial insulator. The value of C k z changesfor a topological phase transition only when the bulk gap closes at W ± = (0 , , ± k c ).From this point of view, the two Weyl points appear to be the critical points for thetopological phase transitions. An important consequence is that the Fermi arc surfacestates arise in Weyl semimetals and terminate at the pair of Weyl points. When eachof the 2D Hamiltonians H k z ( k x , k y ) represents a 2D Chern insulator, if one considers asurface perpendicular to the x direction (still labeled by k y and k z ), each of the 2D Cherninsulators will have a gapless chiral edge mode near the Fermi energy E F = 0. The Fermienergy will cross these states at k y = 0 for all − k c < k z < k c , which leads to a Fermiarc that ends at the Weyl point projections on the surface BZ ( k y - k z plane), as shown inFig. 23. In this particular model, the Fermi arc is a straight line. The Weyl fermions nearthe Weyl points and the Fermi arcs in a Weyl semimetals are fundamentally interestingand can give rise to exotic phenomena absent in fully gapped topological phases, such asanomalous (topological) electromagnetic responses [24, 253–255].To realize this model of Weyl semimetals, a scheme has been proposed by using ul-tracold fermionic atoms in a 2D square OL subjected to experimentally realizable SOCand an artificial dimension from an external parameter space (acting as k z ) [136]. It wasfurther shown that in the cold atom system, the simulated Weyl points can be experi-mentally detected by measuring the atomic transfer fractions in a Bloch-Landau-Zeneroscillation, and the topological invariants of the Weyl semimetals can be measured withthe particle pumping approach [136]. Another proposal to construct a Weyl semimetalwas to stack 1D topological phases in double-well OLs with two artificial dimensions[256]. Similar 3D lattice models for realizing Weyl semimetals with cold atoms fromstacking 2D layers of Chern insulators in checkerboard or honeycomb OLs with syntheticstaggered fluxes were suggested in Refs. [257–260]. The realization of chiral anomaly byusing a magnetic-field gradient in the system was also discussed [258]. In these schemes,the spin degree of freedom can be encoded by two atomic internal states or sublattices,and then the required hopping terms can be realized by synthetic SOC or magneticfields. These ingredients are well within current experimental reach of ultracold gases.It was illustrated that Weyl excitations can also emerge in 3D OL of Rydberg-dressedatomic fermions or dipolar particles [261, 262]. The Weyl points may automatically arisein the Floquet band structure during the shaking of a 3D face-centered-cubic OL withoutrequiring sophisticated design of the tunneling [263]. (a) (b) (c) (d) Figure 24. (Color online) Sketch of the 3D cubic lattice with engineered hopping along x and z directions, whichpossesses Weyl points in momentum space. Dashed and solid lines depict hopping with acquired phase π and 0,respectively. (a) The xy planes of the lattice are equivalent to the lattice of the Hofstadter-Harper Hamiltonianwith α = 1 / xz plane, which is shown in (c) and (d); the hopping between these planes (along y ) is regular. The hopping along z is alternating with phase 0 or π , depending on the position in the xy plane with broken inversion symmetry.Reprinted with permission from Dubcek et al. [264]. Copyright c (cid:13) (2015) by the American Physical Society. An alternative scheme to realize the Weyl semimetal phase by stacking 2D Hofstadter-Harper systems in cubic OLs was proposed in Refs.[264, 265]. A sketch of the 3D latticewith laser-assisted tunneling along both x and z directions is shown in Fig. 24. To63 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 realize this hopping configuration, natural tunneling ( J x , J z ) along these directions isfirst suppressed by introducing a large linear tilt of energy ∆ per lattice site ( J x , J z (cid:28) ∆ (cid:28) E gap ), which can be obtained by a linear gradient potential (e.g., gravity or magneticfield gradient) along the ˆ x + ˆ z direction. The tunneling is then resonantly restored by twofar-detuned Raman beams of frequency detuning δω = ω − ω = ∆ / (cid:126) , and momentumdifference δ k = k − k [29, 30]. It yields an effective 3D Hamiltonian for the system[264] H D = − (cid:88) m,n,l ( K x e − i Φ m,n,l a † m +1 ,n,l a m,n,l + (119) J y a † m,n +1 ,l a m,n,l + K z e − i Φ m,n,l a † m,n,l +1 a m,n,l + h.c. ) . Here, a † m,n,l ( a m,n,l ) is the creation (annihilation) operator on the site ( m, n, l ), andΦ m,n,l = δ k · R m,n,l = m Φ x + n Φ y + l Φ z are the nontrivial hopping phases, dependenton the positions R m,n,l . Next, the directions of the Raman lasers are chosen such that(Φ x , Φ y , Φ z ) = π (1 , , m,n,l = ( m + n ) π (modulo 2 π ), as schematically shownin Fig. 24(b). The 3D system can be viewed as an alternating stack of two types of 2Dlattices, parallel to the xz plane, as illustrated in Figs. 24(c) and 24(d); hopping betweenthese planes is regular (along y ). Another view is stacking of 2D lattices described by theHofstadter-Harper Hamiltonian with α = 1 / z has phases 0 or π , for m + n even or odd, respectively. The 3D lattice has twosublattices (A-B) with broken inversion symmetry and has the Bloch Hamiltonian H ( k ) = − J y cos k y σ x + K x sin k x σ y − K z cos k z σ z ) . (120)The energy spectrum of the Hamiltonian has two bands, E ( k ) = ± (cid:113) K x sin k x + J y cos k y + K z cos k z , (121)which touch at four Weyl points within the first BZ at ( k x , k y , k z ) = (0 , ± π/ , ± π/ H W ± = v q j σ ± v F q · σ , (122)where an additional term with the Fermi velocity v introduces an overall tilt of the Weylcones. This term is forbidden by Lorentz symmetry for the Weyl Hamiltonian in vacuumbut it can generically appear when expanding the Bloch Hamiltonian at the Weyl points.The energy spectrum is given by E ± = v q j ± v F | q | for particle and hole bands. When | v | < | v F | , the energy of the particle (hole) band is still positive (negative) and the Weylpoint is called a type-I Weyl point as discussed previously. When | v | > | v F | , the Weylpoint is still there, but the two bands now overlap in energy in certain regions, formingparticle and hole pockets. In this case, the Weyl point becomes a point at which a particleand a hole pocket touch and is dubbed a type-II Weyl point. These Weyl semimetals withbroken Lorentz symmetry that have no analog in quantum field theory are called type-IIWeyl semimetals [267]. Several schemes have been proposed to realize and detect type-IIWeyl semimetals (points) with cold atoms in 3D OLs [268–270].64 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 There is an additional subclass of Weyl semimetals called the multi-Weyl semimetals[271–273]. In these systems, the Weyl points carry topological charges of higher magni-tude, such as C W = ± C W = ± In 3D lattice systems, there is another kind of topological semimetals in addition toWeyl and Dirac semimetals. In contract to Weyl and Dirac semimetals that have bandtouching at isolated points, they have the band touching along lines in the 3D BZ, termednodal-line semimetals [255, 280, 281]. The nodal lines can be topologically stable undercertain discrete symmetry, and each carries a quantized π Berry phase (a Z topologicalinvariant). The topological nodal-line semimetal state has been predicted and recentlyconfirmed to exist in some materials [280–284].A simple two-band model of nodal-line semimetals in the continuum can be written as h ( k ) = [ k − ( k x + k y + k z )] σ z + k z σ y , (123)which has both inversion symmetry ˆ P = σ z and time-reversal symmetry ˆ T = ˆ K with thecomplex-conjugate operator ˆ K , and thus the combined ˆ P ˆ T symmetry. It is found thatthe gapless points form a closed nodal line on the k x - k y plane with k z = 0, which maybe enclosed by a loop from the gapped region, such as a tiny circle on the k y - k z plane.The circle is parametrized as (0 , k + ρ cos φ, ρ sin φ ), with ρ being the radius and φ theangle. If ρ is sufficiently small, the Hamiltonian restricted on the circle is expanded as h ( φ ) = − k ρ cos φσ z + ρ sin φσ y + O ( ρ ). The Berry phase of the occupied state wavefunction of such a Hamiltonian is quantized in units of π , which is equal to one modulo2, namely γ = 1 π (cid:90) (cid:104) ψ ( φ ) | i∂ φ | ψ ( φ ) (cid:105) dφ = 1 mod 2 , (124)with | ψ ( φ ) (cid:105) being the occupied state of h ( φ ). In a lattice system, the periodicity of themomentum coordinates allows every large circle going inside the nodal loop to havenontrivial topological charge γ = 1. For straight lines inside, each may be regarded ascorresponding to a 1D gapped system that is of topological band structure, leading togapless boundary modes. If particle-hole symmetry is additionally present, these modesform the drumhead-shape surface states as a flat band over the surface BZ enclosed bythe projection of the nodal line. There are still nearly flat surface bands in the absence65 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 of this additional symmetry.Notably, the quantized π Berry phase means that the nodal line is in the topologi-cal class of the Z classification [285, 286]. According to the classification theory, thetopological protection of the stability of the nodal line requires only the combined ˆ P ˆ T symmetry, rather than both ˆ P and ˆ T symmetries. In other words, the nodal line stillexists for topological reasons when perturbations break both ˆ P and ˆ T symmetries butpreserve ˆ P ˆ T symmetry. In general, a σ y term with even functions of k and a σ z term withodd functions of k break both ˆ P and ˆ T , which just change the shape and (or) positionof the nodal line.Several schemes were proposed to realize topological nodal-line semimetals in cold atomsystems. Ref. [285] proposed to realize tunable ˆ P ˆ T -invariant topological nodal-loop stateswith ultracold atoms in a 3D OL, which is described by the Bloch Hamiltonian H ( k ) = f z ( k ) σ z − J s sin k z σ y − f ( k ) σ . (125)Here f z ( k ) = m z − α − (cos k x +cos k y ) − α + cos k z and f ( k ) = α + (cos k x +cos k y )+ α − cos k z ,where m z and α ± ≡ J ↑ ± J ↓ are tunable parameters for adjusting the nodal rings, and J ↑ , ↓ are the natural hopping strengths for two spins. To realize this Hamiltonian, atoms withtwo hyperfine spin states are loaded in a spin-dependent 3D OL and two pairs of Ramanlasers are used to create spin-flip hopping with a site-dependent phase along the z direc-tion. It was also demonstrated that the characteristic nodal ring can be detected fromBloch-Landau-Zener oscillations, the topological invariant may be measured based onthe time-of-flight imaging, and the surface states may be probed via Bragg spectroscopy.A four-band model allowing Dirac or Weyl rings was also suggested to realize with coldatoms in 3D OLs, where the superfluidity of attractive Fermi gases in the model exhibitsDirac and Weyl rings in the quasiparticle spectrum [287]. The atomic topological super-fluid with ring nodal degeneracies in the bulk was proposed in Ref. [288]. An alternativemodel of 3D topological semimetals whose energy spectrum exhibits a nodal line actingas a vortex ring was proposed in Ref. [289], which may be realized with cold atoms.Even in a dissipative system with particle gain and loss, a novel type of topological ringwas theoretically discovered [290], which is dubbed a Weyl exceptional ring consistingof exceptional points at which two eigenstates coalesce. Such a Weyl exceptional ringis characterized by both a Chern number and a quantized Berry phase, and may berealized and measured in a dissipative cold atomic gas trapped in an OL. Recently, itwas theoretically found that there are other possible configurations for 1D nodal linesof band touching, such as a nodal chain [291] containing connected loops and a nodallink [292–294] hosting linked nodal rings in the BZ. A scheme to realize the topologicalsemimetal with double-helix nodal links using cold atoms in an OL was also presented[292].In a very recent experiment [295], a 3D topological nodal-line semimetal phase forultracold fermions with synthetic SOCs in an optical Raman lattice was realized. Thenodal lines embedded in the semimetal bands were observed by measuring the atomicspin-texture. Moreover, the realized topological band structure was confirmed by ob-serving the band inversion lines from the dynamics of the quench from a deep trivialregime to topological semimetal phases. This work demonstrated a promising approachto explore 3D band topology for ultracold atoms in OLs. Z topological insulators Inspired by the study of 2D Z topological insulators [18, 20, 22], three groups of theoristsindependently proposed 3D generalizations of the quantum spin Hall insulators [296–298]. A single Z invariant ν characterizes the topology of a 2D topological insulator;66 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 in contrast, four Z topological invariants ν ; ( ν ν ν ) are needed to fully characterize a3D Z topological insulator. As introduced in Sec. A.3, the mathematical formulationsof the four Z invariants can be obtained from the 2D case, which involves the quantities δ i of 8 distinct inversion invariant points Γ i in the 3D BZ. Points k = Γ i in the BZ areinversion invariant since − Γ i = Γ i + G = Γ i for a reciprocal lattice vector G . Thus, thesepoints are also time reversal invariant ˆ T H (Γ i ) ˆ T − = H (Γ i ) and are called time reversalinvariant momenta. The eight Γ i are expressed in terms of primitive reciprocal latticevectors as Γ i =( n n n ) = ( n b + n b + n b ) / , (126)where n j = 0 , b l are primitive reciprocal lattice vectors. They can be regarded asthe vertices of a cube.In the 2D case with b = 0, the Z invariant can be determined by the quantities δ a = (cid:112) det[ U (Γ a )]Pf[ U (Γ a )] = ± , (127)where Γ a are the four time reversal invariant momenta with the form (126) in the 2DBZ, and U is the so-called sewing matrix defined by U mn ( k ) = (cid:104) u m ( − k ) | U T | u n ( k ) (cid:105) ∗ , (128)which builds from the occupied Bloch functions | u m ( k ) (cid:105) [160]. At k = Γ a , U mn = −U nm ,so the Pfaffian Pf[ U ] satisfying det[ U ] = Pf[ U ] is well defined. At this time, the singleZ invariant ν is given by ( − ν = (cid:81) a =1 δ a . Similarly, the four Z indices ν ; ( ν ν ν ) inthe 3D BZ can be defined in term of δ n n n as( − ν = (cid:89) n j =0 , δ n n n , ( − ν i =1 , , = (cid:89) n j (cid:54) = i =0 , n i =1 δ n n n . (129)One can see that ν can be expressed as the product over all eight points, while the otherthree invariants ν i are given by products of four δ i , with which Γ i reside in the same plane.If the lattice system has inversion symmetry, the problem of evaluating the Z invariantscan be greatly simplified [299]. At the special points Γ i , the Bloch states | u m (Γ i ) (cid:105) are alsothe parity eigenstates of the 2 m -th occupied energy band with eigenvalue ξ m (Γ i ) = ± ξ m = ξ m − with its Kramers degenerate part. Theproduct involves the 2 N occupied bands that can be divided into N Kramers pairs. Inthis case, the Z invariants are still determined by Eq. (129) with δ n n n = N (cid:89) m ξ m (Γ i ) . (130)According to the parity of ν , the system can be divided into two classes of phases. For ν = 0, the system is referred to as a “weak” topological insulator with an even number ofDirac cones at the surfaces, which can be interpreted as stacked layers of the 2D quantumspin Hall insulators. The TRS does not protect their surface states and the system isnot robust against disorder. For ν = 1, the crystal is called a “strong” topologicalinsulator with an odd number of Dirac cones on all surfaces of the BZ. The connectionbetween the bulk topological indices and the presence of unique metallic surface statesis established. The 3D Z topological insulators have been theoretically predicted and67 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 then experimentally discovered in several materials, such as Bi − x Sb x [299, 300] as wellas the “second generation” topological insulators in Bi Te , Sb Tb [301], and Bi Se [301, 302].The realization of the Z topological insulators in ultracold atomic systems will allowinvestigation of interesting properties that cannot readily be explored in solid-state ma-terials in a controlled way, such as the strong correlations and other perturbations inZ topological insulators. Based on the optical flux lattice [303] with synthetic SOC, ageneric scheme was proposed to realize 2D and 3D Z topological insulators with coldatoms [304]. Interestingly, the proposed lattice system work in the nearly free particleregime, which allows for large gaps with the size set by the recoil energy. For an atom of N internal states in an optical potential with position r and momentum p , the genericHamiltonian of the atom-laser system can be written as H = p m N + V ˆ M , (131)where V has dimensions of energy, N is the identity, and ˆ M ( r ) is a position-dependent N × N matrix acting on the internal states of the atom describing the interaction betweenthe atom and the laser field. To realize a Z topological insulator, one requires N tobe even and the Hamiltonian invariant under time reversal: ˆ T ˆ M ˆ T − = ˆ M with ˆ T = i ( ⊗ ˆ σ y ) ˆ K . The smallest nontrivial case has N = 4 with [304]ˆ M = (cid:18) ( A + B ) C − i ˆ σ · D C + i ˆ σ · D ( A − B ) (cid:19) , (132)where A , B and C are real parameters, and D = ( D x , D y , D z ) is a 3D vector.The optical potential in Eq. (132) can be implemented by using four internal states of Yb atom (with nuclear spin 1 / 2) [304]. Both the ground state ( S = g ) and the long-lived excited state ( P = e ) have two internal states. The magnetic field is considered tobe sufficiently small that the Zeeman splitting is negligible, and all four e - g transitionsinvolve the same frequency ω = ( E e − E g ) / (cid:126) . Under this single photon coupling with thestate-dependent potential V am , the optical potential in the rotating wave approximation[305] is given by V ˆ M = (cid:18) ( (cid:126) ∆ d + V am ) − i ˆ σ · εd r i ˆ σ · εd r − ( (cid:126) ∆ d + V am ) (cid:19) , (133)where ∆ d = ω − ω is the detuning, d r is the reduced dipole moment, and ε representsthe electric amplitude vector. For the 3D case, this optical potential can be achievedby three standing waves of linearly polarized light at the coupling frequency ω : two ofequal amplitude with wave vectors in the 2D plane ( K for y polarization and K for z polarization) and one with a wave vector K normal to the 2D plane for x polarizationwith an amplitude smaller by a factor of δ . The corresponding electric field, detuning,and state-dependent potential in Eq. (133) are given by d r ε = V (cid:2) δ (cid:48) , cos( r · K ) , cos( r · K ) (cid:3) , (cid:126) ∆ / V am = V (cid:2) c + δ (cid:48) ( µ + c + c ) (cid:3) , (134)where δ (cid:48) = δ cos( r · K ) and c ij = cos (cid:2) r · ( K i + K j ) (cid:3) , with K = (1 , , κ , K =(cos θ, sin θ, κ , and K = (0 , , κ . The amplitudes are chosen to have a common energyscale V , which can be interpreted as a measure of the Rabi coupling. Since ω (cid:39) ω , themagnitude of the wave vectors is κ (cid:39) π/λ with λ = 578 nm being the wavelengthof the e - g transition. The space-dependent V am ( r ) is set by a standing wave at the68 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 anti-magic wavelength λ am [224], which fixes the angle θ = 2arccos( ± λ /λ am ). For Ybatoms, λ /λ am (cid:39) / 2, so θ (cid:39) ± π (let θ = π ). The optical coupling ˆ M preservesthe symmetry of a monoclinic lattice, where the lattice vectors a = ( √ / , − / , a , a = (0 , , a , and a = (0 , , √ / a , with a ≡ π/ ( √ κ ). Thus, the eight time reversalinvariant momenta of the 3D topological insulator in this lattice are given by Γ mnl =( m K + n K + l K ) / 2, with m, n, l = 0 , K i . Inaddition, the system has inversion symmetry ˆ P : r → − r as the optical coupling is evenunder the spatial inversion. Thus the Z topological invariants in this system take a simpleform based on Eqs. (129) and (130): The product (cid:81) m,n,l =0 , (cid:81) α ∈ filled ξ α (Γ mnl ) = − ν = 1, where ξ α (Γ mnl ) are the parity eigenvalues of the α th Kramers pair of bands atthe momenta Γ mnl . As we discussed in Sec. 4.1, 1D chiral topological insulators classified in the AIII classhave been extensively studied in condensed matter systems and OLs with cold atoms.Similarly, according to the ten-fold classification of topological insulators [123, 161], thereare two distinct classes of 3D topological insulators protected by the chiral symmetry ˆ S ,which is the combination of time-reversal ˆ T and charge-conjugation ˆ C symmetries. Thefirst class is the class AIII in the 3D cases, and the second one is the class DIII whichis invariant under both ˆ T and ˆ C symmetries. The realization of 3D chiral topologicalinsulators in condensed matter materials has been studied [306, 307]. In addition, the ex-perimental schemes for implementing the class AIII and DIII chiral topological insulatorsusing cold atomic gases in 3D OLs have been proposed [308, 309].In the proposal in Ref. [308], an optical potential coupling noninteracting atoms withtwo spin states was constructed, which is described by the model Hamiltonian H ( p , r ) = p m + V (cid:2) cos kx + cos ky + cos kz (cid:3) + B Z ( r ) · σ, (135)with B Z ( r ) = B Z (cid:80) i =1 b i cos( k b i · r ) . Here k = 2 π/a denotes the length scale of thewave vector; p and r are the single-particle momentum and position; ˆx , ˆy , and ˆz areorthogonal unit vectors; and σ represents the Pauli matrices in spin space. Moreover,the tetrahedral vectors b i are represented as b = ( − ˆx + ˆy + ˆz ) / b = ( ˆx − ˆy + ˆz ) / b = ( ˆx + ˆy − ˆz ) / 2, and b = − ( ˆx + ˆy + ˆz ) / 2. The potential V ( r ) creates a spin-independentcubic lattice, while the effective Zeeman term B Z ( r ) · σ creates an alternating magnetichedgehog texture around the wells of the lattice [308], leading to the lattice structurewith the translation symmetry of a face-centered-cubic lattice. Although the Zeemanfield B Z ( r ) breaks ˆ T since σ = − σ y σ ∗ σ y , the symmetry is restored by a translation T / through a along any of the cubic axes. This Hamiltonian then has the combined symmetryΣ = ˆ T T / , which satisfies Σ = − 1, and therefore keeps the necessary topologicalcharacteristic of a nontrivial topological insulator phase.In the deep-well limit, the Hamiltonian (135) reduces to the following tight-bindingmodel on the fcc cubic lattice H tb = (cid:88) r ∈ A (cid:88) e c † r ( J + J M e · σ ) c r + e + H.c. , (136)where J and J M (both are real) are the nearest-neighbor and spin-dependent hop-ping, respectively. ˆ c r = (ˆ c r , ↑ , ˆ c r , ↓ ) is the fermionic annihilation operator at site r , and e ∈ ± ˆx , ± ˆy , ± ˆz ; A labels one of the two sublattices of the fcc cubic lattice. The Bolch69 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 Hamiltonian in momentum space is given by H ( k ) = 2 (cid:18) g ( k ) g † ( k ) 0 (cid:19) , g ( k ) = (cid:88) j ∈ x,y,z ( J cos k j − iJ M sin k j σ j ) . (137)The energy spectrum of this Hamiltonian has two bands (with twofold degeneracy) (cid:15) ( k ) = ± (cid:113) J f ( k ) + J M f M ( k ) , (138)where f ( k ) = (cid:80) j ∈ x,y,z cos k j , and f M ( k ) = (cid:80) j ∈ x,y,z sin k j . In this tight-binding regime,the system are protected by an extra chiral symmetry: ˆ S H ( k ) ˆ S − = −H ( k ) with theoperator ˆ S = τ z ⊗ σ , where τ j are the Pauli matrices in the sublattice space and σ isthe identity matrix. Thus, this model belongs to the symmetry class DIII. The associatedtopological invariant of this system can be characterized by the 3D winding number[161, 310, 311] ν w = π (cid:90) d k (2 π ) (cid:15) abc Tr (cid:0) ˆ SD a D b D c (cid:1) = 1 , (139)where D a = H − ( k ) ∂ k a H ( k ) , and the integral is over the whole BZ. The difficulty in theproposed scheme is to realize the optical potential coupling on the two atomic internalstates B Z ( r ) · σ , which may be achieved with the optical flux lattice method similar asto the one used for the 3D DIII chiral topological insulators [304].Another experimental scheme to realize a 3D AIII chiral topological insulator with coldfermionic atoms in an OL was proposed in Ref.[309]. The proposed model Hamiltonianin momentum space is given by H ( k ) = q − iq q − iq q + iq q + iq , (140)where q = 2 J ( h + cos k x a + cos k y a + cos k z a ), q = 2 J sin k x a , q = 2 J sin k y a , and q = 2 J sin k z a , with h being a tunable parameter. Here H ( k ) anticommutes with ˆ S andthus has a chiral symmetry with the operator ˆ S = diag(1 , , − H ( k )breaks TRS, and thus this model belongs to symmetry class AIII. H ( k ) has three energybands, with a zero-energy flat band protected by the chiral symmetry and the other twobands having energy dispersion E ± = ± (cid:112) q + q + q + q . This model is characterizedby the Z topological invariant (winding number) [312, 313] ν w = 112 π (cid:90) BZ d k(cid:15) αβγρ (cid:15) µντ E q α ∂ µ q β ∂ ν q γ ∂ τ q ρ , (141)where (cid:15) is the Levi-Civita symbol with { α, β, γ, ρ } and ( µ, ν, τ ) labeling respectively thefour components of the vector field q and the three coordinates of the momentum k . Thetopological invariant ν w as a function of h is given by ν w ( h ) = − , | h | < , < | h | < , | h | > . (142)70 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 As shown in Fig. 25(a), the system is gapped when | h | (cid:54) = 1 , | h | < 3. The band gap closes for | h | = 1 , 3, indicatingtopological quantum phase transitions. Figure 25(b) shows the numerical results of theenergy spectrum for the system, which keeps x and y directions in momentum spacewith periodic boundaries and z direction in real space with open boundaries, revealingthe macroscopic flat band as well as the surface states with Dirac cones. Figure 25. (Color online). (a) The topological invariant ν w as a function of h . (b) Energy spectrum for threebulk bands (surface plot) and surface states (mesh plot) at the boundary along the z direction for h = 2. (c)Proposed scheme to realize the Hamiltonian (140). A linear tilt ∆ x,y,z per site in the lattice along each direction.The detuning in x direction matches the frequency offset of the corresponding Raman beams, which are shown inpanel (d). Polarizations of each beam are shown in brackets. Rabi frequencies for each beam are: Ω π = Ω e ikx ,Ω π = Ω e iky , Ω x = i √ e ikz , Ω x = − i √ e ikz , Ω y = −√ e ikz , Ω y = √ e ikz , and Ω z = 2 i Ω e ikz .Reprinted with permission from Wang et al. [309]. Copyright c (cid:13) (2014) by the American Physical Society. To realize this model Hamiltonian, a three-species gas of noninteracting fermionicatoms (denoted by | (cid:105) , | (cid:105) and | (cid:105) ) trapped in a 3D cubic OL is considered. The tight-binding Hamiltonian (140) in the real space has the following form H = J (cid:88) r [(2 ihc † , r c , r + H.c.) + H rˆx + H rˆy + H rˆz ] ,H rˆx = ic † , r − ˆx ( c , r + c , r ) − ic † , r + ˆx ( c , r − c , r ) + H.c. ,H rˆy = − c † , r − ˆy ( c , r − ic , r ) + c † , r + ˆy ( c , r + ic , r ) + H.c. ,H rˆz = 2 ic † , r − ˆz c , r + H.c. . (143)Here c j, r ( j = 1 , , 3) denotes the annihilation operator at the lattice site r with thespin state | j (cid:105) . The major difficulty for implementing this Hamiltonian is realization ofthe spin-transferring hopping terms H rˆx , H rˆy , H rˆz . In principle, the required hoppingcan be realized by using the Raman-assisted tunneling with proper laser-frequency andpolarization selections [26, 29, 30, 54, 114, 238].A scheme to realize this Hamiltonian was proposed in Ref. [309]. As illustrated inFig. 25(c), the atom-laser coupling configuration was suggested to realize the hoppingterms, of which two beams Ω π = Ω e ikx and Ω π = Ω e iky constitute the π -polarizedlights, propagating respectively along the x and y directions, where k = 2 π/a is themagnitude of the laser wave vector. The other five beams Ω x,y,z , are all propagating alongthe z direction with the polarizations shown in Fig. 25(c). Note that the required brokenparity (left-right) symmetry is achieved by titling the lattice with a homogeneous energygradient along the x -, y -, z -directions. A different linear energy shift per site ∆ x,y,z alongdifferent directions is required, such as ∆ z ≈ . y ≈ x . Then the natural hopping issuppressed by the large tilt potential, and the hopping terms are restored and engineeredby applying two-photon Raman coupling with laser beams of proper configurations.71 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 In general, 3D topological insulators should be protected by certain kinds of symmetries[123, 161], such as time-reversal, particle-hole, or chiral symmetries. However, a specialclass of 3D topological insulators without any symmetry other than the prerequisite U (1)charge conservation was theoretically proposed, called Hopf insulators [66, 294, 314–316].The Hopf insulator is topologically characterized by a topological invariant termed Hopfindex (also known as Hopf charge or Hopf invariant) as discussed in Sec. A.7, and has zeroChern numbers. A two-band tight-binding model was first constructed on a cubic latticeto realize a special Hopf insulator with the Hopf index χ = 1 [66]. Subsequently, a class oftight-binding Hamiltonians that realize arbitrary Hopf insulator phases with any integerHopf index χ were suggested [314]. Recently, an experimental scheme to implement amodel Hamiltonian for Hopf insulators and to measure the Hopf topology in ultracoldatomic systems has been proposed in Ref. [317]. The observation of topological links andHopf fibration associated with Hopf insulators in a quantum simulator has been reportedin Ref. [318].The model Hamiltonian in momentum space discussed in Ref. [317] is given by H ( k ) = S ( k ) · σ, S ( k ) = η † ση, (144)where S ( k ) is the pseudospin field. It is defined in terms of the two complex fields as η = (¯ η p ↑ , ¯ η q ↓ ) t with p and q being coprime integers, ¯ η ↑ , ↓ = η ∗↑ , ↓ , where η ↑ and η ↓ arecomplex numbers given by η ↑ ( k ) = sin k x + i sin k y , η ↓ ( k ) = sin k z + i (cos k x + cos k y + cos k z + h ) , (145)with h being a constant parameter. We introduce the standard CP field z ( k ) = η/ (cid:112) | S ( k ) | = ( z ↑ ( k ) , z ↓ ( k )) t and the normalized pseudospin ˆS ( k ) = S ( k ) / | S ( k ) | = z † σz with | S ( k ) | = | η ↑ | p + | η ↓ | q . It is easy to obtain the expression of ˆS ( k ),ˆ S x + i ˆ S y = 2 η p ↑ ¯ η q ↓ /η + , ˆ S z = η − /η + , (146)where η ± = | η ↑ | p ± | η ↓ | q . As we can see, the CP field constructed by a four-componentvector N ( k ) with the configuration N = Re[ z ↑ ( k )], N = Im[ z ↑ ( k )], N = Re[ z ↓ ( k )],and N = Im[ z ↓ ( k )], takes values on the 3D sphere S , together with the normalizationcondition (cid:80) i N i = 1. Therefore, Eq. (145) forms a map g : T → S , where T is a3D torus (describing the first BZ). On the other hand, the normalized pseudospin ˆS ( k )expressed as z † σz defines a mapping f : S → S , where the S coordinates N ( k ) =( N , N , N , N ) are mapped to S coordinates ( ˆ S x , ˆ S y , ˆ S z ). Consequently, the underlyingstructure of the Hamiltonian (144) constructs a composite map ˆS ( k ) = f ◦ g ( k ): T → S from the BZ to the target pseudospin space S .The topological properties of the Hamiltonian in Eq. (144) are characterized by theHopf index (see Sec. A.7), which has a simple integral expression [67, 319] ν H (ˆ S ) = − (cid:90) BZ F · A d k , (147)where F is the Berry curvature defined as F µ = π (cid:15) µντ ˆ S · ( ∂ ν ˆ S × ∂ τ ˆ S ) with (cid:15) µντ being theLevi-Civita symbol and ∂ ν,τ ≡ ∂ k ν,τ ( µ, ν, τ ∈ { x, y, z } ), and A is the Berry connectionthat satisfies ∇ × A = F . One can prove [314] that the Chern number C µ = 0 inall three directions, and the Hopf index takes all integer values Z and has an analytic72 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 expression with ν H (ˆ S ) = ± pq when 1 < | h | < ν H (ˆ S ) = ± pq when | h | < ν H (ˆ S ) = 0otherwise. As we discussed before, ˆ S ( k ) is a composition of two maps ˆS ( k ) = f ◦ g ( k ).The generalized Hopf map f from S → S has a known Hopf index χ ( f ) = ± pq [319].Thus, we can decompose the composition map, ν H (ˆ S ) = ν H ( f )Λ( g ), where Λ( g ) is thetopological invariant classifying the maps g from T → S Λ( g ) = 112 π (cid:90) BZ d k (cid:15) µνρτ (cid:15) αβγ | η | η µ ∂ α η ν ∂ β η ρ ∂ γ η τ . (148)Here η = (Re[ η ↑ ( k )] , Im[ η ↑ ( k )] , Re[ η ↓ ( k )] , Im[ η ↓ ( k )]). Λ( g ) = 1 when 1 < | h | < 3, Λ( g ) = − | h | < 1, and Λ( g ) = 0 otherwise. A geometric interpretation of such compositionis as follows: Λ( g ) counts how many times T wraps around S nontrivially under themap g and ν H ( f ) describes how many times S wraps around S under f . This compositeprocess ultimately gives the Hopf index ν H (ˆ S )[314]. Numerical results show that thetopologically protected surface states and zero-energy modes in these exotic nontrivialphases are robust against random perturbations [66, 314].A geometrical image of the Hopf invariant can be obtained by noting that each pointon S has a preimage that is a circle in T , and that the linking number of two suchcircles taken from different points of S is the Hopf invariant ν H (ˆ S ). To visualize suchcircles and knots more easily, one can work with S rather than T and probe the Hopfindex ν H ( f ). Similarly, the linking number of two preimage contours of distinct spinorientations is equal to the Hopf invariant ν H ( f ). Nevertheless, S is a hypersphere in4D space R where is difficult to visualize the circles in S . So one can visualize the Hopflinks by using a stereographic projection of S to R , where the topological structure isretained [320, 321]. The stereographic projection used in Ref. [317] is defined as( x, y, z ) = 11 + η ( η , η , η ) , (149)where ( x, y, z ) and ( η , η , η , η ) are points of R and S , respectively. Stereographicprojection preserves circles and maps of Hopf fibers as geometrically perfect circles inR , but there is one exception: the Hopf circle containing the projection point (0 , , , − as a “circle through infinity”. Moreover, the preimage map f − (ˆ S ) in S must be determined in order to obtain the stereographic projection ofthe point on S in R by using Eq. (149). A direct parametrization of the 3D-sphereemploying the Hopf map is as follows [319] η ↑ = | η ↑ | e iqθ , η ↓ = | η ↓ | e ipθ , (150)or as follows in Euclidean R η = | η ↑ | cos( qθ ) , η = | η ↑ | sin( qθ ) , η = | η ↓ | cos( pθ ) , η = | η ↓ | sin( pθ ) , (151)where θ , runs over the range 0 to 2 π , with | η ↑ | + | η ↓ | = 1. A mapping of the aboveparametrization to the 2D sphere (according to Eq. (146)) is given byˆ S x = 2 | η ↑ | p | η ↓ | q η + cos[ pq ( θ − θ )] , ˆ S y = 2 | η ↑ | p | η ↓ | q η + sin[ pq ( θ − θ )] , ˆ S z = η − η + (152)For ˆ S = (1 , , (cid:0) ˆ S = (0 , , (cid:1) , we have θ = θ (cid:0) θ = θ + π pq (cid:1) , | η ↑ | p = | η ↓ | q . Bycombining it with the normalization condition | η ↑ | + | η ↓ | = 1, we can obtain the values73 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 of | η ↑ | and | η ↓ | . One can easily verify that | η ↑ | = 1, | η ↓ | = 0 (cid:0) | η ↑ | = 0 , | η ↓ | = 1 (cid:1) forˆ S = (0 , , (cid:0) ˆ S = (0 , , − (cid:1) . With these preparations, we can check that a set of points η = (cos qθ , sin qθ , , (cid:0) η = (0 , , cos pθ , sin pθ ) (cid:1) forming a ring in R is the preimageof the point ˆ S = (0 , , (cid:0) ˆ S = (0 , , − (cid:1) on S . If we denote the stereographic projection s : S \ (0 , , , − → R given in Eq. (149), then s ◦ f − ((0 , , x - y plane, s ◦ f − ((0 , , − z axis, and for any other point on S not equal to(0 , , 1) or (0 , , − s ◦ f − (ˆ S ) is a circle in R when we choose p = q = 1. In Fig. 26(a),the simplest nontrivial spin texture corresponding to ν H ( f ) = 1 is sketched, where theparameters are chosen as p = q = 1 and h = 2. This spin texture twisted with ν H ( f ) = 1is nontrivial and cannot be continuously untwined unless a topological phase transitionoccurs. Following the above ideas, one can find more complex knots and links for larger p and q , such as the well-known trefoil knot ( p = 3 , q = 2) and the Solomon seal knot( p = 5 , q = 2) plotted in Ref. [317] with nonunit knot polynomials [322]. Figure 26. (Color online). (a) Hopf links and spin texture in stereographic coordinates. Spins residing on the red(green) circle point to the x ( z ) direction and those on the z axis all point to the south (negative z direction). Thered (green) circle represents the preimage of ˆ S = (1 , , 0) (ˆ S = (0 , , z axis represent the preimageof (0 , , − k z = 0 layer. The background color scalelabels the magnitude of the out-of-plane component ˆ S z , and the arrows label the magnitude and direction of spinsin the k x - k y plane. (c) Topological links between the preimages from two spin states on the Bloch sphere ˆ S andˆ S . The red (blue) circle denotes the theoretical preimage of ˆ S (ˆ S ) and the scattered red squares (blue dots)are numerically simulated preimage of the (cid:15) -neighborhood of ˆ S (ˆ S ), which can be observed from time-of-flightimages. The parameters are chosen as p = q = 1 and h = 2. Reprinted by permission from Deng et al. [317]. In the previous discussion, a nonvanishing value of ν H (ˆ S ) indicates that the pseudospinfield ˆ S has a nontrivial texture that cannot be continuously deformed into a trivial one.Since the spin textures for the model can be interpreted as ˆ S = (cid:104) σ (cid:105) , they can be observedin cold-atom experiments through time-of-flight imaging [244, 313]. Fig. 26(b) shows aslice of the observed ˆ S with k z = 0 for the simplest case of p = q = 1, which provides aglimpse of the 3D twisting of the Hopfion [323]. With the obtained spin texture, one canreconstruct the topological links and knots by mapping out the preimages of two differentorientations of ˆ S ( k ). However, the various kinds of noises involved in a real experimentmay lead to inaccurate measurement results of ˆ S ( k ). To simulate real experiments, theauthors discussed the Hamiltonian (144) in real space and considered a finite-size latticewith open boundaries, so the spin orientation ˆ S ( k ) in real experiment is always pixelizedwith a finite resolution, which means the observed ˆ S ( k ) can only be approximately ratherthan exactly equal to the specific orientations (such as ˆ S , ) at any momentum point k .To circumvent these difficulties, one need to consider a small (cid:15) -neigborhood of a specificorientation (e.g., ˆ S ): N (cid:15) (ˆ S ) = { ˆ S : | ˆ S − ˆ S | ≤ (cid:15) } , (153)where | ˆ S − ˆ S | = [( ˆ S x − ˆ S x ) + ( ˆ S y − ˆ S y ) + ( ˆ S z − ˆ S z ) ] / represents the distance74 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 between ˆ S and ˆ S . The preimages of all orientations in N (cid:15) (ˆ S ) are then denoted as a setof P (cid:15) (ˆ S ) = ( f ◦ g ) − [ N (cid:15) (ˆ S )] describing the points in T . Due to the finite resolution andthe discrete BZ, P (cid:15) (ˆ S ) contains finite momentum points. Therefore, one should choosean appropriate value of (cid:15) to ensure that P (cid:15) (ˆ S ) contains a proper amount of momentumpoints that could depict the closed loop structure of ( f ◦ g ) − (ˆ S ). Fig. 26(c) shows thesimulated Hopf link with linking number one of real experiments. In order to obtain suchimages, one should first examine the discrete ˆ S ( k ) at each momentum point k and thenappend k to the set of P (cid:15) (ˆ S ) ( P (cid:15) (ˆ S )) while ˆ S ( k ) is in an (cid:15) -neighborhood of ˆ S (ˆ S ).By plotting g ( P (cid:15) (ˆ S )) and g ( P (cid:15) (ˆ S )) in the stereographic coordinate system defined inEq. (149), one can obtain Fig. 26(c) in R .On the other hand, with the observed ˆ S ( k ), one can directly extract the Hopf invariant.Since F µ = π (cid:15) µντ ˆ S · ∂ ν ˆ S × ∂ τ ˆ S , one can obtain the discrete Berry curvature F at eachpixel of the BZ. Berry connection A ( k ) can be extracted from F by solving a discretizedversion of the electrostatics equation ∇ × A = F in momentum space with the Coulombgauge ∇ · A = . Finally, one can attain the value of the Hopf index ν H (ˆ S ) by a discretesum over all momentum points. It was also numerically demonstrated that a finite-sizelattice of 10 × × 10 is already capable of producing highly accurate estimation ofthe quantized Hopf index and the detection method remains robust to experimentalimperfections and the global harmonic trap [317].The physical realization of the Hopf insulators is of great interest but also especiallychallenging. In principle, the model Hamiltonian of the Hopf insulators in Eq. (144) (with p = q = 1 as the simplest case) could be realized using the Raman-assisted hoppingtechnique with ultracold atoms in OLs, which will involve a number of laser beams [317]. After the discovery of the QHE in 2D systems [7, 8], it was shown that if there is a bandgap in a 3D periodic lattice, the integer QHE can also exhibit when the Fermi energy liesinside the gap [324–327]. In the 3D QHE, the Hall conductance in each crystal plane canhave a quantized Hall value defined on a torus spanned by the two quasi-momenta for thecrystal plane. It is hard to obtain the energy spectrum with band gaps for the emergenceof quantized Hall conductivities in 3D periodic lattices since a motion along the thirddirection may wash out the gaps of the perpendicular 2D plane. Therefore up to now,the 3D QHE has been predicted or observed only in systems with extreme anisotropy orunconventional toroidal magnetic fields [327–331].A scheme was recently proposed to realize the 3D QHE in a tunable generalized 3DHofstadter system that can be simulated by engineering the Raman-assisted hoppingof ultracold atoms in a cubic OL [265]. The optical lattice is tilted along the y and z axis, as shown in Fig. 27(a). The atoms are prepared in a hyperfine state of theground state manifold, and the tilt potentials with linear energy shift per lattice site∆ s ( s = y, z ) can be generated by the gravity or real magnetic field gradients B s s .For the case ∆ s (cid:29) J s we considered, where J s denotes the bare hopping amplitudealong the s axis, the atomic hopping between neighboring sites in these two directionsis suppressed. To restore and engineer the hopping terms with tunable effective phases,we can use the Raman-assisted tunneling technique, which has been used to realize theoriginal Hofstadter model in 2D OLs [29, 30, 48]. In order to fully and independentlyengineer the atomic hopping along the y and z axes, one can use three far-detunedRaman beams denoted by their frequencies and wave vectors { ω j , k j } ( j = 1 , , ω − ω = ∆ y / (cid:126) and ω − ω = ∆ z / (cid:126) with ∆ y (cid:54) = ∆ z are required. Themomentum transfers Q = k − k ≡ ( Q x , Q y , Q z ) and P = k − k ≡ ( P x , P y , P z ) can75 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401Figure 27. (Color online) A cold-atom setup for realizing a 3D generalized Hofstadter model [265]. (a) The OLand hopping configuration. Along the s ( s = y, z ) axis, the tilted lattice with large tilt potentials ∆ s can be createdby magnetic field gradients B s s . The natural hopping along the s axis is suppressed and then be restored by usingthree far-detuned Raman lasers denoted by { ω j , k j } ( j = 1 , , T y e ± iφ m,n,l and T z e ± iϕ m,n,l with site indices ( m, n, l ). (b) Laser-assisted tunneling between nearest neighboringsites along the s axis with the frequency differences ω − ω = ∆ y / (cid:126) and ω − ω = ∆ z / (cid:126) and the effectivetwo-photon Rabi frequency Ω s . (c) The effective magnetic fluxes { Φ , Φ , Φ } in the three elementary plaquettesin the { xy, xz, yz } planes, respectively. (d) (e) The energy spectra E as a function of the hopping strength T z for (d) Φ = 1 / 2, Φ = 1 / 3, and T y = 0 . 5; (e) Φ = 1 / 3, Φ = 1 / 5, and T y = 0 . 5. The dashed lines are shownin (d) with T z = 0 . T z = 0 . 5. The Chern numbers C = ( C xy , C xz , C yz ) when the Fermi levellies in each energy gap are shown. Reprinted with permission from Zhang et al. [265]. Copyright c (cid:13) (2017) by theAmerican Physical Society. be independently tunable, for instance, through independently adjusting the angles ofthe second and third Raman lasers with the first Raman laser being fixed, as shownin Fig. 27(a). Therefore, the Raman lasers induce atomic hopping along the y and z axes with tunable, spatially dependent phases φ m,n,l = Q · R = mφ x + nφ y + lφ z and ϕ m,n,l = P · R = mϕ x + nϕ y + lϕ z , respectively, where R = ( ma, na, la ) denotes theposition vector for the lattice site ( m, n, l ), φ x,y,z = aQ x,y,z and ϕ x,y,z = aP x,y,z . Thissystem realizes a generalized 3D Hofstadter Hamiltonian with fully tunable hoppingparameters [265] H = − (cid:88) m,n,l [ J x a † m +1 ,n,l a m,n,l + e iφ m,n,l ( T y a † m,n +1 ,l a m,n,l + T z a † m,n,l +1 a m,n,l ) + H.c.] , (154)where J x is the natural hopping along the x axis, T y e iφ m,n ( T z e iϕ m,l ) denotes the Raman-induced hopping along the y ( z ) axis with the spatially-varying phase φ m,n ( ϕ m,l ) im-printed by the Raman lasers. The hopping strengths T s = Ω s λ s can also be tuned viathe laser intensities, with λ s denoting the overlap integral of Wannier-Stark functions be-tween neighbor sites along the s axis. One can introduce three effective magnetic fluxes { Φ , Φ , Φ } through the three elementary plaquettes in the { xy, xz, yz } planes with thearea S = a , as shown in Fig. 27(c). The effective fluxes, in units of the magnetic fluxquantum, are determined by the phases picked up anticlockwise around the plaquettes.They are obtained as Φ = φ x π , Φ = ϕ x π , and Φ = φ z − ϕ y π , which can be independentlytuned. For certain hopping configurations, the bulk bands of the system can respectivelyhave Weyl points and nodal loops [265], similar as the one proposed in Ref. [264]. Thisallows the study of both nodal semimetal states within this cold atom system. Further-more, the system can exhibit the 3D QHE when the Fermi level lies in the band gaps,which is topologically characterized by one or two nonzero Chern numbers.For simplicity, we consider Φ = 0 and rational fluxes Φ = p /q and Φ = p /q ,with mutually prime integers p , and q , . In this case, the Hamiltonian (154) can beblock diagonalized as H = (cid:76) H x ( k y , k z ), where k y and k z are the quasimomenta along76 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 the periodic directions and the decoupled block Hamiltonian is given by H x ( k y , k z ) = − (cid:88) m ( J x a † m +1 a m + H.c.) − (cid:88) m V m a † m a m , (155)where V m = 2 T y cos(2 π Φ m + k y a ) + 2 T z cos(2 π Φ m + k z a ). The corresponding single-particle wave function Ψ mnl is written as Ψ mnl = e ik y y + ik z z ψ m , and then the Schr¨odingerequation ˆ H x ( k y , k z )Ψ mnl = E Ψ mnl reduces to a generalized Harper equation: − J x ( ψ m − + ψ m +1 ) − V m ψ m = Eψ m . This 1D reduced tight-binding system with two commensurabilities Φ and Φ has aperiod of the least common multiple of integers q and q denoted by ˜ q = [ q , q ]. Underthe periodic boundary condition along the x axis, the wave function ψ m satisfies ψ m = e ik x x u m ( k ) with u m ( k ) = u m +˜ q ( k ). Therefore in a general case, the spectrum of thethree-dimensional system in the presence of the effective magnetic fluxes consists of˜ q energy bands and each band has a reduced (magnetic) BZ: − π/ ˜ qa ≤ k x ≤ π/ ˜ qa , − π/a ≤ k y ≤ π/a , and − π/a ≤ k z ≤ π/a . In term of the reduced Bloch wave function u m ( k ), one has − J x ( e ik x u m − + e − ik x u m +1 ) − V m u m = E ( k ) u m .It was proven that every quantized invariant on a d -dimensional torus T d is a functionof the d ( d − / d by the d ( d − / [332]. In this 3D Hofstadter system, the topological invariants for the QHE are givenby three Chern numbers C = ( C xy , C xz , C yz ) for three 2D planes, with C yz = 0 for thetrivial yz plane since Φ = 0. As with the approach in Refs. [325, 326], when the Fermienergy lies in an energy gap between two bands N and N + 1 in this system, the othertwo Chern numbers C xs with s = y, z are given by C xs = 12 π (cid:88) n (cid:54) N (cid:90) π − π dk s (cid:48) c ( n ) xs ( k s (cid:48) ) , (156)where s (cid:48) denotes replacing s between y and z , and the Chern number c ( n ) xs ( k s (cid:48) ) for the n -th filling band (or n -th occupied Bloch state) is defined on the torus T spanned by k x and k s : c ( n ) xs ( k s (cid:48) ) = π (cid:82) π/ ˜ q − π/ ˜ q dk x (cid:82) π − π dk s F ( n ) xs ( k ), where F ( n ) xs ( k ) is the correspondingBerry curvature as a topological expression as a generalization of the results in 2D [10].In Figs. 27(d) and (e), the three Chern numbers C = ( C xy , C xz , C yz ) when the Fermilevel lies in each energy gap and the spectra are plotted. The results demonstrate that theQHE in this 3D Hofstadter system is topologically characterized by one or two nonzeroChern numbers. Higher and synthetic dimensions An important development in exploration of topological states with ultracold atomicgases is the concept of “synthetic dimensions”. As discussed in Sec. 4.1.5, the topologicalproperties of 1D quasiperiodic OLs described by the Aubry-Andr´e-Harper model can bemapped to the 2D QHE. A key in the mapping is that a cyclical parameter adiabaticallyvarying from 0 to 2 π in the 1D lattice (such as the phase of the lattice potential) plays therole of the quasimomentum of the second dimension. In Thouless pumping, the 1D systemconstitutes a Fourier component of a 2D quantum Hall system at each point of the cycle,where an adiabatic periodic pump parameter also acts as the quasimomentum. In thesecases, the cyclical parameter can effectively be considered a synthetic dimension under77 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 J (a) (b)(c) J Ω J c Figure 28. (Color online) (a) A synthetic gauge field in a synthetic dimension. Yb fermionic atoms are loadedin a hybrid lattice, generated by an OL along a real direction ˆ x with tunneling J , and by a Raman-induced hoppingbetween nuclear spin states along a synthetic direction ˆ m with a complex tunneling Ω , e iϕj . (b) Experimentalobservation of chiral edge currents J c . (c) Experimental observation of edge-skipping orbits in the ladder. Reprintedfrom Mancini et al. [37]. Reprinted with permission from AAAS. the periodic boundary condition along this dimension. This approach can be extended tostudying topological systems in higher dimensions D = d r + d s (cid:62) d r = { , , } and synthetic dimensions d s . For example, it was proposedto simulate 3D Weyl semimetal physics with cold atoms in d r = { , } D OLs that aresubjected to d s = { , } D synthetic dimensions from external cyclical parameters [136,256]. More interestingly, the intriguing 4D quantum Hall physics [16, 64] can be exploredin a 2D topological charge pump in 2D OLs with two cyclical parameters, which togethergive an effective 4D BZ [17, 333, 334].Another kind of synthetic dimension is engineered by a set of discrete internal atomic(spin) states as fictitious lattice sites [335, 336]. In this approach, the atoms loaded intoa d r -dimensional OL can potentially simulate systems of D = d r + 1 spatial dimensions.The hopping processes along the synthetic dimension can be induced by driving transi-tions between different internal states with Raman lasers. The laser-coupling between twointernal atomic states has complex coupling element, which represents the tunneling ma-trix in the synthetic dimension picture. Hence, similar to the Raman-assisted-tunnelingscheme, this fictitious tunneling contains a complex phase-factor, which can then be usedto simulate synthetic gauge fields in the synthetic dimensions and a finite strip of theHofstadter model [336]. For the atoms that have the hyperfine spin F , the Raman laserscouple spin state | m F (cid:105) to | m F ± (cid:105) , where m F takes any value between − F and F witha total of W = 2 F + 1 components. This provides the naturally sharp boundaries in theextra dimension, while it is also possible to create periodic boundary conditions in thesynthetic dimension by using an additional coupling to connect the extremal internalstates. This differs from the cyclical synthetic dimension previously introduced. There-fore, the proposed 1D OL [336] that combines real and synthetic spaces offers a keyadvantage to work with a finite-sized system with sharp and addressable edges, such asthe detection of chiral edge states resulting from the synthetic magnetic flux.The proposed synthetic-dimension scheme [336] was recently realized in two indepen-dent experiments with Yb fermionic atoms [37] and Rb bosonic atoms [38]. As shownin Fig. 28(a), a system of Yb fermionic atoms in an atomic Hall ribbon of tunablewidth pierced by an effective gauge field was experimentally synthesized [37]. One real di-mension is realized by an OL with the tunneling J between different sites along directionˆ x . The synthetic dimension is encoded in the different internal spin states (the sites of78 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 (c) Figure 29. (Color online) (a) The synthetic 4D lattice in the absence of perturbing fields, with atoms in a 3DOL with hopping amplitude J and flux 2 π Φ from x -dependent Peierls phase-factors in z hopping. The fourthdimension w is a synthetic dimension with flux 2 π Φ in the y − w plane by adding a y -dependent phase-factor tothe Raman-induced internal-state transitions. (b) Energy spectrum E ( k x , k y ) for Φ , = 1 / k z,w , with the second Chern number of the lowest band C = − 1. (c) Simulation of the center-of-mass trajectory x c.m. ( t ) after ramping up a perturbing “electric” and “magnetic” field (the blue sold curve), with the predicteddrift for C = − et al. [41]. Copyright c (cid:13) (2015) bythe American Physical Society. the synthetic dimension for the F = 5 / e iϕx between different spin components. The phase amounts to the synthesis of an effectivemagnetic field with tunable flux ϕ/ π (in units of the magnetic flux quantum) per pla-quette, mimicking the 2D Hofstadter model on a three-leg ladder. The Hamiltonian ofthe system is given by H = (cid:88) j (cid:88) α (cid:20) − J ( c † j,α c j +1 ,α − Ω α e iϕj c † j,α c j,α +1 + h.c.) + µ j n j,α + ξ α n j,α (cid:21) , (157)where c † j,α ( c j,α ) are fermionic creation (annihilation) operators on the site ( j, α ) in the real( j ) and synthetic ( α = 1 , , 3) dimension, and n = c † j,α c j,α . The hopping parameters Ω α are typically inhomogeneous due to the Clebsch-Gordan coefficients associated with theatomic transitions. Besides the tunneling terms, µ j describes a weak trapping potentialalong ˆ x , while ξ α accounts for a state-dependent light shift, giving an energy offset alongˆ m . In the experiment, the chiral currents J c in the upper and lower edge chains withopposite sign were observed, while the central leg showed a suppressed net current inthe bulk, as shown in Fig. 28(b). This directly signals the existence of chiral statespropagating along the edges of the system, reminiscent of the edge states in the QHEin the Hofstadter model. In addition, the edge-skipping orbits with the cyclotron-typedynamics in the ladder due to the presence of the synthetic magnetic field were furtherdetected through state-resolved images of the atomic cloud, as shown in Fig. 28(c). Inthe ladders, the finite-size effect (such as the overlap of chiral edge modes in differentedges) is significant because the lattice size along the synthetic dimension is small. Tolimit undesired finite-size effects, one may use other atomic species with more addressableinternal states in the synthetic-dimension approach.The synthetic three-leg ladder with the magnetic flux was also realized for Rb bosonicatoms, and the chiral edge currents and skipping orbits were both observed in the quan-tum Hall regime [38]. In addition, this synthetic-dimension approach was demonstratedin a more recent experiment without two-photon Raman transitions but instead, basedon a single-photon optical clock transition coupling two long-lived electronic states oftwo-electron Yb atoms [39]. These two systems involve less heating, which would beimportant for further studies, such as spectroscopic measurements of the Hofstadter but-terfly and realizations of Laughlin’s charge pump.There is an important difference between ordinary lattice systems and systems in-volving a synthetic dimension. In the synthetic dimension, interactions are genericallylong-ranged, in contrast to the on-site interactions along the physical dimension. With the79 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 effectively nonlocal interactions, the cold atom systems realize extended-Hubbard mod-els, which can display novel phases due to the intriguing many-body effects unattainablein conventional condensed matter setups [337–340]. Such an interacting fermion gas withmulti-spin components coupled through Raman beams in a 1D OL provides an idealsystem to realize the topological fractional pumping reflected by the quantization tofractional values of the pumped charge and to measure the many-body Chern number ina cold-atom experiment [153, 341].The synthetic-dimension technique offers a novel platform for exploring topologicalstates in higher dimensions [16, 40], such as detecting the 4D QHE [41]. In the 2D QHE,the quantized Hall response induced by an external electric field is topologically charac-terized by the first Chern number. As one of the first predictions of the time-reversal sym-metric topological insulators [16, 64], the QHE can be generalized to 4D systems. In the4D QHE, an additional quantized Hall response appears, which is nonlinear and describedby a 4D topological index, the second Chern number. The intriguing 4D QHE with thesecond Chern number was also theoretically studied in other models [64, 333, 342, 343].In the proposal [41], as shown in Fig. 29(a), a synthetic 4D lattice contains atoms hoppingin a 3D OL with Raman-coupling internal states as the fourth dimension w , in which the x − z and y − w planes are penetrated by synthetic uniform magnetic fluxes Φ , , respec-tively. This corresponds to two copies of the Hofstadter model defined in disconnectedplanes, described by the tight-binding Hamiltonian H = − J (cid:88) r (cid:0) c † r + a e x c r + c † r + a e y c r + e i π Φ x/a c † r + a e z c r + e i π Φ y/a c † r + a e w c r (cid:1) +h.c. , (158)where c † r creates a fermion at lattice site r = ( x, y, z, w ). To realize this Hamiltonian,one requires x ( y ) dependent Peierls phases for tunneling along the z ( w ) direction,generating a uniform flux Φ (Φ ) in the x − z ( y − w ) plane. This can be created bycombining the laser-assisted hopping along the z direction and the synthetic gauge fieldin the synthetic dimension. As shown in Fig. 29(b), the bulk energy spectrum E ( k ) ofthe Hamiltonian is reminiscent of the two underlying 2D Hofstadter models defined inthe x − z and y − w planes, where the lowest band can be non-degenerate and well isolatedfrom higher-energy bands for suitable fluxes Φ , . Moreover, the lowest band E ( k ) ischaracterized by a non-zero second Chern number C = − C = 14 π (cid:90) T (Ω xy Ω zw +Ω wx Ω zy +Ω zx Ω yw )d k, (159)where Ω µν is the 4D generalized Berry curvature. If there are additional perturbing“electric” field E y and “magnetic” field B zw = − π ˜Φ /a , the current density along the x dimension as a response for a filled band is given by j x = C π E y B zw = − C πa E y ˜Φ , (160)which reveals a genuine non-linear 4D quantum Hall response and is directly relatedto C . Hence, as shown in Fig. 29(c), the second Chern number in this system canbe measured from the center-of-mass drift along the x direction [41, 344]: x c.m. ( t ) = x c.m. (0) + v c.m. t = x c.m. (0) + j x V cell t , with V cell as the (magnetic) unit cell volume. Forthe neutral atoms, the perturbing “electric” field corresponds to a linear gradient thatcan created either magnetically or optically, and the perturbing “magnetic” field can begenerated by engineering additional Peierls phases.Recently, based on a 2D topological charge pump as proposed in Ref. [333], a dynamical80 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 (a) (b) (c) (d)(e) Figure 30. (Color online) 4D quantum Hall system and corresponding 2D topological charge pump. (a) A 2Dquantum Hall system on a cylinder pierced by a uniform magnetic flux Φ xz , and an electric field E z on the surfaceresulting in a linear Hall response along x with velocity v x . (b) A 4D quantum Hall system can be composed oftwo 2D quantum Hall systems in the xz - and yw -planes. (c) A dynamical version of the 4D quantum Hall systemcan be realized with a topological charge pump in a 2D superlattice (blue potentials). (d) The pumping gives riseto a motion of the atom cloud in the x -direction, corresponding to the quantized linear response of a 2D quantumHall system. (e) The velocity of the non-linear response is determined by the product of the Berry curvaturesΩ x Ω y . The left (right) torus shows a cut at k y = 0, ϕ y = π/ k x = π/ (2 d l ), ϕ x = π/ 2) through the generalized4D BZ spanned by k x,y and ϕ x,y . Reprinted by permission from Macmillan Publishers Ltd: Lohse et al. [17],copyright c (cid:13) (2018). version of the 4D integer QHE was realized by using ultracold bosonic atoms in an angledoptical superlattice, and the bulk quantized response associated with the second Chernnumber was observed [17]. The dynamical 4D quantum Hall system was also experimen-tally realized with tunable 2D arrays of photonic waveguides [334]. For the geometry inFigs. 30(a,b), the 2D subsystem as a Fourier component of a 4D quantum Hall system isa square superlattice in Fig. 30(c). It consists of two 1D superlattices along x and y , eachformed by superimposing two lattices V s,µ sin ( πµ/d s ) + V l,µ sin ( πµ/d l − ϕ µ / µ ∈ { x, y } , d s,µ , and d l,µ = 2 d s,µ denote the lattice periods, and V s,µ ( V l,µ ) is the depthof the short (long) lattice potential. The superlattice phases ϕ µ determined the positionof the long lattices relative to the short ones, and the phase ϕ x was chosen as the pumpparameter in the experiment [17]. This is equivalent to threading ϕ x in the 4D model,which leads to a quantized motion along x as the linear response shown in Figs. 30(c,d).The magnetic perturbation B xw corresponds to a transverse superlattice phase ϕ y thatdepends linearly on x , which is realized by tilting the long y -lattice relative to the shortone by a small angle θ in the xy -plane, with ϕ y ( x ) = ϕ (0) y + 2 πθ x/d l,y to first order in θ .As the two orthogonal axes are coupled, by varying ϕ x , the motion along x changes ϕ y .This is analogous to the Lorentz force in 4D and induces a quantized non-linear responsealong y [333]. For a uniformly populated band in an infinite system, the change in thecenter-of-mass position ∆ r during one cycle ϕ x = 0 → π is given by [17]∆ r = C x d l,x e x + C θ d l,x e y . (161)The first term proportional to the pump’s first Chern number C x describes the quantizedlinear response in the x -direction. The second term is the non-linear response in the y -direction, which is quantified by a 4D integer topological invariant, the pump’s secondChern number C = 14 π (cid:73) BZ Ω x Ω y d k x d k y d ϕ x d ϕ y . (162)The generalized 4D BZ is shown in Fig. 30(e), and the Berry curvature Ω µ ( k µ , ϕ µ ) = i (cid:0) (cid:104) ∂ ϕ µ u | ∂ k µ u (cid:105) − (cid:104) ∂ k µ u | ∂ ϕ µ u (cid:105) (cid:1) , with | u ( k µ , ϕ x ) (cid:105) as the eigenstate of a given non-degenerate band at k µ and ϕ µ . In the experiment [17], the 2D topological charge pumpwas realized with bosonic Rb atoms forming a Mott insulator in the superlattice. The4D-like nonlinear response of the lowest subband with C = +1 was experimentally ob-served from the atomic center-of-mass shift after a pump circle. Furthermore, using a81 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 small cloud of atoms as a local probe, the 4D geometric properties and the quantizationof the response were fully characterized via in situ imaging and site-resolved band map-ping. This work paves the way for exploring higher-dimensional quantum Hall systemswith additional strongly correlated topological phases, exotic collective excitations andboundary phenomena such as isolated Weyl fermions [16, 64].Finally we note that the concept of synthetic dimension has been greatly extended,which can involve other kinds of degrees of freedom. For instance, the synthetic di-mensions may also be engineered by a set of orbital angular momenta for light [345]or harmonic oscillator eigenstates for cold atoms in a harmonic trap [346] as fictitiouslattice sites. Moreover, it was proposed to create an effective synthetic lattice of sitesin momentum space based on discrete momentum states of neutral atoms, which canbe parametrically coupled with interfering Bragg laser fields [143, 144]. The syntheticmomentum-space lattice has been realized with cold atoms and opened up new prospectsin the experimental study of disordered and topological systems [142, 145, 347–349]. Higher-spin topological quasiparticles In the previous sections, we focus on the spin-1/2 systems, such as the Dirac and Weylfermions, which have rich topological features. Quasiparticles with higher spin num-bers are also fundamentally important but rarely studied in condensed-matter physicsor artificial systems [350–352]. These systems can potentially provide a quantum fam-ily to find relativistic quasiparticles that have no high-energy analogs, such as integer-(speudo)spin fermionic excitations. Recently, a series of work theoretically predicted thatunconventional fermions beyond the Dirac-Weyl-Majorana classification (also termed“new fermions”, which means no elementary particle analogs) can emerge in some bandstructures [350, 353]. These works have set off a boom in investigating and realizing “newfermions” in condensed matter and artificial systems [354–361].A recent work to implement the pseudospin-1 fermions in cold-atom systems was pro-posed in Ref. [355]. In this paper, the authors constructed 2D and 3D tight-bindingmodels realizable with cold fermionic atoms in OLs, where the low-energy excitationsare effectively described by the spin-1 Maxwell equations in the Hamiltonian form. TheHamiltonian of these low-energy excitations is given by H M = v x k x ˆ S x + v y k y ˆ S y + v z k z ˆ S z , (163)where ˆ S β = ( ˆ S αγ ) β = i(cid:15) αβγ , and (cid:15) αβγ ( α, β, γ = x, y, z ) is the Levi-Civita symbol. Thisso-called Maxwell Hamiltonian H M originally describes a massless relativistic boson (pho-ton) with spin one. Because quasiparticles in a lattice system are constrained only bycertain subgroups (space groups) of the Poincar´e symmetry rather than by Poincar´e sym-metry in high-energy physics [350], there is the potential to find free fermionic excitationsdescribed by H M in lattice systems. Such relativistic (linear dispersion) excitations withunconventional integer pseudospins are termed Maxwell fermions. For specificity andwithout loss of generality, we describe the topological features of Maxwell fermions in2D and 3D OLs in some detail.The proposed Bloch Hamiltonian for a 2D case in momentum space is as follows [355] H ( k ) = R ( k ) · ˆ S (164)where the Bolch vector R ( k ) = ( R x , R y , R z ) is given by R x = 2 J sin k x , R y = 2 J sin k y , R z = 2 J ( M − cos k x − cos k y ) , (165)82 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401Figure 31. (Color online) The energy spectra and topological features of the 2D Maxwell lattices [355]. (a) Theenergy spectrum for M = 2; (b) The energy spectrum for M = 0; (c) The Berry phase γ as a function of theparameter M , which corresponds to the Chern number C = γ/ π when the 2D system is in the insulating phasewith M (cid:54) = 0 , ± 2. Reprinted with permission from Zhu et al. [355]. Copyright c (cid:13) (2017) by the American PhysicalSociety. with M being a tunable parameter. The energy spectrum of this system is given by E ( k ) = 0 , ±| R ( k ) | , which has a zero-energy flat band in the middle of the three bands.The three bands touch at a single point K ± = (0 , / ( π, π ) when M = ± 2, as shown inFig. 31(a), and touch at two points when M = 0. The low-energy effective Hamiltoniannear K ± can be expanded to linear order as H ± ( q ) = ± v ( q x ˆ S x + q y ˆ S y ) , (166)where v = 2 J is the effective speed of light and q = k − K ± . These fermionic exci-tations are described by the Maxwell Hamiltonian H M in 2D. In this sense, these low-energy excitations can be named Maxwell fermions, and the threefold degenerate pointas Maxwell point. When the Fermi level lies near the Maxwell point, this system is alsonamed Maxwell metal with a zero-energy flat band. To study the topological propertiesof Maxwell metal phase, one can evaluate the Berry phase circling around the Maxwellpoint γ = (cid:72) c d k · F ( k ) , where the Berry curvature F ( k ) for the lower band in the k x - k y space has an expression of F xy = − R R · ( ∂ k x R × ∂ k y R ) . (167)For M = ± 2, the gapless points contribute to non-trivial Berry phase γ = ± π . In otherwords, each pseudospin-1 Maxwell point contributes to an integer Hall conductance whenan external synthetic magnetic field along the z axis is applied [351]. When M = 0 andwith the spectrum depicted in Fig. 31(b), two Maxwell points touch at (0 , π ) and ( π, H ( q ) = ± v ( q x ˆ S x − q y ˆ S y ) . (168)In this case, the Berry phase for both Maxwell points is γ = 0, which corresponds to atrivial metallic state.The system is in an insulating state when M (cid:54) = 0 , ± C n forthe three bands: C n = 12 π (cid:90) BZ dk x dk y F xy ( k x , k y ) = γ/ π. (169)Here n = − , , pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 C − = − C = 2sgn( M ) for | M | < C − = C = 0 for | M | > 2, and thus thezero Chern number C ( M ) = 0 for the flat band. Figure 31(d) shows the phase diagramsof this model characterized by γ/ π of the lowest band as a function of the parameter M , which implies topological phase transition with band closing in this system when M = − , , 2. In addition, a correspondence between the helicity of these edge states andthe polarization of photons was found by investigating the edge modes between the firstgap. For the case 0 < M < 2, the system is a non-trivial insulator with C = 2. Underthe open boundary condition along the x direction, there are two edge states for eachedge and the corresponding effective Hamiltonian is given by H edge = v y k y ˆ S y . This edgeHamiltonian is the 1D Hamiltonian of circularly-polarized photons [355]. The helicityoperator defined as h = ˆ S · k / | k | = sgn( k y ) ˆ S y (170)is the projection of the spin along the direction of the linear momentum [351]. Hence,the edge quasiparticle-streams in this Maxwell topological insulator can be treated asMaxwell fermion-streams with the same helicities h ≡ (cid:104) ˆ h (cid:105) = +1 for opposite momenta.The model Hamiltonian determined by the Bloch vector (165) can be generalized tothe 3D model by adding an external term − J cos k z to the z -component of R ( k ). Thusthe Bloch Hamiltonian preserves inversion symmetry ( ˆ P ) represented as ˆ P H ( k ) ˆ P − = H ( − k ) and breaks the TRS ( ˆ T ) since ˆ T H ( k ) ˆ T − (cid:54) = H ( − k ) , where ˆ P = diag(1 , , − 1) andˆ T = ˆ I ˆ K , with ˆ I = diag(1 , , 1) and ˆ K being the complex conjugate operator. The systemis a Maxwell metal for | M | < | M | > 3. For simplicity, weconsider the typical case of M = 2, where the band spectrum hosts two Maxwell pointsin the first BZ at M ± = (cid:0) , , ± π (cid:1) . The corresponding low-energy effective Hamiltoniannow becomes H M ± ( q ) = vq x ˆ S x + vq y ˆ S y ± vq z ˆ S z . (171)The two 3D Maxwell points have topological monopoles C M ± = ± 2, which is defined interms of a Chern number (defined by the lowest band) on a sphere enclosing the bandtouching point. There are two Fermi arcs connecting the two points under open boundarycondition, which are similar to those in double-Weyl semimetals. The difference betweenthem is that the 3D Maxwell points in Maxwell metals have linear momenta along allthree directions, while the dispersion near double-Weyl points takes the quadratic form.Besides, the topological stability of Maxwell points is weaker than that of Weyl points.In this proposed model, the band gaps will be opened and Maxwell points will disappearwhen the inversion symmetry is broken by introducing a perturbation term with one ofthe other five SU (3) Gell-Mann matrices.Two different schemes were proposed to realize the spin-1 Maxwell fermions in OLs[355]. The first scheme is to use non-interacting fermionic atoms in a square or cubicOL and choose three atomic internal states in the ground state manifold to encode thethree spin states. Using three atomic internal states to form the pseudospin-1 basis leadsto the realization of Maxwell fermions in a lattice of simplest geometry, i.e., a primitivesquare or cubic lattice. Moreover, Maxwell fermions can be alternatively realized by usingsingle-component fermionic atoms in OLs with three sublattices, where the pseudospin-1basis is represented by the three sublattices in a unit cell. Both schemes involve Raman-assisted hopping with proper laser-frequency and polarization selections [355], which issimilar to the method we discussed in Sec. 4.3.5.Inspired by the investigation of type-II Weyl semimetals, a recent work studied thetopological triply-degenerate points [360] in OLs induced by spin-tensor-momentum cou-84 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 pling [358]. These triply-degenerate points possess fermionic excitations with effectiveinteger spins. As we mentioned above, a 3D Maxwell point with threefold degeneracywill be destroyed by a small spin-tensor perturbation ∼ N ij . Here the spin-1 matrices ˆ S are also termed spin-vector-momentum-coupling, and the spin-tensor-coupling matrices N ij = ( ˆ S i ˆ S j + ˆ S j ˆ S i ) / i, j = x, y, z ) are equivalent to the so-called Gell-Mann matri-ces, which form a basis of the SU (3) algebra. A simple Hamiltonian of a stable triply-degenerate point induced by a momentum-dependent term k i N ij is as follows [358] H ( k ) = k x ˆ S x + k y ˆ S y + k z ( α ˆ S z + βN ij ) , (172)where the spin-tensor N ij is coupled to the k z direction. At k = 0, H ( k ) exhibits atriply-degenerate point with a topological charge C . The model Hamiltonian (172) hasa symmetry H ( k ) = −H ( − k ), indicating that the Chern number C = − C − for theupper and lower bands and C = 0 for the middle one. For convenience, we use the lower-band Chern number as an topological invariant for labeling triply-degenerate points, i.e., C = C − . For β = 0, such a triply-degenerate point is the 3D Maxwell point we discussedbefore, which carries the topological monopole C = 12 π (cid:73) S F − ( k ) · d S = 2sgn( α ) , (173)where S is a surface enclosing a triply-degenerate point, and F − ( k ) = ∇ ×(cid:104) u − ( k ) | i∂ k | u − ( k ) (cid:105) = sgn( α ) k / | k | . Hereafter, the simplest triply-degenerate point withtopological monopoles C = ± βN ij term will induce three types of triply-degenerate points [358] in Eq. (172).The monopole charge of a type-I triply-degenerate point will not be changed by the threespin-tensors N xx , N yy , and N xy . The tensor N zz induces a type-II triply-degenerate pointwith C = ± | β | > | α | (cid:54) = 0. A type-III triply-degenerate point with C = 0 can beinduced by the tensor N xz or N yz for | β | > | α | (cid:54) = 0.It was proposed to realize type-II and type-III triply-degenerate points by couplingthree atomic hyperfine states, based on the realization of spin-tensor-momentum cou-pling and spin-vector-momentum-coupling with spin-1 cold atoms [260, 362–364]. Therealization of topological monopoles of three different types of triply-degenerate pointsand investigation of their geometric properties using the parameter space formed by threehyperfine states of ultracold atoms coupled by radio-frequency fields was proposed [361].The Maxwell points has recently been experimentally realized in the parameter spaceof a superconducting qutrit [357], where the other two types of triply-degenerate pointsmay also be realized in this artificial atom system. Furthermore, there is potential tostudy even higher spin qusiparticles with untracold atoms. For instance, the spin-3/2qusiparticles satisfy the so-called Rarita-Schwinger equation in Rarita-Schwinger-Weylsemimetal [352], and Dirac-Weyl fermions with arbitrary spin in 2D optical superlattices[351]. 5. Probing methods Since the atoms are neutral, the traditional transport measurements in solids that can beused to determine the topological bands, such as measuring the Chern numbers via theQHE, are very challenging in cold atomic systems. Therefore, new methods of probing thetopological states of matter in cold atom systems are needed. On the other hand, sometopological invariants, such as the underlying Berry curvature as the central measureof topology, could be directly measured in OL systems, which is not easily accessible in85 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 (a) (b) Figure 32. (Color online) Detecting Dirac points from Bloch-Landau-Zener transition. (a) The quasi-momentumdistribution of the atoms before and after one Bloch oscillation, and (b) the corresponding trajectories. Reprintedby permission from Macmillan Publishers Ltd: Tarruell et al. [77], copyright c (cid:13) (2012). traditional condensed matter materials. In this section, we review the methods developedto reveal intrinsic properties of topological states and phenomena in cold atom systems.We focus our discussion on the noninteracting atomic gases in OLs, highlighting thoseexperimental probes of topological invariants that are specific to the single-particle topo-logical Bloch bands, and will mention extensions to the topological atomic states withinteractions. Detection of Dirac points and topological transition It has been demonstrated that the Dirac points (massless Dirac fermions) in a honeycombOL can be probed from measuring the atomic fraction tunnelling to the upper band inBloch oscillations [77, 215, 216], which is the Landau-Zener transitions between thetwo energy bands. The starting point of the experiment is a ultracold gas of fermionic K prepared in the lowest-energy band of a honeycomb OL. The atomic cloud is thensubjected to a constant force F along the x direction by application of a weak magneticfield gradient, as the effect is equivalent to that produced by an electric field in solid-statesystems. As shown in Fig. 32, the atoms are accelerated such that their quasi-momentum q x increases linearly up to the edge of the BZ, where a Bragg reflection occurs. The cloudeventually returns to the centre of the band, performing one full Bloch oscillation, andthe quasi-momentum distribution of the atoms in the different bands is measured. Fora trajectory far from the Dirac points, the atoms remain in the lowest-energy band. Incontrast, when passing through a Dirac point, the atoms are transferred from the firstband to the second because of the vanishing energy splitting at the linear band crossing.Thus, the points (position) of maximum transfer factions can be used to identify the Diracpoints as the transition probability in a single Landau-Zener event increases exponentiallyas the energy gap decreases. Moreover, the topological transition from gapless (masslessDirac fermions) to gapped bands (massive Dirac fermions) can also be mapped out byrecording the fraction of atoms transferred to the second band. This Bloch-Landau-Zener-oscillation technique can be extended to detect other band-touching points, such as theWeyl points and nodal lines in 3D OLs [258, 275, 285, 365].The Bragg spectroscopy can provide an alternative method to confirm the linear dis-persion relation for the massless Dirac fermions and the energy gap for the massive ones[76]. As shown in Fig. 33(a), two laser beams are shined on the atomic gas in the Braggspectroscopy, by fixing the angle between the two beams, which gives rise to the relativemomentum transfer q = k − k , with k i being the wave vector of each laser beam. Onecan then measure the atomic transition rate by scanning the laser frequency difference ω = ω − ω . From the Fermi’s golden rule, this transition rate basically measures the86 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401Figure 33. (Color online) Detecting Dirac fermions from Bragg spectroscopy [76]. (a) Schematic of the Braggscattering. (b) The dynamic structure factors S ( q , ω ) for the massless Dirac fermions (solid line) and for themassive ones (dotted line). following dynamical structure factor S ( q , ω ) = (cid:88) k , k |(cid:104) f k | H B | i k (cid:105)| δ [ (cid:126) ω − E f k + E i k ] , (174)where H B = (cid:80) k , k Ω e i q · r | i k (cid:105)(cid:104) f k | + h.c. is the light-atom interaction Hamiltonian, and | i k (cid:105) and | f k (cid:105) denote the initial and the final atomic states with the energies E i k and E f k and the momenta k and k , respectively. At the half filling, the excitations aredominantly around the Dirac point, and S ( q, ω ) has the expression for the massless Diracfermions [76] S ( q, ω ) = (cid:40) , ω (cid:54) ω r ; π Ω v F q r − q √ q r − q Υ( ω − ω r ) , ω > ω r . (175)where Υ is the unit step function, ω r = qv F / (cid:126) ( q ≡ | q | ) and q r = (cid:126) ω/v F . For massiveDirac fermions with the dispersion E ≈ ± (∆ g + (cid:126) q x / m x + (cid:126) q y / m y ) with the effectivemass m x,y = (cid:126) ∆ g /v x , y , the Fermi velocity q x,y along x, y axis and the energy gap ∆ g ,the dynamical structure factor becomes [76] S ( q, ω ) = (cid:40) , ω (cid:54) ω r ; π Ω ∆ g v x v y Υ( ω − ω x,yc ) , ω > ω r . (176)where ω x,yc = 2∆ g + (cid:126) q x,y / m x,y . As shown in Fig. 33(b), the dynamical structurefactor for massless Dirac fermions has the lower cutoff frequency ω r that is linearlyproportional to the momentum difference q and vanishes when q tends to zero. Formassive ones, the lower cutoff frequency ω x,yc does not vanish as the momentum transfergoes to zero. This distinctive difference between the dynamical structure factors can beused to distinguish the cases with massive or massless Dirac fermions. Similar Bragg-spectroscopy methods were proposed to probe the edge and bulk states in 2D Cherninsulators in OLs [203, 234, 366, 367], such that the topological phase transition fromtrivial insulating phase to quantum anomalous Hall phase can be detected. Interferometer in momentum space An interferometric method for measuring Berry’s phases and topological properties ofBloch bands for ultracold atoms in 2D OLs was proposed in Ref.[368]. The proposalis based on a combination of Ramsey interference and Bloch oscillations in the BZ to87 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 x k y k G G x k y k G G F -F pulse /2 pulse /2 x k y k G G k F -F pulse /2 A BD C (a) (b) (c) Figure 34. (Color online) An interferometer method to measure the Zak phase and Chern number [368]. A cloudof ultracold atoms with a well-defined quasi-momentum k in the spin-up state is initially loaded into a 2D OL,with G and G as reciprocal lattice vectors. (a) A π/ |↑(cid:105) and |↓(cid:105) states. Then spin-selective forces ± F parallel to G are applied. (b) The two spins meet in the quasi-momentumspace after half a period of Bloch oscillations, following by another π/ measure Zak phases, which can be used to measure π Berry’s phase of Dirac points andthe first Chern number of topological bands.The scheme for measuring the Zak phase consists of three steps, as shown in Figs. 34(a,b). The atoms are initially prepared in a spin-up state with a given quasi-momentum k in the n th band, and a first π/ ψ k n ( r ) ⊗ ( | ↑(cid:105) + | ↓(cid:105) ) / √ 2. The eigenfunctionsin the n th band is written as ψ k n ( r ) = e i kr u k n ( r ), where u k n is the cell-periodic Blochfunction, satisfying u k n ( r + G i ) = u k n ( r ) with two primitive reciprocal lattice vectors G i ( i = 1 , ± F on the spin-up and spin-down are applied parallel tosome reciprocal lattice vector G , which can be created by a magnetic field gradient alongthe y axis. The atoms then exhibit the Bloch oscillations in the momentum space, whichis assumed to be adiabatic. In this case, the evolution under the application of the force ± F is described by the time-dependent wave function (Ψ ↑ ( r , t ) ⊗| ↑(cid:105) +Ψ ↓ ( r , t ) ⊗| ↓(cid:105) ) / √ σ ( r , t ) ( σ = ↑ , ↓ ) obey the Schr¨odinger equation i (cid:126) ∂ Ψ σ ( r , t ) /∂t = H σ Ψ σ ( r , t ) , where the Hamiltonian H ↑ , ↓ = H ∓ Fr ± E Z , H = − (cid:126) m ∇ + V ( r ) , (177)with V ( r ) being the lattice potential and E Z being the Zeeman energy. Under the adia-batic condition, the atoms remain within the Bloch band. The wave functions take theform: Ψ ↑ ( ↓ ) ( r , t ) = e iξ ↑ ( ↓ ) ( t ) ψ k ± ( t ) ,n ( r ) , where k ± ( t ) = k ± f t , f = F / (cid:126) , and the phase ξ ↑ ( ↓ ) ( t ) is given by: ξ ↑ ( ↓ ) ( t ) = i (cid:90) k ± ( t ) k (cid:104) u k (cid:48) n |∇ k (cid:48) u k (cid:48) n (cid:105) d k (cid:48) − (cid:126) (cid:90) t (cid:15) n ( k ± ( t (cid:48) )) dt (cid:48) ∓ E Z t (cid:126) . (178)After half a period of the Bloch oscillations (period is given by T = | G | / | f | ), the twospins meet at the edge of the first BZ, another π/ ξ ↑ ( T / − ξ ↓ ( T / ϕ tot = ϕ Zak + ϕ dyn + ϕ Zeeman , (179)88 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 where the Zak phase is given by: ϕ Zak = i (cid:90) k + G / k − G / (cid:104) u k (cid:48) n |∇ k (cid:48) u k (cid:48) n (cid:105) d k (cid:48) (180)and the dynamical phase and Zeeman phase are given by ϕ dyn = − (cid:126) (cid:82) T/ − T/ sgn( t (cid:48) ) (cid:15) n ( k + f t (cid:48) ) dt (cid:48) and ϕ Zeeman = − E Z T / (cid:126) , respectively. For a band structure with symmetric dis-persion relation, (cid:15) n ( k + f t (cid:48) ) = (cid:15) n ( k − f t (cid:48) ), the dynamical phase vanishes. The Zeemanphase can be also eliminated by a spin echo sequence [125, 368]. Thus, the Ramseyinterferometry can directly give the Zak phase.The interferometer scheme can be further used to measure the Chern number of agapped band for cold atoms in 2D OLs. As shown in Fig. 34 (c), the initial quasi-momentum can be prepared to k ( α ) = α G , where α ∈ [0; 1). Then the Zak phase fora specific α can be measured through the interferometer protocol. The small change ofZak phase as α is increased by δα is equal to the integral of the Berry curvature overthe rectangle δS defined by the corresponding trajectories, which is the Berry’s phasefor the contour ABCDA . To see this, one can choose a smooth gauge for the periodicBloch function in δS , such that the Berry’s phase γ can be represented as the sum ofthe Berry’s phases for the four sides of the rectangle, γ = γ AB + γ BC + γ CD + γ DA .Since the sides AB and CD are equivalent but traversed in the opposite direction, theircontribution vanishes, γ AB + γ CD = 0, then γ BC + γ DA is equal to the difference of theZak phases for trajectories BC and DA . Thus, the change of the Zak phase is related tothe Berry phase and is given by an integral of the Berry curvature: γ = (cid:90) δ S d k Ω( k ) = − ie − iϕ Zak ( α ) ∂ α e iϕ Zak ( α ) δα. (181)As the Chern number C = π (cid:82) BZ d k Ω( k ), we then obtain C from the winding numberof the Zak phase [131], C = − i π (cid:90) dαe − iϕ Zak ( α ) ∂ α e iϕ Zak ( α ) . (182)This relation implies that the Chern number can be extracted from the interferometricmeasurements of the Zak phase across the BZ. The method for a more general latticestructure is introduced in Ref. [368]. In addition, the π Berry phase of a Dirac point canalso be determined from the interferometric measurement over a trajectory enclosing thepoint in the momentum space.The proposed interferometer method has been demonstrated in cold atom experiments[125, 140]. The Zak phase of topological Bloch bands for cold atoms in a 1D dimerizedOL, which realizes the SSH/Rice-Mele Hamiltonians (see Sec. 4.1.1), has been directlydetected from the interferometric measurements [125]. Furthermore, the atomic inter-ferometer to measure π Berry flux of a Dirac point in momentum space has also beendemonstrated [140], which is in analogy to an Aharonov-Bohm interferometer that mea-sures magnetic flux in real space (See Sec. 4.2.1).Based on the proposed measurements of non-Abelian generalizations of Zak phases(the Wilson loops) or time-reversal polarizations, the interferometric method by com-bining the Bloch oscillations with Ramsey interferometry can be generalized to probeZ topological invariants of time-reversal-invariant topological insulators realized in OLs[369]. Moreover, by using an additional mobile impurities that bind to quasiparticles of ahost many-body system, an interferometric scheme for detecting many-body topological89 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 invariants of interacting states with topological order, such as the fractional excitationsin fractional quantum Hall systems, was proposed in Ref. [370]. Hall drift of accelerated wave packets For a wavepacket evolving on a lattice, which is centered at position r with the quasi-momentum k and driven by an external force F , the equations of motion are given by[131] ˙ r n = 1 (cid:126) ∂E n ( k ) ∂ k − ( ˙ k × e z )Ω n ( k ) , (cid:126) ˙ k = F , (183)where Ω n ( k ) is the Berry’s curvature of the n -band, and E n ( k ) is the corresponding bandstructure. Here the equations are valid when the force F is weak enough to precludeany inter-band transitions. Considering a 2D lattice and the force along the y direction F = F e y , the average velocity v n = ˙ r n along the transverse ( x ) direction can be obtainedas v xn ( k ) = ∂E n ( k ) (cid:126) ∂k x − F (cid:126) Ω n ( k ) . (184)The first term in the above equation describes the usual band velocity for Bloch oscil-lations, and the second term related to the Berry curvature is the so-called anomalous velocity, which can produce a net drift transverse to the applied force. It was shown thatthe anomalous velocity can be isolated and observed by canceling the contribution fromthe band velocity through comparing trajectories for opposite forces ± F [371]. Followingthis protocol, a measurement of the averaged velocity of the accelerated wavepacket formany trajectories gives the Berry’s curvature Ω n ( k ) over a “pixelated” BZ, and thus theChern number can be evaluated by properly adjusting the paths. In the experiment ofrealizing the Haldane model with ultracold fermions [28], the drift measurement has beenperformed to probe the nontrivial Berry curvature and map out the topological regimeof the model (see Sec. 4.2.3 ).It was further shown that the Chern number could be directly measured by imagingthe center-of-mass drift of a Fermi gas as an effective Hall response to the external force[372]. Due to the periodicity of the energies in k -space, (cid:90) BZ (cid:0) ∂E n ( k ) /∂k x (cid:1) d k = 0 , (185)so the contribution from the band velocity naturally vanishes by averaging the velocityover the entire first BZ. By setting the Fermi energy within a topological bulk gap, theaveraged anomalous velocity can thus be isolated by uniformly populating the bands,and the displacement along the transverse direction is directly proportional to the Chernnumber [344, 372]. Considering a general square lattice system of size A syst = L x × L y anda unit cell size A cell , the number of states within each band is N states = A syst /A cell , andthe total number of particles is N tot = (cid:80) n N ( n ) , where N ( n ) is the number of particlesoccupying the n -band. By assuming that each band is populated homogeneously, theaverage number of particles in a Bloch state u n ( k ) is uniform over the BZ and is givenby ρ ( n ) ( k ) = ρ ( n ) = N ( n ) /N states , (186)90 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 which acts as the band filling factor. The total averaged velocity along the directiontransverse to the force is given by [372] v x tot = − ( F A cell /h ) (cid:88) n N ( n ) C ( n ) , (187)where C ( n ) = π (cid:80) k Ω n ( k ) ∆ k x ∆ k y ≈ π (cid:82) BZ Ω n ( k ) d k is the band Chern number, with∆ k x,y = 2 π/L x,y . This equation reveals that the averaged transverse velocity of the wavepacket is related to the Chern number, which is a topological invariant and remains aconstant as long as the spectral gaps to other bands do not vanish. For a free spin-polarized Fermi gas at zero temperature loaded in a 2D OL with topological bands, theFermi energy E F within a spectral gap naturally leads to a perfect filling of the bandslocated below the gap with ρ ( n ) = N ( n ) /N states = 1 for E n < E F . Thus one has v x tot = − ( F A syst /h ) (cid:88) E n It was revealed by Strˇeda that the Hall conductivity and thus the first Chern numbercan be represented as the number of occupied states in the QHE [373]. Considering theHofstadter model [218] on a 2D lattice subjected to a uniform magnetic field B withthe butterfly energy spectrum shown in Fig. 17, when the magnetic flux per plaquetteis rational Φ = p/q , this spectrum splits into q sub-bands E n ( n = 1 , . . . , q ). Each bulkband E n ( k ) is associated with a Chern number C ( n ) , which remains constant as long asthe bulk gaps do not close. The quantized Hall conductivity of a 2D electron system canbe obtained from the Strˇeda formula (let h = c = e = 1)[373]: σ H = σ C = σ (cid:88) E n Tomography of Bloch states A method for full tomography of Bloch vectors was first proposed for a specific realizationof a Haldane-Chern insulator in spin-dependent hexagonal OLs [244]. The Haldane model[13] and some other models of Chern insulators can be well described by two-band BlochHamiltonians of the form H ( k ) = (cid:15) ( k ) I × + d ( k ) · σ , (193)with the Bloch vectors d ( k ) = d x,y,z ( k ). The Berry curvature Ω and the Chern number C of the lowest energy band can be expressed in terms of the normalized Bloch vector n ( k ) = d ( k ) / | d ( k ) | [64]:Ω( k ) = 12 n · ( ∂ k x n × ∂ k y n ) , C = 12 π (cid:90) BZ Ω( k ) d k . (194)Based on this equation, when the system is in a phase C (cid:54) = 0, an experimental mea-surement of n ( k ) would depict a Skyrmion pattern on a “pixelated” BZ, leading to anapproximate measurement of the Chern number. For a specific Haldane-like model thatcould be realized with fermionic atoms of two spin states confined on the two triangularsublattices of the honeycomb pattern, the Bloch vector distribution n ( k ) ∝ (cid:104) σ (cid:105) can beexperimentally determined from spin-resolved time-of-flight images [244]. In a typical ex-periment, after the ground state is prepared, switching off the trap in adequate timescalesprojects the atom cloud into the momentum density distributions ρ a,b ( k ), which give thepseudospin component n z ( k ) = [ ρ a ( k ) − ρ b ( k )] / [ ρ a ( k ) + ρ b ( k )]. A fast Raman pulseduring time-of-flight allows one to rotate the atomic states and map n x and n y , whichis the tomography of the whole Bloch vector field. Actual experiments “pixelize” thetime of flight images, counting the number of atoms on each “square” of the effectiveBZ and estimating the averages of n x , n y or n z . Either through repetitions or throughself-averaging in an experiment with multiple copies of the lattice, a set of normalizedvectors { n j } L × Lj =1 , evenly sampled over momentum space can be obtained, which givesthe Chern number with the error O (4 π /L ) expected from the discretization with thesmooth integrand [244].A different scheme for tomography of Bloch states in OLs with two sublattice stateswas further proposed, which is based on the quench dynamics and thus is not re-stricted to a specific system [375]. Consider spinless fermions in a 2D OL with twosublattice states A and B , and the system is described by the two-band Hamiltonianin Eq. (193) with σ acting in the sublattice space. For every quasimomentum k , thesublattice space defines a Bloch sphere, with north and south poles given by | k A (cid:105) and | k B (cid:105) , respectively. The normalized vector on the Bloch sphere is parametrized as n ( k ) = (sin ϑ k cos ϕ k , sin ϑ k sin ϕ k , cos ϑ k ). Then the Bloch state of the lowest band is givenby | k −(cid:105) = sin( ϑ k / | k A (cid:105) − cos( ϑ k / e iϕ k | k B (cid:105) . In order to obtain the full informationof the Bloch state determined by ϑ k and ϕ k , one can measure the momentum distri-bution of the system, which is subjected to an abrupt quench with a potential off-set (cid:15) A − (cid:15) B ≡ (cid:126) ω AB between A and B sites for suppressing tunneling at the measurementtime t m , leading to an observable dynamics in the momentum distribution [375] ρ ( k , t ) = f ( k ) { − sin ϑ k cos[ ϕ k + ω AB ( t − t m )] } . (195)Here f ( k ) is a broad envelope function given by the momentum distribution of theWannier function. The oscillatory time dependence in ρ ( k , t ) directly reveals both ϕ k andsin ϑ k = 1 − | n z ( k ) | . The time-dependence of ρ ( k , t ) allows one to reconstruct n x,y,z ( k )93 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 from the amplitude and the phase of the oscillations. Such a complete tomography ofthe Bloch states is mandatory for a measurement of the Berry curvature. It was shownthat this scheme is applicable to extract the Chern number and topological transitionsfor the Hofstadter model with π flux and the Haldane model on a honeycomb OL [375].An alternative but generic scheme for measuring the Bloch wavefunction based on thetime-of-flight imaging was presented in Ref. [313]. For fermionic atoms with N spin statesreferred as | s (cid:105) ( s = 1 , , ..., N ) in a generic OL, the Bloch state in the non-degenerate n -band can be denoted by | u n ( k ) (cid:105) = (cid:80) Ns =1 c ns ( k ) | s (cid:105) , where c ns ( k ) is the Bloch wavefunctionwith normalization (cid:80) s | c ns ( k ) | = 1. To measure c ns ( k ), one can first separate differentspin components through a magnetic field gradient and directly map out the atomicmomentum distribution ρ ns ( k ) = | c ns ( k ) | for the filled band using the conventionaltime-of-flight imaging. One then measure the phase information of c ns ( k ) by introducinga π/ s and s (cid:48) with an impulsive pulselight before the flight of atoms, which induces the transition c ns ( k ) → [ c ns ( k ) + c ns (cid:48) ( k )] / √ , c ns (cid:48) ( k ) → [ c ns ( k ) − c ns (cid:48) ( k )] / √ . With this pulse, the difference between | c ns ( k ) ± c ns (cid:48) ( k ) | / c ∗ ns ( k ) c ns (cid:48) ( k )]. Theimaginary part Im[ c ∗ ns ( k ) c ns (cid:48) ( k )] can be obtained by the same way with a different ro-tation. The measurement of the population and interference terms determines the Blochwave function up to an arbitrary overall phase c ns ( k ) → c ns ( k ) e iχ ( k ) , where χ ( k ) ingeneral depends on k instead of the spin index. This arbitrary k -dependent phase posesan obstacle to measurement of the topological invariants, which can be overcame by agauge-invariant method to calculate the Berry curvature based on the so-called U (1)-link defined for each pixel of the discrete BZ in experiments [376]. It was shown that theproposed method is generally applicable to probe the topological invariants in varioustopological bands [265, 313, 317] and robust to typical experimental imperfections suchas inhomogeneous trapping potentials and disorders in the systems.Recently, the tomography of Bloch states has been experimentally demonstrated withtwo different approaches [45, 46]. Based on the method proposed in Ref. [375], a fulltomography of the Bloch states across the entire BZ was experimentally demonstrated byobserving the quench dynamics at each momentum point [46]. In the experiment, a cloudof single-component fermionic K atoms in a shaking hexagonal OL formed a tunableFloquet band insulator. Even though the global of the band has zero Chern number inthe system, the measured distribution of Berry curvature showed the rich topology, suchas the phase vortices as topological defects near Dirac points and their chiralities as thesignal of the topological transition due to the shaking [46]. The topological defects ofthe Bloch states in the hexagonal OL were further experimentally studied by mappingout the azimuthal phase profile ϕ k in the entire momentum space and by identifying thephase windings [377].The state tomography methods discussed above are applicable for non-degenerate orisolated bands, where the Berry phase is merely a number. However, some lattice systemshaving multiple bands with degeneracies, such as in topological insulators and graphene,can seldom be understood with standard Berry phases but can instead be described usingmatrix-valued Wilson lines [109, 369, 378, 379]. Wilson lines as non-Abelian generaliza-tion of Berry phase [109] provide indispensable information to identify the topologicalstructure of bands as they encode the geometry of degenerate states, such as the eigen-values of Wilson-Zak loops (i.e., Wilson lines closed by a reciprocal lattice vector) forformulating the Z topological invariants [369, 378, 379].In a recent experiment [45], using an ultracold gas of rubidium atoms loaded in a hon-94 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 (a)(b)(c) (d)(e) Figure 35. (Color online) Realizing the Wilson lines in the honeycomb lattice. (a) In a non-degenerate system(left), adiabatic evolution of a state through parameter space (cid:126)R results in the acquisition of a Berry phase. Ina degenerate system (right), the evolution is instead governed by the matrix-valued Wilson line, which lead topopulation changes between the levels. (b) The band structure of the lowest two bands of the honeycomb latticein effective energy units of | F | d . As the force F is increased, the largest energy scale of the bands becomes small.At large forces (iii), the effect of the band energies is negligible and the system is effectively degenerate. In thisregime, the evolution is governed by the Wilson line operator. (c) The measured population remaining in the firstband for different forces after transport to Γ + 0 . G (green), Γ + 0 . G (red), and Γ + G (blue), where insetnumbers i to iii refer to band schematics in (b). (d) Measuring mixing angles θ q at different final quasimomenta q . (e) Measuring relative phases φ q at different q , lying at angular coordinate α on a circle centered at Γ. Thequantized jumps of π in the phase of the interference fringe each time α is swept through a Dirac point. Reprintedfrom Li et al. [45]. Reprinted with permission from AAAS. eycomb OL, the strong-force dynamics in Bloch bands that are described by Wilson lineswas realized and an evolution in the band populations for revealing the band geometrywas observed. This enables a full tomography of band eigenstates using Wilson lines. Thisapproach can be used to determine the topological invariants in single- and multi-bandsystems. As shown in Fig. 35(a), the Berry phase merely multiplies a state by a phasefactor, while the Wilson line is a matrix-valued operator that can mix state populations.The Wilson line was measured by detecting changes in the band populations under theinfluence of an external force F , such that atoms with initial quasimomentum q (0) evolveto quasimomentum q ( t ) = q (0) + F t/ (cid:126) after a time t . When the force is sufficiently weakand the bands are non-degenerate, the system will remain in the lowest band and thequantum state merely acquires a Berry phase and a dynamical phase. Transitions toother bands occur at stronger forces, and when the force is infinite with respect to a cho-sen set of bands, the effect of the dispersion vanishes and the bands appear as effectivelydegenerate, as shown in Fig. 35(b). The system then evolves according to the formalismfor adiabatic motion in a degenerate system [109], and the dynamics is described by theunitary time-evolution operator as the Wilson line matrix [45]:ˆ W q (0) → q ( t ) = ˆ P exp[ i (cid:90) C d q ˆ A q ] , (196)where the path-ordered ˆ P integral runs over the path C in reciprocal space from q (0)to q ( t ) and ˆA q is the Wilczek-Zee connection for local geometric properties of the statespace. In the honeycomb OL, the Wilson line operator describing transport of a Blochstate from Q to q reduces to ˆW Q → q = e i ( q − Q ) · ˆ r , and thus the Wilson line operator95 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 simply measures the overlap between the cell-periodic Bloch functions denoted | u n Q (cid:105) and | u m q (cid:105) (with the band index n, m ) at the initial and final quasimomenta [379]: W mn Q → q = (cid:104) u m q | u n Q (cid:105) . (197)This enables a tomograph of the cell-periodic Bloch functions over the entire BZ in thebasis of the states | u n Q (cid:105) . In the experiment [45], a nearly pure BEC of Rb was initiallyloaded into the lowest band at the center of the BZ Q = Γ, and an inertial force, createdby accelerating the lattice via linearly sweeping the frequency of the laser beams, wasused to realize the Wilson line. The Wilson line was then verified by transporting theatoms from Γ to different final quasimomenta using a variable force | F | and performingband mapping to measure the population remaining in the lowest band, as shown in Fig.35(c). The saturation value | W → q | = |(cid:104) u q | u (cid:105)| of the population after transport to q is a measure of the overlap between the Bloch functions of the first band at Γ and q .To demonstrate the reconstruct of Bloch states using the Wilson lines, it is convenientto represent the state | u q (cid:105) in the basis of | (cid:105) = | u Q (cid:105) and | (cid:105) = | u Q (cid:105) at a fixed referencequasimomentum Q as | u q (cid:105) = cos θ q | (cid:105) + sin θ q e iφ q | (cid:105) . (198)Obtaining θ q and φ q for each quasimomentum q will map out the geometric structureof the lowest band [244, 375]. As shown in Fig. 35(d,e), mixing angles θ q at differentfinal quasimomenta q was measured from the atom population remaining in the firstband after the transport, and relative phases φ q at different q was measured through aprocedure analogous to Ramsey or St¨uckelberg interferometry [45]. Using the data, theBloch states in the lowest band | u q (cid:105) and the eigenvalues of Wilson-Zak loops can bothbe reconstructed. Spin polarization at high symmetry momenta It was proposed that for a class of Chern insulators, the topological index can be obtainedby only measuring the spin polarization of the atomic gas at highly symmetric points ofthe BZ [380]. The two-band Bloch Hamiltonian of the Chern insulators in square latticesis given by H ( k ) = [ m z − J cos( k x a ) − J cos( k y a )] σ z − J so sin( k x a ) σ x − J so sin( k y a ) σ y , (199)where the m z is an effective Zeeman splitting, J and J so represent the nearest-neighborspin-conserved and spin-flipped hopping coefficients, respectively. Notably, the cold-atomrealization of this Hamiltonian with the needed 2D SOC has been theoretically proposed[117] and then experimentally achieved [36] by a simple optical Raman lattice schemethat applies two pairs of light beams to create the lattice and Raman potentials simul-taneously. The topology of the lowest Bloch band can be characterized by the Chernnumber C = sgn( m z ) when 0 < | m z | < J , and otherwise C = 0.The lattice system has an inversion symmetry defined by the 2D inversion trans-formation ˆ P ⊗ ˆ R , where ˆ P = σ z acting on spin space and ˆ R D transforms Bra-vais lattice vector R → − R . For the corresponding Bloch Hamiltonian, one hasˆ P H ( k ) ˆ P − = H ( − k ), which follows that [ ˆ P , H ( Λ i )] = 0 at the four highly symmetricpoints { Λ i } = { G (0 , , X (0 , π ) , X ( π, , M ( π, π ) } . Therefore the Bloch states | u ± ( Λ i ) (cid:105) in the two energy bands are also eigenstates of the parity operator ˆ P with eigenvalues96 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 P ± = +1 or − 1. The topology of the inversion-symmetric Chern band can also be deter-mined by the following invariant [36, 117]Θ = (cid:89) i sgn[ P − ( Λ i )] . (200)It can be proven rigorously that Θ = − C = − − Θ4 (cid:80) i =1 sgn[ P ( Λ i )]. Since the parityeigenstates are simply the spin eigenstates (here ˆ P = σ z ), with the spin-up and spin-downcorresponding to different atomic internal states, the topological invariants Θ and C ofthe lowest Bloch band can be determined by measuring the spin polarization P ( − ) ( k ) ofan atomic cloud at the four highly symmetric points: P − ( Λ i ) = n ↑ ( Λ i ) − n ↓ ( Λ i ) n ↑ ( Λ i ) + n ↓ ( Λ i ) , (201)where n ↑ , ↓ ( Λ i ) denotes the atomic momentum density of spin states | ↑ , ↓(cid:105) , which canbe measured directly by spin-resolved time-of-flight imaging. This method can be usedto probe other topological bands with specific symmetries by measuring atomic spin po-larization at highly symmetric points in momentum space, such as the chiral topologicalinsulators [165, 381] and double-Weyl semimetals [275].Based on the optical Raman lattice method [117], the Bloch Hamiltonian in Eq. (199)with the topological bands has been experimentally realized for a BEC of Rb atoms[36]. For the condensate in the OL, the spin polarization at the four high symmetrymomenta can be written as (cid:104) σ z ( Λ i ) (cid:105) ≈ P − ( Λ i ) f ( E − , T ) + P + ( Λ i ) f ( E + , T ) , (202)where f ( E ± , T ) = 1 / [ e ( E ± ( Λ i ) − µ ) /k B T − 1] is the BEC with µ and T respectively beingthe chemical potential and temperature. Since P + ( Λ i ) = − P − ( Λ i ), one has (cid:104) σ z ( Λ i ) (cid:105) ≈ P − ( Λ i )[ f ( E − , T ) − f ( E + , T )]. Thus by preparing a cloud of bosonic atoms with thetemperature satisfying f ( E − ( Λ i ) , T ) > f ( E + ( Λ i ) , T ), one can obtainsgn[ (cid:104) σ z ( Λ i ) (cid:105) ] = sgn[ P − ( Λ i )] . (203)Thus, the spin polarization can be precisely measured with a condensate at low temper-ature. In the experiment [36], the spin polarization was measured as a function of thetunable parameter m z to determine Θ and C for the topology of the lowest-energy band,which is topologically nontrivial when 0 < m z < | m cz | , whereas it is trivial for m z > | m cz | ,as show in Fig. 36. The 2D SOC and the band topology for BECs in the optical Ramanlattices have recently been further investigated [260, 382, 383] Topological pumping approach As introduced in Sec. 4.1.2, the topological pumping, geometric pumping, and spinpumping have been recently realized with ultracold atoms in 1D optical superlattices[146, 157–159]. Very recently, a 2D topological charge pump as a dynamical version ofthe 4D integer QHE was realized by using ultracold bosonic atoms in a 2D optical super-lattice [17]. The quantized transported particle (i.e., the atomic center-of-mass changein the experiments) in the adiabatic cyclic evolution of these pumps indicates the under-97 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 (a) C=0 C=0C= -1 C=1 (b) Figure 36. (Color online) (a) Measured spin polarization P ( Λ i ) at the four symmetric momenta Λ i as a functionof m z . (b) Obtained invariant Θ, which determines the Chern number C of the lowest band in Hamiltonian (199).Reprinted from Wu et al. [36]. Reprinted with permission from AAAS. lying band topology, such as the first and second Chern numbers characterizing the 1Dand 2D topological pumping, respectively.Based on the pumping approach in OLs and hybrid Wannier functions in band theory[384], it was shown that the Chern number can be extracted from hybrid time-of-flightimages [385]. In the modern theory of polarization, a 2D insulating lattice system can beviewed as a fictitious 1D insulator along one direction, say along x , subject to an externalparameter k y , where k y is the crystal-momentum along y . The polarization of this 1Dinsulator can be defined by means of hybrid Wannier functions [384], in which the Fouriertransform from Bloch functions is carried out in the y direction only. The polarization ateach k y is then given by the center of the corresponding hybrid Wannier functions, and thechange in polarization from adiabatically changing k y by 2 π is proportional to the Chernnumber of the 2D insulators [384, 386]. This is a manifestation of topological particlepumping with k y being the adiabatic pumping parameter. A generalization of the hybridWannier functions of band theory to the hybrid particle densities in cold atomic gases ρ ( x, k y ), which are the particle densities resolved along the x -direction as a functionof k y , provides a natural way to measure the Chern number in OLs. Experimentally, ρ ( x, k y ) can be measured by combing in situ imaging along x and time-of-flight imagingalong the release direction y . In the measurement, the OL is switched off along the y direction while the system remains unchanged in the x direction. It was shown thatthe hybrid particle density provides an efficient numerical reconstruction of the Chernnumber in topologically-ordered OLs [385], such as the Hofstadter and Haldane models.This method is general and allows the measurement of other topological invariants inOLs, such as the Z topological invariant in time-reversal symmetric insulators and the k z -dependent Chern number C k z in Weyl semimetals [136, 275]. Detection of topological edge states According to the bulk-edge correspondence, the topological index of the bulk bandscorresponds to the number of gapless edge-modes present within the bulk gap [14, 15].In Chern insulators, all the gapless edge states propagate in the same direction, suchthat they are chiral. In the context of the QHE, the chiral edge states are responsiblefor the quantized Hall conductivity. As introduced in Sec. 4.4, the chiral edge states ina quantum Hall ribbon have been experimentally realized and detected with cold atomsthrough the synthetic dimension method [37, 38]. In addition, the chiral currents were alsoobserved in an optical ladder for ultracold bosonic atoms exposed to a uniform artificialmagnetic field [174]. The high-resolution addressing technique in cold atom gases offersthe possibility of directly visualizing the time-evolution of these edge states.In a 2D atomic Chern insulator under an external trapping potential, the direct detec-tion of topological edge states is challenging because the number of occupied edge modeswithin a bulk gap and below the Fermi level contains a very small fraction of the total98 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 number of particles. In addition, these edge states would be washed out by the smoothharmonic trap and thus one may not be able to distinguish them from the bulk states. Tocircumvent these problems in detecting the topological edge states, it was proposed to usea steep confining potential and to image the edge states from optical Bragg spectroscopy[203, 234, 366, 367]. Based on a generalization of Bragg spectroscopy sensitive to angularmomentum [234], the Bragg probe can transfer energy and angular momentum to atomslocated in the vicinity of the Fermi level and simultaneously changes their internal states,which completely removes the edge states from the cloud and allows imaging on a darkbackground unpolluted by the untransferred atoms. In this scheme, the Bragg spectracan provide an unambiguous signature of the topological edge states that establishestheir chiral nature. Another method to directly image the propagating edge states wasproposed by forcing them to propagate in a region that is unoccupied by the bulk statesafter suddenly removing the potential [235]. Other methods to visualize the edge statecurrents were also proposed by quenching the parameters of the system Hamiltonian[387, 388]. It would to interesting to extend these cold-atom schemes to directly imagethe helical edge states in Z topological insulators and the Fermi arc surface states in 3Dtopological semimetals. 6. Topological quantum matter in continuous form In this section, we move beyond topological Bloch bands in lattice systems to describesome of the quantum matter in the continuum that have topologically nontrivial prop-erties. Here we focus on there model systems realized with cold atoms without latticepotentials, which are the topological solitons in Jackiw-Rebbi model, various topologicaldefects in BECs, and the atomic (quantum) spin Hall effect. Jackiw-Rebbi model with topological solitons In relativistic quantum field theory, Jackiw and Rebbi introduced a celebrated model togenerate topological soliton modes with fractional particle numbers [124]. The Jackiw-Rebbi model describes a 1D Dirac field coupled to nontrivial background fields. Therelativistic Dirac Hamiltonian for 1D Dirac fermions subjected to two static bosonicfields ϕ and ϕ can be written as [124, 127, 128] H D = c x σ x p x + ϕ ( x ) σ z + ϕ ( x ) σ y , (204)where c x is the (effective) speed of light and the background field with a kink potentialcan be described as ϕ ( x ) = Γ , ϕ ( x → ±∞ ) = ± ∆ , (205)with positive constants Γ and ∆ . The relativistic Dirac Hamiltonian with such a topo-logically nontrivial background potential supports an nondegenerate soliton state, whichgives rise to fractionalization of particle number. In the original Jackiw-Rebbi modelwith Γ = 0, the nondegenerate zero-energy soliton state is protected by the conjugation(particle-hole) symmetry that connects each state with energy E to its partner locatedat the opposite energy. In the many-particle description, there are two degenerate many-body ground states corresponding to the soliton state being filled or empty, carryingfractional Fermi number N = ± / 2, respectively.With the conjugation-symmetry-breaking term Γ, the soliton mode has non-zero energy99 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 (b)(a) valence bandconduction band Γ= E Γ−= E x a − a kink anti-kink >∆ <∆− Fermi level N N − N N − Figure 37. (Color online) Fractionalization in Jackiw-Rebbi model [389]. (a) A background field with a pair of kinkand anti-kink, both of which support a localized soliton state. (b) The energy spectrum and a pair of solitons withenergies E = ± Γ and fractional particle numbers ±|N | . At the kink, the soliton state picks a fractional particlenumber of |N | from the effective valance band (Fermi sea) and another 1 − |N | from the effective conductionband, and vice versa for the opposite case at the anti-kink. and the particle number is generally irrational [127]: N = 1 π arctan ∆ Γ . (206)One can see that N depends only on the asymptotic value of the kink rather than itsdetailed shape. In this sense, it is topological and is insensitive to local fluctuations of thebackground field. To have a better understanding of arbitrary fractional particle number[389], one can consider another kind of background field with a simple but practicalconfiguration, that is, a pair of kink and anti-kink both with a step-function profile asshown in Fig. 37(a). Solving the energy spectrum of the Dirac Hamiltonian (204) atthe kink potential (near x = − a ) with ϕ ( x ) = ∆ sgn( x + a ) yields a localized in-gapeigenstate in the kink at E = Γ with the wave function decaying as exp ( − ∆ | x + a | / (cid:126) c )and the energy gap E g = 2 (cid:112) ∆ + Γ . It is understood that the soliton state picks upa fractional fermion number of |N | from the effective valance band (Fermi sea) and(1 − |N | ) from the effective conduction band, as shown in Fig. 37(b). Without thesoliton, one can assume N fermions fully occupying the valence band acting as a uniformbackground in the state counting. For filling with N + 1 fermions in the presence of kinkand anti-kink configuration, the expectation value of the fermion number of the solitonmodes at the kink and anti-kink can be | N | and 1 − | N | if they are both occupied. Foran anti-kink potential (near x = a ) with ϕ ( x ) = − ∆ sgn( x − a ), the localized solitonstate is obtained at E = − Γ with the wave function decaying as exp ( − ∆ | x − a | / (cid:126) c x ).It picks up (1 − |N ) | from the valence band and |N | from the conduction band. Theremust be pairs of kink and anti-kink in a periodic system. If both states are unoccupied,the particle numbers are −|N | at the kink and |N | − E = − Γ soliton state is occupied first andthe particle numbers at kink and anti-kink are ∓|N | , respectively. When both states areoccupied, there are particles (1 − |N | ) and |N | at the kink and anti-kink, respectively.The first condensed matter realization of the Jackiw-Rebbi model is the conductingpolymers described by the SSH model for lattice systems, wherein the low-energy effectiveHamiltonian of the Bloch Hamiltonian near Dirac points takes the Dirac form with kink-soliton modes in the continuum. In particular, the Jackiw-Rebbi model and the SSHmodel share many similar features related to topological insulators under a suitableregularisation. The soliton modes and related topological properties in the SSH modelhave been intensively investigated with cold atoms in 1D OLs (see Sec. 4.1.1).The direct realization of the Jackiw-Rebbi model and the detection of the inducedsoliton mode with the fractional particle number by using a 1D atomic Fermi gas inthe continuum were proposed in Ref. [389]. The first procedure is to create strong SOCsfor the ultracold atoms, and the second one is to generate a kink-like potential and100 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 a tunable conjugation-symmetry-breaking term are properly constructed by laser-atominteractions, leading to an effective low-energy relativistic Dirac Hamiltonian with a topo-logically nontrivial background field. The fractionalization of the particle number in theatomic system may be detected through the soliton density and the local density of statesnear the kink by using two standard experimental detection methods for ultracold atomicgases, the in situ absorption imaging technique and spatially resolved rf spectroscopy[389]. The realization of 1D homogeneous Dirac-like Hamiltonian (particles) and relatedrelativistic effects (e.g. Klein tunneling and Zitterbewegung) with cold atoms in the con-tinuum has been studied in theories and experiments [92, 93, 95, 96, 105, 390, 391].See Ref. [52] for a review on relativistic quantum effects of Dirac particles simulated byultracold atoms. Topological defects in Bose-Einstein condensates BECs of atomic gases in a harmonic trap without lattice potentials can host varioustopological defects in real space. These topological defects include solitons in 1D, vorticesand Skyrmions in 2D, monopoles, Skyrmions and knots in 3D. The topological defectshave different physical properties and are classified by homotopy groups of their order-parameter space. Thus, they are distinguished by their topological charges with discretevalues [392] and robust against to external perturbations. In particular, atomic BECswith internal spin degrees of freedom provide unique platforms for investigating differenttopological objects due to the rich structure of their superfluid order parameters, whichare vectors rather than scalar quantities. Moreover, the well-developed manipulationtechniques for atomic motion and spin states enable one to engineer the topologicaldefects of interest in real space for studying their dynamics and stability in a highlycontrollable manner.In early experiments of atomic BECs, the 1D solitons in the atomic density distribu-tions have been created and controlled by a phase imprinting method [393–395]. The 2Dtopological vortices, which are line defects in the superfluid order parameter accompa-nied by a quantized phase winding of an integer multiple of 2 π , have also been generatedin single- and multi-component atomic BECs by an external rotation [396, 397] or thephase imprinting method [398–400].As another kind of topological defects, Skyrmions are first envisioned in field theoryand then extended to condensed matter physics. A 2D Skyrmion is characterized by alocal spin that continuously rotates through an angle of π from the center to the boundaryof the system. In terms of a unit spin vector d , a typical 2D Skyrmion spin texture shownin Fig. 38(e) with z -axis symmetry can be written as in the polar coordinate d ( r, φ ) = cos β ( r )ˆ z + sin β ( r )(cos φ ˆ x + sin φ ˆ y ) , (207)where β ( r ) is the bending angle characterizing the rotation or “bending” of the localspin across the cloud, with the boundary conditions β (0) = 0 and β ( ∞ ) = π . This spintexture has the topological charge ν w = 14 π (cid:90) dxdy d · ( ∂ x d × ∂ y d ) = 1 , (208)which is the 2D winding number representing the number of times that the spin textureencloses the whole spin space. Such a 2D Skyrmion spin texture was first created in aspin-2 condensate of Rb atoms with coherent Raman transitions between the spin states[401]. In the experiment, the BEC was prepared in the | F = 2 , m F = 2 (cid:105) = | (cid:105) state, andthen two Raman beams respectively with first-order Laguerre-Gaussian and Gaussian101 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401Figure 38. (Color online) Creation of 2D Skyrmions in a spinor BEC. The absorption image (a) of a 2D skyrmioncreated in spin-2 Rb. The winding number ν w for each spin state is indicated. (b) Azimuthally averaged lineouts(points) for each spin state. (c) 3D plot of the solid lines in (b), where the colors red, green, and blue correspondto the | (cid:105) , | (cid:105) , and |− (cid:105) states, respectively, with the number of arrowheads indicating the winding number of thespin state. (d) The polarization of the skyrmion. (e) The vector field of the Skyrmion. Reprinted with permissionfrom Leslie et al. [401]. Copyright c (cid:13) (2009) by the American Physical Society. intensity profiles were applied to transfer the population to the | (cid:105) ( |− (cid:105) ) state, whichacquires a ν w = 1 ( ν w = 2) azimuthal phase winding. The Raman interaction effectivelyevolves the order parameter of the spinor BEC to [401]Ψ( r ) = (cid:112) n ( r ) cos ( β ( r ) / √ e iφ sin( β ( r ) / β ( r ) / e iφ sin ( β ( r ) / . (209)Here n ( r ) is the density of the atomic cloud, and the distribution β ( r ) as the form inEq. (207) can be engineered to generate the 2D Skyrmion spin texture. The Skyrmionwas detected from absorption image of the atomic density profile shown in Fig. 38(a-d)and further conformed by matter-wave interference [401]. It was demonstrated that thestate 2D Skyrmions can be created in a spin-1 BEC of Na atoms in a harmonic trapby using a 3D quadrupole magnetic field [402, 403], and moreover, an atomic geometricHall effect in the spinor BEC with a 2D Skyrmion spin texture has been observed [404].In 3D, Skyrmion is a particlelike soliton hypothetically introduced by Skyrme [405].A 3D Skyrmion has a nonsingular texture that can be topologically characterized by a3D winding number. The stability of a 3D Skyrmion in two-component BECs, whichcan be simply viewed as a vortex ring containing a superflow, has been theoreticallystudied and found as metastable solutions of the energy functional [406, 407]. Severalschemes have been proposed to create and stabilize the metastable 3D Skyrmions inmulti-component BECs [408–412]. Recently, it was shown that a fully stable 3D Skyrmioncan spontaneously emerge as the ground state of a two-component BECs coupled witha synthetic non-Abelian gauge field [413]. The 3D Skyrmion spin texture is elusive inexperiments until recently, and it was realized within a spin-polarized ferromagnetic Rb BEC that is exposed to an externally controlled magnetic field [414].On the other hand, knots are another 3D topological objects, which are characterizedby a linking number or a Hopf invariant [415]. The existence of a stable knot soliton wasfirst discussed in the context of a two-component BEC [416]. An experimental schemefor generation of knot spin textures in a spin-1 BEC was proposed [417]. Based on thistheoretical proposal, the creation and observation of knot solitons in the spinor BEC hasrecently been demonstrated [418]. 102 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 (a) (b) (c)(d) (e) Figure 39. (Color online) Creation of Dirac monopoles in a spinor BEC. (a-c) Spin orientation (red arrows) inthe condensate when the magnetic field zero (black dot) is above (a), entering (b) and in the middle of (c) thecondensate. The helix represents the singularity in the vorticity. (d) Azimuthal superfluid velocity v s (red arrowand colour scale by equatorial velocity v e ). Black arrows depict the synthetic magnetic field B ∗ . (e) Experimentalsetup with a controlled 3D magnetic quadrupole. Reprinted by permission from Macmillan Publishers Ltd: Ray et al. [419], copyright c (cid:13) (2014). A fundamentally important and interesting topological defect is monopole, followingDirac’s theory of magnetic monopoles which are consistent with both quantum mechanicsand the gauge invariance of the electromagnetic field [1]. The experimental evidence ofmagnetic monopoles as fundamental constituents of matter is still absent, however, theycan emerge as quasiparticle excitations or other analogies in condensed matter systems,such as topological insulators [420]. It has been proposed that the light-induced gaugepotentials for neural atoms in proper Raman laser fields can provide the realization ofsynthetic magnetic monopoles and even non-Abelian monopoles [92, 421–423]. Alterna-tively, it was theoretically demonstrated that a topological defect as the Dirac magneticmonopole can be imprint on the spin texture of an atomic BEC by using external mag-netic fields [424]. Due to the spin of the condensate aligning with the local magnetic fieldwith nontrivial 3D structures, one can create a pointlike defect to the spin texture ofthe condensate giving rise to a vorticity equivalent to the magnetic field of a magneticmonopole. A synthetic monopole field on a sphere with exact flat Landau levels on curvedspherical geometry in a system of spinful cold atoms could be realized by engineering ofa magnetic quadrupole field [425].Following the method introduced in Ref. [424], the Dirac monopoles have been experi-mentally created in the synthetic electromagnetic field that arises in the order parameterof a ferromagnetic spin-1 Rb BEC in a tailored excited state [419]. The order param-eter Ψ ( r, t ) = ψ ( r, t ) ζ ( r, t ) is the product of a scalar order parameter ψ , and a spinor ζ = ( ζ +1 , ζ , ζ − ) T ≡ | ζ (cid:105) , where ζ m = (cid:104) m | ζ (cid:105) represents the m th spinor component along z , with ζ = (1 , , T at the beginning. The spin texture S = Ψ † F Ψ are given by thecondensate order parameter and the spin-1 matrices F . The spinor order parameter corre-sponding to the Dirac monopole was generated by an adiabatic spin rotation in responseto a time-varying magnetic field [419] B ( r, t ) = b q ( x ˆ x + y ˆ y − z ˆ z ) + B z ( t )ˆ z, (210)where b q > B z ( t ) is a uniform biasfield. As shown in Fig. 39, the magnetic field zero is initially located on the z axis at103 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 z = B z (0) / b q , and the spin rotation occurs as B z is reduced, drawing the magneticfield zero into the region occupied by the superfluid. In the experiment, the condensatespin nearly adiabatically follows the local direction of the field, as shown in Fig. 39(a-c).Using a scaled and shifted coordinate system with x (cid:48) = x , y (cid:48) = y , z (cid:48) = 2 z − B z /b q ,corresponding spherical coordinates ( r (cid:48) , θ (cid:48) , φ (cid:48) ), the applied magnetic field is then B = b q ( x (cid:48) ˆ x (cid:48) + y (cid:48) ˆ y (cid:48) − z (cid:48) ˆ z (cid:48) ). As B z is reduced, each spin rotates by an angle π − θ (cid:48) about anaxis defined by the unit vector ˆ n ( r (cid:48) , θ (cid:48) , φ (cid:48) ) = − ˆ x (cid:48) sin φ (cid:48) + ˆ y (cid:48) cos φ (cid:48) . In the adiabatic limit,the condensate order parameter corresponds to the local eigenstate of the linear Zeemanoperator g F µ B B · F , and this spatially-dependent rotation leads to a superfluid velocity[419, 424] v s = (cid:126) M r (cid:48) θ (cid:48) sin θ (cid:48) ˆ ϕ (cid:48) , (211)and vorticity Ω s = ∇ (cid:48) × v s = − (cid:126) M r (cid:48) ˆ r (cid:48) + 4 π (cid:126) M δ ( x (cid:48) ) δ ( y (cid:48) ) Θ ( z (cid:48) ) ˆ r (cid:48) , (212)where M is the atomic mass, δ is the Dirac delta function and Θ is the Heaviside stepfunction. The vorticity is a monopole attached to a semi-infinite vortex line singularityas an analog of Dirac string, with phase winding 4 π , extending along the positive z (cid:48) axis. The synthetic vector potential arising from the spin rotation can be written as A ∗ = − M v s / (cid:126) , and the synthetic magnetic field of the monopole is B ∗ = (cid:126) r (cid:48) ˆ r (cid:48) . (213)The fields v s and B ∗ are depicted in Fig. 39(d). The created Dirac monopoles were thenexperimentally identified at the termini of vortex lines within the condensate by directlyimaging such a vortex line in real space [419]. Based on this method, a topological pointdefect as an isolated monopole without terminating nodal lines (the Dirac string) wasalso created and observed in the order parameter of the spin-1 BEC [426]. Spin Hall effect in atomic gases Spin Hall effects [427] are a class of SOC phenomena where flowing particles experienceorthogonally directed spin-dependent Lorentz-like forces and give rise spin currents. Thisis analogous to the conventional Lorentz force for the Hall effect, but opposite in signfor two spin states. A quantized spin Hall effect is closely related to the Z topologicalinsulators which preserves time-reversal symmetry (see Sec. 4.2.4 and 4.3.4). The spinHall effects have been observed for electrons in spin-orbit coupled materials [428, 429] andcircularly polarized photons passing through certain surfaces [430, 431]. It was proposedthat the spin Hall effects can be realized for neutral atoms with spin-dependent Lorentzforces [84, 108], which can be achieved by the synthetic gauge potentials as discussedin Sec. 3.4. Following the proposal of Ref. [84], the spin Hall effect was observed ina pseudospin-1/2 Rb BEC subjected to spin- and space-dependent vector potentials[104].Experimentally, two Raman lasers counterpropagating with wave number k R along ˆ x were used to couple the internal states | f = 1; m F = 0 , − (cid:105) = | ↑ , ↓(cid:105) , which comprisepseudospin-1 / (cid:126) Ω < E R with the single-photon recoil energy E R , the 2D effective pseudospin Hamiltonian (ig-104 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 (a) (b)(c) (d) Figure 40. (Color online) Atomic spin Hall effect. (a) Experiment schematic of two controlled Raman beamspropagating along ˆ x coupled two ground states in Rb atoms. (b) The induced double-well dispersion E ( q ) forthe three different y -positions marked in (a), with the synthetic vector potentials A . (c) Spin Hall effect withspin-dependent forces along ˆ x from motion along ˆ y . (d) Acquired momentum along ˆ x versus final momentumalong ˆ x . Reprinted by permission from Macmillan Publishers Ltd: Beeler et al. [104], copyright c (cid:13) (2013). noring the trap, the light shift and the zero-energy shift from the Raman dressing) canbe written as [104] ˆ H = 12 m ( ˆp − A ˆ σ z e x ) , (214)where A = (cid:126) k R (cid:104) − ( (cid:126) Ω / E R ) (cid:105) / is a light-induced spin-dependent vector potentialalong ˆ x . As shown in Fig. 40(b), the modified dispersion features two degenerate wellsand each of them displaces from zero by an amount A . Here A has the spatial dependenceof the Raman lasers’ Gaussian intensity profile since Ω depends on the laser intensity.The spatial dependence of A gives rise to a spin Hall effect in the atomic gas [84, 108]. Inthe experiment [104], the mechanism underlying the spin Hall effect was first probed fromobserving spin-dependent shear in the atomic density distribution by abruptly changing A , which gives rise to a spin-dependent “electric” force − ∂ A /∂t on the atom cloud.Then for a time-independent A , the resulting spin Hall effect was further observed bydetecting a spin-dependent Lorentz-like response along ± ˆ x with atoms propelled in eitherspin state along ˆ y and realizing an atomic spin transistor using a mixture of both spins.As shown in Fig 40(c,d), each spin-polarized BEC acquired a momentum along ˆ x thatwas detected oppositely for the two spins and related to its final momentum along ˆ y ,which demonstrates an intrinsic spin Hall effect.Since the (pseudo)spin here is a good quantum number, the system can be thought ofas two independent subsystems that respond oppositely to temporal and spatial gradientsof the light-induced gauge potential A . By introducing a large non-zero curl for A , eachspin state could be separately driven to the regime of integer QHE [84, 104]. Therefore,one can create a quantum spin Hall effect in this atomic system composed of an equalmixture of both spins. 105 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 7. Topological matter with interactions Until now we have mainly reviewed cold-atom realizations of essentially non-interactingtopological phases. But interactions can lead to new topological phases including in-trinsic topological phases and symmetry-protected topological phases for both bosonicand fermionic systems in all dimensions. While tremendous theoretical efforts have beenrecently paid with fruitful achievements usually by advanced mathematics, the experi-mental realization for most theoretically predicted phases hardly has any solid progressyet except several classic examples. This section is intended not to give a systematicalreview of interacting topological phases, which is beyond the scope of this review article,but much more modestly to introduce a number of interesting models with interactions,which can exhibit interaction-intrinsic topological phases. In particular we focus on howto realize them by recent advances of cold-atom techniques, and wish to convey the ex-pectation that the high-tunability of cold-atom systems would enable us to explore thisopen and deep field further. Spin chains Ultracold atoms in OLs is a promising platform to realize some spin-1/2 models. We hereconsider the well-know anisotropic Heisenberg model (XXZ model) in 1D, which arises inthe context of various condensed matter systems. The Hamiltonian of XXZ spin modelis given by H XXZ = − N (cid:88) j =1 (cid:104) λ z σ zj σ zj +1 + λ ⊥ (cid:16) σ xj σ xj +1 + σ yj σ yj +1 (cid:17)(cid:105) , (215)where λ z ( λ ⊥ ) denotes the nearest neighbor interaction along z -direction ( x − and y -direction), and σ x,y,zj are the Pauli matrices for the j th spin. The phase diagram of thisHamiltonian is pretty rich. In addition, the geometric phase of the ground state in thisspin model is quantized in certain parameters and it obeys scaling behavior in the vicinityof a quantum phase transition[432, 433].When λ ⊥ = 0 and in the presence of an applied magnetic field along x direction, themodel in Eq.(215) becomes the transverse Ising model, which Hamiltonian is given by H Ising = − λ z N − (cid:88) j =1 σ zj σ zj +1 − h x N (cid:88) j =1 σ xj , (216)where the parameter h x is the intensity of the magnetic field applied in the x direction.Consider the projection onto the x -axis of the spin with the fermionic occupation number n : | ↑(cid:105) ↔ n = 0 , | ↓(cid:105) ↔ n = 1, one has σ xj = ( − a † j a j . Employing the string-likeannihilation and creation operators (the Jordan-Wigner transformation): a j = (cid:32) j − (cid:89) k =1 σ xk (cid:33) σ + j , a † j = (cid:32) j − (cid:89) k =1 σ xk (cid:33) σ − j , (217)106 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 where σ + and σ − are the spin raising and lower operators, H Ising can be rewritten as H Ising = J N − (cid:88) j =1 ( a j − a † j )( a j +1 + a † j +1 ) + 2 h z N (cid:88) j =1 (cid:16) a † j a j − / (cid:17) . (218)This shows that the 1D transverse Ising chain is mathematically equivalent to the Kiteav’schain [434] of p-wave superconductor (see the next section) and thus exhibits the sametopological phase, in which the ground state degeneracy is dependent on the boundaryconditions of the chain. For the superconductors, the Z symmetry of fermionic paritycannot be lifted by any local physical operators, because such operators must containan even number of fermion operators. However, the Z symmetry in the Ising modelis given by a global spin flip in the σ z basis: P S = (cid:81) Nj =1 σ xj , such that its degeneracycan be lifted by a simple longitudinal magnetic field h z (cid:80) j σ zj . This indicates that thetopological phase in the transverse Ising chain is much weaker, and thus the creation andmanipulation of Majorana edge modes in this system are more difficult.We now turn to address a scheme proposed in Ref. [435] to realize the spin modelswith ultracold atoms in OLs. Consider an ensemble of ultracold bosonic or fermionicatoms confined in an OL. We are interested in the Mott insulator regime, and the atomicdensity of roughly one atom per lattice site. Each atom is assumed to have two relevantinternal states, which are denoted with the effective spin index σ = ↑ , ↓ , respectively. Weassume that the atoms with spins σ = ↑ , ↓ are trapped by independent standing wavelaser beams through polarization (or frequency) selection. Each laser beam creates aperiodic potential V µσ sin ( (cid:126)k µ · (cid:126)r ) in a certain direction µ , where (cid:126)k µ is the wavevector oflight. For sufficiently strong periodic potential and low temperatures, the atoms will beconfined to the lowest Bloch band, and the low energy Hamiltonian is then given by theBoson- or Fermi-Hubbard Hamiltonian H = − (cid:88) (cid:104) ij (cid:105) σ (cid:16) J µσ a † iσ a jσ + H.c. (cid:17) + 12 (cid:88) i,σ U σ n iσ ( n iσ − 1) + U ↑↓ (cid:88) i n i ↑ n i ↓ , (219)where (cid:104) i, j (cid:105) denotes the near neighbor sites, a iσ are bosonic (or fermionic) annihilationoperators respectively for bosonic (or fermionic) atoms of spin σ localized on site i , and n iσ = a † i σ a i σ .For the cubic lattice ( µ = x , y , z ) and using a harmonic approximation around theminima of the potential, the spin-dependent tunneling energies and the on-site interactionenergies are given by J µσ ≈ E / R ( V µσ ) / √ π e − V µσ /E R ) / , U ↑↓ ≈ (cid:0) π (cid:1) / ( ka s ↑↓ )( E R V ↑↓ V ↑↓ V ↑↓ ) / ,U σ ≈ (cid:0) π (cid:1) / ( ka sσ ) ( E R V σ V σ V σ ) / (for bosons) , U σ ≈ (cid:112) V µσ E R (for fermions) , where V µ ↑↓ = 4 V µ ↑ V µ ↓ / ( V / µ ↑ + V / µ ↓ ) is the spin average potential in each direction, E R = (cid:126) k / m is the atomic recoil energy, and a s ↑↓ is the scattering length between theatoms of different spins. For fermionic atoms, U σ is on the order of Bloch band separation ∼ (cid:112) V µσ E R , which is typically much larger than U ↑↓ and can be taken to be infinite.In writing Eq.(219), overall energy shifts (cid:80) iµ (cid:0)(cid:112) E R V µ ↑ − (cid:112) E R V µ ↓ (cid:1) ( n i ↑ − n i ↓ ) / z direction.From the above expressions, we observe that J µσ depend sensitively (exponentially)107 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 upon the ratios V µσ /E R while U ↑↓ and U σ exhibit only weak dependence. So we caneasily introduce spin-dependent tunneling J µσ by varying the potential depth V µ ↑ and V µ ↓ with control of the intensity of the trapping laser. This simple experimental methodprovides us a powerful tool to engineer many-body Hamiltonians. In the regime where J µσ (cid:28) U σ , U ↑↓ and (cid:104) n i ↑ (cid:105) + (cid:104) n i ↓ (cid:105) (cid:39) 1, which corresponds to an insulating phase, the termsproportional to tunneling J µσ can be considered via perturbation theory. To the leadingorder in J µσ /U ↑↓ , Eq. (219) is equivalent to the following effective Hamiltonian H = − N (cid:88) j =1 (cid:104) λ µz σ zj σ zj +1 + λ µ ⊥ (cid:16) σ xj σ xj +1 + σ yj σ yj +1 (cid:17)(cid:105) , (220)where σ zi = n i ↑ − n i ↓ , σ xi = a † i ↑ a i ↓ + a † i ↓ a i ↑ , and σ yi = − i (cid:16) a † i ↑ a i ↓ − a † i ↓ a i ↑ (cid:17) are the usualspin operators. The + and − signs before λ µ ⊥ correspond, respectively, to the cases offermionic and bosonic atoms. The parameters λ µz and λ µ ⊥ for the bosonic atoms aregiven by λ µz = J µ ↑ + J µ ↓ U ↑↓ − J µ ↑ U ↑ − J µ ↓ U ↓ , λ µ ⊥ = J µ ↑ J µ ↓ U ↑↓ . (221)For fermionic atoms the expression for λ ⊥ is the same as in (221), but in the expres-sion for λ z the last two terms vanish since U σ (cid:29) U ↑↓ . In writing Eq. (220), the term (cid:80) iµ (cid:16) J µ ↑ /U ↑ − J µ ↓ /U ↓ (cid:17) σ zi is neglected, since it can be easily compensated by an ap-plied external magnetic field. When we set V µ ↓ /V µ ↑ (cid:29) 1, so that J µ ↓ becomes negligiblewhile J µ ↑ remains finite. In this case, the Hamiltonian (220) reduces to the Ising model H = (cid:80) (cid:104) i,j (cid:105) λ µz σ zi σ zj , with λ µz = J µ ↑ / (1 / U ↑↓ − /U ↑ ). The transverse field term inEq.(216) can be easily achieved with an applied external magnetic field along the x di-rection. The Ising model has been realized experimentally with atoms in OLs [436, 437].The approach using ultracold atoms to realize the spin models has a unique advantagein that the parameters λ µz and λ µ ⊥ can be easily controlled by adjusting the intensityof the trapping laser beams. They can also be changed within a broad range by tuningthe ratio between the scattering lengths a s ↑↓ and a sσ ( σ = ↑ , ↓ ) by adjusting an externalmagnetic field through Feshbach resonance. Therefore, even with bosonic atoms alone,it is possible to realize the entire class of Hamiltonians in the general form (220) with anarbitrary ratio λ µz /λ µ ⊥ . This is important since bosonic atoms are generally easier tocool. We estimate the typical energy scales for the realized Hamiltonian. For Rb atomswith a lattice constant π/ (cid:12)(cid:12)(cid:12) (cid:126)k (cid:12)(cid:12)(cid:12) ∼ J/ (cid:126) can be chosenfrom zero to a few kHz. The on-site interaction U/ (cid:126) corresponds to a few kHz at zeromagnetic field, but can be much larger near the Feshbach resonance. The energy scale formagnetic interaction is about J / (cid:126) U ∼ . U ∼ J/U ) ∼ / V ij = C/R ij between Rydberg atom pairs at a distance R ij , where C is the van der Waals coefficient.Such interactions have recently been used to explore quantum many-body physics of108 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401Figure 41. (Color online) Experimental platform for realization of the Ising model. (a) Individual Rb atoms aretrapped using optical tweezers (vertical red beams) and arranged into defect-free arrays. Coherent interactions V ij between the atoms are enabled by exciting them to a Rydberg state with strength Ω and detuning ∆. (b) A two-photon process couples the ground state | g (cid:105) = | S / , F = 2 , m F = − (cid:105) to the Rydberg state | r (cid:105) = | S / , J =1 / , m J = − / (cid:105) via an intermediate state | e (cid:105) = | P / , F = 3 , m F = − (cid:105) with detuning δ , using circularlypolarized 420 nm and 1013 nm lasers with single-photon Rabi frequencies of Ω B and Ω R , respectively. Reprintedby permission from Macmillan Publishers Ltd: Bernien et al. [438], copyright c (cid:13) (2017). Ising spin systems in OLs [438–441]. The achieved Hamiltonian ( (cid:126) = 1) is given by˜ H Ising = (cid:88) i Ω i σ xi − (cid:88) i ∆ i n i + (cid:88) i The well-known Kitaev chain was proposed by Alexei Kitaev [469], which is the sim-plest model system that shows unpaired Majorana zero modes. The Kitaev model is atoy model but can be exactly solved, which provides an extremely useful paradigm forMajorana zero modes at the two ends of a quantum wire of p -wave superconductor. TheKitaev chain is the spinless fermion model with nearest-neighbor hopping and pairingbetween the sites of a 1D lattice described by the Hamiltonian H = (cid:88) j [ − J ( c † j c j +1 + c † j +1 c j ) − µ ( c † j c j − 12 ) + ∆ c j c j +1 + ∆ ∗ c † j +1 c † j ] , (226)where µ is a chemical potential, and ∆ = | ∆ | e iθ is a superconducting gap. Consider achain with L sites and open boundary conditions, as shown in Fig.(42), we can rewritethis Hamiltonian in the Majorana representation by using the Majorana operators as H = i (cid:88) j [ − µγ j, γ j, + ( J + | ∆ | ) γ j, γ j +1 , + ( − J + | ∆ | ) γ j, γ j +1 , ] . (227)Here the Majorana operators are defined as γ j, = e i θ c j + e − i θ c † j , γ j, = ( e i θ c j − e − i θ c † j ) /i. (228)which satisfy the relations γ † j,α = γ j,α , { γ j,α , γ k,β } = 2 δ jk δ αβ , (229)for j, k = 1 , , ..., L and α, β = 1 , 2. 111 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 Now we discuss two specific cases. The topological trivial case for | ∆ | = J = 0 isconsidered first. The Hamiltonian becomes H = − µ (cid:88) j ( c † j c j − 12 ) = i − µ ) (cid:88) j γ j, γ j, . (230)The Majorana operators γ j, , γ j, from the same site j are paired together, as shown inFig. 42(a), to form a ground state with the occupation number 0 ( µ < 0) or 1 ( µ > | ∆ | = J > µ = 0, and we have H = iJ L − (cid:88) j γ j, γ j +1 , . (231)Now the Majorana operators γ j, and γ j +1 , from different sites are paired together, asillustrated in Fig. 42(a). The ground state of this Hamiltonian is easily found by definingnew annihilation and creation operators a j = 12 ( γ j, + iγ j +1 , ) , a † j = 12 ( γ j, − iγ j +1 , ) , (232)with iγ j, γ j +1 , = 2 a † j a j − j = 1 , , ..., L − 1. Subsequently, the Hamiltonian (231)can be rewritten in a canonical form H = 2 J L − (cid:88) j a † j a j − J ( L − . (233)As we can see that Hamiltonian (231) does not contain operators γ , and γ L, , i.e.,[ γ , , H ] = [ γ L, , H ] = 0, while all pairs of ( γ j, , γ j +1 , ) for j = 1 , , ..., L − J > a j | g (cid:105) = 0 for all j , and H | g (cid:105) = − J ( L − | g (cid:105) . (234)These represent zero-energy Majorana modes localized at the two ends of the chain.Since [ γ , , H ] = [ γ L, , H ] = 0, the two orthogonal ground states of the Kitaev chainmodel can be constructed as | g (cid:105) and a † | g (cid:105) , where a = ( γ , + iγ L, ) is an ordinaryzero-energy fermion operator. These states have different fermionic parities: one is evenand the other is odd (i.e., it is a superposition of states with even or odd number ofelectrons). Note that the ground states with double degeneracies or not reveal that thesystem is topologically nontrivial or trivial, respectively. Similarly, the considerationswill also yield unpaired Majorana zero modes for the special case | ∆ | = − J and µ = 0.These two specific cases represent two distinct phases of the Kitaev chain: topologicallytrivial or nontrivial, corresponding to different pairing methods without or with unpairedMajorana zero modes localized at the ends of the chain.To study the general properties of the Hamiltonian (226) at arbitrary values of J , µ and ∆, we diagonalize the Kitaev Hamiltonian under periodic boundary condition. Afterthe Fourier transformation with c † j = 1 / √ L (cid:80) k c † k e i k · r j , the Hamiltonian in momentum112 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 space can be written in Bogoliubov-de Gennes form H = 12 (cid:88) k (cid:16) c † k c − k (cid:17) H BdG ( k ) (cid:18) c k c †− k (cid:19) , (235)where the Bogoliubov-de Gennes Hamiltonian is written in terms of Pauli matrices (cid:126)τ as H BdG ( k ) = (cid:15) ( k ) τ z + ∆( k )cos θτ y + ∆( k )sin θτ x , (236)with (cid:15) ( k ) = − J cos k − µ , ∆( k ) = 2 | ∆ | sin k . The energy spectrum is given by E ( k ) = ± (cid:112) (cid:15) ( k ) + | ∆( k ) | . (237)For ∆ (cid:54) = 0, the system is in superconducting states. The energy spectrum always fullygapped except when 2 J = ± µ . As shown in Fig. 42(b), two lines represent gap closing aredefined, which mark the phase transition between the two distinct phases of the model.We can identify that the system in the region | J | > | µ | / H BdG preserves intrinsic particle-hole symmetry. One can checkthat the Hamiltonian (236) satisfies the relationˆ C H BdG ( k ) ˆ C − = −H BdG ( − k ) , (238)where the particle-hole operator ˆ C = τ x ˆ K satisfies ˆ C = +1. According to the topologicalclassifications, Hamiltonian (236) belongs to the symmetry class D ( d = 1) and thus hasa Z -type topological number. The relevant topological invariant of the system describedby the Hamiltonian (226) is the so-called Majorana number M = ± 1, which is actuallythe Z index, first formulated by Kitaev. In Kitaev’s paper [469], it was shown that all 1Dfermionic systems with superconducting order fall into two categories distinguished by M . One is topologically trivial with M = +1 and the other is nontrivial with M = − M , we consider the Hamiltonian that can be written in the Majoranarepresentation as H = i (cid:88) lmαβ B αβ ( l − m ) γ lα γ mβ , (239)where l and m label the lattice sites while α and β denote all other quantum numbers.Then M is defined as M = sgn { Pf[ ˜ B (0)] } sgn { Pf[ ˜ B ( π )] } (240)where ˜ B ( k ) denotes the spatial Fourier transform of B ( l − m ) regarded as a matrixin indices α , β and Pf[ A ] denotes the Pfaffian where Pf[ A ] =det[ A ], with A being anantisymmetric matrix. Thus, we can calculate the Majorana number of the Kitaev modelby using Eq. (240). In momentum space, the Kitaev Hamiltonian (227) can be written113 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 in the following form H = i (cid:88) k (cid:0) γ k, γ k, (cid:1) (cid:18) D ( k ) − D ∗ ( k ) 0 (cid:19) (cid:18) γ − k, γ − k, (cid:19) , (241)with D ( k ) = − J cos k − i | ∆ | sin k − µ . The operators γ k, and γ k, are defined as γ k, = c †− k + c k , γ k, = i ( c †− k − c k ) , (242)which satisfy the relations { γ † k,α , γ k (cid:48) ,β } = 2 δ kk (cid:48) δ αβ , γ † k,α = γ − k,α . (243)Note that γ k, and γ k, are not the Majorana operators except when k = 0. As wecan see that the matrix ˜ B ( k ) here for k = 0 , π is antisymmetric, the Pfaffian of 2 × D ( k ) ,π .It yields the Majorana number M = sgn( D (0))sgn( D ( π )) = sgn( µ − J ) . (244)One can check that the topological superconducting phase occurs when M = − | µ | < | J | , the other phase is trivial when M = +1 for | µ | > | J | , as we have discussedabove.The Kitaev’s model describes a 1D system of spinless fermions but electron spectraare usually degenerate with respect to spin in real system. For this reason it has beeninitially viewed as a somewhat unphysical toy model because the physical realization of aquantum wire M = − p -wave superconductors and Majorana fermions incondensed matter systems, but their unambiguous detection (realization) remains anoutstanding challenge. T he search for observable signatures that identify exoticstates of quantum matter and their fractionalizedexcitations has become a main focus of research inquantum physics. A paradigmatic example is the hunt forMajorana quasi-particles (MQPs) that exist at the ends oftopological superconductors . First experimental evidence consistent with the presence of MQPs has recently beenreported in various superconducting hybrid systems . Whilethe ultimate goal is to probe the existence of non-Abelian anyonssuch as MQPs by performing controlled braiding operations,several possible fingerprints have been proposed that may beeasier to access experimentally.A prominent example hallmarking MQPs is the fraction-alization of the Josephson effect, which can exhibit a 4 p (half frequency) period due to a non-equilibrium population ofexcited states that is protected by fermion parity conservation .However, a similar, though non-protected, fractionalization isalso known to occur in conventional S-wave superconductors,due to the presence of accidental mid-gap states . As anew signature for MQPs, here we show how a dissipationless,non-equilibrium 8 p periodic Josephson effect occurs when twoMQPs are subject to a super-exchange coupling via a controllableenergy level interrupting a Kitaev chain, an effect that is notfound in S-wave superconductors. In addition, we show how ourmodel can be realized in systems of cold atoms in optical lattices,where isolation from the environment creates an ideal platformfor the study of such non-equilibrium phenomena.Our proposal is motivated by remarkable recent experimentalprogress with cold atom systems, including the observation ofthe non-equilibrium Josephson effect , initially demonstratedwith Bose–Einstein condensates , and later observed over theBose-Einstein Condensate (BEC)–Bardeen-Cooper-Schrieffer(BCS) crossover . These results demonstrate not only theability to measure non-equilibrium signals, but in addition, thisrealization of the 2 p Josephson effect will provide a crucial pieceof our implementation. More concretely, in our proposal, thestarting point is an atomic realization of the Kitaev wire , hereusing a system of alkaline earth atoms (AEAs) coupled to a BECreservoir (Fig. 1b). AEAs allow the creation of a controllable extrasite by means of species-dependent potentials , while thereservoir allows both the implementation of the Kitaev wireand the modification of the Josephson phase via an underlyingJosephson effect of the reservoir itself. In addition, we investigatethe visibility of this effect by studying the transient dynamics ofthe Josephson current in the presence of imperfections, includingvarious dissipation mechanisms (single-particle losses anddephasing) captured by a quantum master equation. Our simulations support not only the observability of the 8 p effect,but further underline how this signature is characteristic ofMQPs: while 4 p peaks in the Fourier signal cannot bedistinguished from those arising from mid-gap states in anordinary S-wave SC, and peaks at 4 p , 2 p and zero frequencycan be enhanced from dissipation, the 8 p signal visible in ourset-up provides a signature that cannot be confused with theseundesired effects. Results Model Hamiltonian . We consider spinless fermions with fieldoperators c j , where j ¼ y N (cid:2) H F ð Þ ¼ X N (cid:2) j ¼ (cid:2) t c y j c j þ þ D c j c j þ (cid:2) m c y j c j (cid:2) (cid:2) (cid:3)(cid:4) (cid:5) þ t L c y N (cid:2) c þ t R c y c e i F = þ m c y c þ h : c :; ð Þ which describes a proximity-induced P-wave superconductor with pairing D , interrupted by an extra site at j ¼ 0, which isassumed to be not affected by the pairing (Fig. 1a). The hoppingstrength is denoted by t and the chemical potential relative tohalf-filling by m . The site at j ¼ t L and t R , respectively, and has an energy offset m .The phase factor e i F /2 on the hopping between j ¼ j ¼ F whenmoving around the ring.For | m | o t , | D | t L ¼ t R ¼ 0, in the limit of large N thesystem hosts a single pair of zero-energy MQPs , g L and g R , whichare localized exponentially around j ¼ N (cid:2) j ¼ 1, respectively.All other quasi-particles of the superconductor are gapped, suchthat c along with g L and g R form a subspace that is energeticallydetached from the bulk spectrum. To understand the qualitative F dependence of equation (1) in the physically relevant regime t L , t R oo D , t , we hence consider a minimal model encompassingthe dynamics within this low-energy sector. Decomposing c intothe Majorana operators g x ¼ c þ c y ; g y ¼ c (cid:2) c y i , and setting m ¼ 0, the effective Hamiltonian then reads as H J F ð Þ ¼ i t L g L g x (cid:2) t R g R g x sin F = ð Þ þ g y cos F = ð Þ (cid:6) (cid:7)h i : ð Þ In Fig. 2, we compare the energy spectra of H J ( F ) and H ( F ).The full qualitative agreement confirms that the effectiveHamiltonian H J ( F ) captures the basic Josephson physics of thefull model H ( F ). To understand the various level (avoided) (cid:2) = (cid:2) + Φ (cid:2) (cid:3) γ L γ R (cid:3) γ x γ y jN – 3 J L J , Δ e i Φ J R e i Φ /2 J , Δ N – N – Figure 1 | System Hamiltonian and cold atom setting. ( a ) Schematics of the model Hamiltonian equation (1): the central part of the system is magnified inthe box at the bottom, where the Majorana degrees of freedom included in the simplified model (equation (2)) are highlighted. ( b ) Implementationin a cold atom system. A 1D optical lattice is coupled to a BEC reservoir that gives rise to the Kitaev Hamiltonian in the chain. An optical barrier acts both tocreate the impurity site (red) and triggers the Josephson effect in the reservoir itself. The phase difference across the barrier in the reservoir thenacts as the phase F for the optical lattice. ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12280 (a) (b) Figure 43. (Color online) (a) Schematics of the model Hamiltonian: the central part of the system is magnified inthe box at the bottom, where the Majorana degrees of freedom included in the simplified model are highlighted.(b) Realization in a cold atom system. A 1D OL is coupled to a BEC reservoir that gives rise to the KitaevHamiltonian in the chain. An optical barrier acts both to create the impurity site (red) and triggers the Josephsoneffect in the reservoir itself. The phase difference across the barrier in the reservoir then acts as the phase Φ forthe OL. Rrprinted by permission from Laflamme et al. [472]. Cold atom systems may provide a platform with high controllability to simulate andstudy the Kitaev chain model based on the remarkable advances in recent experiments.114 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 Several experimental proposals for realization of the Kitaev wires with cold atoms inthe past years have been presented [472–477], where the two key ingredients to inducethe unpaired Majorana zero modes can be created. In these proposals, a single-piece gasof cold fermionic atoms which can be regarded as the spinless fermions are considered.Furthermore, the effective p -wave pairing term can be realized by a Raman induceddissociation of Cooper pairs [473] or Feshbach molecules [476] forming an atomic BCS(or BEC) reservoir. The Kitaev chain model is usually considered as a noninteractingsystem; however, the pairing in the superfluid should be formed in an interacting atomicsystem.Without loss of generality, we describes a recent work for realizing the Kitaev wireswith cold atoms [472]. In this experimental scheme, the model Hamiltonian describes thespinless fermions with field operator ψ j in a ring OL reads as [472] H (Φ) = N − (cid:88) j =1 [ − J ψ † j ψ j +1 + ∆ ψ j ψ j +1 − µ ψ † j ψ j − 12 )]+ J L ψ † N − ψ + J R ψ † ψ e i Φ / + µ ψ † ψ + h.c., (245)where j = 0 , ...N − p -wave superconductor with pairing ∆, interrupted by an extra siteat j = 0 (assumed to be not affected by the pairing), J and µ denote the normal nearest-neighbour hopping and the chemical potential relative to half-filling, respectively. Thesite at j = 0 is connected to its neighbours by the hopping amplitudes J L and J R ,respectively, and has an energy offset µ . The phase factor e i Φ / on the hopping between j = 0 and j = 1 represents the phase of a Cooper pair by Φ when moving around thering. For J L = J R = µ = 0, the model Hamiltonian returns to the Eq. (226), whichdescribes the original Kitaev chain model. Thus, for the case | µ | < J and | ∆ | > γ L and γ R , which are localized around j = N − j = 1, respectively.In their proposed setup, three points to realize the model Hamiltonian are required:the realization of a 1D Kitaev chain, the additional single site separating the two endsof the chain, and the time control of the phase Φ. A system of fermionic alkaline earthatoms prepared in their S ground state in a 1D ring lattice was considered. The choiceof alkaline earth atoms allows one to independently trap the S ground state | g (cid:105) and the P metastable excited state | e (cid:105) . To address the first issue, the hopping terms J in thelattice naturally arise and the pairing term ∆ can be induced by coupling the fermionsin the lattice to a BEC reservoir. Here a radio-frequency field is used to break up Cooperpairs into neighbouring sites directly in the lattice [473], as depicted in Fig. 43(b). Thesecond step is to interrupt the chain with a single site. Following this idea, a barrier isengineered to inhibit | g (cid:105) atoms from being at site j = 0, which splits the Kitaev wireinto two wires. It can be done by using a highly focused beam at the so-called anti-magicwavelength, which acts as a sink for | e (cid:105) , and as a source for | g (cid:105) . Consequently, the | e (cid:105) atom only being trapped at site j = 0 acts as the additional site coupling the two endsof the chain. Although the natural tunnelling into and out of this site is deterred bythis barrier, the hopping J L and J R are then can be reintroduced with Raman processesinvolving a clock transition [114, 224, 478]. Finally, the realization of the time control ofphase Φ is related to the Josephson effect where the additional site j = 0 and its nearest-neighbour form a Josephson-like knot [472]. Within these setups, this system will occura non-equilibrium Josephson effect with a characteristic 8 π periodicity of the Josephsoncurrent. At this point, the system is pumped to an excited state after slowly increasingΦ by 4 π , and returns to the ground state after a second 4 π cycle.115 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 1D Anyon-Hubbard model The Hubbard model of 1D lattice anyons with on-site interactions, called Anyon-Hubbardmodel, takes the form H A = − J N (cid:88) j =1 ( a † j a j +1 + h.c. ) + U N (cid:88) j =1 n j ( n j − . (246)Here n j = a † j a j is the number operator for anyons and the operators a j and a † j annihilateor create an anyon on site j , and they are defined by the commutation relations a j a † k − e − iθ sgn( j − k ) a † k a j = δ jk , a j a k = e iθ sgn( j − k ) a k a j , (247)which are parameterized by the statistical angle θ . The sign function in the above equa-tions is sgn( j − k ) = − , , j < k , j = k , j > k , respectively. Thus, two particleson the same site behave as ordinary bosons. Consequently, even for θ = π , these latticeanyons are just pseudofermions: they are bosons on-site and fermions off-site, since manyof them are allowed to occupy the same site.There exists an exact mapping between anyons and bosons in 1D. Define the fractionalversion of a Jordan-Wigner transformation, a j = b j exp (cid:32) iθ j − (cid:88) k =1 n k (cid:33) , (248)with n k = a † k a k = b † k b k the number operator for both particle types. One can check thatthe mapped operators a j and a † j indeed obey the anyonic commutation relations in Eq.(247), provided that the particles of type b are bosons with the bosonic commutationrelations: [ b j , b † k ] = δ jk and [ b j , b k ] = 0. By inserting the anyon-boson mapping (248), theHamiltonian (246) can be rewritten in terms of bosons [479], H B = − J N (cid:88) j =1 ( b † j b j +1 e iθn j + h.c. ) + U N (cid:88) j =1 n j ( n j − . (249)Therefore, the anyonic exchange phase has been translated to an occupation-dependentPeierls phase: when tunneling from right to left ( j + 1 , j ), a boson picks up a phasegiven by θ times the number of particles occupying the site that it jumps to. Underthis condition, the many-body wave function picks up a phase of θ ( − θ ) if two particlespass each other via two subsequent tunneling processes to the right (left). The proposedconditional-hopping scheme is depicted in Fig. 44. Interestingly, the non-local mappingbetween anyons and bosons in Eq. (248) leads to a purely local and thus realizableHamiltonian (249).The occupation-dependent gauge potential can be implemented in OLs with a laser-assisted Raman tunneling scheme [479], generalized the idea proposed in Ref. [114] tocreate an artificial gauge potential, see the Sec.3.4.2. Figure (44) displays the basic con-cept. A non-zero on-site interaction U is required to distinguish between different localoccupational states. The OL is tilted, with an energy offset ∆ between neighbouring sites.For simplicity, we consider lattice site occupations that are restricted to n j = 0 , , 2, buthigher local truncations are also possible. Two different occupational states in either ofthe two sites form in total a 4D atomic ground state manifold, which are coupled to an116 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 JJ J θi Je iθ Je - ( a )( b ) ( c ) e U j j Figure 44. (Color online) The mapping between Anyon- and Boson-Hubbard models and a scheme to realizeoccupation-dependent gauge potential [479]. (a) Anyons in 1D can be mapped onto bosons featuring occupation-dependent gauge potential. (b) Assisted Raman tunnelling proposed for the realization of the Anyon-Hubbardmodel. excited state | e (cid:105) via four external driving fields, labelled L , L , L and L in Fig. 44(b).The excited state can be experimentally realized in at least two alternative ways. First,two spin-dependent lattices can be used. We take Rb as an example. One lattice trapsatoms in the F = 1, m F = − F = 1, m F = 0 hyperfine state is chosen as the excited state | e (cid:105) and trapped ina second lattice. Atoms in the excited state would then be localized between the leftand right wells of the F = 1, m F = − U and ∆, both of the order ofa few kHz, which requires a laser with a linewidth δ (cid:28) U, ∆. This is a necessary conditionfor selectively coupling the four different states in the ground state manifold. Second, onecan use two optical lattices, and trap ground state manifold atoms in the red-detunedlattice, while the excited state would live in the blue-detuned one. The driving fieldsrequired in this case would be typically in the THz frequency regime, making a preciseresolution of U and ∆ more challenging in experiments.The effective tunnelling rates J ab ( a ∈ , b ∈ , 4) between the four different levelsare obtained in terms of the effective Rabi frequencies J ab = Ω ∗ a Ω b / ∆, where an overlapintegral should be included in the Rabi frequencies Ω a,b since ground and excited statesfeel different lattices. It is demonstrated that the model in Eq. (249) can be implementedif the conditions J = J ≡ J and J = J ≡ J e iθ were satisfied [479]. It is notablethat the tilt energy ∆ disappears in the effective Hamiltonian after rotating out time-dependent phase factors, this energy offset is absorbed by the external radiation field,yielding a total Hamiltonian without a tilt term.To realize the model in Eq. (249), one has the parameters to satisfy the followingconditions. (i) The lasers linewidth δ (cid:28) ∆ , U , so that the external driving fields canresolve the different levels of the ground state manifold. (ii) A short-lived excited stateand the validity of the adiabatic elimination require large detunings ∆ (cid:29) | Ω − | . (iii)∆ and U can be in the same frequency regime (a few kHz), but their difference shouldbe much larger than the lasers linewidth δ . As an example, ∆ ≈ U ≈ | J ab | = J ≈ | Ω ab | ≈ 20 kHz would be sufficient if the linewidth of theradiation field were δ ≈ 50 Hz, which is a realistic assumption for typical radio-frequencydriving fields. However, it was shown in Ref. [480] that a further condition U, ∆ (cid:29) δ isalso required in the above scheme. For typical experimental parameters, it would lead tolarge heating. This drawback was solved by a scheme proposed in Ref. [480], where oneground-state component bosonic gas is replaced by two ground-state components atomic117 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 gas.An alternative scheme for the experimental realization of 1D Anyon-Hubbard model,based on time-periodic forcing, was proposed in Ref. [481]. The occupation-dependentPeierls phase can be engineered by means of coherent lattice-shaking-assisted tunnelingagainst potential offsets created by a combination of a static potential tilt and strongon-site interaction. The potential tilt ∆ is added in the Hamiltonian with the term∆ jb † j b j . By shaking the lattice, a similar term F ( t ) jb † j b j can be further added in theHamiltonian, where F ( t ) = F ( t + T ) incorporates a homogeneous time-periodic force ofangular frequency ω = 2 π/T with vanishing cycle average 1 /T (cid:82) T dtF ( t ) = 0 and theresonance condition ∆ = (cid:126) ω . It can be implemented as an inertial force F ( t ) /a = − m ¨ x ( t ),with lattice constant a , by shaking the lattice x ( t ) back and forth.A fully 1D Anyon-Hubbard model introduced here has not yet been experimentally re-alized. However, some relevant ingredients have been achieved, such as the experimentalimplementation of tunable occupation-dependent tunneling with Floquet engineering ofthe on-site interaction energy [482] and the realization of the occupation-number sensi-tivity of the tunneling [483]. These techniques may immediately applied to generatinglow-dimensional anyons. Bosonic quantum Hall states The integer and fractional QHE are among the most important discoveries in condensedmatter physics in 1980s. It is a quantum-mechanical version of the Hall effect, observed in2D electron systems subjected to low temperatures and strong magnetic fields, in whichthe Hall conductance σ H undergoes quantum Hall transitions to take on the quantizedvalues σ H = ν e h , where e is the elementary charge and h is Planck’s constant. Theprefactor ν is known as the filling factor, and can take on either integer or fractionalvalues. The QHE is referred to as the integer or fractional QHE depending on whether ν is an integer or fraction, respectively. Until now, the QHE has been observed only inelectron systems. Can we experimentally observe such important quantum properties inother systems is still a long-standing open question. Recently, there has been consider-able progress towards their realization in cold-atom systems. In this section, we introduceseveral theoretical proposals for realization of the QHE with cold atoms. In principle,both bosonic and fermionic atoms can be used in the experiments; however, the prepa-ration of topological states of matter relies on quick thermalization and cooling belowthe many-body gap, which is hard to achieve in cold atom systems. Since bosonic atomsare generally easier to cool, we focus on the realization of the QHE with bosonic atoms.We will mainly focus on the realizations of bosonic integer QHE with a Chern number C = 2, and the fractional quantum Hall state with the filling factor ν = 1 / 2, since theywill be the most experimentally accessible conditions.Compared with the QHE of fermions, non-interacting boson phases are topologicallytrivial, and integer QHE with bosons can only occur under the strong interactions. Theneeded strong interactions for creating bosonic quantum Hall states makes them harderto study than their fermionic cousins. As a smoking gun of the realization of quantumHall state, one can compute the many-body Chern number of the ground state | Ψ (cid:105) . Inthe theory of the QHE, it is well understood that the conductance quantization is due tothe existence of certain topological invariants, the so-called Chern numbers. The Chernnumbers with the single-particle problem and Bloch waves have been introduced in theprevious sections. For fermions, the Chern number is defined as an integration over theoccupied states in momentum space [10]. This definition cannot be applied to the bosonicsystem as many bosons can occupy the same momentum state. The generalization tomany-body systems has been proposed by Niu et al. [484] by manipulating the phases118 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 under the closed boundary conditions on a torus for both the integer and fractionalquantum Hall systems. Suppose the ground state | Ψ (cid:105) has a gap to the excited state anddepends on the parameters θ x , θ y through the generalized periodic boundary conditions: | Ψ( m + M, l ) (cid:105) = e iθ x | Ψ( m, l ) (cid:105) , | Ψ( m, l + L ) (cid:105) = e iθ y | Ψ( m, l ) (cid:105) , where M × L denotes the system size, and ( θ x , θ y ) are the twist angles vary on the torus.Under this boundary condition, we numerically diagonalize the Hamiltonian of the systemand derive the ground state | Ψ( θ x , θ y ) (cid:105) , and then one can define the many-body Chernnumber C MB as a topological invariant by the following formula [484] C MB = 12 π (cid:90) π dθ x (cid:90) π dθ y ( ∂ θ x A θ y − ∂ θ y A θ x ) , (250)where the Berry connection A µ ≡ i (cid:104) Ψ( θ x , θ y ) | ∂ µ | Ψ( θ x , θ y ) (cid:105) ( µ = θ x , θ y ).As for fractional quantum Hall state, one can also calculate the overlap between theground state and the Laughlin wavefunction. If N is the number of particles in thesystem and N φ is the number of magnetic fluxes measured in units of the fundamentalflux quanta Φ = 2 π (cid:126) /e , we can define the filling factor ν = N/N φ . In the simplest formthe fractional QHE occurs if the number of magnetic fluxes is an integer 1 /ν . At thisvalue of the magnetic field, the ground state of the system is an incompressible quantumliquid which is separated from all other states by an energy gap and is well described bythe Laughlin wavefunctionΨ( z , z , . . . z N ) = e − (cid:80) j | z j | / (cid:89) j 1, andsuperfluid phase corresponds to the opposite limit J/U (cid:29) pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 the single-component Bose-Hubbard model. They argued that the interactions of atomslocalized in the lattices are strongly enhanced compared to the interaction of atoms infree space, so the created states of the quantum Hall type in OLs are characterized bylarge energy gaps. It is a clear advantage from an experimental point of view becausethe state is more robust against external perturbations. There are two energy scalesfor the system in the presence of an artificial magnetic field: the first is the magnetictunneling term, J α , which is related to the cyclotron energy in the continuum limit (cid:126) ω c = 4 πJ α , and the second is the on-site interaction energy U . By using the methodof exact diagonalization [122], it was shown that the overlap of the ground state wavefunction | Ψ (cid:105) of the Hamiltonian in Eq. (252) with the Laughlin wave function is verygood when ν = 1 / α ≤ . 3, but the overlap start to fall off for α ≥ . 3. Furthermore,the Chern numbers for fixed ν = 1 / α ’s were calculated in Ref. [485]. Theresults show that, for higher α , the lattice structure becomes more apparent and theoverlap with the Laughlin state decreases. However, the ground state remains twofolddegenerate and the ground state Chern number tends to remain equal to 1 before reachingsome critical α c ≈ . κ = ∂ρ/∂µ , where the density ρ = (cid:80) m,l (cid:104) Ψ | ˆ n m,l | Ψ (cid:105) / ( M L ). It is incompressible( κ = 0) for the quantum Hall states, and finite for the superfluid states. The compressibil-ity of the Hamiltonian in Eq. (252) was calculated by using the cluster Gutzwiller meanfield theory in Ref. [486]. The results for α = 1 / α = 1 / ρ varies linearlywith the chemical potential µ . However, for specific values of filling factor ν there arestates with constant ρ , represented by the blue horizontal lines, and these incompressiblepositions correspond to the existence of quantum Hall states. In Fig. 45(a), the plateausor the constant ρ values correspond to ν = 1 / , , / , , / , , / , , / ρ values are να . In Fig. 45(b), the plateaus correspond to ν = 1 / , , / σ xy must beeven for any bosonic quantum Hall state without fractional quasiparticle excitations[487]. To have a basic idea about this issue, we consider some excitations created in ageneral bosonic quantum Hall state. Each of them can be considered as a bosonic particleattaching with 2 π flux and has charge σ xy . If we braid one excitation around another,the statistical phase follows from the Aharonov-Bohm effect: θ = 2 πσ xy . Similarly, ifwe exchange two excitations, the associated phase is θ/ πσ xy . On the other hand,if the state does not support fractional quasiparticles, then these excitations must bebosons. Therefore, we conclude that σ xy must be even for any bosonic quantum Hallstate without fractional quasiparticle excitations. Based on this argument, the ν = 1state in Fig. 45(a) cannot be a stable integer quantum Hall state.Notably, the integer quantum Hall state for single component bosons can occur insome lattice structures with Chern number C MB = 2 [488–491]. Recently, two differentlattice versions of bosonic quantum Hall states have been proposed at integer filling ν = 1 of the lowest topological flat-band with C MB = 2. The optical flux lattice hasbeen studied by exact diagonalization of the projected Hamiltonian in momentum space[490] and correlated Haldane honeycomb lattice has been studied by infinite densitymatrix renormalization group of hardcore boson in real space. The authors in Ref. [489]established the existences of the bosonic quantum Hall phase in their model by providingnumerical evidence: (i) a quantized Hall conductance with σ xy = 2; (ii) two counterpropagating gapless edge modes. On the other hand, it was demonstrated that bosonicinteger quantum Hall state emerges in integer boson filling factor ν = 1 of the lowestband in a generalized Hofstadter lattice (including the nearest neighbor hopping) with120 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401Figure 45. (Color online) The variation in the number density ρ in the presence of an artificial magnetic fieldwith α = 1 / α = 1 / ρ varies linearly with µ ,as shown with solid black lines. For specific values of filling factor ν there are states with constant ρ , representedby the blue lines, and these incompressible states correspond to the existence of quantum Hall states. Reprintedwith permission from Bai et al. [486]. Copyright c (cid:13) (2018) by the American Physical Society. gg / ↑↓ ν (composite fermion) (Moore-Read) πγ / -0.25 0 0.25 0.50.5 2 3-1 -0.5 ∞ collapse CFL PSPS Figure 46. (Color online) Ground-state phase diagram in the space of the total filling factor ν and the ratio U ↑↓ /U = tan γ between the intercomponent coupling constant and the intracomponent one. The product states ofa pair of nearly uncorrelated quantum Hall states (Laughlin, composite fermion, and Moore-Read states) appearwhen U ↑↓ < 0. BIQH: bosonic integer quantum Hall state; PS: phase separation; CFSS:composite fermion spin-singlet state; CFL: composite fermion liquid. SU (3) : the Halperin state with an SU (3) symmetry. Reprintedwith permission from Furukawa et al. [492]. Copyright c (cid:13) (2017) by the American Physical Society. C MB = 2 [488]. We further address the quantum Hall states in a two-component Bose-Hubbard model.We consider a system of a 2D pseudospin-1/2 bosonic gas (in the xy plane) subject tothe same magnetic fields B along the z axis for both spin states. In the second-quantizedform, the interaction Hamiltonian is written as H int = (cid:88) αβ U αβ (cid:90) d r ˆΨ † α ( r ) ˆΨ † β ( r ) ˆΨ α ( r ) ˆΨ β ( r ) , where ˆΨ α ( r ) is the bosonic field operator for the spin state α (= ↑ or ↓ ). We set thestrengths of the intracomponent contact interactions U ↑↑ = U ↓↓ = U > U ↑↓ = U ↓↑ . For a 2D systemof area A, the number of magnetic flux quanta piercing each component is given by N φ = | φ | / (2 π (cid:126) ) = A/ (2 (cid:96) ), where (cid:96) = (cid:112) (cid:126) A/ | φ | is the magnetic length. Strongly corre-lated physics is expected to emerge when N φ becomes comparable with or larger than121 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 the total number of particles, N = N ↑ + N ↓ , where N ↑ and N ↓ are the numbers ofpseudospin- ↑ and ↓ bosons, respectively.The ground-state phase diagram of pseudospin-1/2 bosonic gases in a uniform artificialmagnetic field in the space of the total filling factor ν = N/N φ and the coupling ratio U ↑↓ /U were numerically calculated by performing an extensive exact diagonalizationanalysis in the lowest-Landau level based on spherical and torus geometries [492]. Themain results are summarized in Fig. 46. In the figure, the two coupling constants areparametrized as ( U, U ↑↓ ) = G(cid:96) (cos γ, sin γ ) , where G > 0, and γ ∈ [ − π/ , π/ U ↑↓ < | U ↑↓ | is comparable to the intracomponent coupling U .This sharply contrasts with the case of an intercomponent repulsion ( U ↑↓ < U ↑↓ ≈ U . This remarkable dependence on the sign of U ↑↓ can be interpretedin light of Haldanes pseudopotentials on a sphere. More specifically, the stability of thedoubled quantum Hall states for U ↑↓ < SU (2)-symmetric] interactions with U ↑↓ = U .Among those states, relatively large gaps are found for the Halperin state with an SU (3) symmetry at ν = 2 / U (1)symmetry at ν = 2 [487, 494]. At ν = 4 / 3, two types of spin-singlet quantum Hall statescompete in finite-size systems: a non-Abelian SU (3) state and a composite fermion spin-singlet state. Furthermore, a gapless spin-singlet composite Fermi liquid has been shownto appear at ν = 1 [495]. In all these spin-singlet states, the two components are highlyentangled. For small U ↑↓ /U , in contrast, the system can be viewed as two weakly coupledscalar bosonic gases, and the product states of nearly independent quantum Hall states(doubled quantum Hall states) are expected to appear.We can compare the phase diagram in Fig. 46 with that of the two-component bosonicgases in antiparallel magnetic fields [496]. In the latter case, the pseudospin ↑ ( ↓ ) com-ponent is subject to the magnetic field B ( − B ) in the direction perpendicular to the 2Dgas, and the system possesses the TRS [84, 108]. In the regime with ν = O (1), (frac-tional) quantum spin Hall states [21] composed of a pair of quantum Hall states withopposite chiralities are robust for an intercomponent repulsion U ↑↓ > U ↑↓ as large as U . Similar results have also been found in the stability of two coupledbosonic Laughlin states in lattice models. These results suggest that the case of U ↑↓ > U ↑↓ < Kitaev honeycomb model Generalized the previous idea on realization of the unpaired Majorana zero modes in a1D system, in 2006, Kitaev further proposed another model that unpaired zero-energyMajorana modes can appear in a 2D spin-1/2 system on a honeycomb lattice, wherenearest-neighbor interactions can be reduced to a problem of non-interacting Majoranafermions [497]. It is one of the rare examples where a complex system is described by anexactly solvable 2D spin Hamiltonian. Its quantum-mechanical ground state is a quantumspin liquid and supports exotic excitations which obey Abelian or non-Abelian statistics.122 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 (a) xDyDzD B z A x A y A J J J (b) Figure 47. (Color online) (a) Kitaev model on the honeycomb lattice where interactions between nearest neighborsare J ν , depending on the direction of the link D ν . (b) The phase diagram of the Kitaev honeycomb model in the J x + J y + J z = 1 plane in the parameter space. In the three unshaded areas labeled A x , A y , and A z , the system isgapped with Abelian anyon excitations, and in the shaded area labeled B the system is gapless with non-Abelianexcitations. The Kitaev honeycomb model is a spin-1/2 system in which spins are located at thevertices of a honeycomb lattice with a spatially anisotropic interaction between neigh-boring spins, as shown in Fig. 47(a). This lattice consists of two equivalent sublatticeslabelled ‘A’ and ‘B’, which are shown by open and filled circles. A unit cell of the latticecontains both of them. The Hamiltonian is given by [497] H KHM = − (cid:88) ν, (cid:104) j,l (cid:105)∈ D ν J ν σ νj σ νl , (253)where σ νj are the Pauli matrices at the site j , J ν ( ν = x, y, z ) are interaction parameters,and the symbol (cid:104) j, l (cid:105) ∈ D ν denotes the neighboring spins in the D ν directions. Neigh-boring spins in Heisenberg models normally interact isotropically so that the spin-spininteraction does not depend on the spatial direction between neighbors. In the abovemodel, however, neighboring spins along links pointing in different directions interactdifferently.The ground state of the Kitaev honeycomb model has two distinct phases in the pa-rameter space and the phase diagram can be shown in terms of points in an equilateraltriangle satisfying J x + J y + J z = 1 (the value of J ν is the distance from the oppositeside), as shown in Fig. 47(b) [497]. If J x < J y + J z , J y < J z + J x and J z < J x + J y , thesystem is gapless with non-Abelian excitations corresponding to B phase. For all othervalues of ( J x , J y , J z ), the system is gapped with Abelian anyon excitations, labelled, A x where J x > J y + J z , A y where J y > J z + J x , A z where J z > J x + J y . The gapped phases, A x , A y , and A z , are algebraically distinct, though related to each other by rotationalsymmetry. They differ in the way lattice translations act on anyonic states, and thus acontinuous transition from one gapped phase to another is impossible. The two phases A and B are separated by three transition lines, i.e., J x = 1 / J y = 1 / 2, and J z = 1 / B phase.Precise proposals to realize an artificial Kitaev model using atomic OLs have beenmade in the literature [435, 498], where the well-controllable OLs offer the possibility ofdesigning such anisotropic spin lattice models. The main idea is that the two-componentBose-Hubbard model in a honeycomb lattice can reduce to the Kitaev spin model athalf filling and large on-site repulsion. For completeness, we here focus on the proposalin Ref. [435] and its modified implementation scheme for Rb atoms proposed in Ref.[498]. To implement the Kitaev honeycomb model using ultracold atoms, we first get aneffective 2D configuration with a set of independent identical 2D lattice in the xy planeby raising the potential barriers along the vertical direction z in the 3D OL so that thetunneling and the spin exchange interactions in z direction are completely suppressed.And then a honeycomb lattice can be constructed with three trapping potentials of the123 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 forms V j ( x, y ) = V sin [ k (cid:107) ( x cos θ j + y sin θ j + ϕ ) , (254)where j = 1 , , 3, and θ = π/ θ = π/ θ = − π/ 6. Each of the potentials is formed bytwo blue-detuned interfering traveling laser beams above the xy plane with an angle ϕ (cid:107) ,so that the wave vector k (cid:107) projected onto the xy plane has the value k (cid:107) = k sin( ϕ (cid:107) / 2) = k/ √ 3. The relative phase ϕ is chosen as π/ V b ≈ V / 4. Actually,the honeycomb (hexagonal) lattice and its topologically equivalent configuration, thebrick wall lattice, have been experimentally realized by several groups [28, 77, 140].We consider a Rb BEC and two hyperfine ground states | ↑(cid:105) = | F = 2 , m F = − (cid:105) and | ↓(cid:105) = | F = 1 , m F = − (cid:105) are defined as the effective atomic spin. The potentialbarrier between neighboring atoms in the honeycomb lattice is adiabatically ramped upto approximately V b = 14 E R to obtain a Mott insulator state with one atom per latticesite, where E R = (cid:126) k / (2 m ) is the recoil energy for Rb atoms.In this hexagonal lattice, one can engineer the anisotropic spin-spin interactions J ν σ νj σ νm in Eq. (253) using additional spin-dependent standing wave laser beams in the xy plane. To this end, one can apply three blue-detuned standing-wave laser beams inthe xy plane along the tunneling directions denoted by D x , D y , and D z , respectively: V νσ ( x, y ) = V νσ sin [ k ( x cos χ ν + y sin χ ν ) , (255)where χ x = − π/ χ y = π , χ z = π/ 3. With properly chosen laser configurations, aspin-dependent potential V νσ = V ν + | + (cid:105) ν (cid:104) + | + V ν − |−(cid:105) ν (cid:104)−| (256)along different tunneling directions ν can be generated, where | + (cid:105) ν ( |−(cid:105) ν ) is the eigenstateof the corresponding Pauli operator σ ν with the eigenvalue +1( − V ν + and V ν − by varying the laser intensity in the D ν direction so that atoms can virtuallytunnel with a rate t + ν only when it is in the eigenstate | + (cid:105) ν , which yields the effectivespin-spin exchange interactions J ν σ νj σ νm with the interaction strength J ν ≈ − t ν / (2 U ).Here the on-site interactions U ↑↓ ≈ U ↑ ≈ U ↓ ≈ U .As for a typical example, we introduce more detailed on how to generate the spin-spininteraction J z σ zj σ zm in the Hamiltonian Eq. (253) following the proposal in Ref. [498],and the other spin-spin interaction terms can be created using a similar procedure [435].The potentials (255) and (256) do not have influence on the equilibrium positions ofthe atoms, but they change the potential barrier between the neighboring atoms in the D z direction from V b ≈ V / V (cid:48) zσ = V b + V zσ . The standing wave laser beam usedfor generating spin-dependent tunneling is along the z -link direction and has a detuning∆ = 2 π × P / state (corresponding to a wavelength 787 . | ↑(cid:105) , but a red-detuningpotential for | ↓(cid:105) atoms. For instance, with a properly chosen laser intensity, the spin-dependent potential barriers may be set as V ↓ = 8 E R and V ↑ = − E R , which, combinedwith the spin-independent lattice potential barrier V = 14 E R , yield the total effectivespin-dependent lattice potential barriers V ↓ = 22 E R and V ↑ = 10 E R for neighboringatoms in the honeycomb lattice. Therefore, the tunneling rates for two spin states satisfy t ↑ /t ↓ (cid:29) 1, which leads to the spin-spin interaction J z σ zj σ zm with J z ≈ t ↑ /U , as shownin Ref. [435]. For Rb atoms, the time scale for the spin-spin interaction h/J z ≈ pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 ms. By carefully tuning the spin-dependent lattice depth in different directions, one canin principle access all phases of the Kitaev honeycomb model. However, to observe theproperties in this model, the temperature of the system needs to be much lower than thespin-spin interaction strength T (cid:28) J z /k B ≈ 8. Conclusion and outlook In the previous sections, we have reviewed the recent theoretical and experimental ad-vances on exploring topological quantum matter with cold atoms. The cold atom systemsprovide many interesting possibilities of searching for exotic topological states that arecurrently absent or unrealizable in real materials. These include some unconventionaltopological insulators and semimetals, the topological phases with many-body interac-tions, non-equilibrium dynamics, and non-Hermitian or dissipation perturbations. So far,the theoretical understanding of these phases is limited and most experimental studiesare at the single-particle level, but progress is already being made. In this final sectionof the review, we aim to discuss some promising developments for the near future. Unconventional topological bands Ultracold atomic gases in OLs with fully engineered geometries and atomic hopping formsprovide a promising platform for exploring certain unconventional topological bands.These include some unconventional topological insulators and semimetals that are diffi-cult to realize in solid-state materials, such as the chiral topological insulators protectedby the chiral symmetry [163, 308, 309], topological nodal-line semimetals protected bythe combined space-time symmetry [285, 292], and the topological bands with unconven-tional relativistic quasiparticles [350, 355, 358]. The chiral symmetry played by certainsublattice symmetry is typically broken by disorder potential in real materials; however,it naturally arises for cold atoms in OLs with negligible disorder [309]. The topologicalsemimetals or metals with tunable structures of nodal points or lines could be experi-mentally implemented by varying the atom-laser interaction configuration [355, 499].Furthermore, some theoretically predicted topological insulators beyond the ten-fold classification could be implemented in OLs. These include the Hopf insulators[66, 314, 317], the 3D quantum Hall states [265], the topological crystalline insulatorsstabilized by crystalline lattice symmetries [500], the topological Anderson insulators[501–503], and the so-called higher-order topological insulators and semimetals [504–509]. The topological Anderson insulator is a disorder-driven topological phase, wherethe static disorder induces nontrivial topology when added to a trivial band structure[501–503]. Although there are many theoretical studies, the topological Anderson insu-125 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 lator has so far evaded experimental realization due to the lack of precise control overdisorder and topology in real materials. In a recent experiment, evidence for the topolog-ical Anderson insulator phase in synthetic cold-atomic wires with controllable disorderhas been found [349]. As an extension of the topological insulator family, the recentlyproposed higher-order ( n -th order) topological insulators can host quantized multipolemoments in the bulk bands, such as quadrupole and octupole, and has robust gaplessstates at the intersection of n crystal faces (but is gapped otherwise). For instance, a bulk2D topological quadrupole insulator described by nested Wilson loops hosts protectedcorner states with fractional charges [504], and its extension to a layered 3D system cangive rise to a topological quadrupolar semimetal [509]. It was also proposed to realizetopological quadrupole insulators using ultracold atoms in an optical superlattice [504].Cold gases also allow the exploration of band topology in D = d r + d s dimensionshigher than the real dimension d r = 3, through the use of d s synthetic dimensions. Arecent experiment has demonstrated the topological response of an effective 4D systemby using 2D pumping in an OL [17]. Other topological states in 4D would be studied inthe near future, such as 4D intriguing fractional phases induced by interactions [16] andtime-reversal-symmetric 4D QHE. In addition, the 5D generalization of the topologicalWeyl semimetals with Yang monopoles and linked Weyl surfaces in the BZ [510, 511]would be similarly simulated with cold atoms. Very recently, the quantum simulation ofa Yang monopole in a 5D parameter space built from the internal states of an atomicquantum gas was reported [512]. Moreover, its topological charges (the second Chernnumbers) were measured by experimentally characterizing the associated non-AbelianBerry curvatures in the parameter space. Other interacting topological phases Topological superfluids with Majorana bound states. It has been theoretically shown thatthe p + ip -wave or p -wave topological superconductors/superfluids can be effectively in-duced in conventional s -wave superconductors/superfluids by combining the SOC andZeeman splitting [513–516]. The zero-energy Majorana bound states with non-Abelianstatistics can emerge in these systems, but they have not yet been experimentally con-firmed. With the recent advances, all the individual ingredients including the syntheticSOC and effective Zeeman fields for topological superfluids in fermionic quantum gasesare in place. Several concrete proposals for realizing exotic topological superfluids withMajorana bound states for cold atomic gases have been proposed [103, 473, 514–519].Recently, a 2D SOC and a perpendicular Zeeman field have been simultaneously gener-ated in ultracold Fermi gases [35, 113], which paved the way for future exploration oftopological superfluids in ultracold atoms. Once the systems are experimentally realized,the high degree of experimental control over these cold atom systems will enable newapproaches for the direct observation and manipulation of Majorana bound states, suchas non-Abelian braiding. Topological Mott insulators. In general, a strong interaction will open a trivial energygap and break the band topology. However, there exists a class of topological insula-tors called the topological Mott insulators [520], where the many-body interactions areresponsible for topological insulator behaviors. Although the topological Mott insula-tor phase was first reveled in an extended Fermi-Hubbard model on a 2D honeycomblattice [520], it has now been known as a class of interaction-induced topological insu-lators for interacting fermions or bosons. The topological Mott insulators in 1D and 3Dfermion systems have also been investigated [521–523]. Whereas many studies providegrowing evidence for the existence of the topological Mott insulating phase, its exper-imental observation in electron systems is still outstanding. The schemes for realizing126 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 the topological Mott insulator with Rydberg-dressed fermionic atoms in 2D OLs wereput forward [524, 525]. Furthermore, several works have theoretically and numericallyshown that the topological Mott insulator phase can occur for interacting bosonic atoms[189, 526–528], atomic mixtures [529], and fermionic atoms [530] in 1D optical super-lattices. Due to the tunable atomic interactions in OLs (similar as the engineered Bose-or Fermi-Hubbard models for cold atoms), these artificial systems may provide the firstrealization of topological Mott insulators in the near future. Topological Kondo insulators. A class of topological insulators called topological Kondoinsulators for few heavy-fermion materials were recently predicted [531–533], which orig-inates from the hybridization between itinerant conduction bands and correlated elec-trons. Topological Kondo insulators are essentially induced by the strongly correlatedKondo effect that leads to the insulating gap, although they share the same topologi-cal properties with conventional topological insulators. Little evidence for a topologicalKondo insulator state in SmB has been reported [534, 535], and more theoretical andexperimental works are needed to fully understand it. On the other hand, a scheme hasrecently been proposed to realize and observe topological (Chern) Kondo insulators ina 2D optical superlattice with laser-assisted s and p orbital hybridization and a syn-thetic gauge field [536]. The topological Kondo insulator phase was also predicted oninteracting “sp-ladder” models [537, 538], which could be experimentally realized in OLswith higher orbitals loaded with ultracold fermionic atoms. Motivated by experimentaladvances on ultracold atoms coupled to a pumped optical cavity [539], a scheme for syn-thesizing and observing the topological Kondo insulator in Fermi gases trapped in OLswas also proposed [540]. Non-equilibrium dynamics and band topology In a recent experiment [47], the dynamical evolution of the Bloch wavefunction was stud-ied by using time- and momentum-resolved state tomography for spinless fermionic atomsin the driven hexagonal OL. In particular, the appearance, movement and annihilationof dynamical vortices in momentum space after sudden quenches close to the topologi-cal phase transition were observed. Furthermore, it was theoretically proposed [541] andexperimentally demonstrated [542] that the topological Chern number of a static Hamil-tonian can be measured from a dynamical quench process through a rigorous mappingbetween the band topology and the quench dynamics, i.e., the mapping of the Chernnumber to the linking number of dynamical vortex trajectories appearing after a quenchto the Hamiltonian. It was also predicted that a topologically quantized Hall responsecan be dynamically built up from nontopological states [543].Very recently, a different dynamical approach with high precision has been experi-mentally demonstrated to reveal topology through the unitary evolution after a quenchfrom a topological trivial initial state to a 2D Chern band realized in an ultracold Rbatom gas [383]. The emerging ring structure in the spin dynamics uniquely determinesthe Chern number for the post-quench band. The dynamical quantum phase transitionand the topological properties in the quench dynamics have been theoretically studiedfor various topological systems [544–550]. These studies have shown that the cold atomsystems provide a natural and promising platform to explore the connection betweentopological phases and non-equilibrium dynamics. Topological states in open or dissipative systems Topological states in open systems. The topological phases discussed so far are groundstate phases in isolated systems. With an additional dissipative coupling between atoms127 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 and an environment, one may consider the possibility of engineering topological statesusing the concept of dissipative state preparation [551, 552]. For simplicity, here we takethe environment temperature to be T = 0 and restrict our discussion to the cold atomimplementations [474, 553, 554]. For a weak coupling to a Markovian bath, which is agood approximation of atoms coupled to a continuum of radiation modes, the masterequation takes the Lindblad form [555]d ˆ ρ/ d t = i (cid:104) ˆ ρ, ˆ H (cid:105) + (cid:88) j (cid:18) ˆ L j ˆ ρ ˆ L † j − (cid:110) ˆ L † j ˆ L j , ˆ ρ (cid:111)(cid:19) , (257)where ˆ ρ denotes the reduced density matrix of the system, and the incoherently actingLindblad operators ˆ L j (also called jump operators) account for the system-bath couplingwith the dissipative channels being labelled by j . In the open systems, the steady states ˆ ρ s are defined by d ˆ ρ s / d t = 0, and the counterpart to an energy gap is provided by a dampinggap defined as the smallest rate at which deviations from ˆ ρ s are damped out. When thecoupling is engineered so that the system ends up after some relaxation time into a purestate (called a dark state), “topology by dissipation” is achieved where the pure state hasnontrivial topological properties [474, 553]. The specific system studied in Ref. [474] is aquantum wire of spinless atomic fermions in an OL coupled to a bath. The key featureof the dissipative dynamics described by the Lindblad master equation is the existenceof Majorana edge modes, and their topological protection is granted by a non-trivialwinding number of the system density matrix. Such a concept of topology by dissipationhas formally been extended to higher spatial dimensions and various symmetry classes[554]. Furthermore, it was shown that the dissipation can lead to a novel manifestationof topological states with no Hamiltonian counterpart [553], such as spatially separatedMajorana zero modes in the dissipation-induced p -wave paired phase of 2D spin-polarizedfermions with zero Chern number. Topological superradiant states. The experimental advances on ultracold atomic gasescoupled to an optical cavity have shown that the interplay between the atomic motion andthe light fields can give rise to rich dynamical processes and exotic many-body collectivephenomena [539], such as the Dicke superradiant state [556]. Recently, a topologicalsuperradiant state in a 1D spin-1/2 degenerate Fermi gas in a cavity with cavity-assistedRaman processes was predicted [557, 558]. This novel steady-state topological phase of adriven-dissipative system is characterized simultaneously by a local order parameter anda global topological invariant (the winding number of momentum-space spin texture) witha superradiance-induced bulk gap. It was also suggested to detect the topological phasetransition between normal and topological superradiant states from its signatures in themomentum distribution of the atoms or the variation of the cavity photon occupation,due to the nontrivial feedback of the atoms on the cavity field [557]. A superradianttopological Peierls insulator involving transversely laser-driven atoms coupled to a singlemode of an optical resonator in the dispersive regime was also predicted [559]. A fermionicquantum gas in a 2D OL coupled to an optical cavity can self-organize into a state inwhich the cavity mode is occupied and an artificial magnetic field dynamically emerges,such that the fermionic atoms can form steady-state chiral insulators [560] or topologicalHofstadter insulators [561]. Topological states in non-Hermitian systems. Recently, the search for topological statesof matter in non-Hermitian systems has attracted increasing interest (see Ref. [562] fora review). For a dissipative cold atom system with particle gain and loss, a new type oftopological ring characterized by both a quantized Chern number and a quantized Berryphased (defined via the Riemann surfaces) was revealed [290], dubbed a Weyl exceptionalring consisting of exceptional points at which two eigenstates coalesce. Realizing the Weyl128 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 exceptional ring requires a non-Hermitian term associated with particle loss for spin-downatoms. Without this term, the system is in the Weyl semimetal phase that can be realizedwith cold atoms in 3D OLs (see Sec. 4.3.2). To generate the decay term representing anatom loss for spin-down atoms, one may consider using a resonant optical beam to kickthe spin-down atoms out of a weak trap, or alternatively, using a radio frequency pulse toexcite the spin-down atoms to another irrelevant internal state. A possible approach tomeasure the Weyl exceptional ring is probing the dynamics of atom numbers of each spincomponent after a quench [290]. The non-Hermitian Hamiltonian was recently realizedin a noninteracting Li Fermi gas via generating state-dependent atom loss, and thenon-Hermitian term was achieved by an optical beam resonant with the atomic decaycoupling [563]. Acknowledgements For figures with copyright from the American Physical Society: Readers may view,browse, and/or download material for temporary copying purposes only, provided theseuses are for noncommercial personal purposes. Except as provided by law, this materialmay not be further reproduced, distributed, transmitted, modified, adapted, performed,displayed, published, or sold in whole or part, without prior written permission from theAmerican Physical Society. Disclosure statement No potential conflict of interest was reported by the authors. Funding This work was supported by the NKRDP of China [grant no. 2016YFA0301803], theNSFC [grant nos. 11604103, 11474153, 91636218, and 11874201], the NSAF [grant nos.U1830111 and U1801661], the NSF of Guangdong Province [grant no. 2016A030313436],the KPST of Guangzhou [grant no. 201804020055], and the Startup Foundation of SCNU. Appendix A. Formulas of topological invariants The purpose of this Appendix is to provide general discussions and more details of deriva-tions of topological invariants referenced in this review. Mathematically, these topologicalinvariants are defined for vector or principal bundles to characterize the topological (iso-morphism) classes of the bundles and have applications wherever the bundles find theirmanifestations in physical systems. In the course of adapting this mathematical subjectinto physics, condensed matter and high energy physics communities made tremendousendeavors. To serve our main subject of topological cold-atom systems, we do not intenda complete review but focus on the Berry bundle of band theories for an insulator ofnon-interacting fermions. The formulas of topological invariants are applicable to othersuitable bundles of physical systems. 129 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 A.1 Flattened Hamiltonians and Berry Bundles We denote the momentum-space Hamiltonian of the insulator as H ( k ) with k in the firstBrillouin zone (BZ), and assume finite number of bands, namely H ( k ) is a ( M + N )-dimensional matrix at each k , where M and N are numbers of conduction and va-lence bands, respectively. At each k , H ( k ) can be diagonalized and the conductionand valence eigenpairs are ( E + ,a , | + , k , a (cid:105) ) and ( E − ,b , |− , k , b (cid:105) ), respectively, with a = 1 , · · · , M and b = 1 , · · · , N . Therefore the Hamiltonian is now expressed as H ( k ) = (cid:80) a E + ,a | + , k , a (cid:105)(cid:104) + , k , a | + (cid:80) b E − ,b |− , k , b (cid:105)(cid:104)− , k , b | . We further introduce theprojectors onto conduction and valence spaces as Π ± ( k ) = (cid:80) a |± , k , a (cid:105)(cid:104)± , k , a | , whichsatisfy the following relations,1 = Π + + Π − , Π ± = Π ± , Π + Π − = Π − Π + = 0 . (A1)Then, it is clear that H ( k ) can be adiabatically deformed to be the flattened Hamiltonian (cid:101) H ( k ) = Π + ( k ) − Π − ( k ) (A2)without closing the energy gap by smoothly regulating positive and negative energiesconverging to ± 1, respectively. Since the topological properties of an insulator do notchange under gap-preserving continuous deformations, it is sufficient and more convenientto adopt the flattened (cid:101) H ( k ) for studying topological properties.At each k , valence states |− , k , b (cid:105) span an N dimensional vector space that is the imageof Π − ( k ), and these vector spaces spread smoothly over the whole BZ, forming an N Dvector bundle, which is called the Berry bundle of valence bands of an insulator. Since theBerry bundle is generated by the projector Π − ( k ), there exists a canonical Levi-Civitaconnection, called the Berry connection, which is given by A µb,b (cid:48) ( k ) = (cid:104)− , k , b | ∂∂k µ |− , k , b (cid:48) (cid:105) (A3)with µ = 1 , , · · · , d labeling momentum coordinates. To see that Eq. (A3) is indeed aconnection, one may check that under a gauge transformation |− , k , b (cid:105) −→ |− , k , b (cid:48) (cid:105) U b (cid:48) b ( k ) (A4)with U ( k ) being a field of unitary matrices globally defined in the whole BZ, the Berryconnection transforms as A µ ( k ) −→ U † A µ U + U † ∂ µ U. (A5)Accordingly, the Berry curvature of the Berry bundle is F µν = ∂ µ A ν − ∂ ν A µ + [ A µ , A ν ] , (A6)whose gauge transformation is given by F µν −→ U † F µν U. (A7)As an example we discuss the general two-band model for insulators, H b ( k ) = d ( k ) · σ, (A8)130 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 where σ i with i = 1 , , σ term with σ the 2 × | d ( k ) | is not equal to zero for all k , since the spectrum is given by E ± ( k ) = ±| d ( k ) | . The Hamiltonian of Eq. (A8) can be flattened as (cid:101) H b ( k ) = ˆ d ( k ) · σ (A9)with ˆ d ( k ) being the unit vector d ( k ) / | d ( k ) | . Accordingly the projectors for valence andconduction bands are Π b ± ( k ) = 12 [ σ ± ˆ d ( k ) · σ ] . (A10)The valence eigenstates can be represented by |− , k (cid:105) = e − iσ φ ( k ) / e − iσ θ ( k ) / | ↓(cid:105) , where θ ( k ) and φ ( k ) are the standard spherical coordinates of ˆ d ( k ), and | ↓(cid:105) is the negativeeigenstate of σ . The Berry connection can be straightforwardly derived as A µ ( k ) = i θ ( k ) ∂ µ φ ( k ) . (A11)Note that for two-band case the Berry connection is Abelian. Under the U (1) gaugetransformation |− , k (cid:105) → e iϕ ( k ) |− , k (cid:105) , the Berry connection A µ ( k ) is transformed to be A µ ( k )+ i∂ k µ ϕ ( k ). But the Berry curvature is invariant under gauge transformations, andis given from Eq. (A6) by F µν ( k ) = − i sin θ ( k )( ∂ µ θ ( k ) ∂ ν φ ( k ) − ∂ ν θ ( k ) ∂ µ φ ( k )), whichcan be recast in terms of ˆ d ( k ) as F µν ( k ) = 12 i ˆ d · ( ∂ µ ˆ d × ∂ ν ˆ d ) . (A12)Equations (A5), (A6) and (A7) also appear in gauge theory. More specifically the Berryconnection A µ ( k ) and curvature F µν ( k ) corresponds to the gauge field or potential andfield strength tensor, respectively, according to the terminologies of U ( N ) gauge theory,for which the base space is the spacetime. Conventionally the connection of a vectorbundle is not unique (but usually forming a space), and actually the definition of theBerry connection, namely Eq. (A3), is just a canonical way to assign a vector bundle withHermitian metric connection. Therefore a band theory can be regarded as a U ( N ) gaugetheory with a given connection, the Berry connection. Once a connection is assigned fora vector bundle, we can compare two vectors at two separate points y and z throughparallel transport of the one at y along a path C to z . Parameterizing the path C as x ( τ ),where τ ∈ [ τ i , τ f ], x ( τ i ) = y and x ( τ f ) = z , the mutually parallel vectors | ψ ( x ( τ )) (cid:105) along C satisfy the equation, dx µ dτ D µ ψ = 0 , (A13)with the covariant derivative D µ = ∂∂x µ + A µ ( x ). The solution is | ψ ( x ( τ )) (cid:105) = U P ( z, y ) | ψ ( y ) (cid:105) , and the parallel-transport operator U P is given by U P ( z, y ) = ˆ P exp (cid:20) − (cid:90) C dτ dx µ dτ A µ ( x ( τ )) (cid:21) , (A14)where ˆ P indicates that the integral is path ordered. If C is a closed path, the parallel-131 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 transport operator is called the Wilson-loop operator in the context of gauge theory, orthe holonomy along C in mathematics. If τ ∈ [ τ i , τ f ] is divided into N equal intervals,the parallel-transport operator can be approximated by U P ( z, y ) ≈ U N U N − · · · U , (A15)where U j = exp[ −A y (( τ j − + τ j ) / τ /N ] with τ j = τ i + j ∆ τ /N and ∆ τ = ( τ f − τ i ) /N ,so that the equality is recovered taking N to infinity.Performing a gauge transformation | ψ ( x ( τ )) (cid:105) → V ( x ( τ )) | ψ ( x ( τ )) (cid:105) along the path C ,the parallel-transport operator is transformed as U P ( z, y ) −→ V ( z ) U P ( z, y ) V † ( y ) . (A16)For a Wilson loop C , which is a closed circle, it is a unitary transformation given bythe reference point y , namely U P ( y, y ) → V ( y ) U P ( y, y ) V † ( y ). In particular, for Abelianconnection A µ , the path order ˆ P is not important, and therefore the Wilson loop is gaugeinvariant, and is given by the flux inserted over the area surrounded by C , U C = exp (cid:20) − (cid:90) D d x F ( x ) (cid:21) , (A17)where D is any smooth surface with C being its boundary, and F is the correspondingAbelian Berry curvature. In this case, the phase factor U C is called the geometric Berryphase along C as well. A.2 Chern Number and Chern-Simons Term The Chern number can be formulated for any even-dimensional integral domain. For 2 n dimensions, the corresponding Chern number is called the n th Chern number, and thecorresponding integrand is called the n th Chern character. When n = 1, the first Chernnumber for a 2D insulator is explicitly given as C = i π (cid:90) T d k tr F . (A18)Here and hereafter T n represents a 2 n dimensional torus. Noting that the trace overthe commutator in Eq. (A6) vanishes, the first Chern number essentially comes from theAbelian connection a µ = tr A µ , and can be accordingly recast as C = ( i/ π ) (cid:82) T d k f with f µν = ∂ µ a ν − ∂ ν a µ . If n = 2, the second Chern number for a 4D insulator is C = − π (cid:90) T d k (cid:15) µνλσ tr F µν F λσ , (A19)which is essentially non-Abelian.For 2D insulators, the first Chern number of Eq. (A18) is also called the Thouless-Kohmoto-Nightingale-den Nijs (TKNN) invariant, and was shown to be the transverseconductance in the unit of e /h by using the Kubo formula [10]. A nonvanishing trans-verse conductance requires the breaking of TRS. This is consistent with Eq. (A18),because the first Chern number has to be vanishing in order to preserve TRS, since i F isodd under time-reversal, which shall be clear when we discuss TRS. In other words, a 2DChern insulator cannot have TRS. In contrast, the second Chern number of Eq. (A19)132 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 is time reversal symmetric, namely there exist time-reversal-symmetric 4D Chern insu-lators. The meaning of the second Chern number for electromagnetic response can befound in Refs. [63, 64].If a 2 n dimensional sphere S n is chosen to enclose a singular point, where the bundleis not well-defined, in a (2 n + 1)D space, the Chern number may be calculated on theS n , and is referred to as the monopole charge of the singular point. For monopolesin 3D space, the monopole charge can be calculated by the Abelian connection a µ =tr A µ , and therefore are termed as Abelian monopoles. For instance the Weyl points for H W ( k ) = ± k · σ can be interpreted as unit Abelian monopoles in momentum space forthe respective Abelian Berry bundles of valence band restricted on S surrounding theorigin. Monopoles in 5D space are categorized into non-Abelian monopoles, the monopolecharge must be calculated by non-Abelian connections. Accordingly the 5D Weyl points H DW ( k ) = ± k µ Γ µ with µ = 1 , , · · · , 5, where Γ µ are 4 × G ( ω, k ) = 1 / [ iω − H ( k )] [64, 564–567],which for Eq. (A18) is explicitly given by C [ k ] = − π (cid:90) ∞−∞ dω (cid:90) T d k(cid:15) µνλ tr G∂ µ G − G∂ ν G − G∂ λ G − . (A20)Note that G ( ω, k ) is an invertible matrix for each ( ω, k ), namely G ( ω, k ) ∈ GL ( N + M, C),because H ( k ) is invertible.Although there is no global Berry connection A over the base manifold if the Berrybundle is nontrivial, if the base manifold is trivially a disk D n , A can be given over thewhole D n , and furthermore the Chern character can be expressed as a total derivativeof the Chern-Simons form. If n = 1, it is obvious that the first Chern character C ( F ) = dQ ( A ) with Q ( A ) = i π tr A . (A21)For n = 2, the Chern-Simons form is a third form Q ( A , F ) = 12 (cid:18) i π (cid:19) tr( A d A + 23 A ) , (A22)and it is straightforward to check that dQ ( A , F ) = C ( F ). A general formula Q n − ( A , F ) for any n is Q n − ( A , F ) = 1( n − (cid:18) i π (cid:19) n (cid:90) dt tr (cid:16) AF ( n − t (cid:17) (A23)with F t = t F + t ( t − A . The integration of Q n − over a (2 n − 1) dimensional manifold,for instance S n − , is called the Chern-Simons term, ν n − CS [ A ] = (cid:90) S n − Q n − ( A , F ) . (A24)133 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 A significant difference of the Chern-Simons term from Chern number is that it is notgauge invariant. For instance, Q ( A U ) = Q ( A ) + i π tr U † ∂ k U, (A25)and therefore the change of ν CS is just the winding number of U ( k ). Analogous calcu-lations for Q ( A , F ), although are a little more complicated, can also be made straight-forwardly. It is actually a general conclusion that a gauge transformation of U changesthe Chern-Simons term over S n − by a winding number ν n − [ U ] of U , namely ν n − CS [ A U ] − ν n − CS [ A ] = ν n − [ U ] . (A26)We now make a classic application of the mathematics introduced in this subsection,which is of fundamental importance. Consider a Berry bundle on a 2D sphere S . If thebundle is nontrivial, there is no globally well-defined Berry connection A . But one canalways have Berry connections on the north hemisphere D N and the south as A N and A S , respectively, and glue the wave functions along the equator S , which is given by thetransition function from the south hemisphere to the north, namely U ( k ) ∈ U ( N ) with k ∈ S . Then, the Chern number is calculated as C = (cid:90) D N dQ ( A N , F N ) + (cid:90) D S dQ ( A S , F S ) = (cid:90) S Q ( A N , F N ) − (cid:90) S Q ( A S , F S )= ν [ U ] , (A27)where the minus sign in the second equality is due to the opposite orientations of S n − with respect to the north hemisphere and the south, and the third equality has usedEq. (A26). It is concluded that the Chern number of the bundle on the sphere is just thewinding number of the transition function along the equator. The topological invariant for 2D Cherninsulators is the first Chern number of Eq. (A18). There does not exist a complete set ofglobally well-defined valence eigenstates in the whole torus T . But we can find it overthe cylinder, which does not require periodic boundary condition over k y ∈ ( − π, π ], andthen use a transition function from the bundle on the circle S − at k y = − π to that on S at k y = π . The Chern number is just the winding number of the transition function as amapping from S to U ( N ), which can be inferred from the discussions about Eq. (A27).Given such a set of eigenstates over the cylinder, we can use the method of Wilson loopsover k y , which are parametrized by k x ∈ ( − π, π ], to obtain the transition function. Ateach k x , the Wilson-loop operator is given by U ( k x ) = ˆ P exp (cid:90) π − π dk y A y ( k x , k y ) ∈ U ( N ) . (A28)After working out the transition function by Wilson loops, the Chern number is calculatedby the winding number ν w = i π (cid:73) dk x tr U ( k x ) ∂ k x U † ( k x ) . (A29)Note that tr F = tr( d A + A ∧ A ) = tr d A = (cid:80) a d A aa , it implies the fact that whether A is non-Abelian is not important in two dimensions, namely each valence band canbe treated individually. In practice, one can always add appropriate perturbations to134 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 separate valence bands, and for the a th band the corresponding Berry connection A a isjust an Abelian connection. Thus the Chern number is just the summation of windingnumbers for each valence band, C = i π (cid:88) a (cid:73) dk x U a ( k x ) ∂ k x U † a ( k x ) . (A30) Chern Number of Two-band Model. For the two-band model of Eq. (A8) as an example,the Chern number can be expressed explicitly by C = 14 π (cid:90) T d k ˆ d · ( ∂ k x ˆ d × ∂ k y ˆ d ) , (A31)which can be derived by directly substituting Eq. (A12) into Eq. (A18), or alternativelyby substituting the Green’ function with imaginary frequency, G ( ω, k ) = 1 / [ iω − d ( k ) · σ ],into Eq. (A20). The simplified formula of Eq. (A31) is just the winding number of thevector field ˆ d ( k ) as a mapping from T to S . A.3 Topological Invariants for Topological Insulators A.3.1 3D Topological Insulators. The Chern insulators do not require any symmetry for the momentum-space Hamilto-nian, but the 3D topological insulator requires TRS. The TRS is represented in mo-mentum space (even in real space) by ˆ T = U T ˆ K , where U T is a unitary matrix, andˆ K is the complex conjugate operator, satisfying ˆ T = − 1. Note that for electronicsystems, U T = − iσ with σ acting in the spin space. If a system has TRS, thenˆ T † H ( − k ) ˆ T = H ( k ), which may be explicitly expressed as U † T H ( − k ) U T = H ∗ ( k ) . (A32)For a valence eigenstate | a, k (cid:105) with H ( k ) | a, k (cid:105) = E a ( k ) | a, k (cid:105) , U T | a, k (cid:105) ∗ is an eigenstateof H ( − k ) with the same energy E a ( k ), which can be deduced from Eq. (A32). Here weabbreviate |− , a, k (cid:105) to be | a, k (cid:105) for simplicity. Thus, the spectrum is inversion symmetricin momentum space, and U T | a, k (cid:105) ∗ can be expanded by the basis at − k as U T | a, k (cid:105) ∗ = (cid:88) b U ∗ ab ( − k ) | b, − k (cid:105) , (A33)where U ( k ) is a unitary matrix for each k . Due to the constraint of Eq. (A33) exertedby the TRS in the valence bands, the Berry connection satisfies the relation, A ∗ ( k ) = U ( − k ) A ( − k ) U † ( − k ) + U ( − k ) d U † ( − k ) . (A34)In general the Chern-Simons term can take any real number, but symmetry could leadto quantization of the Chern-Simons term. For 3D topological insulators with TRS, therelation of Eq. (A34) can be applied to quantize the Chern-Simons term. Observingthat A ∗ ( k ) is just the gauge transformed A ( − k ) by U † ( − k ) from Eq. (A34), and theChern-Simons term is a real number odd under inversion, we can deduce2 ν CS [ A ] = ν [ U ] , (A35)135 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 in the light of Eq. (A26). The right hand of Eq. (A35) is the winding number of U ( k )over the 3D BZ. Because of the gauge ambiguity described by Eq. (A26), 2 ν CS [ A ] can beregarded as a Z topological invariant for 3D topological insulators, which is explicitlygiven by [64] ν (1)Z = − π (cid:90) T d k(cid:15) µνλ tr( A µ ∂ ν A λ + 23 A µ A ν A λ ) mod 2 . (A36)It is essential important for 3D topological insulators that ˆ T = − 1, which can readilybe seen from another topological invariant given by Fu-Kane-Mele [296]. The fact thatˆ T = − U T is anti-symmetric, namely U tT = − U T , which further implies U t ( k ) = −U ( − k ). In the 3D BZ, there are eight inversion invariant points Γ i with i = 1 , , · · · , U (Γ i ) are anti-symmetric. A significant consequence of TRS is the presence ofthe two-fold Kramers degeneracy for energy eigenstates, and therefore the valence-statenumber is even at Γ i . For an anti-symmetric even-dimensional matrix U , the PfaffianPf( U ) can be defined as a polynomial of entries of U , and the topological invariant isgiven by ( − ν (1)Z2 = (cid:89) i =1 Pf( U (Γ i )) (cid:112) Det( U (Γ i )) , (A37)which is called the Fu-Kane-Mele invariant [296]. Note that the determinants of U (Γ i )as unitary matrices are all definitely positive. Although, the expression of Eq. (A37) islocal at Γ j , the global information is acquired by the requirement that the valence wavefunctions are globally well-defined in the whole BZ. A proof for the equality of Eqs. (A36)and (A37) can be found in Ref. [568]. Despite of using the unfamiliar Pfaffian, Eq. (A37)can be radically simplified in the presence of inversion symmetry ˆ P . Since each Γ i isinversion invariant, each valence state |− , Γ i , a (cid:105) at Γ i with a = 1 , , · · · , N is also aneigenstate of ˆ P with eigenvalue (parity) ξ a (Γ i ) = ± 1. Assuming that (2 m − m th states form Kramers pairs with m = 1 , , · · · , N , we define δ i = (cid:81) Nm =1 ξ m − (Γ i )noticing that two states of each Kramers pair have the same parity because of [ ˆ P , ˆ T ] = 0.Then Eq. (A37) takes the simple expression [299],( − ν (1)Z2 = (cid:89) i δ i . (A38)The convenience of Eq. (A38) lies in that it is entirely determined by the representationof ˆ P , which can be derived from local eigenstates at Γ i , and hence its practice doesnot require global valence eigenstates, which as aforementioned are usually technicallydifficult to obtain. A.3.2 2D Topological Insulators There exists no Chern insulator without TRS, since the first Chern number is odd un-der time-reversal. From Eq. (A5), it is found that F ∗ ( k ) = U ( − k ) F ( − k ) U † ( − k ), but F ∗ ( k ) = −F t ( k ). However, new time reversal invariant topological insulators arise withZ classification, and the corresponding topological properties requires and are protectedby TRS [18]. The Z topology can be characterized by a topological invariant quite sim-ilar to Eq. (A37), but now the product is over four inversion invariant points in the 2D136 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 BZ [160], ( − ν (2)Z2 = (cid:89) i =1 Pf( U (Γ i )) (cid:112) Det( U (Γ i )) , (A39)which again requires globally well-defined wave functions. If the inversion symmetry ispresent, Eq. (A38) is also a usually more convenient alternative for Eq. (A39), where theproduct is now over four inversion invariant points.A topological invariant in terms of the Berry connection may also be formulated forthe time-reversal-symmetric topological phase. For each k , let | I, α, k (cid:105) and | II, α, − k (cid:105) be a pair of states labelled by α , which are related by TRS by | I, α, k (cid:105) = U T | II, α, − k (cid:105) ∗ , | II, α, k (cid:105) = − U T | I, α, − k (cid:105) ∗ . Locally the state label a is further specified to be ( s, α ) with s = I or II . However, if thetopological insulator has nontrivial topological invariant, such a basis | s, α, k (cid:105) does notexist globally. Instead, we can choose such a basis for the 1D subsystem with k y = − π ,and also for the one with k y = 0. Then the topological invariant is given by ν (2)Z = i π (cid:90) π − π dk x tr[ A ( k x , − π ) − A ( k x , − i π (cid:90) π − π dk x (cid:90) − π dk y tr F mod 2 , (A40)noting that the 1D subsystems with k y = − π and 0 are the boundary of the integrationdomain of the second term, which is a cylinder. The gauge freedom of the first termjustifies the Z nature of the topological invariant, recalling Eq. (A26). The equivalenceof the two topological invariants Eqs. (A39) and (A40) can be found in Ref. [160]. A.4 Winding Numbers for Chiral Classes In this section, we consider a Hamiltonian H ( k ) with chiral symmetry Γ, namely H ( k )anti-commutes with Γ, {H ( k ) , Γ } = 0 , (A41)and assume that Γ = 1 and Γ † = Γ. The chiral symmetry implies at each k the valencestates of the insulator H ( k ) have a one-to-one correspondence to conduction states,noting that if H ( k ) | ψ (cid:105) = E | ψ (cid:105) , then H ( k )Γ | ψ (cid:105) = − E Γ | ψ (cid:105) . One can always choose abasis, for which Γ = diag(1 N , − N ) with 1 N being the N × N identity matrix, and thenthe Hamiltonian takes the anti-diagonal form, H ( k ) = (cid:18) q † ( k ) q ( k ) 0 (cid:19) , (A42)where q ( k ) is an N × N invertible matrix for each k , since H ( k ) describing an insulatoris invertible. Thus, the winding number of q from the BZ to GL ( N, C) is a topologicalinvariant for this symmetry class, which is given by ν n +1w [ q ] = η n (cid:90) T n +1 tr( qdq − ) n +1 (A43)137 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 with η n = n ! / [(2 n + 1)!(2 πi ) n +1 ] for a (2 n + 1)D BZ [123]. Note that even-dimensionalinsulators in this symmetry class are all trivial. In physical dimensions, for 1D insulatorswith n = 0, the topological invariant is explicitly given by ν [ q ] = 12 πi (cid:73) dk tr qdq − , (A44)and for 3D insulators with n = 1, ν [ q ] = − π (cid:90) d k(cid:15) µνλ tr q∂ µ q − q∂ ν q − q∂ λ q − . (A45)The definition of q ( k ) or the anti-diagonal form of Eq. (A42) is based on the particularrepresentation of Γ = diag(1 N , − N ), and therefore Eq. (A43) is gauge dependent. Agauge independent expression of the topological invariant can be given for hermitianchiral symmetry Γ with the normalization Γ = 1 as ν n +1w [ H ] = η n (cid:90) T n +1 trΓ( H d H − ) n +1 . (A46) A.5 Quantized Zak Phase In general the Berry phase of valence bands in the unit of π for a 1D gapped system isgiven by ν = iπ (cid:73) dk tr A (A47)with A given by Eq. (A3), may be any real number, and thus cannot be a topologicalinvariant. However, certain symmetries can quantize it into integers, which is similar tothat the quantized Chern-Simons term in three dimensions, recalling from Eq. (A21)that Eq. (A47) is just the Chern-Simons term in one dimension. The quantization ofEq. (A47) was first discussed in 1D band theory by Zak taking into account inversionsymmetry [62], and therefore the Berry phase in band theory is also called the Zak phase.Only the parity of the quantized Berry phase of Eq. (A47) is gauge invariant, since alarge gauge transformation, | k, b (cid:105) → (cid:80) c U bc ( k ) | k, c (cid:105) , can change Eq. (A47) by two timesof the winding number of u ( k ) = Det[ U ( k )] ∈ U (1), namely ν → ν + 1 πi (cid:90) dk u ( k ) ∂ k u † ( k ) , (A48)which is just Eq. (A26) specialized to n = 1. These results are consistent with the physicalmeaning of the Zak phase. The Zak phase of Eq. (A47) is just the center of the Wannierfunctions with the lattice constant normalized to be 2. For a periodic system, of coursethe center of the Wannier functions should be a position modulo the lattice constant. It isalso clear that in order to preserve inversion symmetry the center has to be concentratedat lattice sites or at the midpoints of lattice sites, namely it is an integer for the latticeconstant 2.If a gapped system has chiral symmetry, casting the Hamiltonian into the form of138 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 Eq. (A42), globally well-defined valence states are given readily as | k, b (cid:105) = 1 √ (cid:18) − v b q ( k ) v b (cid:19) (A49)where q = 1 , · · · , N , and v q is a N -vector with all entries being zero except the b th being1. In Eq. (A49), we have assumed (cid:101) H with q ( k ) ∈ U ( N ). Using this set of valence states,one may check that Eq. (A47) is equal to Eq. (A44). Thus, the Berry phase of Eq. (A47)is quantized into integers by chiral symmetry. Since the Zak phase is only a Z invariant,it counts only the parity of the topological invariant of Eq. (A44) for the symmetry class.The Zak phase can also be quantized by charge-conjugate or particle-hole symmetry.In momentum space charge-conjugate symmetry is represented by ˆ C = U C ˆ K ˆ I with U C being a unitary matrix, and is required to satisfy ˆ C = 1, which implies U C U ∗ C = 1 and U C = U tC . The momentum-space Hamiltonian is transformed under charge-conjugatesymmetry as U † C H ( − k ) U C = −H ∗ ( k ) . (A50)The charge-conjugate symmetric 1D gapped systems have a Z topological classifica-tion, and the Zak phase is just the topological invariant. Noting that the Bogoliubov-deGennes Hamiltonians of superconductors are naturally charge-conjugate symmetric, thusthe Zak phase is the topological invariant for 1D topological superconductors. Similarto time-reversal-symmetric topological insulators, the topological invariant can also beequivalently expressed as a product of Pfaffians at two inversion invariant points, whichis given by the Majorana representation of free fermionic systems in Ref. [469]. A.6 Skyrmions in two and three dimensions As discussed in Sec. A.1, a two-band model gives a field of unit vectors n ( k ) over the2D BZ, and the Chern number is just the winding number of the unit-vector field as amapping from T to S . If a similar unit-vector texture n ( x ) occurs in 2D real space R ,the winding number is also called the topological charge ν D w of skyrmions in the vectorfield n ( x ), which is explicitly given by ν D w = 14 π (cid:90) R d x n · ( ∂ x n × ∂ y n ) . (A51)Vectors at infinity are usually assumed to be oriented toward the same direction, namelythe plane R is effectively compactified to be S , and therefore the topological charge ofskyrmions is an integer.In some physical systems, there may exist a field of four-component unit vectors in3D space. For example, SU (2) order parameters after condensation, or two-componentnormalized quantum states. A group element U ( x ) of SU (2) can be expressed as U ( x ) =ˆ d ( x ) σ + i ˆ d i ( x ) σ i with d µ = ( ˆ d , ˆ d ) being a unit vector in 4D Euclidean space, and a two-component normalized quantum state ψ ( x ) can be represented as ψ ( x ) = ( ˆ d + i ˆ d , ˆ d + i ˆ d ) t . If vectors are constant at infinity, the 3D space R is topologically identical to S ,and the vector fields are mappings from S to S . Therefore, there exist 3D skyrmionsas counterpart of 2D ones can exist in such systems, whose topological charge is just thecorresponding winding number. For SU (2) field U ( x ), as afore-mentioned, the winding139 pril 3, 2019 Advances in Physics Manuscript˙AIP˙Final˙20190401 number is given by ν D w = − π (cid:90) d x (cid:15) µνλ tr U ( x ) ∂ µ U † ( x ) U ( x ) ∂ ν U † ( x ) U ( x ) ∂ λ U † ( x ) . (A52)Substituting U ( x ) = ˆ d ( x ) σ + i ˆ d i ( x ) σ i , the formula in terms of the unit vectors is ν D w = 12 π (cid:90) R d x (cid:15) µνλρ ˆ d µ ∂ ˆ d ν ∂ ˆ d λ ∂ ˆ d ρ . (A53)The notion of skyrmion can be readily generalized to any dimension n , where theskyrmion charge is just the winding number of the unit vector field d µ ( x ) as a mappingfrom S n to S n . The formula for topological charge is explicitly given as ν nD w = 1Ω n (cid:90) R n d n x (cid:15) µ µ ··· µ n ˆ d µ ∂ ˆ d µ · · · ∂ n ˆ d µ n , (A54)where Ω n is the geometric angle of ( n + 1) dimensional Euclidean space equal to2 π d/ / Γ( d/ d µ ( x ) ∈ S n . A.7 Hopf Invariant So far, we have mainly concerned with band theories of sufficiently many bands. Wenow consider a topological insulator, which has only two bands and occurs only in threedimensions. For a two-band insulator, the flattened Hamiltonian can always be writtenas Eq. (A9). Hence (cid:101) H ( k ) at each k can be regarded as a point on the unit sphere S , and (cid:101) H gives a mapping from the 3D momentum space to S . Because of the homotopy group π (S ) ∼ = Z, there exist (strong) 3D two-band topological insulators with Z classification,which is termed Hopf insulators. The topological invariant is called the Hopf invariant [66,67], and is given by ν H = − π (cid:90) T d k (cid:15) µνλ A µ ∂ ν A λ , (A55)where A µ = (cid:104)− , k | ∂ k µ |− , k (cid:105) is the Berry connection of the valence band defined inEq. (A3). It is worth noting that we have ignored all cases of weak topological insu-lators, i.e., the Chern number over any 2D sub BZ has been assumed to be zero, suchthat the valence wave function |− , k (cid:105) can be globally well-defined in the whole 3D BZ.We now develop another expression of the Hopf invariant, which also gives an expla-nation of the homotopy group π (S ) ∼ = Z and Eq. (A55). At each k , ˆ d i ( k ) σ i can beobtained from σ by a SU (2) rotation, namely (cid:101) H ( k ) = U ( k ) σ U − ( k ) , (A56)with U ( k ) ∈ SU (2). Then the valence eigenstates are |− , k (cid:105) = U ( k ) | ↓(cid:105) . Note that therotation to the z -axis, e − iσ φ/ , does not change the orientation of the state | ↓(cid:105) , namely U ( k ) and U ( k ) e − iσ φ ( k ) / give the same ˆ d ( k ). Due to the U (1) gauge freedom in thepresentation of ˆ d ( k ) by an element of SU (2), U ( k ) can be made globally well-definedin the whole BZ. 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