Topology-induced phase transitions in quantum spin Hall lattices
TTopology-induced phase transitions in quantum spin-Hall lattices
D. Bercioux ∗ Freiburg Institute for Advanced Studies, Albert-Ludwigs-Universit¨at, D-79104 Freiburg, Germany andPhysikalisches Institut, Albert-Ludwigs-Universit¨at, D-79104 Freiburg, Germany
N. Goldman
Center for Nonlinear Phenomena and Complex Systems - Universit ´ e Libre de Bruxelles (U.L.B.),Code Postal 231, Campus Plaine, B-1050 Brussels, Belgium
D. F. Urban
Physikalisches Institut, Albert-Ludwigs-Universit¨at, D-79104 Freiburg, Germany (Dated: November 3, 2018)Physical phenomena driven by topological properties, such as the quantum Hall effect, have theappealing feature to be robust with respect to external perturbations. Lately, a new class of materialshas emerged manifesting their topological properties at room temperature and without the need ofexternal magnetic fields. These topological insulators are band insulators with large spin-orbitinteractions and exhibit the quantum spin-Hall (QSH) effect. Here we investigate the transitionbetween QSH and normal insulating phases under topological deformations of a two-dimensionallattice. We demonstrate that the QSH phase present in the honeycomb lattice looses its robustnessas the occupancy of extra lattice sites is allowed. Furthermore, we propose a method for verifying ourpredictions with fermionic cold atoms in optical lattices. In this context, the spin-orbit interactionis engineered via an appropriate synthetic gauge field.
PACS numbers: 05.30.Fk,37.10.Jk, 67.85.Fg,73.43.-f
I. INTRODUCTION
In the last three decades the physics community hasbeen fascinated by topological states of quantum mat-ter, initiated by the discovery of the quantum Hall ef-fect (QHE) by von Klitzing in 1981 [1]. Recently it hasbeen discovered that a new class of materials, referredto as topological insulators , show robust conducting edgestates even in the absence of external magnetic fields [2–4]. These materials are two- or three-dimensional bandinsulators with a large spin-orbit interaction (SOI) andexhibit the so-called quantum spin-Hall (QSH) effect [5–8]. Here, spin-orbit effects play a role similar to the ex-ternal magnetic field in the QHE. However, contrary toan external magnetic field, SOIs conserve time-reversalsymmetry (TRS), implying that the edge states appear asKramers doublets or equivalently as counter-propagatingmodes along the system boundary carrying opposite spin.In two-dimensional (2D) systems a Z topological invari-ant ν specifies the robustness of these helical edge-statesin the presence of disorder and interactions [9]. While ν = 1 states a topologically protected ( i.e. stabilized)QSH phase, ν = 0 indicates that the edge states are un-stable with respect to impurity scattering therefore re-ducing the system to a normal insulator (NI).In this Article we explore the interplay between spa-tial lattice topology, which is characterized by the latticeconnectivity, and the topological order associated to the ∗ Electronic address: [email protected] ℓ ℓ xy AB v v v v AB H (a) (b) FIG. 1: (Color online). (a) HCL with lattice vectors v and v and two atoms in the unit cell. (b) T lattice with threeatoms in the unit cell. Rims A and B (open circle and square)have a lower connectivity of 3 compared to 6 for the hub site(filled circle). The T lattice can be viewed as two nestedHCLs (emphasized by the red and blue hexagon). quantum spin-Hall phases. The latter is characterizedby the Z topological invariant defined on a fibre bundleassociated to the wave-functions over the first Brillouinzone [2]. We show that spatial deformations — modi-fying the lattice topology — induce a change in the Z invariant, thus leading to a topological phase transitionfrom a QSH insulator to a NI.Specifically, we study the effect of adding an additionalsite to the unit cell of a honeycomb lattice (HCL) sub-jected to SOI. While this lattice deformation preservesthe relativistic behavior of the low-energy modes, weshow that the system undergoes a topological phase tran-sition that changes its topological class, labeled by the Z a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b index, and its transport properties. The possibility to in-vestigate this topological phase transition with ultra-coldatoms trapped in optical lattices is promising [10, 11]. Inthis context, cold-atom experiments are not only suitablefor engineering the QSH phase [12, 13] and to test its sta-bility with respect to controllable disorder [14], but alsoto induce lattice transformations. II. MODEL AND FORMALISM
Adding an extra site to the unit cell of a HCL givesrise to the so called T lattice, c.f. Fig. 1. The T latticenot only differs from the HCL by the number of inequiva-lent lattice sites but also by their respective coordinationnumbers: while the HCL (Fig. 1a) is characterized bytwo inequivalent lattice sites A and B, each with coordi-nation number 3, the T lattice (Fig. 1b) has a unit cellwith three inequivalent sites. Two of the latter (generallycalled rim A and B) are characterized by a coordinationnumber 3 while one site (referred to as hub
H) is con-nected to 6 nearest-neighbors [15–17].Both lattices are modeled by a tight-binding (TB)Hamiltonian H = t (cid:80) (cid:104) n,m (cid:105) α c † nα c mα with spin inde-pendent nearest-neighbor hopping amplitude t . Here, c † nα ( c nα ) is the creation (annihilation) operator for a par-ticle with spin direction α on the lattice site n . Regardingthe SOI, symmetries allow for two types of interactionsthat preserve TRS, namely the Rashba-type [2] and theHaldane-type [18]. The effects of the Rashba-type SOI onthe transport properties of the HCL and T lattice havebeen reported in Refs. [2, 20]. The Haldane-SOI, which isthe interaction of interest in the present study, is modeledvia a spin-dependent second-neighbor hopping term H SO = i t SO (cid:88) αβ (cid:88) (cid:104)(cid:104) n,m (cid:105)(cid:105) c † n,α ( d i × d j ) · σ αβ c m,β . (1)Here, the σ αβ are matrix elements of the Pauli-matrices σ with respect to the final and initial spin states α and β and d i/j are the two displacement vectors of thesecond-neighbor hopping process connecting sites n and m . Since in 2D lattices hopping is naturally restricted toin-plane processes, the SOI is effectively proportional to σ z .In the absence of SOI the energy spectra of the twolattices are characterized by two identical, electron-holesymmetric branches [2, 19]. Moreover, for the T lattice aunique non-dispersive band is present at the charge neu-trality point [19]. This is rooted in the lattice topology,which allows for insulating states with finite wavefunctionamplitudes on the rim sites and vanishing amplitudes onthe hubs. -4 -2 0 2 4-4-2024 k y ( k x =0) [ ! ] E ( k x , k y ) [ t ] k x k y E (a) (b) FIG. 2: (Color online). (a) Energy spectrum as a functionof the in-plane momentum ( k x , k y ) and for a fixed value ofthe spin-orbit interaction t SO = 0 . t . The first Brillouinzone and its six corners are indicated. (b) Cut of the energyspectrum at k x = 0 (shadowed plane in (a)). III. TRIVIAL
VS.
TOPOLOGICALINSULATORS
We start by analyzing the effect of finite SOI onthe energy spectrum of the infinite T lattice. Becauseof the unequal connectivity of rim and hub sites, theHamiltonian (1) effectively induces hopping between rimsites of the same kind while this is cancelled for hubsites. The spectrum is obtained by exact diagonaliza-tion (cf. App. A) and the result is shown in Fig. 2. Dueto its topological origin the non-dispersive band at thecharge neutrality point is not affected by SOI. On thecontrary, two bulk energy gaps open between the non-dispersive and the electron/hole branches, respectively.This property can be directly deduced from the single-valley long-wavelength approximation of the Hamiltonian(cf. App. B). We find that the general form of this rela-tivistic Hamiltonian — describing both the HCL and the T lattice — reads H = v F Σ · p ⊗ I + ∆ SO Σ z ⊗ σ z . (2)Here v F is the Fermi velocity, p = − i( ∂ x , ∂ y ,
0) themomentum operator ( (cid:126) = 1) in the lattice xy –plane,∆ SO is the spin-orbit coupling constant, and I is thetwo-dimensional identity matrix. The pseudo-spin op-erator Σ for the HCL is given by the Pauli matrices Σ = ( σ x , σ y , σ z ), thereby describing a S = 1 / T the pseudo-spin k x [ ! ] k x [ ! ] E ( k x ) [ t ] (a) (b) FIG. 3: (Color online). One-dimensional energy bands fora strip of (a) HCL and (b) T lattice with t SO = 0 . t .The bands crossing the gap (blue lines) are spin filtered edgestates. operator Σ = (Σ x , Σ y , Σ z ) is given by [19]Σ x = 1 √ , Σ y = 1 √ − i 0i 0 − i0 i 0 , (3)Σ z = − . These matrices satisfy angular momentum commutationrelations and describe an S = 1 pseudo-spin (cf. App. B).A direct way of visualizing the SOI effects is to considera finite piece of the lattice. The associated TB Hamil-tonian can then be diagonalized with periodic boundaryconditions imposed along one of the spatial directions.The HCL with finite SOI is characterized by the QSHphase: for each energy value within the bulk energy-gap exists a single time-reversed pair of eigenstates oneach edge of the lattice (see Fig. 3a) — these form twoKramers doublets, one on each edge. The conservationof TRS prevents the mixing of this couple of states dueto small external perturbations and scattering from dis-order [2]. The QSH phase is intrinsic to the bulk energyspectrum and can be validated by topological constants.Two important topological invariants are the Z index ν [2] and the spin Chern number n σ [21, 22]. In theabsence of spin-mixing perturbations, both are relatedby ν = n σ mod 2, where n σ = ( N ↑ − N ↓ ) / N ↑ , ↓ represent the Chern numbers associated to the individ-ual spins. For the HCL it has been shown that ν = 1therefore classifying it as a topological insulator. On theother hand it was found that SOI of Rashba-type doesnot induce this kind of edge-states [2].The physical picture is more involved in the case ofa T lattice with SOI. Here gapless edge states are alsofound for energies located within the gaps between theelectron/hole band and the non-dispersive band, respec-tively (see Fig. 3b). However, contrary to the HCL case,we find two distinct crossing points for these edge states.For each energy value inside the bulk gaps two couples oftime-reversed Kramers doublets are present, which there-fore give rise to four pairs of states along the two edges.As a result, even if SOI allow for edge states within the Lattice Spin Valley Pseudo RealHCL 1/2 1/2 1/2 Topological insulator1/2 1 1/2 Trivial insulatorKagome 1/2 1/2 1/2 Topological insulatorBilayer HCL 1/2 1 1/2 Trivial insulatorLieb - 1 1/2 Topological insulator
TABLE I: (Color online). In the table the valley-spin isassociated with the topology of the Bravais lattice and thepseudo-spin with the number of lattice sites in the unit cell. bulk energy gap, these are not protected against disor-der by TRS — a scattering potential can couple a left-mover into a right-mover and vice versa . This is con-firmed by the topological invariants (cf. App. C): we findthat the Z -index vanishes and that the spin Chern num-ber n σ = 2. For this reason the T lattice is topologicallytrivial.A transition from the HCL to the T lattice is accom-plished by varying the hopping parameter between huband rim B from 0 to t . The remarkable result is thateven for an infinitesimal coupling, the system is in thetrivial insulating phase. Therefore, we conclude that thedifferent behavior of the HCL and the T lattice with re-spect to the SOI arises from the different topologies ofthe two systems. The T lattice can be viewed as twonested HCLs (see Fig. 1b). This supports the conjecturethat two coupled topological insulators behave as a topo-logically trivial insulator [22, 23]. We can interpret theeffect of lattice topology also from a different perspective.Although both lattices allow for a description in termsof a relativistic Hamiltonian (2) in the long-wavelengthapproximation, only the HCL that is associated with apseudo-spin 1/2 is a topological insulator whereas the T lattice with pseudo-spin 1 is not. Nevertheless, theexistence of other topological insulators associated withpseudo-spin 1 [24] suggests that the key factor is not thedimension of the pseudo-spin but rather the total spin ofthe systems under consideration, c.f. Tab. I. In terms ofthe relativistic approximation (2) the total spin is givenby the combination of the valley-spin, accounting for thetopology of the Bravais lattice, the pseudo-spin, associ-ated with the number of lattice sites in the unit cell, andthe real fermion spin. In the case of the HCL we have tocombine three half-integer spins resulting in a half-integertotal spin. On the contrary, for the T lattice we obtainan integer spin by combining two half-integer spins withan integer one. This conjecture is further supported bythe fact that both the kagome lattice [25] and the Lieblattice [24] are non-trivial topological insulators, whilethe bilayer graphene is trivial [22, 26]. -3 -2 -1 0 1 2 301 (cid:113) E F -2 0 201 (cid:113) E F -4 4ab Honeycomb lattice -3 -2 -1 0 1 2 301 (cid:105) E F -4 -2 0 2 401 (cid:105) E F DiracDirac cd HCL, (cid:113) =1/21 ! lattice "!(cid:113) =1/11/ t / t / t / t (a)(b) (c)(d) RelativisticRelativistic
FIG. 4: (Color online). (a) Phase diagrams in the Fermienergy-phase ( E F − φ )–plane for the honeycomb lattice. Thedark regions represent the energy spectrum. The gaps arecolored according to the Z index: orange [resp. green] gapscorrespond to ν = 1 [resp. ν = 0]. (b) Phase diagrams inthe ( E F − φ )–plane for the T lattice. (c) Cut of the phasediagram for the HCL in (a) for φ = 1 /
21. The orange barsdenote the energy intervals for which ν = 1. (d) Cut of thephase diagram for the T in (b) for φ = 1 /
11. Within therelativistic regime limited by E VHS = 1 [resp. E VHS ≈ . bulkgaps for the HCL [resp. T lattice]. IV. OPTICAL LATTICE VERIFICATION
Several proposals for realizing optical lattices resem-bling the HCL and the T lattice have been put forwardrecently. The HCL lattice can be obtained by three copla-nar plane waves with the same frequencies and a relativeangle of 120 ◦ [27]. On the other hand, the T lattice canbe realized considering three pairs of counter-propagatingplane waves with a relative angle of 120 ◦ and a relativerotation of the polarization plane [28]. Both set-ups sharethe same spatial configuration of the laser beams, there-fore an adequate modulation of the laser intensity cantransform one lattice into the other. The TB Hamilto-nian and their long wave length approximations (2) arevalid for optical lattices populated by fermionic atoms, e.g. K or Li [10].Laser-assisted tunneling is a standard technique for im-plementing gauge fields for cold atoms, however a directimplementation of the Haldane-type SOI (1) is extremelycomplex. On the other hand, the QSH effect in 2D canbe interpreted as two standard QHEs acting on each spinseparately. Therefore, an alternative approach is to de-sign an effective vector potential A = ( A x , A y , A z ) × σ z that preserves TRS and opens a bulk energy gap. Re-cently, this type of synthetic SU(2) field — practicallyrealizable with state of the art techniques — has beenproposed for a fermionic atom-chip forming a square lat- tice [12]. Here we propose to subject the atoms to asynthetic SU(2) gauge field A = (cid:18) , πφ x S , (cid:19) × σ z . (4)that is associated with an opposite magnetic field forspin-up and spin-down electrons. Here S is the area ofthe lattice plaquette and φ is the number of magneticflux quanta per plaquette felt by each spin, i.e. , ± φ forspin-up and spin-down, respectively (cf. App. D). Notethat the gauge field (4) is space-dependent while preserv-ing TR symmetry [29]. The TB Hamiltonian includingthis gauge field for both the T lattice and HCL in thelong-wavelength approximation reads H = v F Σ · p ⊗ I + ∆( x )Σ y ⊗ σ z . (5)Contrary to the Hamiltonian (2), the SOI term con-tains Σ y and the coupling ∆( x ) = 2 πφ v F S − x is space-dependent. While the latter gives rise to a complex en-ergy spectrum as a function of φ , we recover the featuresinduced by the Haldane-type SOI in the “low-flux” limit φ (cid:28)
1. The Z -phase diagrams for the HCL and T lat-tice subject to the synthetic gauge field (4) are shownin Figs. 4 (a) and (b), respectively. For arbitrary valuesof φ , the gauge field leads to complex fractal structuresfeaturing alternations of non-trivial (orange) and trivial(green) phases. However, we note that the HCL has agreater tendency to open QSH gaps than the T lattice.The relativistic regime — described by Eq. (5) — is re-covered in the low-flux limit φ (cid:28) ± E VHS inthe density of states. In Figs. 4 (c) and (d), we showthe Z index ν as a function of the Fermi energy E F for φ (cid:28)
1. In the relativistic regime (yellow area), the HCLis characterized by QSH gaps with ν = 1, while the T lat-tice is characterized by trivial insulating gaps with ν = 0.In this energy range we observe that the synthetic gaugefield (4) reproduces the effects induced by the Haldane-SOI described above. Besides, we note that outside of therelativistic regimes, the HCL and T lattice behave verysimilarly as they both feature alternation of non-trivialand trivial phases as a function of E F . This importantresult emphasizes that the optical lattice topology dra-matically influences the nature of the insulating phaseswithin the relativistic regime. Consequently, engineeredgauge fields offer the unique possibility to explore Z -phase transitions as the optical-lattice topology is modi-fied.Different schemes are available for distinguishing theQSH and normal insulating phases in a cold-atom ex-periment [12]. In absence of spin-mixing perturbationsthe spin is conserved and the Z topological constantreduces to ν = | N ↑ | mod 2 because the individual spin-components are associated to N ↑ = − N ↓ . Therefore ν can be determined through density measurements andby applying the Streda formula [31]. Alternatively, thepresence of helical edge-states can be also probed throughlight-scattering methods by exploiting the method re-ported in Ref. [32]. V. CONCLUSIONS
We have studied the effect of Haldane-type SOI ontwo interrelated two-dimensional lattices: the honey-comb and the T lattices. Both can be transformed intoeach other by selectively switching on/off specific latticesites. In absence of SOI, the main effect of the transitionfrom HCL to T is the appearance of a localized bandat the charge neutrality point. For finite SOI though, atopological phase transition occurs that corresponds tothe destruction of the QSH phase which is present in theHCL but not allowed in the T . Since the occurrenceof the QSH phase is related to the lattice topology, itis already destroyed by adding an infinitesimal weaklycoupled lattice site to the HCL which gives rise to the T lattice. Finally, we have proposed a method for im-plementing this topological phase transition for fermioniccold atoms trapped in optical lattices. In this context,the QSH phases are obtained by synthesizing a gaugefield reproducing the effects of the Haldane-type SOI forelectrons. Acknowledgments
We are in debt with I. Cirac, H. Grabert, M. Lewen-stein, and M. Rizzi for useful discussions. DB is sup-ported by the Excellence Initiative of the German Federaland State Governments. NG thanks the F.R.S-F.N.R.S(Belgium) for financial support.
Appendix A: The bulk energy spectrum
The energy spectrum of the T lattice with Haldane-type [18] SOI is obtained by diagonalization of the tight-binding Hamiltonian H = H + H SO . Here the first termis a nearest-neighbor hopping tight-binding Hamiltonianand includes the spin independent nearest-neighbor hop-ping amplitude t . The second term — defined in Eq. (1)— describes the Haldane-type SOI. Because of transla-tional invariance, this Hamiltonian can be diagonalizedin reciprocal space. The evaluation of the contributiondue to the Haldane-SOI requires some care because ofthe different connectivity of hubs and rims. A second-next neighbor hopping process always interconnects sitesof the same species (either rim A, rim B or hub). Be-cause of the two possible distinct processes that connecttwo given hubs, the contributions arising from these twopaths cancel. Therefore, the Haldane-SOI effectively onlycouples the rims of each species but not the hubs. The Hamiltonian now can be written in matrix form H [ k ] = t A ∗ A A A ∗ ⊗ I + t SO B −B ⊗ σ z (A1)where we have defined the coefficients A = 1 + e i k · v + e i k · v , B = 2 { sin( k · v ) − sin( k · v ) + sin[ k · ( v + v )] } . Here the translational vectors of the T lattice are de-fined as v = (3 / −√ / (cid:96) and v = (3 / √ / (cid:96) .Equation (A1) corresponds to a six component spinorialwave function Ψ k ( r ) = ( ψ A , ψ H , ψ B ) ⊗ σ , where we haveintroduced the components of the wave function on thetwo rims and the hub sites and the spin contribution σ .Diagonalization of H [ k ] yields the energy spectrum forthe T lattice in presence of Haldane-SOI ε ( k ) =0 (A2a) ε ± ( k ) = ± (cid:110) (cid:2) t + t ) + 2 t (cos[ k · v ] (A2b)+ cos[ k · ( v − v )] + cos[ k · v ]) − t (cos[2 k · v ] − k · v ]+ cos[2 k · v ] + cos[2 k · ( v − v )]+ 2 cos[2 k · v − k · v ] + 4(sin[ k · v ]+ sin[ k · ( v − v )] sin[ k · v ])) (cid:3)(cid:111) / where k is the two-dimensional momentum vector andeach band is two-fold degenerate. Appendix B: Long-wavelength approximation
For momenta close to the two independent K -points— K = 2 π/ (cid:96) (1 , −√ /
3) and K (cid:48) = 2 π/ (cid:96) (1 , + √ / H + H SO . This approximation consists in expressingthe spatial part of the wave function as the product of afast-varying part times a slow-varying part. In absence ofperturbations that induce a mixing of the two K points,we consider a single K –point — namely K — and thewave function can be written asΨ α ( R α ) ∝ e i k · R α F K α ( R α ) . (B1)Here α ∈ { A,B,H } and R α is the lattice site coordinate.We substitute this wave-function into the Schr¨odingerequation and use the expansion F K α ( R α (cid:48) ± d j ) (cid:39) F K α ( R α (cid:48) ) ± d j · ∇ r F K α ( r ) (cid:12)(cid:12) r = R α (cid:48) + O ( | d | ) . This approximation is valid for energies close to thecharge neutrality point where the system energy is al-most zero. In this case the associated Fermi wavelength λ F ∼ / | (cid:15) | is bigger than the modulus of the displace-ment vectors λ F (cid:29) | d j | . Collecting all the terms weare left with the expression (2) where v F = 3 (cid:96) t/ SO = √ t SO .The pseudo-spin matrices Σ have been defined in (3).These matrices fulfill the algebra of the angular momen-tum [Σ i , Σ j ] = i (cid:15) ijk Σ k and form a 3-dimensional repre-sentation of SU(2). However, contrary to the Pauli matri-ces, they do not form a Clifford algebra, i.e. , { Σ i , Σ j } (cid:54) =2 δ i,j I . By introducing a rotation operator around the z axis defined by D z ( φ ) = exp ( − iΣ z φ ), a generic state | α (cid:105) is transformed into itself by D z (2 π ) | α (cid:105) → | α (cid:105) , implyingthat the pseudo-spin Σ describes an integer spin S = 1. Appendix C: Topological properties of the T latticewith Haldane-SOI The Z topological invariant characterizes the changein the Kramers degeneracy of a time-reversal symmet-ric system [2]. It can be deduced from the topologi-cal structure of the Bloch wave functions of the bulkcrystal in the first Brillouin zone. Here, there are spe-cial points k = M i that are time-reversal invariant andsatisfy − M i = M i + G for a reciprocal-lattice vector G . For the T lattice we have four of these points.These are defined by M i =( n ,n ) = 1 / n g + n g )with n , n ∈ { , ± } , where we have introduced thereciprocal lattice vectors g = 2 π/(cid:96) (1 / −√ /
3) and g = 2 π/(cid:96) (1 /
3; + √ / T = exp(i πσ y / K with the op-erator of complex conjugation K , the four M points arefound to fulfill H (M i ) = T H (M i ) T − .In the case of the T lattice the system is invariant un-der parity operation, which corresponds to an exchangeof rim A with rim B and is expressed by the operator P = . (C1)This implies that [ H (M i ) , P ] = 0 for all four time-reversalinvariant momenta. According to Fu and Kane [23] theZ topological invariant ν is related to the parity eigen-values ξ m (M i ) of the 2 m -th occupied energy bands atthe four time-reversal invariant points M i through therelation ( − ν = (cid:89) i N (cid:89) m =1 ξ m (M i ) . (C2)We have evaluated the eigenstates of H in the points M i analytically and therefore determined straightforwardlythe parity eigenvalues of the occupied bands. At filling1 / / number ν = 0. A BH(a) v (b) FIG. 5: (Color online). Peierls phases generating the SU(2)gauge field for (a) the honeycomb lattice and (b) in the T lat-tice. Appendix D: Subjecting the Honeycomb and T lattices to an SU(2) gauge field On a lattice, the gauge field A enters the formalismthrough the Peierls (or Berry’s) phases: as a particlehops from a site i to a nearest-neighboring site j , thewave function acquires a non-trivial phase e i θ ij , where θ ij = (cid:82) ji A · d l . The set of Peierls phases { θ ij } , definedon the whole lattice, therefore witnesses the presence ofa specific gauge field A . The physical observables asso-ciated to the gauge field, or equivalently to the set { θ ij } ,are given by the loop operators defined on each plaquette (cid:3) : W ( (cid:3) ) = e i θ e i θ . . . e i θ N , (D1)where the plaquette (cid:3) is delimited by the sites { , , . . . , N } . In the case of an Abelian or U(1) gaugefield, the loop operator is related to the magnetic fluxΦ( (cid:3) ) penetrating the plaquette through the relation W ( (cid:3) ) = e iΦ( (cid:3) ) .In this work, we considered a gauge field of the follow-ing form A = 1 S (cid:0) , πφx, (cid:1) ⊗ σ z = (cid:18) A ↑ A ↓ (cid:19) , (D2)that does not couple the two spin species together. Henceup and down spins feel individual and opposite magneticfluxesΦ ↑ ( (cid:3) ) = (cid:90) (cid:3) ∇ × A ↑ = − (cid:90) (cid:3) ∇ × A ↓ = − Φ ↓ ( (cid:3) ) (D3)= 2 πφ, that are uniform. The loop operators corresponding tothe gauge field (D2) thus yield W = e i2 πφσ z within everyplaquette of the lattice. The helical edge-states, hallmarksignature of the QSH phase, are a direct consequence ofthe double-Hall system produced by the specific gaugefield (D2) which preserves time-reversal symmetry.The gauge field (D2) can be engineered in an optical-lattice experiment by inducing the appropriate set ofPeierls phases { θ ij } such that W = e i θ e i θ . . . e i θ N = e i2 πφσ z , (D4)for all the plaquettes constituting the honeycomb and T lattices. Note that the Peierls phases can be generatedthrough different schemes, based on the Raman transi-tions produced by additional lasers or fields along theappropriate links [12, 33, 34]. An adequate set of Peierlsphases { θ ij } satisfying Eq. (D4), and thus reproducingthe effects of the gauge field (D2) on the honeycomb and T lattices, are represented in Fig. 5.In the case of the HCL, a single space-dependentPeierls phase θ ( m ) = 2 πφmσ z is necessary, as illustratedin Fig. 5a. The single-particle Schr¨odinger equation thentakes the form of two coupled Harper difference equa-tions for the wave-function Ψ = ( ψ A , ψ B ) defined at thesite ( m, n ), ψ A ( m, n ) = ψ B ( m, n −
1) + ψ B ( m + 1 , n −
1) (D5a)+ e i2 πφmσ z ψ B ( m, n ) ,ψ B ( m, n ) = ψ A ( m, n + 1) + ψ A ( m − , n + 1) (D5b)+ e − i2 πφmσ z ψ A ( m, n ) , and the loop operator indeed equals W = e i2 πφσ z ,within all the hexagonal plaquettes.In the case of the T lattice, four space-dependentPeierls phases are required, θ ( m ) = 2 πφ (3 m − / σ z , (D6a) θ ( m ) = 2 πφ (3 m − σ z , (D6b) θ ( m ) = 2 πφ ( − m − / σ z , (D6c) θ ( m ) = 2 πφ ( − m − σ z , (D6d) as represented in Fig. 5b. The Harper equations for thewave-function Ψ = ( ψ A , ψ B , ψ H ) defined at the site ( m, n )yields ψ A ( m, n ) =e − i θ ( m ) ψ H ( m, n ) + ψ H ( m, n −
1) (D7a)+ e − i θ ( m +1) ψ H ( m + 1 , n − ,ψ B ( m, n ) =e − i θ ( m ) ψ H ( m, n ) + ψ H ( m, n + 1) (D7b)+ e − i θ ( m − ψ H ( m − , n + 1) ,ψ H ( m, n ) =e i θ ( m ) ψ A ( m, n ) + ψ B ( m, n −
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