Fractional Quantum Hall Physics in Topological Flat Bands
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Fractional Quantum Hall Physics in Topological Flat Bands
S. A. Parameswaran
Department of Physics, University of California, Berkeley, California 94720
R. Roy
Department of Physics and Astronomy, University of California, Los Angeles, California 90095-1547
S. L. Sondhi
Department of Physics, Princeton University, Princeton, NJ 08544
Abstract
We present a pedagogical review of the physics of fractional Chern insulators with a particular focus on the connectionto the fractional quantum Hall effect. While the latter conventionally arises in semiconductor heterostructures at lowtemperatures and in high magnetic fields, interacting Chern insulators at fractional band filling may host phases withthe same topological properties, but stabilized at the lattice scale, potentially leading to high-temperature topologicalorder. We discuss the construction of topological flat band models, provide a survey of numerical results, and establishthe connection between the Chern band and the continuum Landau problem. We then briefly summarize variousaspects of Chern band physics that have no natural continuum analogs, before turning to a discussion of possibleexperimental realizations. We close with a survey of future directions and open problems, as well as a discussion ofextensions of these ideas to higher dimensions and to other topological phases.
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1. Introduction
The application of a transverse magnetic field to a two-dimensional electron gas breaks time-reversalsymmetry and gives rise to a highly degenerate single-particle spectrum in the absence of electron-electroninterections: the celebrated Landau levels. When the electronic density is commensurate with the macroscopicdegeneracy of a Landau level so that an integer number of such levels are occupied, the resulting insulatingphase exhibits the integer quantum Hall effect (IQHE), with spectacular consequences for charge transport:the Hall resistance is quantized to be an integer multiple of h/e , while the longitudinal resistance vanishes Email addresses: [email protected],
Corresponding Author (S. A. Parameswaran), [email protected] (R. Roy), [email protected] (S. L. Sondhi).
Preprint submitted to CR Physique February 28, 2013 a r X i v : . [ c ond - m a t . s t r- e l ] F e b p/q < analytic functionsof z = x + iy . The built-in analyticity of the LLL wave functions and the structure of the topologicalquantum field theory that describes their topological order has allowed the full panoply of techniques ofconformal field theory to be brought to bear in constructing variational wave functions for various QHE‘filling fractions’, ν = p/q [3,6,7,8]. Various effective field theory techniques, such as statistical transmutationby flux attachment, [9,10] also rely on the simple parabolic dispersion assumed for the electrons.Can the QHE be detached from this idealized limit? There are two independent idealizations that mustbe addressed. First, the solid could affect electronic motion more seriously so that the effective mass approx-imation gives way to the formation of energy bands; does the QHE survive in this case? Second, a uniformmagnetic field both breaks time reversal symmetry but also affects electron dynamics at long wavelengthsin a decidedly unusual fashion: the formation of Landau levels. Are both essential for the QHE?The answer to both questions is known for the I(nteger)QHE, which is mostly a single-particle phe-nomenon. In a landmark paper in 1982, Thouless, Kohmoto, Nightingale, and den Nijs [11] analyzed theuniform-field Hall effect in a strong periodic potential that was known to lead to an intricate spectrum, theso-called Hofstadter butterfly; they showed that it gives rise to an integer QHE under certain conditions,i.e., whenever the chemical potential lies in a gap. Indeed, the Hall conductance was shown to map to atopological invariant associated with filled bands — the (first) Chern number. Six years later, in anotherstriking development, Haldane [12] answered the second question, showing by an explicit construction ofa tight-binding model on a honeycomb lattice that a quantized Hall conductance can arise from a fullyfilled band even in the absence of a net magnetic field. In his model, time-reversal symmetry is broken by aspatially inhomogeneous magnetic field with zero average, and the Hall conductance again equals the Chernnumber of the band.Of course, the next logical question is: can the FQHE, canonically a property of interacting electronsin a fractionally filled Landau level, also be separated from the weak lattice and uniform magnetic fieldlimit? The topological flat band models attempt to ask this question in a specific fashion: if it is true forindependent electrons that a filled Chern band is equivalent to a filled Landau level, then is it also truefor interacting electrons that a fractionally filled Chern band is equivalent to a fractionally filled Landaulevel? Microscopically, these models proceed to construct a (nearly) degenerate low-energy subspace – the‘flat band’. While an exact degeneracy requires electron hopping over arbitrarly large distances, the hoppingamplitudes decay exponentially and thus a relatively flat band can be produced by keeping a small set ofhopping amplitudes. The suitability of a band for realizing correlated states is na¨ıvely quantified by itsflatness parameter: the ratio of the band gap (which sets a bound on the strength of the interactions onecan safely include) to the bandwidth – although, as we will see, this is only one criterion for a ‘good’ Chernband.To understand the significance of a lattice realization of the FQHE it is useful to recall that while intheoretical terms the setting in which the FQHE was discovered is quite simple, experiments must go togreat lengths to achieve a limit where this simplicity emerges. First, a variety of growth techniques must beemployed to confine electrons in a two-dimensional plane,while simultaneously maintaining a high electronmobility – the latter restriction because too much disorder destroys the FQHE. Second, an extremely strong2agnetic field – of the order of tens of tesla – must be applied for Landau quantization to be appreciable.Finally, the systems must be cooled to below at least 10 K to reach the energy scales at which FQH physicsis manifest. All in all, this involves a tour de force in experimental technique, and reliably reproducing thefull sequence of FQH states remains a significant challenge even three decades after the original discovery.In contrast, in a lattice model, the characteristic energy scales can be in the tens or even hundreds of kelvin(assuming a lattice energy of a few meV) and are naturally in the limit where each unit cell sees a largeeffective magnetic field. Therefore such systems may realize robust phases that are concomitantly insensitiveto disorder and thermal fluctuations. The possibility of stabilizing exotic topological phases outside of adilution fridge and without a superconducting magnet, and the avenues it opens for further experimentalprobes of these phases, is among the primary motivations of the activity in this field. On a more theoreticalfront, the interplay of lattice symmetries with topological order leads to new physics, unique to the latticerealizations of the FQHE.The recent flurry of interest in the FQHE in topological flat bands began with the work of three groups[13,14,15]; several authors have since constructed models with nearly flat (non-dispersing) Chern bandswhich allow interactions to dominate at partial fillings while leaving the gap to neighboring bands open.Initial papers [15,16,17] reported evidence for FQH states at ν = 1 /
3, 1 / /
2, in finite-sizestudies of short-ranged interactions projected to these bands. A detailed study of properties of ground stateentanglement [18] allows a rationalization of some aspects of these results in terms of a generalized Pauliprinciple familiar from the LLL problem. The connection to the FQHE was made more explicit by Qi [19]who gave a fairly general recipe for translating familiar model wave functions and Hamiltonians from theLLL to Chern bands on cylinders by elegantly mapping Landau gauge eigenfunctions to particular Wannierfunctions. Finally, work by the present authors [20,21] gave a mapping between Hamiltonians in the LLL andthose in the Chern band by studying the correspondence of the operator algebras that arise in the two cases.The outcome of these three early approaches is to identify three desiderata for a Chern band: (i) that it beenergetically flat with a large gap and thus a large flatness parameter, so that the low-energy descriptionmay be approximated by projecting into the band (ii) that it be
Berry flat , i.e. have near-uniform Berrycurvature, so that the algebra of densities projected to the Chern band resembles that of similar operatorsin the LLL and (iii) that it satisfy a set of conditions on the Fubini-Study metric of band eigenstates thatquantify the connection to the LLL. Building on this, a reformulation of the Hamiltonian theory of theFQHE has been applied to study Chern bands [22,23].This is as good a place as any to note that on a lattice the distinction between having a net magnetic fieldand not having it at all is not as sharp as it may seem. Essentially, it is always possible to stick a full fluxquantum through some subset of loops on the lattice to shift the average magnetic field without affectingthe actual physics. From this perspective, the physics in these flat-band models has a family resemblance toearlier studies of lattice versions of the FQHE [24,25] with uniform magnetic fields. In this earlier work, theauthors studied a fixed filling factor while varying the flux per plaquette from small values and large unitcells, where the standard Landau level description holds, to somewhat larger flux values and smaller unitcells, where that description broke down. As they were able to change this parameter without any evidenceof encountering a phase transition, the latter limit constituted an observation of the FQHE in the presence ofstrong lattice effects. As we demonstrate below, a more analytic approach clarifies the equivalence betweenthis earlier Hofstadter and the current Haldane versions of the FQHE. Furthermore, this connection permitsthe construction of closed algebra of one-body operators [20,21,22,23] that gives a Chern band realizationof the Girvin-Macdonald-Platzmann algebra [26] of the LLL.In addition, several authors have examined aspects of Chern band physics where the presence of the latticeintroduces an additional layer of richness and complexity. We discuss two examples in this review. The firstof these is the intriguing question of the transition from a kinetic-energy dominated regime where the inter-particle interactions are weak and the system is in a gapless metallic phase (superfluid, if bosonic) to thestrong-coupling regime where the flat band FQHE sets in.
Continuous bandwidth-tuned transitions betweenthese two limits provide instances of topological phase transitions [27,28], which we discuss briefly below.A second example of the reappearance of lattice physics is the case of bands with higher Chern number.Closely related to ‘bilayer’ quantum Hall states in a Landau level, higher Chern bands both provide a simpleroute to non-abelian FQH states [29]. In addition, the lattice physics reappears via a nontrivial interplay of3rystalline point defects with the topological order, endowing them with a nontrivial ‘quantum dimension.’At the time of writing, FQH physics in Chern bands is numerically well-tested, with the proposed can-didates having passed a host of detailed checks by various groups. However, an experimental realization ofa fractional Chern band remains elusive, although several candidates have been suggested. We will discussa few promising ones below: these include solid-state realizations in oxide interfaces [30], as well as opticallattice examples both with short-range [31] as well as dipolar interactions [32,33].1.1.
This Review
Before proceeding, we list a few disclaimers as to the scope of this review. As with any rapidly progressingfield, there is a risk that some of what we discuss will be out of date by the time of publication, or thatcurrently active directions fail to realize their early promise. To avoid this problem of obsolescence and inthe interests of pedagogy our primary focus is on the case where the characterization at the present timeis most complete: Chern bands in two dimensions. We will eschew a discussion of other topological bands— such as those relevant to fractionalized analogs of time reversal invariant Z topological insulators —and higher dimensions. The bulk of the review will focus on bands with Chern number C = 1, althoughwe will briefly discuss bands with C = 2. We will also restrict our detailed discussion to the early work onthe subject, and devote a significant fraction of this review to elucidating the relationship between FQHEstates realized in the continuum, and their putative analogs in Chern bands. We will attempt to present abroad summary of the subsequent direction of the field, with pointers to the literature for the interestedreader. We will also briefly discuss new aspects of FQHE physics unique to Chern bands, as well as potentialexperimental realizations. Also, while we describe the important numerical results, we will not discuss thespecifics of the numerics in this review, referring the reader to the original literature for further details. Ourintention is that by the end of this article, a novice to the subject will emerge familiar with the broaderaspects of fractionally filled topological bands, and equipped to tackle the (rapidly growing) literature; wetrust that the reader will not find our approach too idiosyncratic.
2. Survey of Models
We begin with a brief introduction to Chern insulators in general and flat band models in particular,that will serve both to fix notation and provide background for the remainder of this review. Following ageneral discussion of tight-binding models without interactions, we proceed to the question of interacting flatband models where we review the numerical results that initiated the systematic study of fractional Cherninsulators. We also briefly discuss measures such as ground state entanglement, that can be unambiguouslyused to identify the topologically ordered phases in these models.2.1.
Lattice Models of Chern Insulators
Consider a tight-binding lattice model of the form H = (cid:88) i,j,a,b t abij c † i,a c j,b , (1)where the { t abij } are the (in general complex) hopping matrix elements, the i, j label Bravais lattice sites,and a, b = 1 , , . . . , N are internal indices that label different orbitals or sites within a unit cell. InvokingBloch’s theorem and working in momentum space, we may write the Hamiltonian as H = (cid:88) k ,a,b c † k ,a h ab ( k ) c k ,b (2)where a, b are indices that label sites within a unit cell, and k is the crystal momentum restricted to thefirst Brillouin zone (BZ). (Here and below, we will explicitly indicate when repeated indices are summed4ver, and unless otherwise specified will work in two dimensions.) The solution of the N × N eigenvalueproblem (cid:80) b h ab ( k ) u αb ( k ) = (cid:15) α ( k ) u αa ( k ) defines the Bloch bands (cid:15) α ( k ) and Bloch states u αa ( k ). We will takethe corresponding eigenvectors to be normalized, (cid:80) a | u αa ( k ) | = 1. The corresponding eigenstates are givenby | k , α (cid:105) = γ † k ,α | (cid:105) ≡ (cid:88) a u αa ( k ) c † k ,a | (cid:105) (3)and in terms of the operators γ † k ,α we have H = (cid:80) N α =1 (cid:15) α ( k ) γ † k ,α γ k ,α .The Chern number of a given band α is a topological invariant which can be defined only if the band isisolated from all other bands, and is computed as C α = 12 π (cid:90) BZ d k B α ( k ) . (4)Here, B α ( k ) is the Chern flux density (Berry curvature), defined as the curl of the Berry connection (Berrygauge potential), B α ( k ) = ∇ k × A α ( k ). In terms of the Bloch states, we have A α ( k ) = i N (cid:88) b =1 u α ∗ b ( k ) ∇ k u αb ( k ) . (5)A filled band with Chern number C α yields a Hall conductance σ H = C α e /h regardless of whether it arisesin a system with a net magnetic field [11] (“Hofstadter band”) or zero net magnetic field [12] (“Haldaneband”). We shall refer to both as Chern bands. In addition, we will assume that we are considering bandswith C α = 1 unless otherwise specified.2.2. Engineering Flat Bands
A Chern band shares one important feature with a Landau level: a nonzero Chern number. A Landaulevel has the additional feature that it is an exactly degenerate manifold, so that the only energy scalewithin a fractionally filled Landau level is provided by the interparticle interactions. In a typical Chernband, however, the kinetic energy from the band dispersion is a competing energy scale. In order to increasethe efficacy of interactions the kinetic energy can be ‘quenched’ by flattening the dispersion, as was notedby several authors; it is important that the single-particle Hamiltonian remain local even after the band-flattening procedure. Consider a lattice model of the form above, and let us assume the lowest band hasChern number 1 and gap ∆ = min( (cid:15) ( k )) − max( (cid:15) ( k )). Simply performing the transformation h ab ( k ) → ˜ h ab ( k ) = h ab ( k ) /(cid:15) ( k ) (6)and Fourier transforming to return to real space defines a tight-binding model in which the hopping matrixelements decay asymptotically as ˜ t abij ∝ e −| i − j | / ∆ and the lowest band is flat. Thus, as a point of principle, any energy band that is gapped away from all other bands can be made flat, at the cost of adding exponentiallydecaying longer-range hoppings; while the resulting tight-binding model is not strictly local (in the sense ofhaving matrix elements with compact support), most arguments based on topological order remain valid forthe exponentially local Hamiltonian that results in this fashion. Of course, truncating the hopping elementsfor sufficiently large | i − j | renders the Hamiltonian strictly local at the cost of introducing some dispersion,characterized by a bandwidth t . Correlations then play the dominant role only if the interaction scale ismuch larger than the bandwidth, U (cid:29) t . The maximum interaction scale is itself limited by the band gap,since the approximation of restricting attention to the lowest band breaks down unless U (cid:28) ∆. Thereforean appropriate quantitative measure of the suitability of a Chern band for realizing correlated phases istheflatness parameter f = ∆ /t . Particles in bands with large f have correlation energies much higher than
1. The assumption of the lowest band is for pedagogical reasons; the generalization to the case when there are several filledbands below the one of interest is trivial and left as an exercise to the reader.
Interacting Flat Band Models
Once a noninteracting Chern insulator hamiltonian H has been obtained that hosts a flat band α withlarge flatness parameter, the next step is to add inter-particle interactions. The full interacting hamiltoniantakes the form H = H + V where V is an interaction term, which typically (but not always) takes ageneralized Hubbard form: V = 12 (cid:88) i,j U ij ˆ n i ˆ n j (7)where ˆ n i = c † i c i is the number operator for electrons on site i . For such a density-density interaction, assuminga translationally invariant Hamiltonian, we have U ij = U ( r i − r j ) = (cid:82) d q (2 π ) U ( q ) e i q · ( r i − r j ) . The interactionterm can then be written V = (cid:80) q U ( q )ˆ n q ˆ n − q where ˆ n q is the Fourier transform of ˆ n i . As a measure of theinteraction scale U we can take (for instance) the maximum value of U ( r i − r j ). Note that the full space ofinteracting Hamiltonians is somewhat more general than the density-density interaction assumed above –for instance it could include ring exchange, pair hopping and other non-density contributions. Neverthelesswe will mostly restrict ourselves to such Hubbard-type models as they are a reasonable starting point andhave a well-founded physical origin in an onsite Coulomb repulsion.2.4. Projecting to the Flat Band
Since we assume U (cid:28) ∆ we may safely neglect the mixing between α and the remaining bands. Thisallows us to project to the partially filled band; such a projection is implemented by the operator P α = (cid:80) k | k , α (cid:105)(cid:104) k , α | . In second-quantized form, the resulting Hamiltonian takes the form H = P α H P α = (cid:88) q (cid:15) α ( q ) γ † q ,α γ q ,α + 12 (cid:88) q U ( q )¯ ρ q ; α ¯ ρ − q ; α (8)where we have defined the projected density operator in the Chern band,¯ ρ q ; α = P α ˆ n q P α = (cid:88) k ,b u α ∗ b (cid:16) k + q (cid:17) u αb (cid:16) k − q (cid:17) × γ † k + q ,α γ k − q ,α (9)If the bandwidth is small compared to the scale of the interactions, t (cid:28) U , (cid:15) α ( k ) may be treated asconstant and thus ignored. With this approximation, we finally arrive at the low energy effective flat bandHamiltonian, which takes the form H effCB ,α = 12 (cid:88) q U ( q )¯ ρ q ; α ¯ ρ − q ; α (10)One encounters a similar Hamiltonian in the treatment of interactions in the LLL in a large magnetic field.In that case, the effective Hamiltonian of a clean system obtained by projecting density-density interactionsto the LLL has the form H effLLL = 12 (cid:88) q V ( q ) e − q (cid:96) B / ρ q ρ − q (11)6here in this case ρ q differs from the projected density, P ρ q P by a q -dependent constant, P ρ q P = e − q (cid:96) B / e i q · R ≡ e − q (cid:96) B / ρ q where P is the operator that projects to the LLL, R is the ‘guiding center’coordinate and (cid:96) B = ( (cid:126) c/eB ) is the magnetic length. V ( q ) is once again the Fourier transform of a two-body density-density interaction.The similarity between Hamiltonians projected to a Landau level and to a Chern band, coupled with thefact that both the LL and the Chern band have a nontrivial Chern number suggests that similar fractionalizedphases may arise in the Chern band as in the FQHE. The missing piece in making this analogy precise isto match the commutation relations of the projected densities, which we discuss at length below. First, wegive a brief overview of the numerical studies that have driven much of the early progress in the field.
3. Numerical Results
Once a flat band Hamiltonian has been constructed, the next logical step is to ascertain the properties ofits ground state at a given filling. Analogously to the study of the FQHE in the LLL, the approach of choiceis to exactly diagonalize the flat band Hamiltonian for small numbers of electrons, and try and extract fromthis the ground state properties in the thermodynamic limit. Since such analysis is built from the existingintuition gleaned from LLL numerics, it is useful to first review those original aspects. We should prefacethis discussion by noting that while much of the numerical intuition for the LLL is obtained in the sphericalgeometry where even topologically ordered ground states are non-degenerate, here we discuss continuumresults that study FQH states on a torus as they can be directly compared to lattice Chern bands.On a torus, FQH states are expected to exhibit a ground state degeneracy in the thermodynamic limit:they should have a low-lying multiplet of topologically degenerate states, e.g. there are m such levels forthe 1 /m Laughlin state. The gap between these and the remaining states in the spectrum should scaleto a constant in the thermodynamic limit. Furthermore, upon ‘flux insertion’ through a handle of the torus– implemented operationally by diagonalizing the problem with twisted boundary conditions – the statesshould exhibit ‘spectral flow’ within the low-lying multiplet: as the flux is increased, the levels should evolveinto each other, with a periodicity in m flux quanta for the 1 /m Laughlin states. For each degenerate groundstate, a many-body Hall conductance can also be computed by suitably twisting boundary conditions, andis expected to match that of the FQH phase. An additional test is to study the uniformity of the particledensities, as the FQH state is expected to be a uniform incompressible liquid. The overlap of a numericalground state with trial wave functions — which can be analytically demonstrated to be in a particular FQHphase – provides further characterization of its properties. Similarly, several of the FQH wave functions aregapped exact ground states of local model ‘pseudopotential Hamiltonians’; adiabatic continuity betweenthese and the actual Hamiltonian can be used to establish the ground state topological order. The fractionalstatistics of the quasiparticle excitations of the FQH also follow a ‘generalized Pauli principle’ that governsthe structure of low-lying many-body eigenstates with a fixed number of excitations in a precise mannerthat can be tested numerically. Finally, a more sophisticated tool is to study ground state entanglement ,whose properties can be rationalized in terms of topological field theoretic considerations and the bulk-edgecorrespondence of the FQHE. Each of these approaches, to a greater or lesser degree, has been transcribedto the problem of fractionally filled Chern bands.3.1.
First Results: Overlaps, Hall Conductances and Spectral Flow
The earliest evidence for FQH physics took the form of studying ground state degeneracies as well asspectral flow, as well as a computation of ground state Hall conductance For instance, Neupert et. al. [15]
2. However, it is possible that other states, with broken symmetries – for instance those with charge density wave order–also exhibit a similar degeneracy on a torus as first noted by Haldane [38] for the case of continuum LL problem, and so somecare must be taken in using the ground state degeneracy alone to judge if the ground state is in a FQH phase.3. We stress that this was at that point state-of-the-art, since the analytic understanding of the FQH in Chern bands wasthen insufficient to implement various other checks. /E b γ x a)b) |t | < μ s g|t | > μ s ̟ E/E b ̟ c) γ x ̟ E/E b ̟ g = g = Figure 1.
Results from Ref. [15] , illustrating (a) energy spectrum for 6 electrons at ν = 1 /
3; (b) spectral flow within theground state multiplet in the fractional Chern insulator; and (c) the lack of such spectral flow in a fractionally filled trivial insulator. studied systems with 6-electrons at 1 / et. al. [16] – on a slightlydifferent flat band model, slightly larger system sizes, and appropriately tuned interactions – found gaps atfillings 1 / / / / Quasihole Counting and Generalized Exclusion Statistics
A careful study of the quasihole excitations of fractional Chern insulators was used by Bernevig andRegnault [18] to argue extremely convincingly for the existence of FQH states in the Chern band. Afterfirst establishing that the gap, ground state degeneracy and Hall conductance all remained robust withincreasing system size, and that the ground state wave functions described uniform-density fluids (as opposedto modulated charge density waves), they proceeded to study the excitations of the putative FQH groundstate. In additional to being fractionally charged, these are expected to exhibit fractional statistics both intheir exchange with each other, as well as in the generalized ‘exclusion principle’ that governs the countingof their low-energy many-body eigenstates [40]. While fractional charge and exchange statistics are difficultto study in finite-size numerics, the counting of quasihole states can be directly verified.Assuming the system at commensurate filling to be in a particular FQH phase places strong constraintson the precise number of low-lying energy eigenvalues of the quasihole states in each generalized momentumsector. In the conventional FQHE, this counting can be rationalized in terms of a generalized exclusion
4. This was revisited in the context of fractional Chern insulators by Shankar and Murthy [22,23]. y =0 1 E ( k x , k y ) U=0 (a) =1/3,V=0,N=48
U=1
U=0.2 k x +N x k y (b) =1/5,U=1,N=60 V=0 k y =0 2 4 E ( k x , k y ) V=0.1
V=1
Figure 2.
Low energy spectrum of fractional Chern insulators and other phases in fractional Chern bands fromRef. [16] . The parameters U and V control the scale of the nearest neighbor and next-nearest neighbor repulsion terms in theHamiltonian. The top panels in (a) and (b) clearly illustrate the difference in scale between the energy spread of the groundstate multiplet and the energy gap. principle [40]. For example, working in the one-dimensional basis characteristic of the continuum Landaulevels, the generalized exclusion principle for the 1 /m Laughlin states forbids more than 1 particle in m adjacent orbitals. This leads to a precise prediction for the counting in each momentum sector (here by‘momentum’, we mean the appropriate conserved quantum number conserved by the symmetries of thechosen gauge-fixing.) Such a principle can be applied to each conserved momentum sector, and at a morecoarse grained level the total number of low-energy states in a finite system with a given, fixed number ofquasiholes is also given by the exclusion principle. Note that these statements are true for the full spectrumonly in the case of model Hamiltonians designed to render a particular FQH wave function and its quasiholesexact eigenstates. For a generic Hamiltonian, they are expected to apply to the low-lying energy spectrumthat is believed to capture the universal topological content of the phase, which is typically separated by agap from spurious content at higher energies – although such a gap may be small or absent for small systemsizes or infelicitously chosen aspect ratios.In order to rationalize the results for the Chern band, Bernevig and Regnault [18,41] gave an ‘unfold-ing’ prescription to obtain a one-dimensional generalized momentum parameter to label the many-bodyeigenstates: for an N x × N y system, for instance, they labeled states in terms of k ≡ k x + N x k y . Withthis procedure, it is possible to compare the counting of quasihole states both in total as well as withineach momentum sector; such states were generated by diagonalizing the problem at a filling which deviatesslightly from commensuration , corresponding to studying the system with a fixed number of quasiholes:for instance, diagonalizing the problem of 9 electrons on a 5 × /
5. The small system size means that this choice can be delicate; see [18] for details. E / U K x + N x K y ⇥ ⇥ ⇤ ⌅ ⇧ ξ k x + N x * k y ⇠ E / U k x + N x k y k x + N x k y te i' '/⇡ ⇠ E , ⇠ E ( a )( b ) E Figure 3.
Results from Refs. [18,41] for N = 8 bosons on an N x = N y = 4 lattice ( ν = 1 / bosons). Clockwise fromtop left: (i) kagom´e lattice model, with tunable hopping phase ϕ ; (ii) (a) energy gap (∆ E ) and finite-size splitting of degenerateeigenstates ( δ ) and (b) entanglement gap (∆ ξ ), as function of ϕ ; (iii) particle entanglement spectrum keeping N A = 4 bosons,showing entanglement gap (counting of levels below gap match predictions from quasihole counting); and finally (iv) low-lyingenergy spectrum at ν = 1 / provided the first evidence that the FCI excitations obeyed similar generalized exclusion statistics as theircousins in the LLL.3.3. Entanglement Spectrum
A powerful tool that has emerged in studying correlated topological phases, particularly in numerics, is thestudy of entanglement properties of the ground state [42,43]. Various aspects of ground state entanglementhave been discussed in the literature, and it is well beyond the scope of this review to give an exhaustivesummary; here we will only provide a telegraphic account of the details needed to discuss the numerics. Thestudy of the entanglement encoded in a ground state | Ψ (cid:105) proceeds first by constructing the density matrixof the state, ˆ ρ = | Ψ (cid:105)(cid:104) Ψ | . The next step is to divide (‘cut’) the Hilbert space into two disjoint subsystems A and B : H = H A ⊗ H B , and construct the reduced density matrix of one subsystem, say A , by tracing overthe degrees of freedom in the remainder:ˆ ρ A = Tr B [ˆ ρ ] (12)10he reduced density matrix captures the various properties of ground state entanglement. If we define an ‘entanglement Hamiltonian’ H A we may write ˆ ρ A = e − ˆ H A , and the set of eigenvalues of ˆ H A constitutethe entanglement spectrum of the ground state [44]. Another commonly used measure is the entanglemententropy [45], which is the von Neumann ( S vN = Tr A [ˆ ρ A ln ˆ ρ A ]) or Renyi ( S ( n ) R = Tr A [ˆ ρ nA ]) entropy computedfrom the reduced density matrix.Devoid of any other information, the entanglement spectrum does not offer significantly more informationthan the entanglement entropy; typically the best that can be hoped for is to understand the level statisticsof the eigenvalue distribution. Where a knowledge of the entanglement spectrum comes into its own iswhen the system is endowed with an additional symmetry that can serve as an organizing principle for theentanglement eigenvalues; if the ‘cut’ commutes with the symmetry, the entanglement spectrum is blockdiagonal in the corresponding symmetry sectors, and various additional properties can be extracted. Forinstance, in pioneering work Li and Haldane demonstrated that the entanglement spectrum of rotationallyinvariant quantum Hall states on the sphere could be organized using their conserved angular momentum.They matched the counting of the low-lying entanglement levels in each angular momentum sector withthat of the chiral conformal field theory that describes gapless excitations at the physical edge of thecorresponding FQH state. An enormous body of knowledge about the entanglement spectrum of FQH statesin the LLL has been built by analytical and numerical studies.Bernevig and Regnault [18] computed the ground-state particle entanglement spectrum – which is obtainedmy making a ‘cut’ in particle number space, i.e. producing a reduced density matrix by tracing out acertain number of particles rather than over a geometric region. The resulting entanglement spectrumwas organized using a similar ‘unfolding’ procedure as that used for the true spectrum. While the real-space entanglement spectrum is tied to the behavior of edge excitations of a FQH droplet, a particle spacecut encodes the structure of quasihole states (at a filling corresponding to the number of particles leftafter the cut is made.) Thus, the particle entanglement spectrum should exhibit similar counting as theactual quasihole spectrum for the same number of quasiholes (see Fig.3, bottom right, for an example).Verification of this property, which has been demonstrated in detail in the LLL, for the fractionally filledChern band constitutes another important indication that the two systems exhibit the same topologicalorder. Furthermore, both the energy and entanglement spectra can be rationalized more precisely with anunderstanding of the emergent many-body symmetries (originally introduced by Haldane for the LLL) inthe context of the Chern band [41].3.4.
Extensions: Trial wave functions, Pseudopotentials, and Adiabatic Continuity
Since the early results that established the existence of FQH phases in Chern bands, there has beenan explosion of numerical studies on fractional Chern insulators, an exhaustive list of which would beimpossible to provide. Limitations of space prevent us from discussing several interesting extensions, such asto bosonic systems, the inclusion of disorder and numerical studies of bands of higher Chern number. Onedevelopment we do wish to briefly summarize, especially as it connects naturally to the next section, is thestudy of trial wave functions and pseudopotential Hamiltonians. Here, Qi provided a route to constructingpseudopotential Hamiltonians and corresponding model wave functions in the Chern band that built on theunderstanding of those in the LLL via a mapping of Wannier states, discussed below. Subsequent detailed
6. Note that ˆ ρ A is positive definite so ˆ H A is Hermitian.7. Here and wherever we discuss entanglement spectra, by ‘low-lying’ we mean states below the so-called ‘entanglement gap’that is generically observed to separate the universal content of the entanglement spectrum from spurious behavior at higherentanglement ‘energies’.8. Note that in a periodic system with topological order there is some ambiguity as to the treatment of degenerate eigenstates;Bernevig and Regnault chose to study a uniform incoherent superposition of the q degenerate Laughlin states, but the phasedependence of the entanglement entropy of coherent superpositions can be used to extract additional topological properties[46].9. Although we briefly discuss a variational wavefunction analysis for phases in C = 2 bands, below, and comment on someanalytical results on lattice defects in such bands.10. We present this discussion here for reasons of organization; to the uninitiated, its clarity may be much improved byreturning to it after reading the next section.
4. Landau Levels and Chern Bands: A Critical Comparison
As noted in the previous section, a variety of different models realize topological flat bands that hostinsulating phases at fractional fillings. On one level, this is extremely satisfying as it vindicates a certainprejudice: that since the IQHE has a lattice realization in a Chern band, so should the FQHE by analogy.This is bolstered by evidence such as the fractional Hall conductance, that are consistent with FQH physics.However the evidence we have reviewed so far has been predominantly numerical; it is clearly desirable tohave a more analytical approach underpinning our understanding of the connection between Landau levelsand Chern bands. Ideally this would achieve two goals:i establish some notion of adiabatic continuity between the FCI ground state and model FQH wavefunctions in a Landau level.ii provide a quantitative measure of the role of the lattice potential in the FCI.To see why such an analytical understanding is important, note that the restriction of the problem to thelow-energy Hilbert space corresponding to the lowest Chern band follows from the first of our three criteriafor a ‘good’ Chern band. In simple terms, we have shown that it is reasonable to study the problem restrictedto a space of states that have the same global topology (as quantified by their Chern number) as the orbitalsin a single Landau level. If we can also demonstrate that a simple local Hamiltonian takes a similar formwhen written in the basis of Chern band eigenstates as in the Landau orbital basis, then we can argueusing the basic notion of adiabatic continuity that the two problems represent essentially the same physics.There are two ways to accomplish this: the first, taken by Qi, is to construct a one-one-mapping betweenLandau gauge orbitals and (linear combinations of) Chern band eigenstates, and transcribe a Hamiltonianappropriate to a particular FQH phase from the LLL to the lattice Chern band. If this mapping is local, thiswould establish the existence of an FQH phase in the Chern band. An alternative approach, put forth by thepresent authors, is to begin with the same model of a projected density-density interaction in both cases,and demonstrate that the algebra of projected density operators in the Chern band has the same featuresas the Girvin-Macdonald-Platzman algebra in the LLL. While the former approach permits a constructionof trial wave functions, an advantage of the latter approach is that makes manifest the importance of thesecond criterion of Berry flatness for a Chern band to reproduce LLL physics. A more detailed study of thealgebra also suggests the various constraints on the structure of the single-particle states in the Chern bandthat constitute the third and final criterion.4.1.
Wannier Orbitals, wave functions, and Pseudopotential Hamiltonians
The first concrete correspondence between Chern bands and Landau levels was proposed by Qi [19], whoconstructed lattice analogs of the Landau gauge single-particle eigenstates familiar from the continuum.Recall that the latter are given by ψ K y ( r ) = 1 π / (cid:112) (cid:96) B L y e iK y y e − (cid:96) B ( x − K y (cid:96) B ) (13)which are centered at x = K y (cid:96) B , and K y = πnL y with n an integer. Clearly, the center of this wavefunctionshifts as x → x + πL y (cid:96) B under transforming K y → K y + πL . In the Chern band, Qi constructed Wannierstates that have a one-dimensional momentum label with respect to which their centers evolve with a similar‘shift’ property. We briefly summarize his Wannier construction for a Chern insulator placed on an infinite12ylinder, periodic in y and infinite in x . We will follow the notation of the previous section and consider aChern band α . It is always possible to perform a unitary (gauge) transformation on the single-particle statesso that we fix one component of the Berry vector potential: ( A α ) y = 0. In this case the Wannier states inthe Chern band are given by (suppressing the index α ) | W ( k y , x ) (cid:105) = 1 L x (cid:88) k x e − i (cid:82) kx A x ( p x ,k y ) e − ik x ( x − θ ( k y ) / π ) | k (cid:105) (14)where k = ( k x , k y ), θ ( k y ) = (cid:82) π A x ( p x , k y ) dp x , and x is an integer labeling the lattice sites. The phase factorinvolving θ ( k y ) guarantees that the symmetry of the Bloch functions under k x → k x + 2 π is properly ac-counted for. Under a gauge transformation that multiplies the Bloch states by a phase factor | k (cid:105) → e iφ ( k ) | k (cid:105) ,the Wannier function is invariant upto a phase: | W α ( k y , x ) (cid:105) → e iφ (0 ,k y ) | W ( k y , x ) (cid:105) . Most significantly, thecenter-of-mass position of the Wannier function | W ( k y , x ) (cid:105) is given by (cid:104) ˆ x (cid:105) = (cid:104) W ( k y , x ) | ˆ x | W ( k y , x ) (cid:105) = x − θ ( k y ) / π (15)– in other words, θ ( k y ) / π is the shift of the Wannier function from the lattice site, often referred to as thecharge polarization. However, the integral of this quantity gives the Chern number, C = − π (cid:82) π θ ( k y ) dk y ;thus, if C (cid:54) = 0, the Wannier function is not periodic in k y . The center-of-mass position shifts from x → x + C when k y evolves from 0 to 2 π , so that the Wannier functions satisfy twisted boundary conditions: | W ( k y + 2 π, x ) (cid:105) = | W ( k y , x + C ) (cid:105) (16)We can define a generalized momentum coordinate K y = k y +2 πx for 0 ≤ k y < π , in which case if we define | W ( K y ) (cid:105) = | W ( k y , x ) (cid:105) , we have a continuously defined Wannier function for K y ∈ R . The center-of-massposition of the Wannier function now evolves continuously for K y , which thus plays a role analogous to theconserved momentum of the Landau gauge eigenfunctions in the continuum. It may be verified that underthe weak-field (Hofstadter) limit (cid:96) B (cid:29)
1, the Wannier functions (cid:104) r | W ( K y ) (cid:105) → ψ K y ( r ).The next step is to recall that many model wave functions representing quantum Hall states in the LLL areexact ground states of local pseudopotential Hamiltonians [6,49]. Working on the cylinder, we can representthe model wave functions in the basis of occupation numbers of the single-particle states (13); using thecorrespondence demonstrated above, this can be transcribed directly into a wavefunction on the latticewritten in the basis of Wannier orbital occupation numbers. For instance, consider the LLL wavefunctionfor the m th Laughlin state on the cylinderΨ /m LLL ( { z i } ) = (cid:104){ z i }| Ψ /m LLL (cid:105) = Ω (cid:89) i An alternative perspective to Qi’s Wannier orbital approach is to consider the algebra of densities pro-jected into the Chern band which will allow a quantitative comparison between the LLL and the Chernband [20]. Recall that in the quantum Hall problem, the magnetic translation operators obey the so-called Girvin-Macdonald-Platzmann Algebra [26],[ ρ q , ρ q ] = 2 i sin (cid:18) q ∧ q (cid:96) B (cid:19) ρ q + q (20)Consider the projected densities in a Chern band, given by (9); at long wavelengths qa (cid:28) 1, we mayexpand (cid:80) b u α ∗ b (cid:0) k + q (cid:1) u αb (cid:0) k − q (cid:1) ≈ − i q · (cid:80) b u α ∗ b ( k ) ∇ k i u αb ( k ) ≈ e i (cid:82) k + q / k − q / d k (cid:48) ·A α ( k (cid:48) ) , so that (here andbelow we suppress band indices) ρ q | k , α (cid:105) ≈ e i (cid:82) k + qk d k (cid:48) ·A α ( k (cid:48) ) | k + q , α (cid:105) . (21)In other words, for small q , ρ q implements parallel transport described by the Berry connection A α ( k ). Eitherfrom this observation or via a gradient expansion, we may show that at long wavelengths, the commutatorof projected density operators at different wavevectors is (cid:2) ρ q , ρ q (cid:3) ≈ i q ∧ q (cid:88) k (cid:34) B α ( k ) (cid:88) b u α ∗ b ( k + ) u αb ( k − ) × γ † k + ,α γ k − ,α (cid:35) (22)where we define k ± = k ± q + q . Finally, let us assume that the local Berry curvature B α ( k ) can be replacedby its average B α = (cid:82) BZ d k B α ( k ) (cid:82) BZ d k = 2 πC α A BZ (23)over the BZ; here A BZ = c /a is the area of the BZ, with a the lattice spacing and c a numerical constantdepending on the unit cell symmetry. This yields (cid:2) ρ q , ρ q (cid:3) ≈ i q ∧ q B α ρ q + q . (24)which is identical to the long-wavelength limit of the density algebra (20) for the LLL, with B α / = √ πC α c a playing the role of the magnetic length (cid:96) B . We may draw several inferences from Eqn. (24), as detailed inprevious work. We summarize them briefly here. First, we observe that a coarse grained, projected, positionoperator may be defined via r cg ≡ x cg ˆ x + y cg ˆ y = lim q → ∇ q i ρ q . It follows from Eqn. (24) that[ x cg , y cg ] = − i B α (25)which identifies the r cg with the guiding center position operator in the LLL. 11. We remind the reader that these are simply the projected densities stripped of a Gaussian factor e − q (cid:96) b / . N unit cells there are N points in the BZ.If B α ( k ) is truly constant we can define a set of N parallel translation operators T q for which (21) holdsexactly: T q | k , α (cid:105) = e i (cid:82) k + qk d k (cid:48) ·A α ( k (cid:48) ) | k + q , α (cid:105) . (26)The algebra of the T q is thus exactly of the W ∞ form (20) without a long-wavelength restriction. We notethat the T q are trivially isomorphic to magnetic translation operators for a system with N sites and flux 1 /N per unit cell and it is straightforward to check that the states in the band form an N dimensional irreduciblerepresentation of their algebra [51]. From this perspective the idealization of a constant curvature Chernband hosts a W ∞ algebra whose long-wavelength generators coincide with the physical density operators.Finally, note that both the “Hofstadter” and “Haldane” problems give rise to (24), which unifies earlierlattice FQHE studies [24,25] with the ones considered here. In this sense, those earlier works can be seenas the first demonstration of lattice FQH phases, albeit those in which the Berry curvature is extrinsic anddue to a magnetic field rather than an intrinsic property of the lattice model. This last observation can beused to get nearly constant curvature bands by approaching the Landau level limit on the lattice, i.e. bypicking flux 1 /m per plaquette and working the lowest subband at large m . While it is impossible to find aconstant-curvature Chern band in models of Chern insulators with N = 2 bands, it is possible to constructmodels with N > W ∞ algebra in a system with N Φ states isgenerated by N density operators while in a Chern band for an N × N lattice system there are N statesthere are only N densities (or T q if one wishes to work with a closed algebra). This distinction arises asthe LLL is formally defined on a continuous space but is without fundamental dynamical significance as therelevant momenta, q(cid:96) B < O ( N Φ ) in the LLL as well. For example, it was shown in [54] that keepingonly this set of momenta keeps the entire physics of the quantum Hall localization transition in the LLL.However this counting discrepancy does have the consequence that the algebra of the densities themselves must close in the LLL at all q which is not the case in the Chern band. We will encounter this issue ofoperator counting again later in this review, when we discuss the Shankar-Murthy ‘Hamiltonian’ approachto the fractional Chern insulator problem.4.3. Geometry of the Chern Band The algebraic approach to the Chern band discussed can be extended further to include effects of the band geometry in addition to the topology [21]. This allows further constraints to be placed on models in orderthat they stabilize fractionalized phases. While important to a complete understanding of the structure ofChern bands, this section uses somewhat more technical machinery than the remainder, and can be skippedon a first reading. In the preceding section, we showed that to order q , and in the uniform Berry curvatureapproximation the algebra of projected densities obeys an analog of the GMP algebra of the LLL. A naturalextension of this result is motivated by examining higher order terms in q in the expansion of [¯ ρ q , ¯ ρ q ], anddemanding that they in turn vanish. At order q , after some algebra we can demonstrate [21] that this occursif and only if a certain gauge-invariant quantity, known as the Fubini-Study (FS) metric tensor , g αij ( k ) is aconstant across the Brillouin zone. The FS metric is a rank two symmetric tensor, g αij ( k ), which for band α has components [55,56,57,58] g αij ( k ) = 12 (cid:88) a (cid:34)(cid:18) ∂u α ∗ a ∂k i ∂u αa ∂k j + ∂u α ∗ a ∂k j ∂u αa ∂k i (cid:19) − (cid:88) b (cid:18) ∂u α ∗ a ∂k j u αa u α ∗ b ∂u αb ∂k i + ∂u α ∗ a ∂k i u αa u α ∗ b ∂u αb ∂k j (cid:19)(cid:35) (27)where the u αa ( k ) are the single-particle eigenstates defined previously. It is worth pausing to expand alittle on the meaning of this rather formidable expression. Recall that the Berry curvature of a band isa natural geometric quantity that emerges from considering the evolution of the eigenstates of the single-particle Hamiltonian h ab ( k ) as the parameter k evolves (adiabatically) across the BZ. Just as we defined the15rojector onto band α we can define the orthogonal projector Q α ( k ) = − (cid:80) β (cid:54) = α | k , β (cid:105)(cid:104) k , β | . It is convenientto introduce the (complex) tensor R αij ( k ) ≡ [ ∂ k i (cid:104) k , α | ] Q α ( k ) (cid:2) ∂ k j | k , α (cid:105) (cid:3) . (28)By rewriting the eigenstates in terms of the u αa ( k ) it is readily verified that the Berry curvature B α ( k ) = − i(cid:15) ij R αij ( k ) = − (cid:2) R αxy ( k ) (cid:3) . A little more work shows that the FS metric is simply the correspondingreal part: g αij ( k ) = Re (cid:2) R αij ( k ) (cid:3) . Note that the gauge-invariance of both the metric and the curvature aremanifest in the above form.The importance of the FS metric is that it introduces a natural notion of ‘distance’ in the single-particleHilbert space that can induced from the k -space evolution of the single-particle eigenstates. Furthermore,since the FS metric and the Berry curvature are the real and imaginary parts of the same complex tensor R α ( k ), it is also possible to show that at any k in the BZ, the trace of the FS metric is bounded from belowby the magnitude of the Berry curvature at k ,tr( g α ( k )) ≥ |B α ( k ) | . (29)This geometrical constraint applies to any insulator. In the remainder of this section, we will use this inconjunction with the existence of a nonzero Chern number, to sharpen our understanding of the specialproperties of the restricted Hilbert space of a Chern band.We were led to consider the metric tensor by studying higher order terms in the expansion of the algebraof projected density, and we identified its uniformity as a criterion for this to match the W ∞ algebra ofthe LLL to order q . Thus, we have found an additional criterion for identifying “good” band structuresfrom the point of view of hosting interacting topological phases. A natural question to ask is whether oneobtains an infinite set of such constraints, order-by-order in q ; it would be somewhat deflating if this werethe case. Remarkably, we will demonstrate that when the band structure satisfies one additional constraint,the Chern band projected densities satisfy the W ∞ algebra of projected LLL densities at all orders in q .To this end, it is useful to consider the transformation properties of R αij under unimodular transformationsof the coordinates, i.e. those of the form x a → x (cid:48) a = Λ ab x b with det Λ = 1. The FS metric transforms in theusual way, ( g αab ) (cid:48) ( k ) = Λ ac Λ bd g αcd ( k ), while the Berry curvature is invariant. Moving to ‘primed’ coordinatesin which g α ( k ) is diagonal, using the invariance of the determinant under unitary transformations andapplying the inequality (29) we obtain2 (cid:112) det g α ( k ) = 2 (cid:112) det( g α ( k )) (cid:48) = tr( g αij ( k )) (cid:48) ≥ | [ B α ( k )] (cid:48) | = |B α ( k ) | (30)Since this is true across the BZ, we can integrate this to find (cid:90) BZ d k det g α ( k ) ≥ (cid:90) BZ d k |B α ( k ) | ≥ A BZ ¯ B α = π C α A BZ (31)where we use the fact that the square-averaged Berry curvature is bounded from below by the square of theChern number per unit area of the BZ. Eq. (31) is the key relation between the Chern band geometry toits nontrivial topology: namely, the integral of the determinant of the Fubini-Study metric is bounded frombelow by a number proportional to the square of the Chern number of the band.We now consider the case when (i) the inequality (31) is saturated and (ii) the FS metric is uniform inthe BZ. It is easily seen that these conditions amount to requiring, in addition to the uniformity of theBerry curvature obtained previously, the single additional constraint that det( g α ( k )) = |B α ( k ) | at all k 12. Note that the curvature also has indices, but in d = 2 there is only one nonzero component so we can suppress them.13. Observe that owing to the equivalence between the position operator ˆ x i and the k -space derivative − i∂/∂k i it is possibleto rewrite R αij = (cid:104) k , α | ˆ x i Q ( k )ˆ x j | k , α (cid:105) ; in this form we see that the non-vanishing of the Berry curvature roughly correspondsto the non-commutativity of the x - and y -components of the position operator.14. This is intimately connected to the geometry of K¨ahler manifolds as well as to the geometric theory of the insulatingstate, but unfortunately we cannot discuss these beautiful subjects here [59].15. For d = 2 in this basis simple algebra shows that the square of the trace is four times the determinant. 16n the BZ. In this limit and in the ‘primed’ coordinate system used previously, in which the FS metric isdiagonal, we have [ g αij ( k )] (cid:48) = ¯ B α δ ij . Writing the density operators in the rotated coordinate system, it isstraightforward to demonstrate that they satisfy a generalized, metric-dependent version of the W ∞ algebra:[¯ ρ q , ¯ ρ q ] = 2 i sin (cid:18) q ∧ q ¯ B α (cid:19) e ( q ) l g αlm ( q ) m ¯ ρ q + q = 2 i sin (cid:18) q ∧ q ¯ B α (cid:19) e ¯ B α ( q + q ) ¯ ρ q + q (32)There are three important aspects of this result worth reflecting on. First, note that Berry curvature andthe FS metric both appear in the form of the W ∞ algebra obtained in (32), which thus algebra also appliesto bands which have a higher Chern number and therefore differ fundamentally from Landau levels with C = 1. Second, observe that the conditions under which we get a closed algebra of the projected densityoperators can be stated purely in terms of the FS metric. Third, we observe that for the C = 1 case,(32) looks remarkably similar to the algebra of projected densities that obtains in a Landau level, with thecombination e ( q ) l g αlm ( q ) m playing a role analogous to the LL-dependent form factor that relates projecteddensities to the magnetic translation operators (which are identical in every Landau level) for the continuumFQHE. A heuristic interpretation of the additional constraints is therefore that they quantify not simply howclose Chern band projected densities are to realizing the continuum Landau level algebra, but in additionhow close they are to realizing the lowest Landau level.For a system where the ideal conditions under which this algebra is obtained do not hold, the degree ofdeviation from these conditions provides an additional criterion to quantify how favorable a Chern bandis for hosting FQH-like physics. Conversely, if one finds fractional topological phases in systems where thedeviations from these conditions is considerable, one could argue that the physics of those systems is newand distinct from the conventional fractional quantum Hall effect. The effects of disorder also enter theHamiltonian through terms that involve the projected density operator. This suggests that the effects ofdisorder in the Chern band are likely to be the same as in the LLL when the conditions stated above forthe FS metric are satisfied. Extensions of the algebraic approach to higher dimensions and to time-reversalsymmetric Z topological insulators have also been discussed in the literature [60,61,62].4.4. Operator Counting, Smoothed Densities, and the Shankar-Murthy Approach We now return to the question of operator counting that we discussed earlier. As motivation, we firstobserve that in cases where the deviation of the Berry curvature from its average value is bounded, |B α ( k ) −B α | < |B α | − (cid:15) , we may define a ‘smoothed’ density operator which may be regarded as the projection of anoperator ρ s ( r ) local in position space; for qa (cid:28) ρ s q | k , α (cid:105) = B α B α ( k ) e i (cid:82) k + qk d k (cid:48) ·A α ( k (cid:48) ) | k + q , α (cid:105) . (33)At long wavelengths, the algebra of smoothed densities closes, and in this limit (24) is an exact equalitywhen ρ q is replaced by ρ s q . This provides a clue that there are observables in the Chern band that indeedsatisfy a well-behaved GMP algebra, related to the projected density operators in some nontrivial fashion.Building on the different operator counting in the LLL versus the Chern band, Shankar and Murthywere able to take a slightly different perspective on the operator algebra approach to FCIs. Their approach(described in Ref. [23,22]) consists of three separate steps:(i) They construct a set of one-body operators ρ GMP ( q + G ) which act only within a Chern band, andsatisfy the full GMP algebra (20). Here, q is a crystal momentum (restricted to lie in the first BZ) while G is a reciprocal lattice vector. For an N × N lattice, the number of independent crystal momenta is N ,and Shankar and Murthy demonstrated that the ‘umklapp’ operators ρ GMP ( q + G ) are independentonly for N choices of reciprocal lattice vector G . Thus, together there are N independent GMPoperators ρ GMP ( q + G ) which thus form a complete basis in terms of which one may express any ofthe N one-body operators in a Chern band.(ii) They then demonstrate that the projected density operators in the Chern band may be rewritten interms of the ρ GMP ( q + G ): 17 attice strength CB limitLLL limit H model deviation from 32 14 H LLLmodel H LLLrealistic H CBrealistic H CBmodel Figure 4. Mappings within and between the LLL and the Chern band. Path 1: Within the LLL, there exist special‘model Hamiltonians’ designed to a render a certain FQH state an exact ground state; a wealth of numerical evidence suggeststhat in many cases these are adiabatically connected to the ground states of more realistic Hamiltonians, for instance thoseobtained by projecting a density-density interaction to the LLL. Path 2: Qi and Shankar-Murthy both construct mappingsfrom the LLL to the Chern band that allow them to construct model Hamiltonians in the Chern band, with the same topologicalorder in the LLL. Path 3: The present authors instead considered realistic density-density interactions projected to the Chernband, and gave conditions for this to be adiabatically connected to the FQH ground state of a similarly realistic Hamiltonian inthe LLL . Path 4: Recent numerical studies suggest that the model Hamiltonians are adiabatically connected to these morerealistic projected density-type interactions. ρ q = (cid:88) G c ( G , q ) ρ GMP ( q + G ) (34)where the expansion is over reciprocal lattice vectors and the coefficients are determined from dataon the lattice realization of the original Chern band. Thus the projected densities are a specific linearcombination of operators satisfying the GMP algebra – which explains why the ρ q themselves onlysatisfy (20) in an idealized limit. They also noted a distinction between the projected density and theexponential of the guiding center density in a Chern band (in the LLL these are the same, up to anormalization depending only on q ). Recall that an analogous situation arose in our discussion of theChern band geometry in the preceding section.(iii) The preceding observations allow them to express a Hamiltonian projected to the Chern band in termsof operators that obey the GMP algebra. Note that since one may naturally formulate the full LLLproblem in terms of the GMP operators, this suggests another route to constructing model Hamilto-nians for a given FQH phase. Furthermore, Shankar and Murthy demonstrate that the ‘Hamiltoniantheory’ that they pioneered for the FQH – in which Chern-Simons flux attachment is placed on anoperator footing – can be ported to the Chern band by working with the ρ GMP ( q ). This in turn permitsthe computation of gaps and response functions within a Hartree-Fock mean field theory.For reasons of brevity we cannot give a detailed elucidation of these points, and encourage the reader tostudy the original articles [23,22] for a pedagogical discussion of the Hamiltonian theory of fractionally filledChern bands.4.5. Adiabatic Continuity and Establishing Topological Order We have provided three different analytical perspectives on the correspondence between Chern bands andthe (lowest) Landau level. We now place these in context by describing how they are related to the problemof establishing the topological order of the ground state of an interacting, fractionally filled Chern band. Forthis, it is useful to recall that at a filling for which a quantum Hall state is known to exist, there are typicallytwo points in the space of all LLL-projected Hamiltonians that are of especial interest. The first is the model18nteraction, which we denote H LLLmodel , that renders a certain FQH wavefunction an exact gapped groundstate . The second is the one that is reached by projecting a realistic electronic interaction Hamiltonianinto the LLL, which we shall call H LLLrealistic . There is much numerical evidence establishing that in manycases the ground states of H LLLrealistic are connected adiabatically to those of H LLLmodel , thus establishing that forexperimentally relevant and reasonable interactions the ground state has the same topological order as thatencoded by the model system.What about the Chern band? Here, we must first produce model Hamiltonians H CBmodel on the lattice, sincethose constructed in the continuum rely intimately on the analytic structure of the LLL wave functions andcannot be ported directly to a Chern band. Here, Qi’s prescription for mapping Landau gauge eigenstatesfrom the LLL to the Chern band provides one route. Another is that of Shankar and Murthy; by suitablypicking a pseudopotential interaction in the LLL and replacing the corresponding projected density operatorsby the ρ GMP s, they can construct model Hamiltonians with similar topological properties in the Chernband. However, both these approaches can lead to Hamiltonians which may look somewhat unnatural onthe lattice – Qi’s construction because the Wannier states may not be related by a simple transformation tothe densities, they are not and Shankar and Murthy’s approach because of the nontrivial relation betweenthe ρ GMP s and the projected density.A different approach, suggested by the present authors, attempts to simply study the properties of H CBrealistic , defined by projecting a density-density interaction of some fixed range to the Chern band. Here,the mismatch between the CB and the LLL is encoded in the structure of the algebra of projected den-sities: as we have emphasized, they obey the GMP algebra only in certain idealized limits. Nevertheless,this suggests that when conditions are such that this idealization is not too far-fetched, one can argue thatthe ground states of H LLLrealistic and H CBrealistic will be in the same phase, and draw on the continuity between H LLLrealistic and H LLLmodel in the continuum Landau problem to establish the topological order of the Chern bandground state for realistic interactions. Alternatively, one may view the idealization as placing constraints onvarious parameters, such as Berry curvature fluctuations, for there to exist an adiabatic path in the Chernband connecting H CBrealistic to H CBmodel . We note that numerical studies have demonstrated that when theseconstraints are violated, the stability of the Chern band ground state is adversely affected; also, the adiabaticcontinuity between Qi’s model Hamiltonians, and more realistic interactions, has recently been verified [48]. 5. Parton Constructions and Phase Transitions Thus far, we have focused on matching the physics of a Chern band with that of the lowest Landaulevel, and in establishing the existence of lattice analogs of familiar FQH states. Our survey of the existinganalytical and numerical evidence should at this point have convinced the reader that at least some of thephases seen in the lowest Landau level survive to the lattice limit. In this section and the next, we changeperspective slightly and ask whether the presence of the lattice can lead to additional physics beyond thatfamiliar from the Landau level. A natural method to analyze such question is to use the ‘parton’ constructionor projective construction, a mean field theory that has had considerable success in analyzing both continuumFQH phases as well as lattice spin liquid models. As has been our theme overall, we will attempt only togive a flavor of this vast area, and identify a few interesting results that have emerged from such studies.5.1. Partons and Projective Symmetry Groups We will give a very simplified account of a parton construction [63] of the ν = 1 /m Laughlin state, first inthe continuum and then its lattice analog. In the continuum, the basic observation is that as a consequenceof the dependence of the quantum of flux on the electric charge, a Landau level that is at 1 /m filling for 16. Note that not all FQH states have parent Hamiltonians of this form; here we shall restrict ourselves to those that do.17. Of course, there may be other details, such as the range of the interaction, etc., but we shall ignore this subtlety anddiscuss all ‘realistic’ Hamiltonians in a unified fashion.18. Recall that in a Chern band, Wannier orbitals can be chosen to be eigenstates of only one position coordinate. e , is fully filled for particles of charge e/m . In other words, an identical number of e/m -charged particles would have a unique, Slater determinant ground state. In order to put this observation touse, we divide each electron into m fermionic ‘partons’, each of charge e/m , distinguished by a flavor index,i.e. we write the electron operator as c ( r ) = m (cid:89) α =1 f α ( r ) (35)We can now write mean-field states for each parton flavor independently. However, in splitting apart theelectron and treating the resulting partons independently, we artificially enlarged the Hilbert space. Toamend this, the final step of the construction is to project the mean-field wavefunction back to the physicalsubspace, by enforcing the requirement that any valid many body wavefunction for the physical electronshave m partons at the same position coordinate. In other words, we write for the electron wavefunctionΨ /me ( { r i } ) = (cid:104) | N (cid:89) i =1 m (cid:89) α =1 f α ( r i ) | MF (cid:105) . (36)Implementing this procedure for the LLL in the disc geometry, and using the fact that the charge e/ /me ( { r i } ) = (cid:2) Ψ ν =1parton ( { r i } ) (cid:3) m = (cid:89) i A fractionally filled Landau level in the continuum limit has no unique ground state in the absence ofinteractions. In contrast, a Chern band (alternatively, a Landau level with a strong periodic potential)typically has a nonzero dispersion, characterized by its bandwidth t (while perfect flat bands have t = 0,these are fine-tuned.) This weak dispersion can select a ground state in the U/t → U is asusual the interaction scale; at partial filling, this ground state is expected to be compressible. As U/t isincreased, eventually we enter the flat band limit where the kinetic energy can be ignored; we assume thesystem forms an incompressible FQH ground state in this limit. A critical point separating the compressibleand incompressible phases is a possibility unrealized in the ideal limit of the FQHE. Although such a Motttransition in a Chern band could always be first-order, there is the additional intriguing possibility of a continuous transition into FQH states.Several examples of such transitions at half-filling were studied by Barkeshli and McGreevy [27,28] andcan be divided into three sets of phases which can have continuous transitions between each other:(i) between superfluid, Mott insulating and ν = 1 / p + ip paired fermion superfluid, the B-phase of Kitaev’s honeycomb model, and a latticeanalog of the Moore Read Pfaffian phase.All these were studied within a parton mean-field theory which is expected to capture the gross features ofthe phases as well as the critical behavior between them. Their results may be summarized as follows. Inessence, through the parton decomposition, transitions between the different phases can be rewritten in termsof Chern number-changing transitions between insulators: | ∆ C | = 1 between successive members of each ofthe three sets above. A general critical theory for Chern-number changing transitions in two dimensions canbe formulated in terms of massless Dirac fermions; a change in the sign of a Dirac mass changes the Chernnumber by 1. In the cases studied by Barkeshli and McGreevy, the massless Dirac fermions are also coupledto a Chern Simons gauge field, which emerges from the parton construction. For all three sets of phases,a direct transition between the first and the third phase requires | ∆ C | = 2 and is generically multicritical,unless an additional symmetry, such as inversion, is present to force two Dirac fermions to undergo asimultaneous change in the sign of their masses. Furthermore, the parton construction describing (i) isthe parent example; the remaining two follow from a further parton decomposition in which an additionalneutral fermionic degree of freedom is attached to each boson in (i) and allowed to form either a Fermisurface of gapless excitations (ii) or a paired state (iii). We direct the interested reader to Refs. [28,27] for adetailed discussion of these transitions, along with a list of physical manifestations in crossover physics andscaling behavior. 6. Higher Chern Numbers and Nonabeliana Another possibility that can be realized in lattice models is a band with higher Chern number; the interestin these examples is primarily concerned with the possibility of realizing non-Abelian statistics in a mannerintrinsically tied to the lattice limit of the FQHE in a Chern band. In this context we note two intriguingdirections we believe are particularly worthy of further attention. The first is a numerical study by Zhang 19. Note that this is the expectation whether the particles are bosons (a superfluid state) or fermions (a filled Fermi sea).20. An alternative possibility that gives ∆ C = 2, a quadratic band-touching, is perturbatively unstable in d = 2 and thusruled out. N copies of a Slater determinant fora fully filled Chern number C band and Gutzwiller projecting to a Hilbert space that obeys a single-site-occupancy constraint. A careful variational Monte Carlo study of the entanglement entropy of degenerateground states on a torus for the case N = C = 2 provides strong evidence that the resulting wavefunctionhas the same topological order as the Moore-Read Pfaffian state, described by an SU (2) Chern-Simonstheory. Combined with a previous analysis of lattice analogs of bosonic Laughlin states along similar lines[66], this suggests that, in general, projecting the N th power of a Slater determinant of a filled Chern number C = k band provides a lattice version of a quantum Hall state whose topological order is identical to that ofthe SU ( N ) k Chern-Simons theory coupled to fermions (which also describes a system of k ‘layers’ at totalfilling ν = k/N .)The interplay between the lattice-scale physics and topological order in a band with C > x → x + C when k y evolves from0 to 2 π . If C > 1, it is therefore possible to define C distinct families of Wannier functions that evolveindependently under k y → k y + 2 π . For concreteness, consider the C = 2 case. Here, the Wannier functionson even and odd sites form distinct families: in the notation introduced previously, we define | W K y = k y +2 πn (cid:105) = | W ( k y , n − (cid:105) , | W K y = k y +2 πn (cid:105) = | W ( k y , n ) (cid:105) (42)On its own, the observation of two families of eigenfunctions is not particularly remarkable; a similar structureobtains, for instance, in multicomponent quantum Hall systems such as quantum Hall ferromagnets andbilayer systems. The key difference is the action of the lattice symmetry operations on the ‘internal’ index.For instance, the translations along x interchange the internal index, which are invariant under translationsalong y : ˆ T x | W K y (cid:105) = | W K y (cid:105) , ˆ T x | W K y (cid:105) = | W K y (cid:105) , ˆ T y | W aK y (cid:105) = e iK y | W aK y (cid:105) , a = 1 , C = 2 band, and showed that such defects that act nontrivially on the this internal ‘ Z ’ degree of freedomhave a nontrivial topological degeneracy – analogous to quasiparticle states in a non -Abelian state. In effect,by acting nontrivially on the layer index, these ‘twist’ defects simulate the action of a change of genus ofthe underlying space – and are hence dubbed ‘genons’. A subsequent detailed analysis [69] of the projectivenon-trivial braiding statistics obtained in this fashion, while beyond the scope of the present article, suggestsan interesting new line of enquiry with possible implications for topological quantum computation. 7. Proposed Realizations Thus far our discussion has focused on lattice models and numerical results. As we emphasized earlier,an appealing feature of the lattice FQHE phases discussed above is that the characteristic energy scaleat which new physics sets in is tied to the microscopic (lattice) scale, which is quite large; if they canbe experimentally realized, it is possible that Chern bands will exhibit fractionalization and associatedphenomena at significantly higher temperature scales and be considerably more robust than in 2DEGs. Weflag a few especially promising examples below.In a solid-state context, Xiao and collaborators suggested that topological bands may be engineeredin oxide interaces [30]. From a combination of first-principles band structure calculations and tight-bindingmodeling, they predict that bilayer [111] heterostructures of transition-metal perovskite oxide host relativelyflat bands with a nontrivial Z topological index. Apart from providing a route to room-temperature quantum 21. Recent work [68] has shown that a slightly more involved definition of the Wannier orbitals and the associated actionof the translation operator is required in order to impose boundary conditions that fully respect various lattice symmetries.Nevertheless, the interplay of lattice defects with topology and the field theory describing this structure remain, so we willignore these subtleties. d -orbitals, suggests that fractional quantumHall physics is a plausible scenario in these systems; see also [36].Another direction is to directly engineer spin-orbit coupled models in optical lattices of ultracold atoms.Cooper [31] has shown that the strong-lattice limit can be achieved in two-species ultracold atom systems(where for instance the two species are the ground and long-lived excited states of an alkaline earth atom)effectively described by the single-particle Hamiltonianˆ H = (cid:20) p m + V ( r ) (cid:21) + V s ( r ) · σ (44)where the ‘spin’ σ refers to the level index. Here, V ( r ) is a conventional scalar potential term, for instancefrom a conventional optical lattice, while V s ( r ) reflects a laser pattern that generates an ‘optical fluxlattice’, where the diagonal components reflect a species-dependent scalar potential, and the off-diagonalterms represent a Raman coupling that tunes the rate of interspecies conversion as a function of position.A judicious choice of laser parameters ( V , V s ) leads to a lowest band with a non-zero Chern number. Itis also possible to adjust parameters such that the band is relatively flat [70]. This and similar proposalssuggest that the high range of tunability afforded by optical lattice models will allow us to study a varietyof features of Chern bands.A different bosonic cold atom implementation is to consider the rotational degrees of freedom of ultracoldmolecules, whose actual spatial motion is pinned by a deep optical lattice. Specifically, Yao et. al. [32,33]consider a system of such pinned, three-level dipoles in a two-dimensional lattices. A fractional Cherninsulator Hamiltonian can be shown to describe the spin flips of this problem which are effective hardcore bosons. The time reversal breaking necessary for a nonzero Chern number is provided by ellipticallypolarized, spatially modulated electromagnetic radiation; the nontrivial spatial dependence of the hoppingrelies on the the long-range dipole-dipole interactions characteristic to polar molecules, which naturallyintroduces orientation-dependent hopping amplitudes. Appropriately tuning these also allows control of thehopping interference and permits bands with f (cid:38) 10 [33]. Numerical studies of these models show a varietyof phases at ν = 1 / 2, including fractional Chern insulators for a range of parameters that are not too farfrom current experimental capabilities [32]. 8. Concluding Remarks We hope that the reader who has persevered to this point has acquired an impressionistic perspectiveof several quite different aspects of this rather active area of research. We have taken a more or less his-torical approach, first explaining how to design topological flat band models and providing an overviewof the numerical results, before drawing analytical parallels between fractional Chern insulators and FQHphysics in the lowest Landau level. We then briefly touched upon a host of interesting topics: construct-ing and classifying fractional Chern insulator wave functions using parton mean-field theories, the theoryof bandwidth-tuned transitions into FQH phases, and various aspects of lattice physics in bands of higherChern number. Each of these has spawned its own numerical and analytical offshoots, and we have attemptedto provide pointers to the relevant literature wherever possible. We also gave an overview of recent progressin developing experimentally realistic scenarios where these correlated phases emerge.In closing this review, we take a somewhat longer view, and outline several directions which seem (to us,at least) worthy of further study, although we hope that the preceding discussion has already prompted ourreaders to form similar questions independent of these. Once again, we sound a note of caution: given therapid progress in the field, it is quite likely that this list of future directions might quite rapidly becomedated as new results emerge on various fronts. Disorder.— The first of these is the important and (at this point) relatively little-studied problem ofdisorder in the Chern band. Here it is worth noting that in addition to the square root of the mean Berrycurvature (which as we have seen plays a role analogous to the magnetic length in the LLL), we have24 second (inverse) length scale: the characteristic momentum-space scale k σ for variations of the Berrycurvature. Na¨ıvely we might expect that the role of disorder depends on how its strength – as quantified bythe mean free path, l MF – compares to one or both of these scales. For instance, in a problem with sufficientdisorder, most notably the problem of localization within the CB, we expect that impurity scattering willeffectively lead to an averaging of the curvature over the band and give results similar to those in the LLL.This is consistent with what is known about the (integer) QH transition in Chern insulators [71] although adirect test in the projected CB would be desirable. A simple estimate for the requisite strength of disordercan be made by comparing l MF to k − σ ; for disorder sufficiently strong that l MF ∼ k − σ , the random potentialwill scatter between points in the BZ that are k σ apart, and thereby average their Berry curvature. Moreambitiously, it would be quite instructive to construct a global phase diagram for fractional Chern insulators,akin to that proposed by Kivelson, Lee and Zhang [72] for the FQHE in the LLL. Here specifically it is worthrecalling that the full global phase diagram for the continuum Landau problem emerges only when mixingbetween different LLs is taken into account; for fractional Chern bands it is possible that the mixing betweenBloch bands leads to qualitatively different physics. Long-Range Interactions and Competing Phases.— Turning for a moment to the continuum problem, weobserve that there fractionalized phases emerge in a comparatively narrow window of parameter space: theycompete with integer QH liquids as well as electronic crystalline phases, the latter in particular drawingenhanced stability from long-range Coulomb physics. We note that so far numerical studies in the Chernband have been largely confined to Hamiltonians with relatively short-ranged interactions. For the FQHEthis is typically justified by invoking screening to argue that the Coulomb repulsion is operational over nomore than a few magnetic lengths – although we note that even in the LLL there are well-known exampleswhere the long-range tail of the Coulomb interactions carries a sting [73]. In a CB in contrast, the onlylength scale is the inter-atomic spacing, and even a screened interaction can be quite long-ranged on thelattice scale. The role of such longer-ranged interactions in stabilizing or inhibiting the formation of FQHphases, and in particular in supporting competing crystalline phases, may be quite relevant to experimentalsystems. We note that some numerical work has begun to explore competition between phases in the Chernband [74]. Edge States.— An integral aspect of the theory of the FQHE is the Luttinger liquid description of thegapless edge modes [75]. The structure of the edge of a fractional Chern insulator has received comparablylittle attention. While presumably the universal edge properties are captured by the standard chiral Luttingerliquid theory, it is possible that the ability to produce a lattice-scale edge and the interplay of latticesymmetries with the edge may provide a source of additional complexity, worth exploring. Experimental Detection.— As we have described already, experiments on fractional Chern insulatorsremain in their infancy, although several proposals seem quite compelling and suggest that a realization evenwith existing technology is not too far off. In this context it is important to develop techniques to characterizethe microscopic properties of topological band structure, which as we have seen play an important role instabilizing correlated phases. In light of this, a recent proposal [76] to measure the Zak phase in Blochbands using a combination of Ramsey intereference and Bloch oscillations are promising, and have beendemonstrated in experiments on 1D optical lattices [77]. Equally important is the development of methodsto identify correlated topological phases, particularly since the presence of the lattice leads to ambiguitiesin interpreting standard measures such as the Hall conductance. Matrix Product and Tensor Network Representations.— The study of gapped phases in one dimensionhas benefited greatly from the observation that they have ground states that can be efficiently representedas ‘matrix product states’ [78]. The power of the matrix product form is twofold: first, the density-matrixrenormalization group (DMRG) algorithm [79,80] efficiently obtains the best matrix product representationof a ground state of a given local Hamiltonian; and second, the structure of the MPS allows efficient analysisof its behavior under symmetry operations. Similar tensor network constructions exist in higher dimensions[81] but a full understanding of these is still being built. The FQH phases are gapped correlated phases,and recently several groups have reformulated the model wavefunctions as matrix product states [82,83]. Itwould be quite instructive to develop a similar understanding for FQH phases in Chern bands, as well as todevelop efficient algorithms to simulate them. Extensions.— We turn finally to the important and interesting question of what other correlated topolog-25cal phases might emerge in lattice models. One obvious extension that has already received quite some nu-merical and analytical attention is the problem of constructing interacting analogs of time-reversal invarianttopological insulators in two and three dimensions. Unlike the FQHE, this is a problem with little or no priorhistory, so there are many open questions; some work along these directions already exists [84,85,86,87,88,89].A more recent development is the classification of ‘symmetry-protected topological phases’ (SPTs) [90]. Theseare gapped phases which have no topological order (they lack ground-state degeneracy and fractionalizedexcitations) but still host gapless edge states, and cannot be adiabatically continued to a trivial gappedphase as long as a certain symmetry is left unbroken – for instance, topological band insulators of non-interacting fermions may be thought of as SPTs protected by time-reversal symmetry, while the Haldanephase of a spin-1 chain is protected by π -spin rotation, time reversal, or inversion symmetry[91,92] . Sincethe understanding of SPTs grew out of very similar considerations as the integer and fractional QHE, itis quite natural to consider Chern bands as a natural venue where they might be realized. A final note ofcomplexity emerges when one considers the interplay of lattice symmetries and topological order, as in afew examples above; work in this direction has potential to reveal signatures of hard-to-detect topologicalphenomena in more conventional probes of the solid state. 9. Acknowledgements Our understanding of various topics discussed here has benefited greatly from discussions with severalcolleagues, and it would be impossible to thank them each individually. However, we would be remiss ifwe did not acknowledge B. A. Bernevig, N. Regnault, S. H. Simon, X.-L. Qi, M. Barkeshli, R. Thomale,R. Shankar, G. Murthy, G. M¨oller, M. Zaletel, and F. D. M. Haldane for several conversations and patientexplanations of their work. In addition, SLS thanks B. A. Bernevig for numerous insightful discussions onthe feasibility of studying correlated phases emerging from Hamiltonians projected into topological bandssome time prior to the recent explosion of work, which inspired our own work [20]. We also thank the theauthors of Refs. [15,16,18] for kindly providing us with figures of their numerical results. We acknowledgesupport from the Simons Foundation (SAP) and the NSF through grants DMR-1006608 and PHY-1005429(SAP, SLS), the EPSRC through grant EP/D050952/1 and UCLA Startup Funds (RR). References [1] K. von Klitzing, G. Dorda, M. Pepper, New method for high-accuracy determination of the fine-structure constant basedon quantized Hall resistance, Phys. Rev. Lett. 45 (6) (1980) 494. doi:10.1103/PhysRevLett.45.494.[2] D. C. Tsui, H. L. St¨ormer, A. C. Gossard, Two-dimensional magnetotransport in the extreme quantum limit, Phys. Rev.Lett. 48 (22) (1982) 1559. doi:10.1103/PhysRevLett.48.1559.[3] R. Laughlin, Anomalous quantum hall effect: an incompressible quantum fluid with fractionally charged excitations, Phys.Rev. Lett. 50 (18) (1983) 1395–1398.[4] X.-G. Wen, Topological order in rigid states, Int. J. Mod. Phys. B4 (1990) 239. doi:10.1142/S0217979290000139.[5] R. B. Laughlin, Anomalous quantum Hall effect: An incompressible quantum fluid with fractionally charged excitations,Phys. Rev. Lett. 50 (18) (1983) 1395. doi:10.1103/PhysRevLett.50.1395.[6] F. D. M. Haldane, Fractional quantization of the Hall effect: a hierarchy of incompressible quantum fluid states, Phys.Rev. Lett. 51 (7) (1983) 605. doi:10.1103/PhysRevLett.51.605.[7] B. I. Halperin, Statistics of quasiparticles and the hierarchy of fractional quantized Hall states, Phys. Rev. Lett. 52 (18)(1984) 1583. doi:10.1103/PhysRevLett.52.1583.[8] G. Moore, N. Read, Nonabelions in the fractional quantum Hall effect, Nucl. Phys. B 360 (2-3) (1991) 362.[9] S.-C. Zhang, T. H. Hansson, S. A. Kivelson, Effective field theory model for the fractional quantum Hall effect, Phys. Rev.Lett. 62 (1) (1989) 82.22. Note that while SPTs are not topologically ordered, they may still be correlated, as exemplified by bosonic SPT insulators. 10] S. C. Zhang, The chern-simons-landau-ginzburg theory of the fractional quantum hall effect, Int. J. Mod. Phys. B 6 (1)(1992) 43–77.[11] D. J. Thouless, M. Kohmoto, M. P. Nightingale, M. den Nijs, Quantized Hall conductance in a two-dimensional periodicpotential, Phys. Rev. Lett. 49 (6) (1982) 405. doi:10.1103/PhysRevLett.49.405.[12] F. D. M. Haldane, Model for a quantum hall effect without landau levels: Condensed-matter realization of the ”parityanomaly”, Phys. Rev. Lett. 61 (18) (1988) 2015–2018. doi:10.1103/PhysRevLett.61.2015.[13] E. Tang, J.-W. Mei, X.-G. Wen, High-temperature fractional quantum hall states, Phys. Rev. Lett. 106 (23) (2011) 236802.doi:10.1103/PhysRevLett.106.236802.[14] K. Sun, Z.-C. Gu, H. Katsura, S. Das Sarma, Nearly flatbands with nontrivial topology, Phys. Rev. Lett. 106 (23) (2011)236803. doi:10.1103/PhysRevLett.106.236803.[15] T. Neupert, L. Santos, C. Chamon, C. Mudry, Fractional quantum hall states at zero magnetic field, Phys. Rev. Lett.106 (23) (2011) 236804. doi:10.1103/PhysRevLett.106.236804.[16] D. Sheng, Z. Gu, K. Sun, L. Sheng, Fractional quantum hall effect in the absence of landau levels, Nat. Commun. 2 (2011)389.[17] Y. F. Wang, H. Yao, Z.-C. Gu, C.-D. Gong, D. N. Sheng, Non-abelian quantum hall effect in topological flat bands, Phys.Rev. Lett. 108 (2012) 126805. doi:10.1103/PhysRevLett.108.126805.URL http://link.aps.org/doi/10.1103/PhysRevLett.108.126805 [18] N. Regnault, B. Bernevig, Fractional chern insulator, Phys. Rev. X 1 (2) (2011) 021014.[19] X.-L. Qi, Generic wave-function description of fractional quantum anomalous hall states and fractional topologicalinsulators, Phys. Rev. Lett. 107 (2011) 126803. doi:10.1103/PhysRevLett.107.126803.URL http://link.aps.org/doi/10.1103/PhysRevLett.107.126803 [20] S. Parameswaran, R. Roy, S. Sondhi, Fractional chern insulators and the w ∞ algebra, Phys. Rev. B 85 (24) (2012) 241308.[21] R. Roy, Band geometry of fractional topological insulators, arXiv:1208.2055.[22] G. Murthy, R. Shankar, Composite fermions for fractionally filled chern bands, arXiv:1108.5501.[23] G. Murthy, R. Shankar, Hamiltonian theory of fractionally filled Chern bands, arXiv:1207.2133.[24] A. S. Sørensen, E. Demler, M. D. Lukin, Fractional quantum hall states of atoms in optical lattices, Phys. Rev. Lett. 94 (8)(2005) 086803. doi:10.1103/PhysRevLett.94.086803.[25] G. M¨oller, N. R. Cooper, Composite fermion theory for bosonic quantum hall states on lattices, Phys. Rev. Lett. 103 (10)(2009) 105303. doi:10.1103/PhysRevLett.103.105303.[26] S. M. Girvin, A. H. MacDonald, P. M. Platzman, Magneto-roton theory of collective excitations in the fractional quantumHall effect, Phys. Rev. B 33 (4) (1986) 2481–2494. doi:10.1103/PhysRevB.33.2481.[27] M. Barkeshli, J. McGreevy, Continuous transitions between composite fermi liquid and landau fermi liquid: A route tofractionalized mott insulators, Phys. Rev. B 86 (2012) 075136. doi:10.1103/PhysRevB.86.075136.URL http://link.aps.org/doi/10.1103/PhysRevB.86.075136 [28] M. Barkeshli, J. McGreevy, eprint arxiv:1201.4393 (1 2012).[29] Y. Zhang, A. Vishwanath, Establishing non-Abelian topological order in Gutzwiller projected Chern insulators viaEntanglement Entropy and Modular S-matrix, ArXiv e-printsarXiv:1209.2424.[30] D. Xiao, W. Zhu, Y. Ran, N. Nagaosa, S. Okamoto, Interface engineering of quantum hall effects in digital transition metaloxide heterostructures, Nat Commun 2.URL http://dx.doi.org/10.1038/ncomms1602 [31] N. R. Cooper, Optical flux lattices for ultracold atomic gases, Phys. Rev. Lett. 106 (2011) 175301.doi:10.1103/PhysRevLett.106.175301.URL http://link.aps.org/doi/10.1103/PhysRevLett.106.175301 [32] N. Y. Yao, A. V. Gorshkov, C. R. Laumann, A. M. L¨auchli, J. Ye, M. D. Lukin, Realizing Fractional Chern Insulatorswith Dipolar Spins, ArXiv e-printsarXiv:1212.4839.[33] N. Y. Yao, C. R. Laumann, A. V. Gorshkov, S. D. Bennett, E. Demler, P. Zoller, M. D. Lukin, Topological flat bands fromdipolar spin systems, Phys. Rev. Lett. 109 (2012) 266804. doi:10.1103/PhysRevLett.109.266804.URL http://link.aps.org/doi/10.1103/PhysRevLett.109.266804 34] X. Hu, M. Kargarian, G. A. Fiete, Topological insulators and fractional quantum hall effect on the ruby lattice, Phys. Rev.B 84 (15) (2011) 155116.[35] S. Yang, Z.-C. Gu, K. Sun, S. Das Sarma, Topological flat band models with arbitrary chern numbers, Phys. Rev. B 86(2012) 241112. doi:10.1103/PhysRevB.86.241112.URL http://link.aps.org/doi/10.1103/PhysRevB.86.241112 [36] F. Wang, Y. Ran, Nearly flat band with chern number c=2 on the dice lattice, Phys. Rev. B 84 (24) (2011) 241103.[37] M. Trescher, E. J. Bergholtz, Flat bands with higher chern number in pyrochlore slabs, Phys. Rev. B 86 (24) (2012) 241111.[38] F. D. M. Haldane, Many-particle translational symmetries of two-dimensional electrons at rational landau-level filling,Phys. Rev. Lett. 55 (1985) 2095–2098. doi:10.1103/PhysRevLett.55.2095.URL http://link.aps.org/doi/10.1103/PhysRevLett.55.2095 [39] A. Kol, N. Read, Fractional quantum hall effect in a periodic potential, Phys. Rev. B 48 (1993) 8890–8898.doi:10.1103/PhysRevB.48.8890.URL http://link.aps.org/doi/10.1103/PhysRevB.48.8890 [40] F. D. M. Haldane, “fractional statistics” in arbitrary dimensions: A generalization of the pauli principle, Phys. Rev. Lett.67 (1991) 937–940. doi:10.1103/PhysRevLett.67.937.URL http://link.aps.org/doi/10.1103/PhysRevLett.67.937 [41] B. A. Bernevig, N. Regnault, Emergent many-body translational symmetries of abelian and non-abelian fractionally filledtopological insulators, Phys. Rev. B 85 (2012) 075128. doi:10.1103/PhysRevB.85.075128.URL http://link.aps.org/doi/10.1103/PhysRevB.85.075128 [42] A. Kitaev, J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96 (2006) 110404.doi:10.1103/PhysRevLett.96.110404.URL http://link.aps.org/doi/10.1103/PhysRevLett.96.110404 [43] M. Levin, X.-G. Wen, Detecting topological order in a ground state wave function, Phys. Rev. Lett. 96 (2006) 110405.doi:10.1103/PhysRevLett.96.110405.URL http://link.aps.org/doi/10.1103/PhysRevLett.96.110405 [44] H. Li, F. D. M. Haldane, Entanglement spectrum as a generalization of entanglement entropy: Identification oftopological order in non-abelian fractional quantum hall effect states, Phys. Rev. Lett. 101 (1) (2008) 010504.doi:10.1103/PhysRevLett.101.010504.[45] M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666–669. doi:10.1103/PhysRevLett.71.666.URL http://link.aps.org/doi/10.1103/PhysRevLett.71.666 [46] Y. Zhang, T. Grover, A. Turner, M. Oshikawa, A. Vishwanath, Quasiparticle statistics and braiding from ground-stateentanglement, Phys. Rev. B 85 (2012) 235151. doi:10.1103/PhysRevB.85.235151.URL http://link.aps.org/doi/10.1103/PhysRevB.85.235151 [47] C. H. Lee, R. Thomale, X.-L. Qi, Pseudopotential Formalism for Fractional Chern Insulators, ArXiv e-printsarXiv:1207.5587.[48] Z. Liu, E. J. Bergholtz, From fractional chern insulators to abelian and non-abelian fractional quantum hall states: Adiabaticcontinuity and orbital entanglement spectrum, Phys. Rev. B 87 (2013) 035306. doi:10.1103/PhysRevB.87.035306.URL http://link.aps.org/doi/10.1103/PhysRevB.87.035306 [49] S. A. Trugman, Localization, percolation, and the quantum Hall effect, Phys. Rev. B 27 (12) (1983) 7539.doi:10.1103/PhysRevB.27.7539.[50] Y.-L. Wu, N. Regnault, B. A. Bernevig, Gauge-fixed wannier wave functions for fractional topological insulators, Phys.Rev. B 86 (2012) 085129. doi:10.1103/PhysRevB.86.085129.URL http://link.aps.org/doi/10.1103/PhysRevB.86.085129 [51] E. Brown, Bloch electrons in a uniform magnetic field, Phys. Rev. 133 (4A) (1964) A1038–A1044.doi:10.1103/PhysRev.133.A1038.[52] R. Roy, unpublished (2011).[53] N. Regnault, B. A. Bernevig, personal communication (2011).[54] S. Boldyrev, V. Gurarie, The integer quantum Hall transition and random su(N) rotation, J. Phys. Cond. Mat. 15 (2003)L125–L132. doi:10.1088/0953-8984/15/4/103.[55] D. Page, Geometrical description of berry’s phase, Phys. Rev. A 36 (1987) 3479–3481. 56] J. Anandan, Y. Aharonov, Geometry of quantum evolution, Phys. Rev. Lett. 65 (1990) 1697–1700.doi:10.1103/PhysRevLett.65.1697.[57] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry: Vol.: 2, Interscience Publishers, New York, 1969.[58] A. K. Pati, Relation between “phases” and “distance” in quantum evolution, Physics Letters A 159 (3) (1991) 105 – 112.doi:10.1016/0375-9601(91)90255-7.[59] R. Resta, The insulating state of matter: a geometrical theory, European Physical Journal B 79 (2011) 121–137.arXiv:1012.5776, doi:10.1140/epjb/e2010-10874-4.[60] T. Neupert, L. Santos, S. Ryu, C. Chamon, C. Mudry, Noncommutative geometry for three-dimensional topologicalinsulators, Phys. Rev. B 86 (2012) 035125. doi:10.1103/PhysRevB.86.035125.URL http://link.aps.org/doi/10.1103/PhysRevB.86.035125 [61] C. Chamon, C. Mudry, Magnetic translation algebra with or without magnetic field in the continuum or on arbitrarybravais lattices in any dimension, Phys. Rev. B 86 (2012) 195125. doi:10.1103/PhysRevB.86.195125.URL http://link.aps.org/doi/10.1103/PhysRevB.86.195125 [62] B. Estienne, N. Regnault, B. A. Bernevig, d -algebra structure of topological insulators, Phys. Rev. B 86 (2012) 241104.doi:10.1103/PhysRevB.86.241104.URL http://link.aps.org/doi/10.1103/PhysRevB.86.241104 [63] Y. M. Lu, Y. Ran, Symmetry-protected fractional chern insulators and fractional topological insulators, Phys. Rev. B 85(2012) 165134. doi:10.1103/PhysRevB.85.165134.[64] X.-G. Wen, Quantum orders and symmetric spin liquids, Phys. Rev. B 65 (2002) 165113. doi:10.1103/PhysRevB.65.165113.URL http://link.aps.org/doi/10.1103/PhysRevB.65.165113 [65] J. McGreevy, B. Swingle, K.-A. Tran, Wave functions for fractional chern insulators, Phys. Rev. B 85 (2012) 125105.doi:10.1103/PhysRevB.85.125105.[66] Y. Zhang, T. Grover, A. Vishwanath, Topological entanglement entropy of Z spin liquids and lattice laughlin states, Phys.Rev. B 84 (2011) 075128. doi:10.1103/PhysRevB.84.075128.URL http://link.aps.org/doi/10.1103/PhysRevB.84.075128 [67] M. Barkeshli, X.-L. Qi, Topological nematic states and non-abelian lattice dislocations, Phys. Rev. X 2 (2012) 031013.doi:10.1103/PhysRevX.2.031013.URL http://link.aps.org/doi/10.1103/PhysRevX.2.031013 [68] Y.-L. Wu, N. Regnault, B. A. Bernevig, Bloch Model Wavefunctions and Pseudopotentials for All Fractional ChernInsulators, ArXiv e-printsarXiv:1210.6356.[69] M. Barkeshli, C.-M. Jian, X.-L. Qi, Genons, twist defects, and projective non-Abelian braiding statistics, ArXiv e-printsarXiv:1208.4834.[70] N. R. Cooper, J. Dalibard, Reaching Fractional Quantum Hall States with Optical Flux Lattices, ArXiv e-printsarXiv:1212.3552.[71] M. Onoda, N. Nagaosa, Quantized anomalous hall effect in two-dimensional ferromagnets: Quantum hall effect in metals,Phys. Rev. Lett. 90 (20) (2003) 206601. doi:10.1103/PhysRevLett.90.206601.[72] S. A. Kivelson, D.-H. Lee, S.-C. Zhang, Global phase diagram in the quantum Hall effect, Phys. Rev. B 46 (4) (1992) 2223.[73] S. L. Sondhi, S. A. Kivelson, Long-range interactions and the quantum Hall effect, Phys. Rev. B 46 (20) (1992) 13319.[74] A. M. L¨auchli, Z. Liu, E. J. Bergholtz, R. Moessner, Hierarchy of fractional Chern insulators and competing compressiblestates, ArXiv e-printsarXiv:1207.6094.[75] X.-G. Wen, Chiral Luttinger liquid and the edge excitations in the fractional quantum Hall states, Phys. Rev. B 41 (18)(1990) 12838. doi:10.1103/PhysRevB.41.12838.[76] D. A. Abanin, T. Kitagawa, I. Bloch, E. Demler, Interferometric approach to measuring band topology in 2D opticallattices, ArXiv e-printsarXiv:1212.0562.[77] M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin, T. Kitagawa, E. Demler, I. Bloch, Direct Measurement of the Zakphase in Topological Bloch Bands, ArXiv e-printsarXiv:1212.0572.[78] M. Fannes, B. Nachtergaele, R. F. Werner, Finitely correlated states on quantum spin chains, Communications inMathematical Physics 144 (1992) 443–490. doi:10.1007/BF02099178. 79] S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69 (1992) 2863–2866.doi:10.1103/PhysRevLett.69.2863.URL http://link.aps.org/doi/10.1103/PhysRevLett.69.2863 [80] S. ¨Ostlund, S. Rommer, Thermodynamic limit of density matrix renormalization, Phys. Rev. Lett. 75 (1995) 3537–3540.doi:10.1103/PhysRevLett.75.3537.URL http://link.aps.org/doi/10.1103/PhysRevLett.75.3537 [81] F. Verstraete, J. I. Cirac, Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions,eprint arXiv:cond-mat/0407066arXiv:arXiv:cond-mat/0407066.[82] M. P. Zaletel, R. S. K. Mong, Exact matrix product states for quantum hall wave functions, Phys. Rev. B 86 (2012) 245305.doi:10.1103/PhysRevB.86.245305.URL http://link.aps.org/doi/10.1103/PhysRevB.86.245305 [83] B. Estienne, Z. Papic, N. Regnault, B. A. Bernevig, Matrix Product States and the Fractional Quantum Hall Effect, ArXive-printsarXiv:1211.3353.[84] M. Levin, A. Stern, Fractional topological insulators, Phys. Rev. Lett. 103 (2009) 196803.doi:10.1103/PhysRevLett.103.196803.URL http://link.aps.org/doi/10.1103/PhysRevLett.103.196803 [85] J. Maciejko, X.-L. Qi, A. Karch, S.-C. Zhang, Fractional topological insulators in three dimensions, Phys. Rev. Lett. 105(2010) 246809. doi:10.1103/PhysRevLett.105.246809.URL http://link.aps.org/doi/10.1103/PhysRevLett.105.246809 [86] B. Swingle, M. Barkeshli, J. McGreevy, T. Senthil, Correlated topological insulators and the fractional magnetoelectriceffect, Phys. Rev. B 83 (2011) 195139. doi:10.1103/PhysRevB.83.195139.URL http://link.aps.org/doi/10.1103/PhysRevB.83.195139 [87] T. Neupert, L. Santos, S. Ryu, C. Chamon, C. Mudry, Fractional topological liquids with time-reversal symmetry andtheir lattice realization, Phys. Rev. B 84 (2011) 165107. doi:10.1103/PhysRevB.84.165107.[88] L. Santos, T. Neupert, S. Ryu, C. Chamon, C. Mudry, Time-reversal symmetric hierarchy of fractional incompressibleliquids, Phys. Rev. B 84 (2011) 165138. doi:10.1103/PhysRevB.84.165138.URL http://link.aps.org/doi/10.1103/PhysRevB.84.165138 [89] M. Levin, F. J. Burnell, M. Koch-Janusz, A. Stern, Exactly soluble models for fractional topological insulators in two andthree dimensions, Phys. Rev. B 84 (2011) 235145. doi:10.1103/PhysRevB.84.235145.URL http://link.aps.org/doi/10.1103/PhysRevB.84.235145 [90] X. Chen, Z.-C. Gu, Z.-X. Liu, X.-G. Wen, Symmetry protected topological orders in interacting bosonic systems, ArXive-printsarXiv:1301.0861.[91] F. Pollmann, A. M. Turner, E. Berg, M. Oshikawa, Entanglement spectrum of a topological phase in one dimension, Phys.Rev. B 81 (2010) 064439. doi:10.1103/PhysRevB.81.064439.URL http://link.aps.org/doi/10.1103/PhysRevB.81.064439 [92] F. Pollmann, E. Berg, A. M. Turner, M. Oshikawa, Symmetry protection of topological phases in one-dimensional quantumspin systems, Phys. Rev. B 85 (2012) 075125. doi:10.1103/PhysRevB.85.075125.URL http://link.aps.org/doi/10.1103/PhysRevB.85.075125http://link.aps.org/doi/10.1103/PhysRevB.85.075125