Topological Bands for Ultracold Atoms

REVIEWS OF MODERN PHYSICS, VOLUME 91, JANUARY–MARCH 2019 Topological bands for ultracold atoms

N. R. Cooper T.C.M. Group, Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom

J. Dalibard Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-Université PSL, Sorbonne Université, 11 place Marcelin Berthelot, 75005, Paris, France

I. B. Spielman Joint Quantum Institute, National Institute of Standards and Technology and University of Maryland, Gaithersburg, Maryland 20899, USA

(published 25 March 2019) There have been significant recent advances in realizing band structures with geometrical and topological features in experiments on cold atomic gases. This review summarizes these developments, beginning with a summary of the key concepts of geometry and topology for Bloch bands. Descriptions are given of the different methods that have been used to generate these novel band structures for cold atoms and of the physical observables that have allowed their characterization. The focus is on the physical principles that underlie the different experimental approaches, providing a conceptual framework within which to view these developments. Also described is how specific experimental implementations can influence physical properties. Moving beyond single-particle effects, descriptions are given of the forms of interparticle interactions that emerge when atoms are subjected to these energy bands and of some of the many-body phases that may be sought in future experiments.

DOI: 10.1103/RevModPhys.91.015005

CONTENTS 3. Topological bands and spin-orbit coupling 22 4. Momentum distribution and edges states 23 I. Introduction 2 B. Wave-packet analysis of the BZ topology 24 II. Topology of Bloch Bands 3 1. Bloch oscillations and Zak phase in 1D 24 A. Band theory 3 2. Measurement of the anomalous velocity 24 B. Geometrical phase 3 3. Interferometry in the BZ 25 C. Topological invariants 4 4. Direct imaging of edge magnetoplasmons 26 1. The Zak phase 4 C. Transport measurements 27 2. The Chern number 6 1. Adiabatic pumping 27 D. Edge states 7 2. Center-of-mass dynamics in 2D 28 1. SSH model 8 V. Interaction Effects 28 2. Haldane model 8 A. Two-body interactions 28 E. Topological insulators 9 1. Beyond contact interactions 28 1. Discrete symmetries 9 a. Continuum models 29 2. The Rice-Mele model 9 b. Tight-binding models 29 F. Adiabatic pumping 10 c. Synthetic dimensions 30 III. Implementations of Topological Lattices 11 d. Current-density coupling 30 A. Iconic models 11 2. Floquet heating 30 1. Harper-Hofstadter model 11 B. Many-body phases 31 2. Haldane model 12 1. Bose-Einstein condensates 31 B. Realization of SSH model 13 2. Topological superfluids 32 C. Inertial forces 15 3. Fractional quantum Hall states 34 D. Resonant coupling: Laser-assisted tunneling 17 a. Bosons 34 E. Synthetic dimensions 18 b. Lattice effects 34 F. Flux lattices: Intrinsic topology 19 c. Symmetry-protected topological phases 34 1. Flux lattices 19 d. Ladders 35 2. Fluxless lattices 20 4. Other strongly correlated phases 35 IV. Experimental Consequences 21 a. Chiral Mott insulator 35 A. Characterization of equilibrium properties 21 b. Chiral spin states 35 1. Time-of-flight measurements 21 c. Number-conserving topological superfluids 35 2. Local measurement of the Berry curvature 21 C. Experimental perspectives 35

1. Equilibrium observables 35 bands, a result that follows from the existence of a conserved 2. Collective modes 36 crystal momentum via Bloch’s theorem. Remarkably, under 3. Edge states 36 certain circumstances, each Bloch energy band can be 4. rf excitation 36 assigned a robust integer-valued topological invariant related 5. Adiabatic pumping 36 to how the quantum wave function of the electron “twists” as a “ ” 6. Hall conductivity from heating 36 function of crystal momentum. This integer is invariant under VI. Outlook 37 continuous changes of material properties, “continuous” A. Turning to atomic species from the lanthanide family 37 meaning that the energy gaps to other bands should not close. B. Topological lattices without light 37 The first example of such a topological invariant for Bloch C. Other topological insulators and topological metals 37 energy bands arose from a highly original analysis of the D. Far-from-equilibrium dynamics 38 E. Invariants in Floquet-Bloch systems 39 integer quantum Hall effect in a two-dimensional (2D) lattice F. Open systems 41 (Thouless et al., 1982). Recent theoretical breakthroughs have VII. Summary 42 shown that, once additional symmetries are included, topo- Acknowledgments 42 logical invariants are much more widespread. These ideas Appendix A: Topological Bands in One Dimension 43 underpin a recent revolution in our understanding of band 1. Edges states in the SSH model 43 insulators and superconductors (Hasan and Kane, 2010; Qi a. Semi-infinite chain 43 and Zhang, 2011). The topological nature of these materials b. Finite chain 43 endows them with physical characteristics that are insensitive 2. Gauge invariance and Zak phase 44 to microscopic details, a notable example being the exact 3. Time-reversal symmetry of the SSH model 44 quantization of the Hall resistance in 2D irrespective of the 4. Adiabatic pumping for the Rice-Mele model 44 presence or form of a random disorder potential. 5. The Kitaev model for topological superconductors 44 A great deal of current research focuses on understanding Appendix B: Floquet Systems and the Magnus Expansion 46 the physical consequences of these new materials, and Appendix C: Light-matter Interaction 47 experimental studies of topological insulators and supercon- Appendix D: Berry Curvature and Unit Cell Geometry 48 ductors in solid state systems continue apace. Furthermore, References 49 there is significant activity in exploring the nature of the strongly correlated phases of matter that arise in these I. INTRODUCTION materials, notably to construct strong-correlation variants of these topological states of weakly interacting electrons. Topology is a mathematical concept that refers to certain Theory suggests many interesting possibilities, which are still properties that are preserved under continuous deformations. seeking experimental realization and verification. One familiar example is the number of twists put into a belt Such questions are ideally addressed using realizations with before its buckle is fastened. Usually we aim to fasten a belt cold atomic gases. Cold atomic gases allow strongly interact- without any twists. But if we were to introduce a single twist ing phases of matter to be explored in controlled experimental we would produce a Möbius strip. No continuous deformation settings. However, a prerequisite for quantum simulations of of the closed belt would get rid of this uncomfortable twist. such issues is the ability to generate topological energy bands The number of twists is said to be a “topological invariant” of for cold atoms. This poses a significant challenge, even at this the closed belt. single-particle level. Realizing topological energy bands The importance of topological invariants in stabilizing typically requires either the introduction of effective orbital spatial deformations and defects is also well known in physics magnetic fields acting on neutral atoms and/or the introduc- in diverse areas ranging from cosmology to condensed matter. tion of a spin-orbit coupling between the internal spin states of For a superfluid confined to a ring, the number of times that an atom and its center-of-mass motion. This is an area of the superfluid phase ϕ changes by 2π around the ring research that has attracted significant attention over the last I years, both theoretical and experimental. Much progress has 1 been made in developing techniques to generate artificial N ∇ϕ l ¼ 2π ·d ð1Þ magnetic fields and spin-orbit coupling for neutral atoms (Zhai, 2015; Dalibard, 2016; Aidelsburger, Nascimbene, and is a topological invariant. This winding number cannot change Goldman, 2017). The use of these techniques in the setting of under smooth deformations of the superfluid: a change would optical lattices has led to the realization and characterization require the superfluid density to vanish somewhere—such that of topological Bloch bands for cold atoms. ϕ is ill defined—requiring the motion of a quantized vortex In this review we describe how topological energy bands line across the ring. Here the topological stability arises from can be generated and probed in cold atom systems. We focus the interplay of the underlying space (the ring) and the form of on existing experimental studies, for which the essential the local order parameter (the phase of the superfluid wave behavior can be understood in terms of noninteracting function). particles. We start by explaining the concepts underpinning In recent years it has come to be understood that topology topological energy bands in Sec. II. We describe the key enters physics in another, very fundamental way through the physical effects that are required to generate these bands and nature of the quantum states of particles moving through how these can be engineered in cold atom systems in Sec. III. crystalline lattices. The energy eigenstates of electrons mov- Our emphasis is on recent experimental developments. In ing through periodic potentials are well known to form energy Sec. IV we describe the principal observables that have been

Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019 015005-2 N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

n used to characterize the geometrical and topological charac- one readily finds that the Bloch states uqðrÞ are eigenstates of ters of the resulting energy bands. In Sec. V we move beyond the q-dependent Hamiltonian single-particle physics to discuss some of the theoretical 1 understanding of the consequences of interactions in these ˆ 2 Hq ¼ − ð∇ þ iqÞ þ VðrÞ : ð4Þ novel optical lattices and to describe some of the interacting 2m many-body phases that can be sought in future experiments on If there are internal degrees of freedom, e.g., spin, then one these systems. We conclude in Sec. VI with comments on the n n should replace uqðrÞ by uqðα; rÞ with α labeling these addi- outlook for future work and point out connections to broader research areas. tional degrees of freedom. States that differP in crystal momentum q by a reciprocal Throughout this review we explain just the essential physics G G lattice vector ¼ imi i are physically equivalent. (The underlying recent developments, so the content is necessarily G incomplete. The review should be used as a starting point from reciprocal lattice is constructed from basis vectors f ig defined by the condition G · a 2πδ .) Thus, q can be which to explore the literature, rather than as a comprehensive i j ¼ ij survey of the field. We note that we focus on topological bands chosen to be restricted to the first Brillouin zone (BZ): the q for atoms in periodic lattices. Related phenomena can appear locus of points that are closer to the origin than to any reciprocal lattice vector G. For example, for a 2D lattice with basis vectors for photons and hybrid light-matter objects (cavity polaritons) a 0 a 0 in novel optical materials, and we refer the interested readers 1 ¼ða1; Þ and 2 ¼ð ;a2Þ the reciprocal lattice has basis G1 2π=a1;0 G2 0;2π=a2 to the review by Ozawa et al. (2018). We note also that all our vectors ¼ð Þ and ¼ð Þ, and the BZ is the −π

Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019 015005-3 N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms tight-binding models with one orbital per unit cell M ¼ 1 (i.e., be imposed in physical systems by the application of external one-band models). However, topological features arise already forces, inducing adiabatic dynamics of the Bloch states. for two orbitals per unit cell, M ¼ 2. We describe topological classifications of Bloch bands in 1D and 2D and illustrate C. Topological invariants these using two-band tight-binding models. Our presentation relies heavily on the concept of the 1. The Zak phase geometrical phase (Berry, 1984), which we briefly review. ˆ Owing to the periodicity of the BZ under the addition of any Consider a Hamiltonian HðXÞ which depends on a set of G q reciprocal lattice vector , a trajectory in wave vector from i parameters X and with nondegenerate spectrum q q G to f ¼ i þ is a closed loop. Since the Hamiltonian is ˆ ˆ periodic, Hq Hq G, the ideas of Berry apply directly. The Hˆ ðXÞjΨðnÞðXÞi ¼ EðnÞðXÞjΨðnÞðXÞi: ð7Þ ¼ þ integral of the Berry connection along such closed loops The system is prepared in an eigenstate jΨðnÞðXÞi at an initial Z q G X ðnÞ iþ time ti, and the parameters t are changed slowly in time t ϕ ðnÞ ∂ ðnÞ q Zak ¼ ihu j qu i ·d ð14Þ q such that the state evolves adiabatically, following an instan- i ˆ taneous eigenstate of HðXtÞ. The parameters are taken around was proposed by Zak as a way to characterize the energy a cycle such that Xt ¼ Xt . Since the Hamiltonian returns to f i bands (Zak, 1989). the initial form at tf , so too must any eigenstate up to an overall phase factor (in this case of a nondegenerate spec- The first glimpse of how the mathematics of topological trum). This phase has both dynamical and geometrical invariants can arise in band theory is provided by computing contributions the Zak phase for simple two-band tight-binding models in one dimension (1D). We illustrate this for the Su-Schrieffer- ðnÞ i½γðnÞ þγðnÞ ðnÞ Heeger (SSH) model. jΨ i¼e dyn geo jΨ i; ð8Þ f Z i The SSH model is a tight-binding model in which there 1 t ðnÞ f ðnÞ 0 are two sites in the unit cell labeled A and B. The sites are γ ¼ − E ðX 0 Þdt ; ð9Þ dyn ℏ t connected by alternating tunnel couplings J and J0; see Fig. 1. ti I The single-particle Hamiltonian reads ðnÞ ðnÞ ðnÞ γgeo ¼ ihΨ j∂XΨ i ·dX: ð10Þ X ˆ − 0 ˆ † ˆ ˆ † ˆ H ¼ ðJ aj bj þ Jaj bj−1 þ H:c:Þ; ð15Þ The geometrical phase is the integral of the Berry connection j

AðnÞ X ≡ ΨðnÞ ∂ ΨðnÞ ˆ † ˆ † ð Þ ih j X ið11Þ where aj and bj create a particle on the A and B sites of the jth unit cell. For a bulk system with periodic boundary conditions, X around the closed loop in parameter space . The Berry we look for the energy eigenstates using the Bloch wave form connection plays a role similar to that of the vector potential X for a magnetic field. It is gauge dependent, varying under local jψ i¼ eijaqðuAjA iþuBjB iÞ; ð16Þ ΨðnÞ → iΦðXÞ ΨðnÞ X q q j q j gauge transformations j i e j i.However,if j has more than one component, one can define a gauge- invariant Berry curvature with a the lattice constant. The problem reduces to finding the A B T eigenvectors ðuq ;uq Þ of the Hamiltonian in reciprocal space, ΩðnÞ ≡∂ AðnÞ − ∂ AðnÞ: ð12Þ ij Xi j Xj i which has matrix representation If the closed loop X can be viewed as the boundary of a 2D 0 J0 þ Je−iqa surface X1;X2 on which the Berry curvature is everywhere ˆ ð Þ Hq ¼ − ð17Þ well defined, then the geometrical phase (10) is the flux of the J0 þ Jeiqa 0 Berry curvature within the BZ −π=a < q ≤ π=a. ΩðnÞ ≡ ϵ ∂ ðnÞ ϵ ∂ ΨðnÞ ∂ ΨðnÞ The Hamiltonian (17) can be written in the general form ij Xi Aj ¼ iji Xi h j Xj ið13Þ ϵ ˆ ˆ through this 2D surface. Here ij is the antisymmetric tensor Hq ¼ −hðqÞ · σ; ð18Þ of two indices ϵxy ¼ −ϵyx ¼ 1 and the summation over repeated indices is assumed. We apply these concepts to physical situations in which the role of external parameters X is played either by the crystal momentum q or by the real-space position r. In both cases, the Berry connection and Berry curvature define local geometric properties of the quantum states. The integrals of these FIG. 1. Tight-binding (SSH) model of the polyacetylene mol- geometric quantities over a closed manifold—the BZ for q, ecule. For the SSH model (17), the on-site energies for A and B or the unit cell of the lattice for r—give rise to topological are supposed to be equal. This constraint will be relaxed for the properties. As discussed in later sections, trajectories of q can Rice-Mele model (40).

Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019 015005-4 N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms where σˆ α are the Pauli operators. Throughout this review, we use the conventional matrix representation of the Pauli operators in which σˆ z is diagonal. Then the SSH model has

0 iqa hxðqÞþihyðqÞ¼J þ Je ;hz ¼ 0; ð19Þ with hx;yðqÞ real periodic functions of q. The energy spectrum is composed of the two bands

ðÞ 2 02 0 1=2 Eq ¼jhðqÞj ¼ ½J þ J þ 2JJ cosðqaÞ ; ð20Þ

h which are separated by a gap provided j j does not vanish at FIG. 2. The winding number for the SSH model. The curves any q. Assuming J, J0 > 0, the gap 2jJ − J0j closes only when 0 plot the locus of ðhx;hyÞ as q runs over the BZ. For J =J > 1 the J J0 q π=a 0 ¼ for the quasimomentum ¼ . vector ðhx;hyÞ does not encircle the origin, but for J =J < 1 it Provided there is a nonzero gap, i.e., jhj ≠ 0 for all q, one encircles the origin once, indicating that these two cases have iϕq can write hx þ ihy ¼jhje with a well-defined ϕq, and the winding numbers (23) of N ¼ 0 and 1, respectively. Bloch states are ! A perceptive reader will notice that the topological invariant uA 1 1 q ffiffiffi [Eq. (23)] we constructed appears to be rather unphysical. The ¼ p ϕ : ð21Þ B ∓ i q uq 2 e two parameter regimes of the SSH model which have different winding numbers, at J0=J > 1 and J0=J < 1, could be trivially In this pseudospin representation, the fact that hz ¼ 0 in related by reversing the labeling of the A and B sublattices in Eqs. (18) and (19) for the SSH model entails that these Fig. 1, at least deep in the bulk of the system. In what sense are eigenstates lie on the equator of the Bloch sphere. The these two parameter regimes topologically distinct? A key resulting Zak phases of the bands (14) are point to note is that the winding number (23) has a basis- Z dependent offset. In place of Eq. (16) we could just as well 1 ∂ϕ ϕðÞ − q have sought an energy eigenstate of the form Zak ¼ dq: ð22Þ 2 BZ ∂q X ψ ijaq ˜ A ˜ B j qi¼ e ðuq jAjþNA iþuq jBjþNB iÞ ð24Þ Thus, j Z X 1 1 ∂ϕ − − ≡ − ϕðÞ q ¼ eijaqðu˜ Ae iNAaqjA iþu˜ Be iNBaqjB iÞ; ð25Þ N Zak ¼ dq ð23Þ q j q j π 2π BZ ∂q j ϕ 2π is the number of times q changes by as q runs over the associating site A in the cell j þ N with site B in cell j þ N . ϕ ϕ 2π A B BZ. Since the Hamiltonian is periodic, q ¼ qþG modulo , Comparing Eq. (25) with (16) shows that this amounts to − − N A B T ˜ A iNAaq ˜ B iNBaq T is an integer winding number, analogous to that for the replacing ðuq ;uq Þ by ðuq e ; uq e Þ , i.e., to a q- phase of a superfluid around a ring (1). It measures the solid dependent unitary transformation Uˆ . In this new basis ðÞ q angle drawn by the pseudospins uq along the equator of the A B T ðu˜ q ; u˜ q Þ , the eigenstates (21) are replaced by Bloch sphere when q spans the BZ. ! The winding number (23) is a topological invariant of 1D A u˜ 1 eiqaNA band insulators arising from Hamiltonians of the form (18): N q ffiffiffi ¼ p ϕ : ð26Þ ˜ B i q iqaNB cannot be changed unless jhðqÞj vanishes at some q, i.e., uq 2 ∓ e e unless the band gap closes. The case of the SSH model is illustrated in Fig. 2, which shows the locus of ðhx;hyÞ as q For NA ¼ NB this transformation may be viewed as a runs over the BZ. For J0=J < 1 the curve encircles the origin reciprocal-space gauge transformation. A direct calculation once, N ¼ 1; while for J0=J > 1 the curve does not encircle of the Zak phase for Eq. (26) shows that the winding number 0 ≡ −ϕ π the origin, N ¼ . These two curves cannot be smoothly N Zak= is increased by NA þ NB as compared to that interconverted without crossing hx ¼ hy ¼ 0, i.e., without the for Eq. (21). This example shows that the absolute value of the band gap closing. One could evade this conclusion by winding number cannot be physically meaningful. Instead, including terms proportional to σˆ z in the Hamiltonian. physical consequences can involve only differences of wind- Then two gapped states with different N could be continu- ing numbers, which are well defined when computed within ously deformed into each other. [We explore this later for the the same reciprocal-space basis choice. In Sec. II.D we Rice-Mele (RM) model.] However, as explained in Sec. II.E, describe how the winding number difference at a boundary including σˆ z terms would break an underlying “chiral” between two regions influences the spectrum of states on the symmetry of the SSH model. The winding number in 1D edge. However, note that we already found one physical is an example of a topological invariant whose existence relies consequence of the winding number difference: the parameter on an underlying symmetry. space of microscopic couplings becomes disconnected, in the

Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019 015005-5 N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms sense that there is no way to continuously change the physical exploits internal atomic states, labeled by index α. Quite parameters to evolve between regions of different winding generally, the nondissipative action of laser light on an atom is numbers without crossing a critical point at which the band to couple state α to state α0 with a well-defined momentum gap closes. Such gap closings between topologically distinct transfer κ. An optical lattice is defined by a set of such α0α regions have direct physical consequences in measurements couplings Vκ , which to preserve periodicity must build up a α0α of the bulk excitation spectrum. In particular, for a set of regular lattice in momentum space. The couplings Vκ noninteracting fermions filling one such band, the gap closing therefore define a tight-binding model in momentum space implies a thermodynamic phase transition between two with amplitudes and phases determined by the laser fields. For insulating regimes, separated by a semimetallic state. We shallow lattices, the net phase acquired on encircling a closed illustrated this gap closing for the SSH model. However, this loop on this momentum-space lattice determines the inte- h 0 also holds for any 1D model in which ¼ðhx;hy; Þ, i.e., grated Berry curvature through that loop, allowing lattices with chiral symmetry. The requirement of the closing of a with broken TRS ΩðqÞ ≠−Ωð−qÞ to be directly constructed band gap in order to change the winding number is a defining (Cooper and Moessner, 2012). feature of a topological invariant of the energy band. To illustrate how a nonzero Chern number arises in tight- binding models, we discuss here the key features of the 2. The Chern number Haldane model (Haldane, 1988). This is a two-band model, so 2 2 A topological classification of (nondegenerate) energy the Bloch Hamiltonian is a × Hermitian matrix bands in 2D exists without requiring any underlying sym- ˆ −h q σˆ metry and without ambiguities related to the choice of basis Hq ¼ ð Þ · ð29Þ in reciprocal space. The topological invariant is the Chern h q number with ð Þ a three-component vector coupled to the Pauli matrices σˆ . Just as in the SSH model, the energy bands Z x;y;z 1 ðÞ h q h C ϵ ∂ ∂ 2 are Eq ¼j ð Þj and there is a gap provided j qj does ¼ i ij qi huqj qj juqid q: ð27Þ 2π BZ not vanish at any q. However, now the Bloch state of the lower band depends on the three-component unit vector We suppress the band index n, but note that a Chern number eðqÞ ≡ hðqÞ=jhðqÞj, e.g., exists for each band. This topological invariant, and terminol- ogy, arises from the mathematics of fiber bundles (Stone and cosðθq=2Þ uq 30 Goldbart, 2009). However, it can be readily interpreted more j i¼ iϕ ð Þ sinðθq=2Þe q physically in terms of the Berry curvature (13) of the Bloch states for eðqÞ¼ðsin θq cos ϕq; sin θq sin ϕq; cos θqÞ. The Chern number can be written in terms of this unit vector as ΩðqÞ ≡ ∇q × AðqÞ · ez ¼ iϵij∂q huqj∂q juqi: ð28Þ i j Z 1 ∂e ∂e 2 The Chern number is related to the flux of the Berry curvature C ¼ N2 ≡ − ϵ e · × d q ð31Þ D 8π ij ∂q ∂q ΩðqÞ through the BZ. Just as the Dirac quantization condition BZ i j requires the magnetic flux through a closed surface to be N2 counts the number of times that the unit vector e q wraps quantized (in units of h=e), the flux of the Berry curvature D ð Þ over the unit sphere as q spans the BZ; it is the 2D analog of through the BZ (a closed surface with the topology of a torus) the 1D winding number (23). is quantized (in units of 2π). This follows by using Stokes’s The Haldane model is described in detail in Sec. III.Itis theorem to relate the integral of ΩðqÞ over the BZ to the line defined on a honeycomb lattice, for which the unit cell integral of AðqÞ around its boundary. Since the BZ is a closed contains two sites, which we label A and B, as in Fig. 3(a). surface this line integral must be an integer multiple of 2π. Nearest-neighbor tunneling is off diagonal in the sublattice This integer C is the number of flux quanta of Berry curvature index and leads to through the BZ. Since the Berry curvature is a gauge-invariant quantity, so too is the Chern number. q q iq·a1 iq·a2 1 The Chern number vanishes for systems with time-reversal hxð Þþihyð Þ¼Jðe þ e þ Þð32Þ symmetry (TRS), for which uqðrÞ ∝ u−qðrÞ and hence ΩðqÞ¼ in Eq. (29), where a1 2 are the lattice vectors marked on −Ωð−qÞ. The realization of Chern bands therefore requires a ; Fig. 3(a). The Hamiltonian has been chosen to be periodic means to break TRS. In Sec. III we describe ways in which under the addition of reciprocal lattice vectors. Note, however, this can be achieved for cold atom systems. One class of that other choices can be made, related to the Hamiltonian via implementation involves tight-binding models, in which a ˆ → Uˆ ˆ Uˆ † spinless particle hops on a lattice with complex tunneling unitary transformations Hq qHq q. Indeed, in Sec. III we matrix elements, e.g., to represent the Peierls phase factors for replace a charged particle in a magnetic field. TRS is broken if, for q q → iq·ρ1 iq·ρ2 iq·ρ3 some closed loop on the lattice, the phases acquired on hxð Þþihyð Þ Jðe þ e þ e Þð33Þ ϕ encircling the loop in the clockwise (þ AB) and anticlockwise −ϕ 2π ϕ Uˆ − σˆ q ρ 2 ρ ( AB) directions differ modulo , i.e., provided AB is not which arises for q ¼ expð i z · 3= Þ, with 3 the nearest- an integer multiple of π. Another class of implementation neighbor lattice vector in Fig. 3(a). This transformation

Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019 015005-6 N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

interparticle interactions by defining a Chern number for the many-body ground state in a geometry with periodic boundary conditions (Niu, Thouless, and Wu, 1985). For noninteracting systems without translational invariance, a local Chern marker can be defined (Bianco and Resta, 2011). The quantization of the Hall conductance is intimately related to the existence of edge states, which we now discuss.

(a) (b) D. Edge states

FIG. 3. (a) Real-space honeycomb lattice for the Haldane Topological band insulators have the generic feature that model. (b) Illustration of the unit vector eðqÞ in the topological although they are bulk insulators—owing to the energy gap phase of the Haldane model. The dashed line shows the conven- between the filled and empty bands—they host gapless states Q tional BZ, whose corners are the points . The arrows show the on their surfaces. e q projection of ð Þ on the plane ðex;eyÞ, while the colors and The existence of gapless edge modes for 2D systems with contours indicate ez. The unit vector wraps once over the sphere nonzero Chern number is well known from studies of the C 1 within the BZ, indicating the topological character with ¼ . integer quantum Hall effect (Halperin, 1982). Each filled Landau level gives rise to a chiral edge mode. This can be does not change the energy spectrum nor the topology of understood semiclassically in terms of the skipping orbit of the band. However, it can be more helpful to work with the cyclotron motion around the edge of the sample; see Eq. (33) when considering physical observables (Bena and Fig. 4. These semiclassical skipping orbits consist of two Montambaux, 2009). As discussed in Appendix D, unitary features. The rapid skipping motion at the cyclotron frequency transformations of the form Uˆ affect the definition of the force is a feature which in a quantum description arises when the q wave function has nonzero amplitude in more than one and current operators as well as the local Berry curvature. Landau level, such that its time dependence involves the The resulting band structure is well known from studies of Landau level energy spacing (cyclotron energy); it is therefore graphene: there are two points at the corners of the BZ q ¼ Q related to the inter-Landau level excitation known as a with “magnetoplasmon.” The drift of the guiding center of the 4π orbit around the perimeter of the sample is a feature that exists Q ¼ pffiffiffi ð1; 0Þ; for quantum states within a single Landau level and that 3 3 a represents the chiral edge mode of that Landau level. The existence of these edge modes is required for consistency of at which the bands touch, i.e., hx þ ihy ¼ 0. Close to either q˜ ≡ q − Q the quantized bulk Hall conductance (Laughlin, 1981). band-touching point, with , the Hamiltonian has More generally, gapless edge states occur at the boundary the 2D Dirac form between two insulating regions with different values of a topological invariant. A simple semiclassical view of these ˆ ≈ ℏ ˜ σˆ ˜ σˆ σˆ Hq v½qx x þ qy yþhz z ð34Þ gapless regions is provided by considering the boundary to arise from a smooth spatial variation in the parameters of the 3 2 ℏ with velocity v ¼ð = ÞJa= . The terms hz arise from effects Hamiltonian, between two phases with different topological other than nearest-neighbor tunneling and open gaps at the indices. Since the two insulators far to the left and far to the Dirac points between the two bands. One such effect is an right of the boundary are topologically distinct, at some point energy splitting Δ between A and B sublattices. In this case in space the gap between the filled and empty bands must þ − Δ the coefficients hz and hz are both equal to and the gap close. This gap closing motivates the existence of gapless edge openings at the two Dirac points are equivalent. This causes states. While this semiclassical argument applies only for the resulting Bloch bands to have vanishing Chern number, so smooth variations in space, the result is a robust feature for the two resulting bands are nontopological. This is consistent any form of boundary, referred to as the bulk-boundary with the fact that this model has TRS. Introducing next- correspondence (Hasan and Kane, 2010). The only restriction nearest-neighbor hopping with the Aharonov-Bohm phase ϕ ≠ 0 π þ AB mod breaks TRS and provides a term with hz ¼ − −hz for which the Chern numbers of the bands are 1 and −1. Figure 3 shows how eðqÞ varies in reciprocal space in this topological phase. The preceding discussions are somewhat abstract, focusing on mathematical aspects of the energy bands in 2D. However, the Chern number has direct physical consequences. As first shown by Thouless et al. (1982), a band insulator exhibits the FIG. 4. Skipping orbits for a charged particle in a uniform integer quantum Hall effect if the total Chern number of the magnetic field. In the bulk, the semiclassical dynamics of a wave filled bands is nonzero, with a Hall conductivity quantized packet leads to a circular cyclotron orbit. The reflection of such 2 at C times the fundamental unit of conductance e =h. This an orbit from the (hard wall) edge leads to a skipping motion result can be extended to systems subjected to disorder and around the sample edge.

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is that, in cases where the topological invariant relies on an In the last line we replaced q˜ by −i∂x to give a 1D Dirac underlying symmetry, this symmetry must be preserved also Hamiltonian in which MðxÞ can be viewed as a spatially in the boundary region. This is illustrated later for the edge varying “mass.” This was a model studied by Jackiw and state of the SSH model. Rebbi (1976), who showed that, provided MðxÞ changes sign, We demonstrate the emergence of edge states in the SSH there is a solution that is localized at the boundary with exactly 0 0 and Haldane models within continuum approximations for zero energy. For Mþ > and M− < , this is which the edge state wave functions take simple analytic R x − Mðx0Þdx0=a π forms. e 0 ei x=a jΨi¼ : ð37Þ 0 1. SSH model σˆ σ 1 Consider a 1D band insulator formed by filling the lower That the state is an eigenstate of z with j z ¼þ i indicates energy band of a Hamiltonian of the form (18), which is that it has nonzero amplitude only on the A sublattice. For 0 0 purely nondiagonal in the sublattice basis and parametrized by Mþ < and M− > , the nonzero amplitude would be on the the two-component vector ðh ;h Þ. This restriction on the B sublattice. This is the gapless edge mode of the SSH model x y M form of the Hamiltonian arises from the chiral symmetry of which arises because describe different topological E 0 the model as discussed further in Sec. II.E. Let the properties phases. That the state has energy ¼ is a consequence of a chiral symmetry that protects the relevant 1D topological of the system depend on some quantity M such that h þ x invariant, as discussed in Sec. II.E. If the Hamiltonian were ≡ iϕðq;MÞ ihy jhðq; MÞje is a function of both wave vector q and to depart from this chiral form, for example, by including M. We introduce a boundary between two gapped phases at terms hz that couple to σˆ z, the energy of this subgap state position x ¼ 0 by allowing MðxÞ to vary slowly in space, as would depart from E ¼ 0. This could occur either by a compared to the lattice constant a and setting Mðx ≪ 0Þ¼ change of the bulk Hamiltonian or due to breaking of the ≫ 0 M− and Mðx Þ¼Mþ. The two phases are characterized symmetry near the edge: for example, arising from an on-site ϕ by winding numbers N (23) computed from ðq; MÞ.Itis potential that shifts the energy of the A site relative to the B − straightforward to show that hx þ ihy must have N− Nþ site near the edge. ≠ vortices within the boundary region; see Fig. 5.SoifNþ Although we derived this edge state for a continuum N− then jhðq; MÞj must vanish at certain points ðq; MÞ: these model, the properties are robust to lattice effects provided are the gap-closing points, discussed semiclassically, which the chiral symmetry is preserved. A derivation of the edge lead to gapless edge states. state of the SSH model for a sharp boundary is provided in 0 iqa For the SSH model, hx þ ihy ¼ J þ Je , the gap-closing Appendix A.1. point (J0=J ¼ 1, qa ¼ π) hosts a single vortex. Defining M ¼ 1 − J0=J and q˜ ¼ q − π=a, we expand the 2. Haldane model ˜ 0 ˜ Hamiltonian around M ¼ q ¼ to first order in q (suitable One can use this solution for the edge state of the SSH ˜ ≪ 1 for the continuum limit, jqj =a)togive model to construct a 1D band of edge states on a surface of the topological band insulator formed from the Haldane ˆ SSH H =J ≈ qa˜ σˆ y þ MðxÞσˆ x ð35Þ model. Consider the low-energy theory for the Haldane model (34) in a topological phase with hz ¼∓ jHj. We impose a ¼ −iaσˆ ∂ þ MðxÞσˆ : ð36Þ boundary to a nontopological phase at x>xR by adding a y x x spatially dependent energy offset ΔðxÞσˆ z, such that hz ðxÞ¼ Δ ∓ Q ðxÞ jHj. The low-energy Hamiltonians close to become

ˆ ≈ ∓ ℏ σˆ ∂ σˆ ℏ ˜ σˆ Hq i v x x þ hz ðxÞ z þ vqy y ð38Þ ˜ → − ∂ where we replaced qx i x. Translational invariance is maintained along the y direction, such that qy (and therefore ˜ − qy ¼ qy Qy ) is conserved. For ΔðxÞ an increasing function of x at x>0 there is a boundary between topological (xxR) regions where ΔðxRÞ¼jHj. At this point, hz ðxRÞ¼ FIG. 5. An edge of the SSH model is represented by þ ϕ 0 so the gap at Q vanishes. The low-energy Hamiltonian h þ ih ¼jh jei ðq;MÞ, which depends both on the wave x y q;M close to Qþ (38) takes the form of the Jackiw-Rebbi vector q and on some control parameter M that varies ≪ 0 ≫ 0 Hamiltonian (36), just under a permutation of the Pauli smoothly between M− at x and Mþ at x .The σˆ σˆ σˆ → σˆ σˆ σˆ ℏ ˜ σˆ matrices [ð x; y; zÞ ð y; z; xÞ], plus a term vqy y number of vortices of hx þ ihy within this region can be computed by integrating ∇ϕðq; MÞ around the boundary. The for which the zero mode (37) is also an eigenstate (noting integrals along q ¼π=a cancel (since the BZ is periodic), the permutation of the Pauli matrices), with eigenvalue ER ℏvq˜ . This is the chiral edge mode, which propagates leaving only the integrals along M ¼ M, which give the qy ¼ x − difference of winding numbers N− Nþ. along the right-hand boundary xR at velocity vey. For a

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Here we explore the physics of symmetry-protected topological band insulators in the context of the SSH model previously described. This provides an example of a topological band insulator that arises in 1D when there is an underlying chiral symmetry. This chiral symmetry arises from the existence of a unitary operator Uˆ that anticommutes with the Hamiltonian Hˆ

Hˆ Uˆ ¼ −Uˆ H:ˆ ð39Þ (a) (b) Then for an energy eigenstate jΨi of eigenvalue E, the state FIG. 6. The low-energy spectrum of the Haldane model on a ˆ Ψ finite-width strip. (a) The strip is bounded in the x direction, but Uj i is readily shown to be an energy eigenstate with energy − uniform along y such that the wave vector q˜ ¼ q − Qþ is E. Thus, the chiral symmetry enforces a symmetry on the y y y 0 conserved. (b) The spectrum has a continuum of states in the spectrum about E ¼ . This rather formal construction arises bulk, shown shaded. The bulk bands are topological, with unit in tight-binding models, such as the SSH model, in which the Chern number, so a single edge state connects between these bulk Hamiltonian involves only terms that hop between two ˆ bands: the red solid line (green dashed line) shows the band different sublattices (labeled A and B). Defining PA=B as corresponding to the edge mode on the left (right) boundary. We ˆ ˆ ˆ projectors onto the A=B sublattices, then U ≡ ðPA − PBÞ show only the part of the spectrum close to the Qþ point, at which ˆ ˆ ˆ ˆ ˆ ˆ satisfies Eq. (39) if PAHPA ¼ PBHPB ¼ 0, i.e., provided the boundaries force a gap closing and at which the edge states ˆ appear. the Hamiltonian H couples only A and B sublattices. The chiral symmetry constrains the Hamiltonian for the SSH model (18) to be off diagonal in the sublattice basis. As finite-width strip, with another boundary to a nontopological described in Sec. II.C.1 this form ensures that the winding phase at x

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γ 2 ð−Þ sinð q= Þ of the loop in parameter space. Note that at this stage no juq i¼ ϕ ð42Þ intuition has been provided on the mechanism at the basis of −ei q cosðγ =2Þ q the transport, nor its direction. For a physical discussion see − − ϕ Sec. IV.C.1, in particular, Fig. 22. with γ and ϕ defined by −ðJ0 þ Je iqaÞ=Δ ≡ tan γ e i q. q q q This quantization has the same topological origin as the The resulting Zak phase (14) is shown in Fig. 7 (left) as a integer quantum Hall effect. To see this, consider tracing out function of J0=J and Δ=J. Along the line Δ ¼ 0 the Zak phase the closed loop in parameter space as time t varies from t ¼ 0 steps by π across J0=J ¼ 1, consistent with the change in to t ¼ T. The Zak phase at any given time t is defined by the winding number by one at the topological transition of the integral of the Berry connection along the BZ −π=a < q ≤ SSH model, and Eq. (23). However, note that the gapped band π=a at that instant. The change in Zak phase between any two insulators at Δ ¼ 0, J0=J > 1 and Δ ¼ 0, J0=J < 1 can now times t1 and t2 can be written as a line integral be continuously connected by tracing out a path that has Δ ≠ 0 when J0=J ¼ 1 (Fig. 7, right). I ϕ − ϕ ≡ A q Zakðt1Þ Zakðt2Þ ·d ð43Þ F. Adiabatic pumping L – Consider again the Rice-Mele model, whose Zak phase is where we label the position in the q t plane by a two- q illustrated in Fig. 7. There is a vortex in ϕ around the component vector ¼ðq1;q2Þ¼ðq; tÞ and the associated Zak ≡ ∂ ∂ ≡∂ ∂ gap-closing point (Δ ¼ 0 and J0=J ¼ 1) at which the Zak Berry connection Ai ihuqj ijuqi, with 1 = q and ϕ 2π ∂2 ≡∂=∂t. The integration contour L is shown in Fig. 8. phase is undefined. The winding of Zak by around a closed loop encircling the gapless point, such as the loop shown in The horizontal lines at fixed t ¼ t1 and t ¼ t2 recover ϕ − ϕ Fig. 7, is a topological invariant of the model: this winding Zakðt1Þ Zakðt2Þ, while the integrals on the lines at q ¼ is preserved under smooth variations of the Rice-Mele π=a cancel as a result of the periodicity of the BZ. Applying Hamiltonian that do not cause the gap to close on this loop. Stokes’s theorem, the line integral in Eq. (43) can be written as Ω ϵ ∂ The existence of this invariant is at the basis of the concept the integral of the Berry curvature ¼ ij iAj over the area of a quantized pump (Thouless, 1983). It describes the generic bounded by L. The fact that the parameters of the Hamiltonian situation of a crystal with filled bands, which is characterized return to their original values as t ¼ 0 → T enforces perio- by parameters that can be externally controlled. When these dicity also in t, such that the q–t plane has the topology of a parameters are varied around a closed loop, the number of torus. Thus, extending the contour to enclose the full region 0 ϕ 0 − ϕ particles that are transported is quantized. It is thus a robust from t ¼ to t ¼ T, thereby computing Zakð Þ ZakðTÞ, quantity that is not affected by a small change of the geometry recovers 2π times a Chern number. The relevant integral is entirely analogous to Eq. (27) with the measure d2q replaced by dqdt. 1.0 ϕ The link between the winding of Zak and the transported particle number for a filled band can be established using 0.5 arguments detailed in Appendix A.4. The transported particle number is determined by computing the change in the 0.0 mean particle position Δx over one cycle of the pump, averaged over all states in the band. The result is that this net −0.5

−1.0

0.0 0.5 1.0 1.5 2.0

FIG. 7. Left: Zak phase of the lower band of the Rice-Mele model, represented by the unit vector ðcos ϕZak; sin ϕZakÞ. The phase changes by 2π around the point Δ ¼ 0 and J0=J ¼ 1 at which the gap closes. The loop denotes a possible pumping cycle. Right: Representation of the eigenstates (42) on the Bloch sphere. Blue dashed line: Closed trajectory on the equator for the SSH model in the topological case (Δ ¼ 0, J0=J < 1), obtained when q scans the Brillouin zone. The Zak phase, given by half the FIG. 8. The relationship of the change in the Zak phase for a 1D subtended solid angle, is equal to π. Green solid line: Trajectory band insulator, with momentum −π=a < q ≤ π=a, under adia- in the nontopological case of the SSH model (Δ ¼ 0, J0=J > 1) batic variation of parameters in time around a cycle of period T, with a zero subtended solid angle. Black dotted line: Closed i.e., with time 0

Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019 015005-10 N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms displacement of particles over one cycle, for example, the in a single, nontopological, cosinusoidal band with full closed loop in Fig. 7,is width 8J. As shown in Fig. 9(a), our task is to imbue the tunneling Δ a ϕ − ϕ 0 matrix elements with nonzero Peierls phase factors so that the x ¼ ½ ZakðTÞ Zakð Þ: ð44Þ 2π phase accrued by a particle encircling a single plaquette is ϕ , equivalent to the Aharonov-Bohm phase ϕ ¼ qBA=ℏ Because ϕ ðTÞ equals ϕ ð0Þ plus 2π times the winding AB AB Zak Zak acquired by a particle with charge q moved around a plaquette number corresponding to the vortex of Fig. 7, the displace- of area A. The resulting complex tunneling matrix elements ment is quantized in units of the lattice period a. When the are required to break time-reversal symmetry and allow a band is filled with exactly one particle per state, this entails nonzero Chern number. As can be confirmed by the phases the quantization of the number of transported particles. depicted in Fig. 9(a), the associated Harper-Hofstadter III. IMPLEMENTATIONS OF TOPOLOGICAL LATTICES Hamiltonian expressed in the Landau gauge X ϕ † † To this point in this review, we have developed our ˆ − im AB ˆ ˆ ˆ ˆ H ¼ J ðe ajþ1;maj;m þ aj;mþ1aj;m þ H:c:Þ; ð46Þ understanding of lattices and discussed how topology presents j;m additional “labels” tied to individual Bloch bands: providing a new way to categorize band structure. This section takes the gives a phase ϕ for tunneling around each plaquette. For next step and describes the currently implemented techniques AB ϕ =2π p=q by which 1D and 2D band structures with nontrivial topology rational AB ¼ , expressed in reduced form, the single have been created. band of Eq. (45) fragments into q (generally) topological bands, with zero aggregate Chern number. A. Iconic models We can focus in on the essential features of this model by considering the special case of one-third flux per unit cell, i.e., ϕ ϕ 2π 3 im AB Cold atom experiments are often able to nearly perfectly AB ¼ = . First take note of the tunneling phase e for realize iconic topological models from condensed matter motion along j: As shown in Fig. 9(a), this tunneling phase theory. Here we briefly describe two such models—the depends on m and has a spatial period of three lattice sites, Harper-Hofstadter model (Harper, 1955; Hofstadter, 1976) implying that the lattice’s unit cell is enlarged beyond the and the Haldane model (Haldane, 1988)—as particularly plaquettes of the underlying square lattice (the unit cells ϕ 2π 3 simple examples of topological lattices and explore what is without magnetic flux) to three plaquettes at AB ¼ = .In essential about each of these models. This allows us to place Fig. 9(b), these unit cells are graphically indicated by the gray experimental approaches in context and to identify what types dashed lines, with a representative unit cell shaded (light blue) of new terms must be added to standard optical lattices. for clarity. Each of these unit cells is identified by integers j and M. In order to distinguish between the three inequivalent 1. Harper-Hofstadter model sublattice sites within each unit cell, we introduce also the site −1 0 1 The Harper-Hofstadter model, describing charged particles index s ¼ ; ; þ , related to the individual plaquette index 3 in a square lattice with a uniform of magnetic field, derives m via m ¼ M þ s. from the simple 2D tight-binding Hamiltonian As a result of this expanded unit cell, the associated 1 3 X Brillouin zone is reduced to = of its initial size along the ˆ − ˆ † ˆ ˆ † ˆ m direction, and the number of bands correspondingly H ¼ J ðajþ1;maj;m þ aj;mþ1aj;m þ H:c:Þð45Þ j;m increases from 1 to 3. Following textbook techniques, we express this Hamiltonian in the Fourier representation, i.e., q for particles hopping in a square lattice with tunneling strength giving states labeled by their crystal momentum ¼ðqj;qmÞ ϕ 2π 3 J. Each individual site of this lattice is labeled a pair of along with the sublattice index s, i.e., for AB ¼ = the integers j and m. The first term in Eq. (45) denotes tunneling states are fjq; −1i; jq; 0i; jq; þ1ig. For each crystal momen- along the j direction (horizontal) and the second term marks tum q the Hamiltonian matrix coupling these sublattice sites tunneling along the m direction (vertical). This model results together is

0 1 2 cos q a − 2π=3 1 exp iq 3a B ð j Þ ½ mð sÞ C Hq ¼ −J@ 12cos ðqjaÞ 1 A; ð47Þ − 3 12 2π 3 exp½ iqmð asÞ cos ðqja þ = Þ

where a and as denote the nearest-neighbor lattice spacings matrix define three separate bands. Figure 9(c) shows the in the directions of increasing j and m, respectively. (Even resulting band structure, where each of the three bands is if m denotes a “synthetic dimension” the notion of length as endowed with a nonzero Chern number. The expansion of remains a useful bookkeeping device, by which 3as is the the unit cell to contain three sublattice sites is essential for side of the expanded unit cell.) The three eigenvalues of this the formation of topological bands. Recall that Chern

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(a) (b) topological band structure but without the presence of an overall magnetic field which requires the expansion of the unit cell size. Figure 10(a) plots the Haldane lattice in the conventional honeycomb geometry. We also show its deformation to a “brick-wall” geometry most relevant for its experimental realization with cold atoms. The tunneling matrix elements (black solid lines) with strength J define the underlying honeycomb lattice, along the three nearest- neighbor bonds. As indicated, even the simple honeycomb lattice describes a dimerized lattice with jAi and jBi sublattice sites (possibly offset in energy by Δ), making it an ideal starting point for realizing topological band structures. Figure 10(b) shows the resulting two bands kissing at a pair of Dirac points. Haldane’s addition of next-nearest-neighbor 0 ϕ tunneling with strength J and phase AB [pink dashed lines in Fig. 10(a), along bonds connecting sites of the same sublattice] renders this model topological. This Hamiltonian too can be readily expressed in a crystal momentum-dependent matrix, now with two contributions. First the energy offset and nearest-neighbor tunneling from the (c) underlying honeycomb lattice contribute the matrix 0 P 1 Δ −J expð−iq · ρiÞ B i C Hq;0 ¼ @ P A; ð48Þ −J expðiq · ρiÞ −Δ i

and the next-nearest-neighbor links contribute a second term 0 P 1 cosðq · A − ϕ Þ 0 B i AB C 0 i Hq 1 ¼ −2J @ P A: ; 0 q A ϕ cosð · i þ ABÞ i ð49Þ

We have defined A1 ¼ a1, A2 ¼ −a2, A3 ¼ a2 − a1, with the FIG. 9. The Harper-Hofstadter model. (a) Harper-Hofstadter vectors ρi and ai labeling nearest-neighbor and next-nearest- ϕ lattice geometry with symmetric hopping J and a flux AB in each neighbor separations as indicated in Fig. 3(a). (For practical plaquette. (b) Harper-Hofstadter lattice geometry with flux realizations of the brick-wall lattice, the next-nearest-neighbor ϕ 2π=3 AB ¼ per plaquette. The individual magnetic unit cells coupling, along A3, is suppressed compared to couplings are delineated by gray dashed lines with a representative along A1 2. However, we use equal strengths for all in Fig. 10.) magnetic unit cell set off in blue (dark shading) for clarity. ; Figure 10(b) plots the topological phase diagram associated (c) Computed band structure with ϕAB ¼ 2π=3 showing the three 1 −2 1 with this model as a function of the Aharonov-Bohm tunnel- topological bands with Chern numbers þ , , and þ built ϕ 2Δ from the three inequivalent sites within the magnetic unit cell. ing phases AB and tilt . This system supports three distinct topological regions: zones with Chern number 1, with the majority of parameter space in the topologically trivial phase numbers are derived from the integrated Berry curvature with Chern number 0. over the Brillouin zone. For a tight-binding model, the The Haldane model is particularly amenable to experimen- ϕ Berry curvature can only be nonzero when each Bloch tal study because tuning experimental parameters such as AB wave function has a spin or pseudospin degree of freedom, can directly drive topological phase transitions. While for the ϕ here provided by the sublattice degree of freedom. Harper-Hofstadter lattice, tuning AB does lead to different In Sec. III.D we describe how to experimentally imprint these Chern numbers, the size of the unit cell also changes, leading hopping phases using tailored laser fields and will comment to more dramatic changes in the band structure. In Sec. III.C on the limitations of different experimental approaches. we show how strongly driving the parameters of a brick-wall lattice can break time-reversal symmetry and imbue the lattice’s two bands with nontrivial topology. 2. Haldane model In these examples of topological band structure, we The Haldane model (Haldane, 1988), an extension of the identified two common elements that experimentalists need well-known honeycomb lattice, was an early model of to introduce to create nontrivial topology: complex-valued

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(a) (c)

(d)

(b)

(e)

FIG. 10. Haldane model. (a) Haldane lattice geometry showing overall honeycomb lattice structure with the addition of next-nearest- neighbor hopping with Aharonov-Bohm tunneling phases ϕAB. The shaded (blue) box marks the two sites comprising a single unit cell. We also show the deformation of the underlying lattice to the brick-wall geometry, used to plot the dispersions with qxa ¼ q · a1 and qya ¼ q · a2. (b) Haldane model phase diagram showing the two topological lobes immersed in a nontopological background. (c) Band structure computed with only nearest-neighbor tunneling [black solid lines in (a), showing the familiar pair of Dirac points from this “brick-wall” lattice. (d) Band structure computed in the topological phase at the marked red square in (b), with ϕ π=2 and “tilt” AB ¼ pffiffiffi 0 0 Δ ¼ 0, J ¼ J=10. (e) Band structure at the topological transition, marked by the blue circle in (b), with ϕAB ¼ π=2 and Δ ¼ −3 3J , showing the formation of a single Dirac point.

tunneling matrix elements and unit cells with more than one 2λ0 [pink dashed curve in Fig. 11(b)], gives a combined underlying lattice site or spin degree of freedom. potential [black solid curve in Fig. 11(b)] with generally staggered energy minima and alternating tunneling. This gives B. Realization of SSH model the overall potential

The 1D SSH model of polyacetylene and its generalization 2 2 ϕ 2 the Rice-Mele model are among the most simple topological VðxÞ¼Vshortsin ðkRxÞþVlongsin ½ðkRx þ Þ= ; ð50Þ models to realize. As described in Sec. II.E.2, the Rice-Mele model, Eq. (40), consists of a bipartite 1D lattice with ℏ tunneling strengths alternating between J and J0, and energies where we defined the single-photon recoil momentum kR ¼ 2πℏ λ ℏ2 2 2 of the two sublattice sites, staggered by Δ. = 0, the associated recoil energy ER ¼ kR= m, and the Figure 11(a) depicts a typical laser system required to atomic mass m of the atom under study. ϕ approximate this idealized model, similar to the experimental Figure 11(c) shows the Zak phase Zak, Eq. (14), computed realization of Atala et al. (2013). Here a conventional 1D for this physical system in terms of the experimental control optical lattice with period λ0=2 is generated by a pair of parameters. This figure depicts the singularity expected when 0 ϕ π 2 counterpropagating lasers each with wavelength λ0. In this Vlong ¼ , which at ¼ = corresponds to the location of 0 lattice, the tunneling is uniform with strength J0 and the the topological transition in the SSH model when J ¼ J . energy minima of the lattice sites are degenerate, as indicated Figure 12 shows the bottom part of the infinite band 6 1 by the pale blue dotted curve in Fig. 11(b). A second, weaker, spectrum for Vshort ¼ ER and Vlong ¼ ER, with the relative lattice with period λ0, generated by a laser with wavelength phase set to ϕ ¼ π=2. The right part of this figure shows a

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(a)

(b) (c)

FIG. 12. Bipartite lattice band structure compared to the Rice- Mele model. Left: Band structure computed for the superlattice 6 potential given by Eq. (50) with Vshort ¼ ER, Vlong ¼ ER, and a relative phase ϕ ¼ π=2 corresponding to the particular case of the SSH model. Right: Zoom on the lower pair of bands (continuous lines). The energy offset between adjacent minima and a fit to the FIG. 11. Implementation of the SSH and Rice-Mele models. prediction (41) for the band structure of the Rice-Mele model (a) Representative laser configuration: A pair of overlapping (dashed lines) allow one to extract the practical values of Δ, J, 0 Δ 0 0 069 0 0 037 lasers with wavelengths λ0 and 2λ0 subject a cloud of ultracold and J . (Here ¼ , J ¼ . ER, and J ¼ . ER.) The 0 atoms to 1D optical lattices, with periods λ0=2 and λ0, dotted lines show the (folded) lowest band for Vlong ¼ ,uptoa respectively, and a spatial relative phase ϕ. (b) Energies of global energy shift. these two lattices and combined potential, showing the long- period lattice shifted in position with respect to the short period ϕ lattice by a controllable phase shift . The SSH model is Model parameters: The V ¼ 6E short period lattice ϕ π 2 π short R realized for ¼ = þ n (for integer n), in which case all depth used in these simulations, a typical laboratory scale, sets minima have the same energy and are separated by potential the nominal tunneling strength of J ≈ 0.05E . Intuitively, we barriers with staggered height. An example of band structure in R ϕ π expect that the energy difference between the minima to be this case is shown in Fig. 12. The choices ¼ n (for integer n) ϕ about Vlong cosð Þ. Similarly, the barriers between the sites lead to equal barrier heights between adjacent sites, and ϕ staggered site energies, but with J0 ¼ J, Δ ≠ 0. The displayed differ in height by roughly Vlong sinð Þ. The tunneling strength data are for an intermediate case of ϕ ¼ π=4 where J0 ≠ J and in the effective SSH model has a nontrivial, but monotonically Δ ≠ 0. (c) Zak phase for the lower pair of bands controlled by decreasing, exponential-type behavior in the barrier height. tuning the phase shift ϕ between the long and short period This then begs the question of obtaining the parameters of the lattices and the strength of the long-period lattice, as repre- Rice-Mele model in Eq. (40), including the two tunneling ϕ ϕ 0 Δ sented by the unit vector ðcos Zak; sin ZakÞ. strengths J and J along with the energy difference between sublattice sites. First, recall that the band structure of simple 1D optical 2 zoom on the lowest pair of bands, which are the only relevant lattice potential Vsin ðkRxÞ only approaches that of a tight- ones when the temperature and the interaction energies are binding model with nearest-neighbor tunneling when V ≫ E . 0 R comparable to or lower than ER. The result for Vlong ¼ is For nearest-neighbor tunneling strength J, the resulting −2 π indicated for comparison with dotted lines. The BZ is reduced dispersion is simply J cosð q=kRÞ. As a result the effective in size by a factor of 2 as compared to that of the short period nearest-neighbor tunneling strength can be directly obtained 0 lattice only. This is reflected by the Vlong ¼ bands touching from the lowest term in a Fourier expansion of the band at the edge of the BZ; in 1D this marks each such linked pair as structure of the physical 1D optical lattice. [The higher terms truly being one band in a doubled BZ. in the series describe longer range tunneling, which becomes Then the additional independent control of both J, J0, and Δ negligible for deep lattices; see Jim´enez-García and requires two additional experimental degrees of freedom. In Spielman (2013) for an introduction.] This approach is insuffi- this realization these parameters are the relative phase ϕ of the cient for the Rice-Mele band structure ½Δ2 þ δJ2 þ 4¯ 2 − δ 2 2 π 1=2 long and short period lattices (displacing one lattice with ð J J Þcos ð q=kRÞ , because fits to this dispersion respect to the other) and the depth of the long period lattice alone cannot effectively disentangle Δ from δJ ¼ J − J0,nor ϕ π 2 ¯ 0 Vlong. Figure 12 has been calculated for ¼ = , in which δJ from J ¼ðJ þ J Þ=2. One practical resolution to this case all minima of VðxÞ have the same energy while the difficulty is to employ symmetry and evaluate the band barrier heights between adjacent minima alternate between structure of the bipartite lattice for two cases: first compute two values. This realizes the SSH model and the dashed lines the band structure for ϕ ¼ 0 where J and J0 are manifestly in the right panel of Fig. 12 show a fit of the SSH prediction equal, and then compute the band structure for ϕ ¼ π=2 where (20) to the two lowest bands, providing thus the relevant Δ ¼ 0. This then allows for the independent determination of 0 Δ 0 6 values of J and J . , J,andJ . For example, for Vshort ¼ ER and Vlong ¼ ER this

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¯ 0 053 ϕ (a) (b) procedure gives J ¼ . ER, almost independent of .For ϕ 0 Δ 0 43 δ 0 ¼ ,wefurtherfind ¼ . ER and J ¼ , while for ϕ π 2 Δ 0 δ 0 032 ¼ = , this becomes ¼ and J ¼ . ER.

C. Inertial forces

The common experience of starting water in a pail spinning by moving the bucket in a circular manner, not rotating the (c) bucket, suggests that applied inertial forces might produce effects akin to those present in rotating systems: described best by effective Lorentz forces. We see later how these ideas are implemented for ultracold atoms in optical lattices and also come to understand the limitations of these approaches. The tight-binding model depicted in Eq. (45) is representative of the tunnel-coupling structure present for atoms confined in optical lattices. It is of particular importance (d) that the tunneling matrix elements J are real valued (more specifically, transformations between different gauges can introduce “trivial” complex amplitudes to the tunneling matrix elements, but in these simple lattices there always exists a gauge choice for which the amplitudes are real valued). In this section we develop a simple model illustrating how inertial forces (linear potential gradients or equivalently spatially shaking the lattice potential) can add tunable complex hopping phases to these matrix elements. From a quantum mechanical perspective the essential concept is to engineer nontrivial phases acquired by the unitary evolution of a time-periodic Hamiltonian which can be cast as complex hopping amplitudes in an effective time- independent Hamiltonian. This physics is minimally captured by the tunnel-coupled pair of lattices sites shown in Fig. 13, essentially comprising a single unit cell of the Rice-Mele FIG. 13. Inertial forces. (a) A simple double-well lattice subject model. We aim for a two-site model described by the to modulation, creating (b) experimentally tunable hopping phases. (c) Shaking or tilting in 1D gives rise to a uniform Hamiltonian Peierls phase factor that shifts the minima of the tight-binding ϕ − ϕ band structure (Struck et al., 2012). (d) Shaking in 2D can break ˆ − i P i P Δ − H ¼ Jðjrihlje þjlihrje Þþ ðjrihrj jlihljÞ time-reversal symmetry giving rise to topological lattices. − ϕ σˆ ϕ σˆ Δσˆ ¼ J cosð PÞ x þ J sinð PÞ y þ z; ð51Þ J and phase ϕ . In our two-site model, this modulation (i.e., ϕ P including a laboratory-controllable tunneling phase P, detuning) is described by ΔðtÞ¼Δðt þ TÞ, with period T, ϕ 0 alas, our lattice is born with P ¼ . In the second line we angular frequency ω ¼ 2π=T, and with zero per-cycle aver- expressed this Hamiltonian in terms of the Pauli operators age hΔðtÞi ¼ 0. σˆ 1 2 T x;y;z, allowing us to follow a simple analysis of a spin- = It is straightforward to eliminate the time-dependent ΔðtÞ ϕ 0 system (Haroche et al., 1970). term in Eq. (51) (initially, with P ¼ ) by making the unitary We earlier noted that gauge transformations can introduce transformation complex-valued tunneling phases. A gauge transformation is Z t simply a position-dependent unitary transformation that 0 i 0 0 jψ ðtÞi ¼ exp Δðt Þσˆ zdt jψðtÞi; ð52Þ adjusts the local phase of the wave function and compensates ℏ 0 the Hamiltonian accordingly; in our double-well model, a gauge transformation in the spatial picture becomes a σˆ z rota- in the language of quantum optics this is akin to the trans- ϕ tion in the spin picture. Evidently, the Peierls phase factor P formation into the time-dependent “interaction” picture. Since can be fixed to a nonzero value by the choice of gauge. Since this transformation is a σˆ z rotation it is equivalent to a time- such a choice is of no physical consequence, it is instead the dependent gauge transformation, leading to a nonzero, time- ϕ ability to change P (either spatially to induce Aharonov- dependent Peierls phase factor Bohm fluxes or temporally to induce artificial electric fields) Z that is the essential content of this discussion. Here we adopt 2 t ϕ Δ 0 0 ϕ 0 PðtÞ¼ ðt Þdt : ð53Þ the most straightforward gauge choice that sets P ¼ for ℏ 0 unadulterated lattices. In the following, we show how a time-periodic linear When the modulation frequency’s associated energy ℏω is gradient provides control over both the tunneling amplitude greatly in excess of the tunneling J, we make a rotating wave

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ˆ 2 (ˆ − δ ) 2 approximation to replace the time-dependent terms introduced Vðx; tÞ¼V0 cos½ kR x xðtÞ = .HereV0 is the lattice by this rotation by their time averages, giving the time- depth and kR is the two-photon recoil momentum from the averaged interaction picture Hamiltonian wavelength λ of the lasers creating the lattice potential. Although this shaking process is physically quite differ- ˆ − ϕ σˆ ϕ σˆ H ¼ Jhcos PðtÞiT x þ Jhsin PðtÞiT y; ð54Þ ent from applying a time-dependent gradient, they are functionally equivalent. The connection between the two with a potentially nonzero dc Peierls phase factor can be seen clearly in terms of a pair of time-dependent transformations. We begin by using the spatial displacement ϕ t ˆ −δ ˆδ ϕ hsin Pð ÞiT operator Dx½ xðtÞ ¼ exp½ik xðtÞ to transform to the non- tanð P;dcÞ¼ ϕ : ð55Þ ˆ −δ ˆ hcos PðtÞiT inertial frame comoving with the lattice, i.e., Dx½ xðtÞx ˆ † Dx½−δxðtÞ ¼ xˆ þδxðtÞ. This exchanges the lattice’smotion Physically, the time-dependent gauge transformation in for a new time-dependent contribution to the Hamiltonian Eq. (52) allows the system to sample a range of Peierls phase ˆ −ℏk∂tδxðtÞ. The once-transformed Hamiltonian factors and retain a nonzero average. In effect, our task is to ϕ 2 2 make hsin PðtÞiT nonzero, and because sin is an odd function ℏ m V0 ˆ ð1Þ ˆ − ∂ δ 2 ˆ we seek a waveform ϕ ðtÞ that takes on positive and negative H ðtÞ¼ k t xðtÞ þ cosð kRxÞ P 2m ℏ 2 values in an “imbalanced” manner. m The most simple example to deploy in the laboratory is a − ∂ δ 2 2 ½ t xðtÞ ð57Þ monochromatic sinusoidal modulation ΔðtÞ¼Δ cosðωtÞ of the tilt. In this case, the Bessel series expansion gives contains a new time-dependent vector potential and a ϕ J 2Δ ℏω ϕ 0 hcos PðtÞiT ¼ 0ð = Þ and hsin PðtÞiT ¼ . Because global time-dependent energy shift that does not the integrated sinusoidal waveform takes on positive and impact the system’s dynamics. Evoking Hamilton’s equa- negative values with equal frequency the average tunneling _ ˆ tion x ¼ ∂ℏkH, we see that the appearance of this vector phase is zero; however, this modulation does renormalize the potential simply describes the fact that in the moving frame → J 2Δ ℏω tunneling strength J J × 0ð = Þ as was observed the velocity of an object differs from that in the lab frame by experimentally (Lignier et al., 2007). While monochromatic the instantaneous velocity of the moving frame −∂tδx. sinusoidal modulation is simple to deploy, it can obscure the We complete our argument using the time- underlying physics, rapidly becoming a tangle of Bessel dependent momentum displacement operator Dˆ ½−δkðtÞ ¼ functions. k exp½−iδkðtÞxˆ, a gauge transformation, that converts the time- Instead consider a waveform consisting of two delta- dependent vector potential into a potential gradient function “kicks” in each drive cycle (Sørensen, Demler, Vˆ ¼½m∂2δxðtÞxˆ. This reminds us that inertial forces are and Lukin, 2005; Anderson, Spielman, and Juzeliūnas, t present in accelerating frames and informs us that experi- 2013), the first with strength Δ=ω at time t ¼ 0 and the menters are free to use either shaken lattices or potential second with strength −Δ=ω at time f × T, a fraction f through gradients equivalently to produce the inertial forces required the drive period T. Integrating this waveform gives the time- to induce ϕ . dependent Peierls phase factor P While this nonzero and uniform Peierls phase factor is an essential first step for emulating Aharonov-Bohm fluxes, a 2Δ 1 − f for 0 ≤ t ≤ fT; ϕ PðtÞ¼ × ð56Þ uniform Peierls phase factor in 1D can be eliminated via a ℏω −f for fT < t ≤ T; gauge transformation (although any temporal change can still lead to effective electric fields). In contrast by moving to 2D a pulse-width modulated waveform with zero average, with systems such as in modulated and shaken honeycomb- duty cycle f ∈ ½0; 1Þ. For f ∉ f0; 1=2g, the resulting asym- geometry lattices, this technique has nontrivial alterations to metric and skewed waveform leads to a nonzero average band structure (Struck et al., 2013) and including those topo- ϕ of hsin PðtÞiT . logically equivalent to the Haldane model (Jotzu et al., 2014). For the special case Δ=ℏω ¼ π, the time-averaged Peierls We now extend our discussion to 2D to understand ϕ −2π − 1 2 phase factor is P;dc ¼ ðf = Þ with tunneling Aharonov-Bohm fluxes. Figure 13(d) depicts a minimal strength unchanged at J. The basic physical picture is that model of shaking in 2D; the top panel illustrates the two after an atom tunnels between sites it acquires a phase triangular plaquettes that make up a single unit cell of a different from what it would have acquired on its initial site, triangular lattice, while the bottom panel graphs a shaking ϕ and P;dc expresses the differential phase acquired upon protocol that first accelerates parallel to the A-B link of the left ϕ returning to its initial site. The time-dependent phase PðtÞ plaquette (i.e., along ex), then accelerates parallel to B-C, and must break time-reversal symmetry to give a nonzero finally accelerates parallel to C-A. For each link, this protocol ϕ average of sin PðtÞ. Figure 13(c) depicts the first exper- leads to the same time-dependent Peierls phase factor given in imental realization of a nonzero Peierls phase factor Eq. (53), with each phase shifted in time by 2π=3, giving the ϕ imprinted using inertial forces (Struck et al., 2013). same nonzero tunneling phase P;dc to each side of the Rather than tilting the lattice potential, Struck et al. plaquette. This then leads to an overall Aharonov-Bohm Φ ϕ (2013) found it more convenient to spatially shake the phase AB ¼ P;dc. lattice potential by modulating the phase of the lasers It might appear that our task is complete, but have we truly creating the optical standing wave, giving the potential created a uniform Aharonov-Bohm phase over all plaquettes?

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Unfortunately Struck et al. (2013) demonstrated that this is not D. Resonant coupling: Laser-assisted tunneling the case. Following the same argument for the second (inverted) plaquette that completes a single unit cell shows that accumu- The previous section outlined the broad range of engineered Φ −ϕ tunnel couplings possible via temporal modulation of the lated phases give a negative flux AB ¼ P;dc, leading to a staggered flux. Therefore on average the Aharonov-Bohm flux parameters of the lasers underlying the lattice potential. While through this lattice is zero. One way to understand this is that it was possible to create complex-valued tunneling, it was not the Peierls phase factors are created with uniform amplitude possible to independently control the phase and amplitude of throughout the lattice, rather than with the linear dependence on tunneling on each lattice link: more control is required. position as for our Landau-gauge example of the Harper- Following the ideas of Jaksch and Zoller (2003), we describe Hofstadter Hamiltonian. Sørensen, Demler, and Lukin (2005) how such a fine-grained control is in principle possible using proposed remedying this by applying a potential whose laser-assisted tunneling and how experimental implementa- gradient itself increased away from the systems center, which tions have approached this task. As we shall see, although this they termed a quadrupole potential. laser-assisted tunneling is effected by temporal modulation, Evidently shaking of this type does not provide a straight- the modulation results from additional lasers, rather than the forward route for realizing Hamiltonians such as the Harper- lasers from which the underlying lattice is assembled. Hofstadter model with uniform fields, but it has proven a The essential concept of this technique is straightforwardly successful route for creating a Haldane-type Hamiltonian that illustrated in the 2D square lattice depicted in Fig. 14(a): the has a zero average flux but still with topological bands (Jotzu native tunneling along the vertical direction is first eliminated et al., 2014). Important to this realization is the fact that the by applying a potential gradient (i.e., tilting the lattice), then effective Hamiltonian for the shaken lattice acquires a next- coupling between neighboring lattices sites is reestablished nearest-neighbor hopping with a nonzero Peierls phase factor. with a traveling wave potential. Here the spatially nonuniform Beyond the time-averaged Hamiltonian discussed, perturba- phase of the traveling wave is imprinted upon atoms as they tive corrections from the time-varying part of the nearest- are moved from site to site described by complex-valued neighbor tunneling lead to next-nearest-neighbor tunneling tunneling amplitudes. Because local optical phases are rela- of order J0 ∼ J2=ℏω, arising from a second-order process tively easy to control [for example, by creating higher-order through an intermediate virtual state detuned by ℏω. Such optical modes such as Laguerre-Gauss modes, as was done by terms are conveniently obtained from the Magnus expansion Chen et al. (2018)], these techniques in principle allow for of this tight-binding model described in Appendix B.An more subtle engineering of the local Aharonov-Bohm phases analysis of the effective model for the shaken lattice that goes than is possible with whole-scale modulation of lattice beyond this Magnus expansion approach was provided by parameters. Still, current implementations (Aidelsburger Modugno and Pettini (2017). et al., 2013; Miyake et al., 2013) rely only on the uniformly

(a) (b)

FIG. 14. Laser-induced hopping. (a) 2D square lattice (right) with a potential gradient along em (vertical) illuminated by a traveling wave potential. The coupling of any pair sites of this lattice jj; mi and jj; m þ 1i is qualitatively described as a two-level system with detuning 2Δ coupled by the traveling wave. (b) Technique for creating the half-flux Harper-Hofstadter Hamiltonian in tilted spin- dependent lattices as implemented in MIT (Miyake et al., 2013) similar to Aidelsburger et al. (2013).

Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019 015005-17 N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms changing phase from plane waves to generate homogeneous with no spatial dependence. As a result, the RWA Hamiltonian fields. is gauge equivalent to the Landau-gauge Harper-Hofstadter The basic principle can be understood in terms of the same Hamiltonian sort of two-level system discussed in Sec. III.C, but from a X † ϕ † − ˆ ˆ ij AB ˆ ˆ perspective in which the rotating wave approximation (RWA) H ¼ Jajþ1;maj;m þ JRWAe aj;mþ1aj;m þ H:c:: ð64Þ is valid. In the present case, we focus on two neighboring j;m 1 lattice sites in Fig. 14(a), labeled by jj; mi and jj; m þ i This Hamiltonian was realized in the manner described by both x 2k coupled by the traveling wave potential VRð Þ¼VR sinð R · the Munich and the Massachusetts Institute for Technology x − ω e tÞ that locally modulates the potential intersecting m (MIT) groups (Aidelsburger et al., 2013; Miyake et al., 2013), with angle θ. Physically this is directly realized (Aidelsburger illustrated in Fig. 14. The MIT group used a lattice derived from et al., 2013; Miyake et al., 2013) by a pair of interfering lasers a 1064 nm laser, with a traveling wave generated by beams from giving rise to a moving standing wave with periodicity λ =2, R the same laser intersecting the em axis at θ ¼ π=4. This k 2π=λ λ Φ 2π 1 2 and recoil wave vector j Rj¼ R (the wavelength R geometry gives a flux AB= ¼ = per plaquette and incorporates all geometric factors present from the intersection illustrates an important practical point of this technique. In a angle between these lasers). similar manner, but with a different laser geometry the Munich Midway between these two sites, at position group realized 1=4 flux per plaquette. In both cases, the laser- x0 ¼ aðj; m þ 1=2Þ, this potential is induced tunneling strength is proportional to cos θ, while the Aharonov-Bohm phase is proportional to sin θ, requiring a VR 2k x −ω x ≈ ið R· 0 tÞ 1 2 k x − x − compromise dependent on the experimental goals. Following VRð Þ fe ½ þ i R · ð 0Þ c:c:g; ð58Þ 2i this initial experiment, the Munich group retooled their technique as pictured in Fig. 14(c) by using the staggered to first order in position. In this expression (1) the first term potential inside individual four-site plaquettes and laser- describes a modulated, but spatially uniform shift in the induced hopping to establish tunneling along all the lattice potential with no physical consequence that may therefore directions, enabling the measurement of the Chern number e be neglected; and (2) the horizontal, j dependence drops out (Aidelsburger et al., 2015); see Sec. IV.C.2. at this order because the localized wave functions, both centered at aj, are symmetric and compact in square geometry E. Synthetic dimensions optical lattices. The remaining terms add a modulated contribution to the detuning The concept of synthetic dimensions is rooted in the fact that a lattice is no more than a set of states labeled by integers, e.g., j Δ ≈ θ 2k x − ω ∶ ðtÞ ½kRa cosð ÞVR cosð R · 0 tÞ ð59Þ and m in the preceding discussions labeled the atoms at sites described by wave functions jj; mi. This insight allows the the same sort of shaking potential we studied in Sec. III.C, creation of lattices that use the atoms’ internal or “spin” degrees now with an overall phase dependent on the center position x0. of freedom as additional synthetic dimensions. Boada et al. Here we focus on the limit in which ℏω ¼ 2Δ ≫ J that leads (2012) and Celi et al. (2014) described how the techniques to the time-independent RWA Hamiltonian discussed in Sec. III.D can be used to create a lattice with one spatial dimension (denoted by j) and one synthetic dimension ϕ ˆ − 1 i RWAðm;jÞ (denoted by m to evoke the atomic m states from which it is HRWA ¼ JRWA½jj; m þ ihj; mje þ H:c; ð60Þ F built). Large artificial magnetic fields using synthetic dimen- with double-well tunneling strength sions were simultaneously realized at the National Institute of Standards and Technology (NIST) and the European Laboratory for Nonlinear Spectroscopy (LENS) (Manciniet al., − VR θ JRWA ¼ J 2Δ kRa cosð Þð61Þ 2015; Stuhl et al., 2015) using hyperfine ground states of bosonic 87Rb and fermionic 173Yb, respectively. and phase Both synthetic dimension experiments then replaced photon-assisted tunneling with two-photon Raman transitions. 1 Physically, these transitions simultaneously change the inter- ϕ ðm; jÞ¼2k a½j sin θ þðm þ Þ cos θ: ð62Þ RWA R 2 nal atomic state and impart the two-photon recoil momentum. For a 1D optical lattice—essentially one chain along ej Here the local phase of the traveling wave potential 2k · x0 at R of the 1D lattice in Fig. 14—the spatially imprinted phase the double well is directly imprinted onto the atoms as they is ϕ ¼ 2k aj sin θ. This expression is equivalent to Eq. (62) tunnel in the m direction, but not when they tunnel in the j syn R derived for photon-assisted tunneling, but without any direction. The result of this double-well analysis can be dependence on m since the Raman laser’s k vector is always extended to the whole lattice, where the expression for “perpendicular” to the synthetic m direction, rendering ϕ is unchanged, and J is qualitatively the same but RWA RWA cos θ → 0. This thereby eliminates the geometric compromise quantitatively altered. required to maximize the laser-assisted tunneling strength at Summing the tunneling phase around any plaquette gives simultaneously large flux. an Aharonov-Bohm flux Although synthetic dimension and photon-assisted tunneling experiments can produce the same sort of magnetic Φ 2 θ AB ¼ kRa sin ; ð63Þ lattice geometrics, the techniques have important practical

Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019 015005-18 N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms differences. For example, spin selective measurements allow 1. Flux lattices the synthetic dimension lattice site to be measured with near- Before turning to our discussion of flux lattices, we pause “ ” perfect spatial resolution. In addition the limited number to reflect on our discussion of topological lattices to this of spin states (typically 3 to 5) produce synthetic dimension point. Section II introduced the concept of topological lattices with striplike geometrics with perfect hard-wall invariants in terms of the Berry connection (Zak phase) boundary conditions, rather than extended planes as for or Berry curvature (Chern number) integrated over the BZ. conventional 2D optical lattices. In addition synthetic dimen- In the latter case, the Chern number can be cast as a sion lattice sites with the same spatial index j but with momentum-space statement of Gauss’s law, in which the different internal index m are in reality spatially overlapping, Berry curvature integrated over the toroidal BZ counts an so that the spatially local atom-atom interactions become integer number of topological “charges” in the inside of the anisotropic: long ranged in m and short ranged in j. 87 torus yielding the Chern number. Both experimental synthetic dimension realizations [ Rb “ ” 173 A flux lattice is an optical lattice potential that instead is and Yb in Mancini et al. (2015) and Stuhl et al. (2015), defined as a lattice in which the integrated Berry curvature in respectively] created large flux, highly elongated strips along each spatial unit cell is nonzero, suggesting that the atoms j, with just three sites in width along m. More recently two-leg might behave as if large magnetic fields are present. This ladder implementations of the synthetic dimension concept provides an intuitive framework in which to link the physics of have been realized (Livi et al., 2016; Kolkowitz et al., 2017) Landau levels and lattice bands. Indeed, in general, the atom 173 87 using the optical clock transition of both Yb and Sr, and experiences a combination of a periodic magnetic field (with synthetic dimensions have even been constructed using nonzero average) and a periodic scalar potential. Figure 15(b) momentum states in lieu of spin states (Meier, An, and depicts a bichromatic laser configuration that gives the same κ Gadway, 2016). used in the initial flux lattice proposals (Cooper, 2011; Cooper and Dalibard, 2011), with four frequency degenerate in-plane F. Flux lattices: Intrinsic topology lasers and with a single down-going laser at a different frequency. As shown by Juzeliūnas and Spielman (2012), Each of the topological lattices discussed in the preceding this geometry can be tuned to produce the desired effective sections was engineered beginning with a nontopological magnetic field with vector strength lattice to which modulation or light-assisted tunneling was added to engineer a desired topological model. In this section πx πy we focus on a different approach for generating a topological κ ¼ κ⊥ cos ex þ cos ey a a lattice without the need for modulation or light-assisted πx πy tunneling. This approach applies to an atom with several þ κ sin sin e ; ð66Þ internal states and subjected to a combination of three light- k a a z matter interactions: those that are independent of internal κ κ atomic states, those that depend on internal atomic states, and where ⊥ and k are set by the intensity and polarization of the those that couple between the different internal atomic states. laser fields, and a ¼ λ is equal to the laser wavelength λ. This In particular, as we show in Appendix C, the far-detuned light- result follows directly from the expressions in Appendix C. matter interaction for alkali atoms takes the form The right panel of Fig. 15(b) shows the spatial distribution of the Berry curvature in a single unit cell with spatial extent a, ˆ r 1ˆ κ r Fˆ κ κ HRWA ¼ Uð Þ þ ð Þ · ; ð65Þ with a clear non-negative mean, evaluated for ⊥ ¼ k. Thus, it achieves the goal of having a net nonzero effective magnetic ˆ where UðrÞ and κðrÞ · F describe the rank-0 (scalar) and rank- field piercing the unit cell. In both the square geometry and a 1 (vector) light shifts acting on an atom with internal angular- similar three-beam setup with 2π=3 intersection angle, the momentum operator Fˆ. The possible lattices formed from resulting band structure can have topological bands (Cooper, Eq. (65) have a very rich range of structures, characterized by 2011; Cooper and Dalibard, 2011). the spatial variations of UðrÞ and the three components of This discussion hides one subtle point: the integrated κðrÞ. Because this approach is not tied to a preexisting tight- curvature over each unit cell must be a multiple of 2π.In binding lattice, it is not limited to deep optical lattices and can the present case the integrated curvature is 8πmF over the have topological properties even for very shallow or weak complete unit cell. The circles in Fig. 15(b) mark the locations optical coupling strengths. of the minima of the adiabatic potential ℏmFjκj. For a deep We discuss the essence of this approach in two ways: (1) we lattice, in which this potential is large compared to the recoil explore “flux lattices” in which the Aharonov-Bohm flux energy, the atoms are strongly confined close to these minima. emulated by a Berry phase in real space leads to lattices with Their locations define effective lattice sites of a tight-binding nonzero effective magnetic fields, and (2) we explore the description, with the unit cell containing four such sites and connection between spin-dependent lattices and established therefore divided into four plaquettes. For a spin-1=2 system, topological models. Indeed in our initial discussion of the with mF ¼1=2, each of these plaquettes will have a flux Haldane model, we identified the two sublattice sites in each Φ ¼ π, while for bosonic alkali atoms, mF takes on integer unit cell with a pseudospin degree of freedom and arrived at a values implying Φ ¼ 2πmF is a multiple of 2π. In either case, spin-dependent band structure, Eq. (48); we now make this nearest-neighbor hopping on this square lattice geometry literal. would not break time-reversal symmetry, so the Berry

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(a) (b)

FIG. 15. Spin-dependent topological lattices. (a) Level diagram for three-level total angular-momentum f ¼ 1 case with mF states labeled, as is applicable for the common alkali atoms 7Li, 23Na, 39K, 41K, and 87Rb. For reference, the diagram shows the decomposition σ π of these optical fields into and , but as discussed in the text, this is not an overly useful way of considering this problem. (b) Laser geometry for a typical optical flux lattice (left) producing a real-space Berry curvature with nonzero average (right). (c) Experimental geometry for spin-dependent topological lattice (Sun et al., 2017). (d) Directly observed topological band structure (left) and computed (right). From Sun et al., 2017.

curvature of the bands must vanish. As a result, for topological 2. Fluxless lattices band structure to emerge in a straightforward way, longer A key insight from the Haldane model is that a net magnetic range hopping as in the Haldane model is required. These field is not a prerequisite for topological band structure. It is considerations indicate that, for this square geometry, the flux sufficient to break time-reversal symmetry, a condition that lattice would not lead to topological bands in the deep-lattice can readily be achieved for lattices that couple internal states. limit. However, we reiterate that the flux lattice approach is The appearance of topological bands for zero net flux is not restricted to deep lattices, but applies also for shallower readily established for shallow lattices (Cooper and Moessner, lattices in which the atoms can move throughout the unit cell 2012). Here we focus on the regime of deep lattices, where an and a restriction to nearest-neighbor hopping is inappropriate. effective (spin-dependent) tight-binding model can be devel- We demonstrated that flux lattices can give rise to a net oped. As we made explicit in our discussion of the Haldane nonzero magnetic field piercing the real-space unit cell. The model, the two basis sites consisting of a single unit cell can be comparison with free particles in a uniform magnetic field, assigned a pseudospin label, and these pseudospins are then which form Landau levels, suggests the appearance of arrayed to form a honeycomb or brick-wall lattice. Building topological Chern bands. However, this is not guaranteed: from this understanding, we conclude by discussing the such lattices may or may not be topological, as defined by the topology of state-dependent lattices, without any explicit usual Chern number computed in momentum space. (The reference to Berry phases. deep-lattice limit of the square flux lattice described above Figure 15(c) displays the phase-stable lattice geometry provides an example in which the bands are not topological realized by Sun et al. (2017), which is closely related to despite the nonzero effective magnetic flux.) Moreover, as we the flux lattice geometry, absent the vertical beam and with the now discuss, there can be cases in which the net flux through in-plane beams driving Raman transitions. A predecessor the unit cell vanishes, yet the bands are topological (Cooper of this setup described by Wu et al. (2016) generated a 2D and Moessner, 2012). In general, to determine the band optical lattice with spin-orbit coupling with 87Rb atoms [a first topology requires a full calculation of the band structure, realization of 2D spin-orbit coupling was reported by Huang including the atom’s kinetic energy. et al. (2016) for a bulk geometry]. The two pseudospin states

Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019 015005-20 N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms are the Zeeman levels jF ¼ 1;m¼ −1i and jF ¼ 1;m¼ 0i discussed by Möller and Cooper (2010) and LeBlanc et al. of the lowest hyperfine state of the ground state manifold. (2015) these differences arise because the synthetic vector The essential concept of this lattice is to first use the scalar potential vanishes during TOF when the laser fields are light shift from the detuned, counterpropagating beams to removed: there are physical differences in how exactly the create a conventional 2D optical lattice operating in the synthetic vector potential returns to zero when TOF begins tight-binding regime, and then to use the vector contribution and physically different effective electric fields present during of the light shift to give a combination of local effective this turn-off. All physical observables remain invariant to local magnetic fields and spin-dependent tunneling. The resulting gauge transformations, provided these are applied consis- tight-binding model is very closely related to that of the tently, i.e., changing the vector potential both before and Haldane model and shares its topological properties. The left during TOF. panel of Fig. 15(d) shows the experimentally measured band structure in good agreement with the predictions of theory 2. Local measurement of the Berry curvature displayed in the right panel. We consider here a cold atomic gas at equilibrium in a 2D IV. EXPERIMENTAL CONSEQUENCES optical lattice. With the so-called band-mapping technique one can precisely measure the distribution N ðqÞ of the This section presents recent experimental investigations quasimomentum q for each energy band. With this technique of nontrivial (global or local) topological properties of energy the laser beams forming the lattice are turned off in a bands, in either 1D or 2D geometries. Interactions play a controlled manner, so that the populations of the Bloch states nonessential role for the experiments described next; hence forming the various energy bands in the presence of the lattice phenomena addressed here correspond to single-particle (or are transferred to states with a well-defined momentum in the ideal gas) physics. absence of the lattice (Greiner et al., 2001). The measurement This section is divided into three parts. In the first one we of N ðqÞ in combination with a suitable temporal variation of describe measurements that are performed on an atomic the lattice parameters before the complete turn-off allows one system at equilibrium, using local probes in momentum space to characterize the band topology, i.e., to access not only the that allow one to reconstruct the topology of the occupied energies, but also the Berry curvature ΩðqÞ. band(s). In the second part we present analyses performed by To illustrate this point we consider a 2D lattice and use looking at the dynamics of wave packets. These wave packets the same two-band model as in Sec. II, assuming a unit cell are well localized at the scale of the Brillouin zone and one can with two nonequivalent sites labeled A and B. The generic ˆ ˆ bring them close to some points of specific interest, Dirac Hamiltonian in reciprocal space is Hq ¼ h0ðqÞ1 − hðqÞ · σˆ points, for example, using an external force. The last part where ðh0; hÞ is a 4-vector with real components that are is devoted to transport measurements, which are closer in periodic over the BZ. The energies of the two bands are Eq ¼ spirit to the techniques that are commonly used in condensed h0ðqÞjhðqÞj and the Bloch states juq i can be written as matter physics. linear combinations of jq;Ai and jq;Bi. Using the expression A. Characterization of equilibrium properties (30) for these Bloch states, one finds

1. Time-of-flight measurements Ω ðqÞ¼∇qðhuq jÞ × ∇qðjuq iÞ 1 Before entering into a discussion of specific measurements, ¼2∇qðcos θqÞ × ∇qϕq; ð67Þ we briefly comment on implications of the time-of-flight (TOF) measurements commonly used in experiment. In the where ðθq; ϕqÞ defines the direction of hðqÞ in spherical vast majority of cold atom experiments, the measurement coordinates. procedure begins with the rapid removal of all applied fields A procedure to determine θq and ϕq, hence the curvature Ω, (both those involved in trapping the ensemble and those in such a two-band situation was proposed by Hauke, required to create the topological lattice of interest). After Lewenstein, and Eckardt (2014) and implemented by this abrupt, projective, turn-off, the atoms then undergo a Fl¨aschner et al. (2016). The starting point is the momentum period of ballistic expansion followed by a measurement of distribution N qðkÞ associated with a Bloch state their density (often in a spin-resolved manner). In many X X experiments, this procedure gives a direct measurement of ð−Þ iq·R ψ q ðrÞ¼ αq wðr − R Þe s : ð68Þ the momentum distribution as it was when the applied fields ;s s s¼A;B R were just removed. s While this sounds simple in principle, this procedure can Here wðrÞ is the Wannier function of the band, supposed to be appear to give measurement results that are gauge dependent. identical for the two sublattices s ¼ A, B, and ðαq;A; αq;BÞ¼ Fortunately any supposed contradiction with general princi- iϕq ( cosðθq=2Þ; e sinðθq=2Þ). The momentum distribution ples of local gauge invariance is illusory. For example, two ð−Þ different experiments might well create a Harper-Hofstadter N qðkÞ is the square of the Fourier transform of ψ q ðrÞ. lattice with the same flux using very different laser geometries, For k inside the first BZ, it is peaked around k ¼ q and given ˜ 2 2 ˜ which naturally define the synthetic vector potential in two by jwðkÞj jαk;A þ αk;Bj , where wðkÞ is the Fourier transform different gauges. The observed momentum distribution will in of wðrÞ. When the lowest band is uniformly filled with general differ in these two cases (Kennedy et al., 2015). As independent fermions, the momentum distribution of the

Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019 015005-21 N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms gas is obtained by summing the contributions N qðkÞ over all quasimomenta q of the BZ:

2 N ðkÞ¼jw˜ ðkÞj ½1 − sin θk cos ϕk: ð69Þ

This distribution can be measured using a ballistic expansion after a sudden switch-off of the lattice. The key point of Eq. (69) is that the measured distribution N ðkÞ is sensitive to the relative phase of the contributions αq;s of the two sites − s ¼ A, B in the expression of each Bloch state jψ q i. More FIG. 16. (Left column) Amplitude ∝ sin θq and (middle col- precisely, although not sufficient to determine unambiguously umn) phase ϕq obtained from the fits to the oscillation (70) of the the angles θ and ϕ, this measurement already provides the momentum distribution. From those fit results, one can recon- value of the product sin θ cos ϕ. To go one step further, Hauke, struct the (right column) momentum-resolved Berry curvature Lewenstein, and Eckardt (2014) suggested to apply an abrupt given in units of the inverse reciprocal lattice vector length jbj quench to the lattice parameters so that the Hamiltonian squared. From Fl¨aschner et al., 2016. ˆ 0 becomes Hq ¼ðℏω0=2Þσˆ z. One then lets the gas evolve in the lattice during a time interval t before measuring the momen- The other two components of h=jhj can be obtained by tum distribution. Since the evolution during this time simply inducing a coherent transition between jai and jbi (Raman consists of adding the phase ω0t=2 to αq;s, the momentum pulse) with an adjustable phase and duration in order to rotate distribution at time t reads the pseudospin during the time of flight. Once the direction of hðqÞ is known at all points of the Brillouin zone, the value of 2 N ðk;tÞ¼jw˜ ðkÞj ½1 − sin θk cosðϕk þ ω0tÞ: ð70Þ the Berry curvature follows from Eq. (67).

By repeating this procedure for various times t and measuring 3. Topological bands and spin-orbit coupling the amplitude and the phase of the time-oscillating signal, one When the considered lattice has some specific geometrical can determine simultaneously ϕ and θ at any point in the BZ. symmetries, the assessment of the topological nature of a band This procedure was implemented by Fläschner et al. (2016) 40 can be notably simplified with respect to the procedure using a hexagonal lattice of tubes filled with fermionic K outlined above. One does not need to characterize the atoms. The unit cell for this graphenelike geometry contains eigenstates of the Hamiltonian at all points of the Brillouin two sites and the lattice parameters are chosen such that there zone to determine if the integral of the Berry curvature (67) ℏω is initially a large energy offset AB between the A and B over this zone is nonzero, and it is sufficient to concentrate on sites, corresponding to essentially flat bands with no tunnel- some highly symmetric points. The basis of this simplifica- ing. As explained in Sec. III.D, the dynamics in the lattice can tion, which is discussed in Appendix A for the case of a 1D be restored by a resonant, circular shaking of the lattice at a lattice [see, in particular, Eq. (A6)], was outlined by Liu et al. Ω ≈ ω frequency AB. In the experiment of Fläschner et al. (2013) for a 2D square optical lattice for pseudospin 1=2 (2016), the shaking was produced by a phase modulation of particles in the presence of spin-orbit coupling terms. the three laser beams forming the lattice, and it also resulted in Let us briefly outline the main result of Liu et al. (2013). a non-negligible value for the Berry curvature. Once the atoms Suppose that the Hamiltonian is invariant under the combined equilibrated in this lattice, the abrupt quench needed for the action of the spin operator σˆ z and the spatial operator trans- procedure of Hauke, Lewenstein, and Eckardt (2014) was forming a Bravais lattice vector R into −R. Consider the obtained by simply switching off the modulation. Measured energy eigenstates (Bloch states) at the four points of the BZ: N k;t amplitudes and phases of the time oscillation of ð Þ are fΛig¼fð0; 0Þ; ð0; πÞ; ðπ; 0Þ; ðπ; πÞg, i ¼ 1; …; 4. The two plotted in Fig. 16, together with the Berry curvature recon- Bloch states jψ ðΛ Þi at each of these locations are also Ω i structed from Eq. (67). One then expects the integral of q to eigenstates of σˆ , and the corresponding eigenvalues ξ can C 2π C z i; be an integer times , where is the Chern index of the take only the values þQ1 or −1. Now one can show that the sign populated band. Here the reconstructed Berry curvature leads of the product Pη ¼ ξ η of the four ξ η of a given subband C i i; i; to a value of compatible with 0 (Fläschner et al., 2016). This η ¼is directly related to the Chern number of this subband. is in agreement with the expected band topology in this case. More specifically, if the sign of Pη is negative, the Chern The method outlined by Hauke, Lewenstein, and Eckardt number of the subband η is odd, hence nonzero: this (2014) is reminiscent of a previous proposal by Alba et al. unambiguously signals a topological band. If Pη has a positive (2011). There the two lattice sites A and B are supposed to be sign, the Chern number of the subband η is even, and this most a occupied by two different internal (pseudospin) states j i and likely signals a nontopological character for this subband b j i. The momentum distribution measurement can be done in (zero Chern number). a spin-resolved way, which provides the local spin polariza- This procedure was realized experimentally by the type tion for the lowest band [see Eq. (30)]: of lattice we described in Sec. III.F.1 involving two pairs of retroreflected Raman lasers intersecting at right angles. This q N q − N q hzð Þ θ bð Þ að Þ configuration can produce the reciprocal-space tight-binding h q ¼ cos q ¼ N q N q : ð71Þ j ð Þj að Þþ bð Þ Hamiltonian with the following generic form (29):

Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019 015005-22 N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

ˆ 2 σˆ σˆ Hq ¼ tSO½sinðaqyÞ x þ sinðaqxÞ y bulk (H¨ugel and Paredes, 2014). For narrow systems with synthetic extent less than q for flux p=q, the states behave þ½m − 2t0( cosðaq Þþcosðaq Þ)σˆ ; ð72Þ z x y z more like those in a continuum [with a single guiding center, see the supplementary material in Stuhl et al. (2015)], and as where m is proportional to the detuning from Raman z the width further increases, the familiar topological edge resonance. The two amplitudes t0 and t characterize the SO modes emerge, but remain slightly gapped at the edge of the tunneling amplitudes without and with spin flip, respectively. 1D Brillouin zone from the bulk bands. They can be controlled independently by varying the inten- In addition, recent experiments have instead turned to using sities of the two pairs of laser beams. In particular the term momentum states for a synthetic dimension, in principle proportional to t can be simplified in the limit of low SO allowing for more extended systems. Indeed the edge states momenta into ∝ k σˆ þ k σˆ , corresponding to the usual form y x x y associated with the 1D SSH model have already been of the Rashba-Dresselhaus spin-orbit coupling for a bulk observed using a momentum-space lattice (Meier, An, and material (Galitski and Spielman, 2013). Gadway, 2016). An interesting feature of this Hamiltonian is the possibility We show in Fig. 17 the results of an experiment performed to control the topology of the lowest band: It is nontrivial if 173 by Mancini et al. (2015). A gas of fermionic Yb atoms can and only if jm j < 4t0. Wu et al. (2016) tested this prediction z tunnel between the sites of an optical lattice along the x 87 T ∼ 100 by loading the Rb gas at a temperature nK such that direction, with an essentially frozen motion along the two the lowest band was quasiuniformly filled, whereas the other (real) y and z directions. The synthetic direction consists population of higher bands remains small. The polarization of three Zeeman substates m ¼ −5=2, −1=2, þ3=2 selected defined in Eq. (71) was measured by a spin-resolved imaging among the six Zeeman states of the ground level. The of the atomic cloud after time of flight, and the product P− for “tunneling” along this synthetic direction is provided by a the lowest band was found to be negative in the expected pair of light beams. These beams induce stimulated Raman range of values of m . z processes between the Zeeman states, hence the desired laser- induced hopping. These beams also provide an artificial gauge 4. Momentum distribution and edges states field thanks to the space-dependent phase φðxÞ printed on Δ 2 It is well known from integer quantum Hall physics the atomic state in a m ¼ transition. that the nontrivial topology of a band can give rise to a The atoms are prepared in a metallic state (less than one atom/lattice site) in the lowest band of the single-particle quantized Hall conductance σxy. Applying a voltage differ- Hamiltonian. Figure 17 shows the momentum distributions ence Vy between the two opposite edges of a rectangular 2D n ðkÞ along the x direction for the three values m of the sample gives rise to a global current I ¼ σ V along the x m x xy y pseudospin. Here the central value of the pseudospin direction. This current flows on the edges of the sample in a (m ¼ −1=2) plays the role of the bulk of the material. The ðþÞ 0 chiral way, with a positive value Ix on the y> side of momentum distribution in this internal state is thus symmetric ð−Þ the sample and a negative value Ix on the y<0 side around 0, corresponding to a null net current. In contrast, a (Hatsugai, 1993). In the absence of an applied voltage nonzero current is associated with the side values of the (Vy ¼ 0), the edge currents are still nonzero but they exactly pseudospin: The distribution for the largest (smallest) value of ð−Þ ðþÞ m is displaced toward positive (respectively, negative) values. compensate each other: Ix ¼ −Ix . The possibility to engineer a 2D atomic gas with one real The displacement is made even clearer in the second row of dimension and one synthetic dimension that we discussed in Sec. IV.B.4 offers a way to directly access these edge currents m=-5/2 m=-1/2 m=+3/2 in a cold atom experiment. Indeed when the second dimension 0.6 0.6 0.6 ) 0.4 0.4 0.4 k

(labeled y) is synthetic, i.e., associated with an internal degree (

n 0.2 0.2 0.2 of freedom (pseudospin), one expects a given sign of Ix for the 0 0 0 largest value of the pseudospin and the opposite sign for its -3 -2 -1 0 123 -3 -2 -1 0 123 -3 -2 -1 0 123 smallest value. kkk 0.2 0.2 0.2 +0.079(6) +0.018(5) –0.062(4) The advantage of a synthetic dimension for observing these 0.1 0.1 0.1 ) k edge states is clear: It provides a sharp boundary to the sample, ( 0 0 0 h whereas a standard 2D optical lattice would lead to edge states -0.1 -0.1 -0.1 -0.2 -0.2 -0.2 that would be smeared over several lattice sites, and hence -3 -2 -1 0 123 -3 -2 -1 0 123 -3 -2 -1 0 123 much more difficult to observe. In addition, the momentum kkk distribution can be measured individually for each pseudospin FIG. 17. Direct visualization of an edge-state current for a two- value. One thus obtains the value of the current on each “site” dimension (one real, one synthetic) lattice in the presence of an of the synthetic direction. The transposition of such a artificial magnetic field. Upper row: Momentum distributions measurement to a real direction y implies a single-site nmðkÞ along the real direction x for the three values of the resolving imaging, which is much more demanding from pseudospin m corresponding to the synthetic direction. The value an experimental point of view. However, the use of spin states m ¼ −1=2 can be viewed as the bulk, whereas m ¼þ3=2 and for synthetic dimensions greatly limits the potential extent of m ¼ −5=2 corresponding to the opposite edges of the sample the synthetic dimension: in the extreme limit of just two spin along the synthetic dimension. Lower row: Function hmðkÞ¼ states, the system can even behave as if it were all edge with no nmðkÞ − nmð−kÞ. From Mancini et al., 2015.

Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019 015005-23 N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

Fig. 17 where the function hmðkÞ¼nmðkÞ − nmð−kÞ is realization of the Hamiltonian proposed by Rice and Mele plotted. Mancini et al. (2015) also checked that the sign of (1982), reducing to the SSH model (Su, Schrieffer, and the edge current is reversed when the sign of the artificial Heeger, 1979) for ϕ ¼ π=2. Here we restrict for simplicity magnetic field is changed. to the SSH case and we refer the reader to Atala et al. (2013) for a discussion of the general case. First we recall that, as B. Wave-packet analysis of the BZ topology discussed in Sec. II.C.1, the Zak phase itself is not invariant under gauge transformations in momentum space. Indeed it Cold atom experiments offer the possibility to prepare the depends on the choice of the relative phases between Wannier particles in a state described by a wave packet that is well states, i.e., the arbitrariness in deciding if the unit cell is localized in momentum space, in comparison with the size of formed by a pair fAj;Bjg [NA ¼ NB in Eq. (24)] or by a pair the Brillouin zone. In this case, the dynamics of the wave fBj−1;Ajg [NA ¼ NB þ 1 in Eq. (24)]. However, once this packet directly reveals the local properties in the BZ: energy D1 choice is made, the gauge-invariant quantity δϕ ≡ ϕð Þ− landscape EðqÞ and Berry curvature ΩðqÞ. These properties Zak Zak ϕðD2Þ π can be encoded either on the dynamics of the center of the Zak ¼ , where D1 and D2 correspond to the two possible wave packet or on the interference pattern that occurs when dimerizations of the system, obtained by choosing either several paths can be simultaneously followed. ϕ ¼ 0 or ϕ ¼ π. δϕ In order to measure Zak, Atala et al. (2013) first prepared 1. Bloch oscillations and Zak phase in 1D the atoms in a wave packet localized at the bottom of the lowest band. Then using a π=2 microwave pulse the atoms Conceptually the simplest example of a wave-packet were placed in a superposition of two spin states j↑i and j↓i, analysis of the band topology is found in a 1D lattice of which underwent Bloch oscillations in opposite directions in −π period a, for which the BZ extends between q ¼ =a and the presence of a magnetic gradient (Fig. 18). At the moment π =a. When an atom initially prepared with the quasimomen- when the two wave packets reach the edges of the BZ, the Zak tum q0 is submitted to an additional uniform force F, its phase is encoded in the relative phase between these wave quasimomentum qðtÞ periodically spans the Brillouin zone packets. In principle it could thus be read using a second π=2 ℏ (Bloch oscillation) at a frequency aF=h: qðtÞ¼q0 þ Ft= microwave pulse, closing the interferometer in quasimomen- 2π (mod =a). The phase that is accumulated in this periodic tum space. However, the magnetic field fluctuations in the lab motion contains relevant information about the topology of cause a random dephasing between the two arms of the the energy bands. interferometer and prevent one from performing this direct Since Bloch oscillations play a key role in several instances measurement. To circumvent this problem, Atala et al. (2013) in the following, we briefly outline the principle of their used a spin echo technique. A second microwave pulse flipped theoretical description. The Hamiltonian of the particle in the the spins as the atoms reached the edges of the BZ and at the presence of the periodic lattice potential VðxÞ and of the force same moment, the dimerization was changed by switching ϕ ˆ 2 F reads H ¼ pˆ =2m þ VðxˆÞ − Fxˆ. Let us write the initial from 0 to π, corresponding to an exchange of states between n n ψ 0 ixq0 ð Þ ð Þ state of the particle ðx; Þ¼e uq0 ðxÞ, where uq0 ðxÞ is the the lower and the upper bands. Finally the two paths were periodic part of the Bloch function associated with the nth recombined when the wave packets reached the top of the band and the quasimomentum q0. One can look for a solution upper band, and the accumulated phase revealed the value of δϕ δϕ π 0 97 2 of the time-dependent Schrödinger equation under the usual Zak. The experimental result Zak= ¼ . ð Þ was in Bloch form ψðx; tÞ¼eixqðtÞuðx; tÞ. The evolution of u is excellent agreement with the expected value. ˆ ˆ ℏ 2 governed by the periodic Hamiltonian HqðtÞ ¼½p þ qðtÞ = 2m þ VðxˆÞ; therefore the solution uðx; tÞ remains spatially 2. Measurement of the anomalous velocity periodic at any time. If we add the assumption that the force We now turn to the case of a 2D lattice and we F is weak enough so that interband transitions play a investigate how the semiclassical dynamics of a wave packet negligible role, the state of the particle will adiabatically follow the corresponding Bloch state of the nth band, i.e., iφðtÞ ðnÞ uðx; tÞ¼e uqðtÞðxÞ, and the relevant information is encoded in the phase φðtÞ. Let us focus on the state of the particle after one Bloch period. At this moment, the quasimomentum is back to its initial value q0 and the phase is theR sum of two contributions: (i) the dynamical phase − E½qðtÞdt=ℏ and (ii) the Zak phase of the band, Eq. (14). FIG. 18. Two-band spectrum of the SSH Hamiltonian and Zak phase measurement. An initial wave packet is prepared at the The first measurement of the Zak phase in a cold atom bottom of the lowest band (left column). Each atom is in the context was performed by Atala et al. (2013). A 1D super- superposition of the ji spin states, which experience opposite lattice was made out of two standing waves with a long and a forces in the presence of a magnetic field gradient. After a short period [see Sebby-Strabley et al. (2006) and Trotzky combination of exchanges of the atomic internal state and of the et al. (2008) for two methods for creating double-well dimerization (middle column), the two wave packets recombine superlattice potentials], generating the potential (50) where at the top of the highest band (right column). The measurement of ϕ both the relative phase and the amplitudes Vlong;short are the accumulated phase provides the value of the Zak phase. From control parameters. As detailed in Sec. III.B, this constitutes a Atala et al., 2013.

Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019 015005-24 N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms can reveal the topological features of a given energy band. (a) The starting point is the set of equations that govern the evolution of the average quasimomentum q and average position r when a constant force F is superimposed to the lattice potential

ℏq_ ¼ F; ð73Þ

ℏr_ ¼ ∇qEðqÞþΩðqÞ × F: ð74Þ (b) (c) This set of equations is valid when the applied force F is weak enough, so that transitions to other bands can be neglected. With the first equation we recover the Bloch oscillation phenomenon: The momentum drifts linearly in time in response to the applied force F. The second equation provides the value of the velocity of the wave packet at a given position in the BZ. It contains two contributions: The first one is the well-known expression for the group velocity [see, e.g., Ashcroft and Mermin (1976) for the case of a Q periodic potential]. The second contribution, which is some- FIG. 19. (a) Brillouin zone and Dirac points for the brick- times called the “anomalous velocity” (Xiao, Chang, and wall lattice of Fig. 10. (b) When the degeneracy is lifted by Niu, 2010), couples the wave-packet dynamics to the local introducing an energy offset between the A and B sites, the subbands are topologically trivial. For the lowest band, the Berry value of the Berry curvature ΩðqÞ. As a result of this curvature has opposite signs in the vicinity of Q . The anomalous contribution, the recording of a given trajectory inside the velocity (white arrows) thus has opposite chirality at these points. Brillouin zone allows one to reconstruct the Berry curvature (c) Lifting of degeneracy obtained by adding complex next- along this trajectory (Price and Cooper, 2012). nearest-neighbor (NNN) couplings (Haldane, 1988). The Berry This method was implemented by Jotzu et al. (2014) in curvature then keeps a constant sign over the BZ, and the order to analyze the topology of a 2D optical lattice in the anomalous velocity has the same chirality at both points. Adapted vicinity of Dirac points. The experiment was performed with a from Jotzu et al., 2014. brick-wall lattice [Fig. 10(a)], which is topologically equivalent to the hexagonal lattice of graphene, with two sites A and B per unit cell (Tarruell et al., 2012). In such a lattice, when 3. Interferometry in the BZ only nearest-neighbor couplings A → B and B → A are taken We now come back to the simple case of a graphenelike into account, the spectrum consists of two bands touching at lattice with only nearest-neighbor (NN) couplings, in which Q Q Q two Dirac points þ and − in the BZ [Fig. 19(a)]. As case the two subbands touch at two Dirac points . In this case explained in Sec. III.C, an additional circular shaking of the it is not possible to calculate the Berry curvature Ω for each Q lattice breaks the time-reversal symmetry of the system and subband because the definition (67) is singular in . The allows one to lift the degeneracy at the Dirac points, with the situation can be viewed as an equivalent (for momentum space) two subbands acquiring a nontrivial topology. Another pos- of an infinitely narrow solenoid (in real space) providing a finite sibility to lift this degeneracy consists of simply introducing magnetic flux. In the latter case it is known that the presence of an energy offset between sites A and B. However, in this case the solenoid can be probed by interferometric means. This is each subband is topologically trivial (Sec. II.C.2). With this indeed a paradigmatic example for the Aharonov-Bohm setup one can thus explore the phase diagram of the Haldane effect: Using a two-path interferometer such that the solenoid model shown in Fig. 10(b). passes through its enclosed area, there exists between the two Using an analysis of wave-packet dynamics close to the paths a phase difference proportional to the magnetic flux. In Q points , Jotzu et al. (2014) studied the transition between the case of a two-path interferometer enclosing a Dirac point the topological and trivial cases. Starting from a wave packet in momentum space, the phase difference between the two Q Q π at the center of the BZ, they dragged it close to þ or − paths is (Mikitik and Sharlai, 1999). using the force F created by a magnetic gradient. The This phase shift was measured in the cold atom context by curvature of the motion of the wave packet in the BZ Duca et al. (2015). The graphenelike structure was generated revealed the sign of the anomalous velocity, hence of the using an optical lattice with three beams at 120° angles. Ω Q Ω Q 87 Berry curvature. In the trivial case, ð þÞ and ð −Þ have Initially the external state of the Rb atoms is a wave packet opposite signs, hence the Chern number which is propor- located at the center of the BZ, and their internal state is a tional to the integral of ΩðqÞ over the BZ is zero [Fig. 19(b)]. given Zeeman state j↑i. A microwave π=2 pulse prepares a Ω Q ↑ ↓ In the topologically nontrivial case, ð Þ have the same coherent superposition of j i and j i, where the magnetic sign [Fig. 19(c)]. The measurements of the drift of the wave moment of j↓i is opposite to that of j↑i. Then the displace- packet performed by Jotzu et al. (2014) quantitatively ments of the two corresponding wave packets are controlled confirmed this scenario. using simultaneously: (i) a lattice acceleration along y which

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FIG. 20. Aharonov-Bohm interferometer around a Dirac point. Starting from a wave packet localized at the center of the BZ of a graphenelike optical lattice, one measures the phase difference φ between the two paths of an interferometer ending at the 0 quasimomentum (kx ¼ 0, ky). When one of the Dirac points K or K is inside the enclosed area of the interferometer, the measured value for φ is in good agreement with the prediction φ ¼ π. From Duca et al., 2015. creates the same inertial force on the two Zeeman states; (ii) a allow the near-perfect resolution of “position” in the spin magnetic gradient which creates opposite forces on them direction. along x; and (iii) a π microwave pulse which exchanges the Recent synthetic-dimension experiments (Mancini et al., atomic spins in the middle of the trajectories, in order to close 2015; Stuhl et al., 2015) used these techniques to directly the interferometer in momentum space. The result obtained image the evolution of edge magnetoplasmons. In a quantum by Duca et al. (2015), shown in Fig. 20, shows that a phase Hall system, an edge magnetoplasmon is the quantum analog difference of ∼π between the two paths appears if and only if to the skipping cyclotron orbits which “bounce” down the the enclosed area contains one of the two Dirac points. Using a edge of a system in a chiral manner, essentially following similar technique Li et al. (2016) could also access the Wilson cyclotron orbits that are interrupted by the system’s edge; see line regime, which generalizes the Berry phase concept to Fig. 4. These dynamical excitations evolve with a character- the case where the state of the system belongs to a (quasi) istic frequency given by the cyclotron frequency and are degenerate manifold. Li et al. (2016) investigated the case superposition states between different Landau levels. These where the transport from one place to another in the BZ is should not be confused with the chiral edge modes that done in a time much shorter than the inverse of the frequency underlie quantized conductance: these modes are built from width of the bands. In that experiment the transport was states all within the same Landau level. These modes have not characterized by a single phase factor and could be analyzed been observed in two-dimensional experiments; however, within the framework of St¨uckelberg interferometry (Lim, their analog has been observed in 1D experiments, also using Fuchs, and Montambaux, 2014, 2015). However, in more synthetic dimensions, where the end modes of 1D systems complex situations, interferometry can also reveal non- have been observed (Meier, An, and Gadway, 2016). Abelian features of the transport (Alexandradinata, Dai, and In condensed matter systems, edge magnetoplasmons Andrei Bernevig, 2014). certainly have been launched and detected, both in steady state, in magnetic focusing experiments (van Houten et al., 1989), and directly in the time domain using electrostatic gates 4. Direct imaging of edge magnetoplasmons (Ashoori et al., 1992). Cold atom experiments complete the Although the synthetic-dimension systems can be described picture by allowing for direct space and time resolution of by the same Harper-Hofstadter Hamiltonian as with real-space these skipping orbits. laser-assisted tunneling approaches, using the spin degree of In these experiments (Mancini et al., 2015; Stuhl et al., freedom to encode one spatial direction enables new prepa- 2015), the system was initialized with no hopping along the ration, control, and measurement opportunities. For example, synthetic dimension, and with all atoms in either one or the the finite number of spin states in effect generates infinitely other edge along the synthetic dimension. Once this initial sharp hard-wall boundaries in the synthetic dimension; the state was prepared, tunneling was instantly turned on. The synthetic-dimension tunneling can be applied and removed on highly localized initial state, described by a superposition of any experimental time scale; atoms can be initially prepared in different Landau levels, then began to evolve, skipping down any initial synthetic-dimension site; and conventional time-of- the system’s edge. Following a tunable period of evolution flight measurements, along with Stern-Gerlach techniques, time-of-flight measurements directly imaged the position

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FIG. 21. Edge-magnetoplasmon trajectories, where the displacement was obtained by integrating the velocity. From Stuhl et al., 2015. along the synthetic direction along with velocity in the spatial direction,1 which gives position by direct imaging. As shown in Fig. 21, this prescription allows for direct imaging of edge magnetoplasmons.

C. Transport measurements

In the last part of this section, we turn to experimental procedures that are closer in spirit to well-established condensed matter techniques. Starting from a uniformly filled FIG. 22. Topological pumping with a 1D superlattice described band, one can perform a transportlike experiment, measure the by the potential (50). Left: The phase ϕ of the long lattice [see displacement of the whole cloud in the lattice after a certain Eq. (50)] is varied from 0 to 2π from top to bottom. Initially the ϕ 0 duration, and infer nontrivial topological aspects from this phase ¼ and the particle is supposed to be localized on dynamics (Dauphin and Goldman, 2013). the site Aj of a given lattice cell j. In the limiting case where the energy difference between A and B sites is large compared to the tunnel matrix elements, this state would be stationary if ϕ was 1. Adiabatic pumping kept at the value 0. When ϕ is increased up to π=2, the sites Aj The concept of a quantized pump introduced by Thouless and Bj have the same energy and the particle is adiabatically (1983) has been described in general terms in Sec. II.F.It transferred to Bj. Note that we neglect here the tunneling of the requires a lattice whose shape is controlled by at least two particle from Aj to Bj−1, assuming that it is inhibited by the large Δ 0 − effective parameters, such as and J J for the Rice-Mele barrier between these two sites. The particle then remains in Bj model [see Eq. (40)]. One starts with a gas that uniformly fills until the phase reaches the value 3π=2, when the particle again a band of the lattice. Then one slowly modifies the lattice undergoes an adiabatic transfer, now from Bj to Ajþ1. (Here again shape in a way that corresponds to a closed loop in parameter we neglect tunneling across the large barrier now present between space. As shown in Appendix A, the resulting displacement of Bj and Aj.) When the phase ϕ ¼ 2π the potential is back to its the center of mass of the gas is then quantized in units of the initial value and the particle has moved by one lattice site. Note lattice spacing. that a motion in the opposite direction occurs if the particle starts This concept can be addressed in a 1D geometry, using a for the site Bj when ϕ ¼ 0. Right: In the two-band approximation superlattice with the potential (50). As explained in Sec. III.B, corresponding to the Rice-Mele model, the system performs a the motion of atoms in the two lowest bands of this potential closed loop around the origin in the parameter space ðJ0 − J;ΔÞ. for a deep enough lattice can be described by the Rice-Mele Hamiltonian. Suppose that the relative phase ϕ of the long- the second half of the cycle π < ϕ < 2π, one now finds J0 J. During this time following of the instantaneous energy levels; see Fig. 22. period, Δ increases and changes sign when ϕ ¼ π=2. During A Thouless pump was implemented in cold atom setups (Lohse et al., 2016; Lu et al., 2016; Nakajima et al., 2016). 1Because TOF measurements yield the momentum distribution, The experiment by Nakajima et al. (2016) used a superlattice some effort is required to derive the velocity in the lattice from this potential similar to that represented in Fig. 22, with Vlong ¼ 30 20 ℏ2 8 2 information. ER and Vshort ¼ ER, respectively, with ER ¼ =ð ma Þ.

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(a) (b) 4 1 t 2x(t)/a f

shi 2

r 2 e f f i 0 D

0 50 100 150 200 (d) BO time (ms)

FIG. 24. Transport measurement in a square lattice with a flux or ϕ π 2 plaquette AB ¼ = (data 1). A Hall-type current is observed through the displacement of the center of mass of the cloud along (e) the x direction, when a uniform force inducing Bloch oscillations (BO) is applied along y for an adjustable time. For short BO times, only the lowest subband is populated, resulting in a linear variation of xðtÞ with time, in agreement with the expected Chern index of this subband. At longer times, heating processes equalize the populations of the subbands and the Hall drift stops. When the flux is zero, no Hall current is observed (data 2). Adapted from FIG. 23. Quantization of the displacement of a cloud of Aidelsburger et al., 2015. fermionic 171Yb atoms placed in a 1D optical superlattice described by the Rice-Mele Hamiltonian. Depending on the closed loop in parameter space ðJ0 − J ≡ 2δ; ΔÞ, the displace- It provided a measurement of the anomalous velocity in good ment can be positive, zero, or negative. The time T represents the agreement with the expected value for the Chern number of duration of a pump cycle. From Nakajima et al., 2016. the lowest subband. For longer times t, heating originating from the resonant modulation applied to the lattice induced Using a gas of noninteracting 171Yb atoms (fermions), transitions between the various subbands, which eventually Nakajima et al. (2016) verified that the displacement of the get equally populated. Since the sum of all four Chern cloud when ϕ varies from 0 to 2π is equal to one lattice period, numbers is zero, the drift of the center of mass then stopped. as expected; see Fig. 23. They also checked that it is A careful analysis of this dynamics, associated with an topologically robust, i.e., it does not change if one slightly independent measurement of the population of each subband, deforms the path in parameter space by adding a time variation provided a measurement of the individual Chern numbers with a 1% precision. modulation of Vlong and Vshort. However, if the modification is such that the closed trajectory in parameter space ðJ0 − J; ΔÞ does not encircle the origin point anymore, the displacement V. INTERACTION EFFECTS per pump cycle drops to zero. Some of the most interesting directions for future work on cold atoms in topological optical lattices involve studies of 2. Center-of-mass dynamics in 2D collective effects that arise from interparticle interactions. As a last example of a cold atom probe of band topology, let Such studies hold promise for the exploration of novel phases us briefly describe the analysis of the dynamics associated of matter and to elucidate the role of topology in strongly with the Harper-Hofstadter Hamiltonian (46) by Aidelsburger correlated many-body systems. et al. (2015). This Hamiltonian was implemented using the laser-induced hopping method explained in Sec. III.D, pro- A. Two-body interactions ϕ π 2 ducing the flux per plaquette AB ¼ = . As a result of this applied gauge field, the lowest band of the lattice was split in The methods previously described for generating topo- 1 −1 −1 1 logical lattices consist either of periodic modulation of four subbands, with Chern numbers ðþ ; ; ; þ Þ, with “ ” the two intermediate subbands touching at Dirac points. site energies, forming a so-called Floquet system (see 87 Appendix B), or of Raman coupling of internal spin states, Bosonic Rb atoms were loaded in majority in the lowest leading to optically “dressed states” of the atoms. Both of subband of the lattice, at a temperature such that this subband these methods lead to effective interactions between particles was filled quasiuniformly. Then a weak uniform force along in the resulting energy bands that have some novel features. the y axis originating from a gradient of the light intensity of an auxiliary beam was applied to the atoms for an adjustable duration t. The position of the center of mass of the atom cloud 1. Beyond contact interactions was monitored as a function of time and it revealed the For ultracold atoms in the continuum, the typical two-body topology of the bands. At short times (typically less than interactions are dominated by the s-wave scattering, which 50 ms), the drift of the cloud along the x direction, similar to a can be represented by a short-range (contact) interaction Hall current, was found to be linear with t; see Fig. 24. gδðr1 − r2Þ, where the Dirac δ distribution is supposed to

Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019 015005-28 N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms be properly regularized. However, for particles “dressed” by a nonadiabatic corrections to the optical dressing, allowing laser field as in the optical lattices described, the interactions two particles to coincide in real space by being in different typically acquire nonlocal character. internal dressed states, dependent on their momentum (Cooper and Dalibard, 2011). The momentum dependence a. Continuum models of the dressed-state wave functions converts the contact For the continuum setting of optical flux lattices of interactions into a momentum-dependent interaction for atoms, Sec. III.F with laser plane waves inducing Raman couplings thereby allowing effective p-wave scattering even at ultralow between internal spin states, nonlocal interactions can arise temperatures. Similarly, dressed-state bosons can acquire a from the momentum dependence of the dressed-state wave momentum-dependent interaction allowing the appearance of functions. For the two-state system described by the d-wave and higher angular-momentum channels in the scatter- Hamiltonian (72), an energy eigenstate in a given band can ing (Williams et al., 2012). be written as b. Tight-binding models Ψ ψ ⊗ Σ j qi¼j qi j qi; ð75Þ For tight-binding lattice systems, we described in Sec. III how periodic driving at frequency ω can be used to tailor the i.e., the product of an orbital Bloch state jψ qi of quasimo- amplitudes and phases of the hopping matrix elements mentum q and of a q-dependent spin state jΣqi written in the between the sites. We demonstrated how, at rapid driving Σ basis jiz. The dependence of the spin component j qi on a frequencies, an effective Hamiltonian with potentially com- wave vector has direct consequences on the two-body scatter- plex modified tunneling amplitudes arises. This is the effective ing matrix elements within this band. Floquet Hamiltonian for the modulated system, which governs Let us first recall the simple case of spinless or fully the dynamics of the particles on time scales large compared to polarized particles. For the contact interaction Vˆ ¼ gδðrˆÞ, the the period of the modulation, T ¼ 2π=ω, as described in matrix element of Vˆ for a pair of free distinguishable particles Appendix B. In the presence of interparticle interactions, e.g., transitioning from ðq1; q2Þ → ðq1 þ Q; q2 − QÞ is equal to g Hubbard interaction U between particles on the same lattice (up to a normalization factor) and it is independent of the site, new terms appear in this effective Floquet Hamiltonian, momentum transfer Q. Here we neglect for simplicity energy- including nonlocal interactions (Eckardt, 2017). To under- dependent corrections of the s-wave scattering amplitude, stand the origin of the nonlocal interactions, consider the which is valid when Q−1 is larger than the s-wave scattering modulated two-site system described in Sec. III.C under a length, a criterion usually satisfied in quantum gases. For two harmonic drive of the energy offset between the sites indistinguishable particles, physical observables involve the ΔðtÞ¼ðΔ=2Þ cos ωt. The unitary transformation Eq. (52) sum over the two permutations leads to a Hamiltonian

ϕ − ϕ ˆ 0 − i PðtÞ i PðtÞ ðq1; q2Þ → ðq1 þ Q; q2 − QÞ or ðq2 − Q; q1 þ QÞð76Þ H ¼ J½e jrihljþe jlihrj; ð77Þ Δ with a relative sign of ε ¼1 for bosons or fermions. The ϕ ω PðtÞ¼ℏω sinð tÞ: ð78Þ matrix element of Vˆ is doubled for polarized bosons and vanishes for polarized fermions: the latter result simply It is convenient to expand this Hamiltonian in its harmonics reflects the fact that single-component fermions are insensitive to the contact interaction, since the Pauli principle precludes X∞ Δ ˆ 0 − J imωt them from having the same spatial position. H ¼ J m ℏω e jrihljþH:c ð79Þ Consider now the scattering of atoms in the dressed-state m¼−∞ band with wave functions (75) and assume for simplicity that J 0 the contact interaction is independent of the internal states where mðzÞ are Bessel functions of the first kind. The m ¼ ji. For bosons and fermions (ε ¼1), the transition matrix term gives the time-averaged Hamiltonian discussed in element of Vˆ for the two-path process (76) is now Sec. III: it describes intersite tunneling with an effective tunneling rate that is modified from the bare rate J by ˆ J 0ðΔ=ℏωÞ. For ℏω large compared to all other energy scales hψ q Q; ψ q −QjVjψ q ; ψ q ihΣq QjΣq ihΣq −QjΣq i 1þ 2 1 2 1þ 1 2 2 (tunneling J and any on-site-interaction energy U) the higher- ε ψ ψ ˆ ψ ψ Σ Σ Σ Σ 0 þ h q2−Q; q1þQjVj q1 ; q2 ih q2−Qj q1 ih q1þQj q2 i. order terms jmj > are far off resonant, so are rapidly oscillating and have small effects on the wave function. For fermions, this matrix element no longer vanishes in Still these terms do perturb the atomic wave function, leading Q 0 ∝ 1 − Σ Σ 2 to the new features we are interested in here. For a particle that general. For example, for ¼ ,itis gð jh q2 j q1 ij Þ. This is an important result that shows that starting from is initially in the right well jri, a rapidly oscillating perturba- ˆ imωt ≠ 0 contact-interacting fermions occupying a nondegenerate band, tion Vme (i.e., m ) causes a first-order correction to the the optical dressing leads to an interacting one-component wave function of Fermi gas, more precisely to effective p-wave interactions ω 2 et al. ω 2 sinðm t= Þ (Zhang , 2008). The possibility of interaction between jψ 0i ≈ jriþjlihljVˆ jrieim t= ð80Þ two identical fermions can be viewed as arising from m mℏω

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ω 2 energy shifts. The imposed energy offset between two imωt=2 sinðm t= Þ ≈jri − JJ mðΔ=ℏωÞe jli. ð81Þ adjacent lattice sites is modified by the on-site interactions mℏω U in such a way that the photon-assisted tunneling can be Thus, these terms induce a small nonzero probability for the brought into or out of resonance depending on the occupations particle to be in the left well, of the sites by other particles. This leads to a density- dependent tunneling term of the form X J2 Δ 2 X J † pl ¼ 2 m ; ð82Þ ˆ ˆ ˆ ˆ 2 ℏω ℏω ½Jðni; niþ1Þbi biþ1 þ H:c:; ð84Þ m≠0 ðm Þ i after averaging over the rapid oscillations and summing over shown here for spinless bosons. Such situations were all such terms. If in addition to the tunneling (78) the static analyzed by Aidelsburger et al. (2015), Bermudez and Hamiltonian has an on-site interaction between particles in the Porras (2015),andRačiūnas et al. (2016). The density- left well, which we consider to be a Hubbard interaction dependent corrections are typically a small modification of ˆ ˆ − 1 2 ˆ Unlðnl Þ= (with nl the number operator in the left well), the photon-assisted hopping in current experimental setups then these rapidly oscillating terms that move particles from of gauge fields on optical lattices [see supplementary the right to the left well will give rise to effective nonlocal information in Aidelsburger et al. (2015)]. However, they eff ˆ ˆ interactions Ulr nlnr, with can be made to dominate in regimes where U is large, such ω 2 that the photon drive frequency is not resonant with the eff UJ bare detuning Δ but with Δ U. Such situations have been U ¼ Upl ∝ : ð83Þ lr ðℏωÞ2 explored in experiments on two-component Fermi systems (Görg et al., 2018; Xu et al., 2018). Analogous coupling A similar nonlocal term will arise from interactions in the right between current and density arises also in the continuum if well. A full analysis of such effects is best achieved through a interactions lead to detunings that influence the local construction of the Floquet Hamiltonian at high drive fre- dressed states, thus causing the induced vector potential quency via the Magnus expansion (Goldman and Dalibard, A to be density dependent (Edmonds et al., 2013). Density- 2014) or other systematic approach (Eckardt and Anisimovas, mediated hopping can also arise in settings where the 2015). An overview of the Magnus expansion is given in single-particle energy bands are flat (dispersionless). Appendix B. Then individual particles do not move, being localized to a region of size of the Wannier orbital, and particle motion c. Synthetic dimensions arises only through interparticle interactions. Such settings An extreme example of nonlocal interactions arises for include the flat energy bands of frustrated lattices, such as systems involving synthetic dimensions (Celi et al., 2014; the Creutz ladder (Creutz, 1999; Mazza et al., 2012; Mancini et al., 2015; Stuhl et al., 2015). There interactions are J¨unemann et al., 2017) in 1D, and the kagome (Huber and Altman, 2010) and dice lattices (Möller and Cooper, typically long ranged among the set of s ¼ 1, Ns internal states which form the synthetic dimension and short ranged in 2012) in 2D, as well as for particles in dressed states of the d spatial coordinates. For a synthetic dimension formed internal spin states in which the direct nearest-neighbor from an internal spin degree of freedom, the interactions are of tunneling can be made to vanish (Bilitewski and infinite range in the synthetic dimension and of conventional Cooper, 2016). short range in position. For example spin-exchange interactions can be viewed as a correlated tunneling, such as 2. Floquet heating m; m0 → m − 1;m0 1 . Such systems realize interesting ð Þ ð þ Þ The explicit time dependence of the Hamiltonian in intermediate situations in which the single-particle physics periodically driven systems relaxes energy conservation and can be viewed as d 1 dimensional (for N is large) while the þ s leads to forms of inelastic scattering and heating not present in interactions remain d dimensional. For a synthetic dimension the time-independent case. For a Floquet system at frequency formed from simple harmonic oscillator subband states (Price, ω, these correspond to the absorption (or emission) of an Ozawa, and Goldman, 2017) the interactions fall off with integer number of “photons” of energy ℏω from the external increasing spacing along the synthetic dimension. Interactions drive. For the “Floquet-Bloch” waves of a spatial- and time- in an effective two-leg ladder, formed from spin-orbit coupled periodic potential, such inelastic scattering could occur even strontium atoms, have been studied experimentally by for a single particle that scatters from a defect in the lattice, Bromley et al. (2018). which allows a momentum transfer. However, an important source of potential inelastic scattering is for pairs of particles d. Current-density coupling via the interparticle interactions, i.e., inelastic two-body Finally, we note that the novel two-body interactions that collisions. are induced by Raman coupling or Floquet modulation can A general description of the inelastic scattering of Floquet- also include terms that are not just density-density inter- Bloch waves was provided by Bilitewski and Cooper (2015b). actions, but that couple the particle motion to particle density. Consider a time-periodic Hamiltonian of frequency ω and These effects arise naturally in photon-assisted tunneling denote the single-particle energy band, with band index ν,by between lattice sites as a consequence of interaction-induced the energies ϵνðqÞ defined as continuous functions of q over

Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019 015005-30 N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms the Brillouin zone. Owing to the time periodicity, this can be MBL” phase is the Floquet time crystal (Moessner and viewed as one member of a sequence of Floquet energy bands Sondhi, 2017). ðmÞ ϵν ðqÞ¼ϵνðqÞþmℏω (Eckardt, 2017). We define inelastic scattering to be those processes in which m or ν, or both, B. Many-body phases change under a scattering event. For weak scattering from state i to state f, the inelastic rate can be computed through a The topological optical lattices described support an array “Floquet Fermi golden rule” (Kitagawa et al., 2011) of many-body phases. Novel features arise for weakly interacting gases through the geometrical and topological 2π X ðmÞ ˆ ð0Þ 2 characters of the underlying band structure. Furthermore, γ → ⟪Φ VΦ ⟫ δ E − E − mℏω ; i f ¼ ℏ j f j i j ð i f Þ ð85Þ there can arise interesting strongly correlated phases, driven m by strong interparticle interactions. ΦðmÞ ≡ −imωt Φð0Þ ⟪Φ Φ ⟫≡ where R j f;i ðtÞi e j f;i ðtÞi and 1j 2 T ˆ 1. Bose-Einstein condensates ð1=TÞ 0 hΦ1jΦ2idt.HereV can denote a one-body potential (e.g., a lattice defect) or a two-body interaction. Processes For an optical lattice loaded with a gas of noninteracting with nonzero Δm correspond to the exchange of Δm quanta bosons one expects the ground state to be a Bose-Einstein from the drive field, changing the energy of the atom by mℏω. condensate (BEC), in which all particles condense at the This provides a simple prescription by which to calculate the minimum of the lowest energy band. This expectation applies inelastic (Δm ≠ 0) contribution to the two-body scattering just as well to the topological optical lattices as to regular of Floquet-Bloch states. Stepping beyond the two-body optical lattices. However, topological optical lattices can bring calculation to a many-body setting can be achieved by several novel features. analyzing instabilities within the Gross-Pitaevskii description (1) For systems without time-reversal symmetry (as required (Choudhury and Mueller, 2014, 2015; Lellouch et al., 2017). to generate a Chern band), the individual Bloch wave functions Calculations for the parameters used in the experimental have phase variations which in general give rise to nonzero local studies of weakly interacting bosons in the Harper-Hofstadter current density. Since the Bloch wave functions are stationary model (Aidelsburger et al., 2015) show that the heating rates states, these currents must be divergenceless, but this still allows 1 observed in those experiments are consistent with the the BEC to support circulating currents in dimensions d> . expected inelastic two-body scattering processes dominated These currents take the form of the local current density of a by single-photon absorption (Bilitewski and Cooper, 2015a). vortex lattice. An example is shown in Fig. 25. 1 This analysis emphasizes the role played by the motion (2) For topological optical lattices in d> the energy transverse to the 2D plane along the weakly confined third minimum in which the BEC forms will, in general, be dimension. The application of an optical lattice to confine this characterized by a nonzero Berry curvature. As described motion and open up gaps at the one-photon resonance is in Sec. V.C this Berry curvature affects the collective modes of expected to significantly reduce the heating rate in such the BEC. experiments (Bilitewski and Cooper, 2015a; Choudhury (3) There can arise situations in which there are a set of and Mueller, 2015). The excitation of motion along tubes, degenerate energy minima. An important example is provided transverse to the optical lattice, has been argued to also be by the Harper-Hofstadter model, for which the energy band responsible for heating rates in an experimental study of a periodically modulated 1D lattice (Reitter et al., 2017). In strongly driven systems (Weinberg et al., 2015) particle transfer to higher bands can arise from multiphoton reso- nances at the single-particle level (Str¨ater and Eckardt, 2016). Floquet heating in the full, many-body system presents an interesting theoretical issue which remains an active area of investigation. That energy is not conserved leads to the expectation that the system will be driven to an infinite temperature state at long times (Lazarides, Das, and Moessner, 2014). This expectation relies on the assumption that energy is redistributed between all degrees of freedom through the interparticle interactions. There can, however, FIG. 25. A vortex lattice configuration for weakly interacting bosons in the Harper-Hofstadter model at flux ϕ ¼ 2π=3. The arise situations in which many-body systems show steady AB gauge-invariant currents flow on the links between the sites of the states that are not at infinite temperature (D’Alessio and square lattice. The pattern of current breaks the translational Polkovnikov, 2013; Chandran and Sondhi, 2016), or in which invariance of the system. The strongly circulating currents there form prethermalized states on intermediate time scales (shown as solid lines and arrows) are around plaquettes that (Bukov et al., 2015; Canovi, Kollar, and Eckstein, 2016). lie along diagonal lines. These are the lattice equivalents of the Furthermore, in settings involving disorder, it has been shown “vortex cores,” now distorted from the triangular lattice expected that many-body localized (MBL) phases can be robust to in the continuum models, to be pinned to the plaquettes of the Floquet modulations, allowing the existence of nonthermal square lattice. There are weak countercirculating currents around steady states to arbitrarily long times (Abanin, De Roeck, and other plaquettes, such that the net particle flow vanishes, when Huveneers, 2016). A striking example of such a “Floquet coarse grained on scales large compared to the lattice spacing.

Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019 015005-31 N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms 1 cˆ typically has multiple degenerate minima. For example, at ˆ − μ ˆ ˆ† ˆ HBdG ϕ 2π H N ¼ ðc cÞ † ; ð87Þ flux AB ¼ =q the band has q degenerate minima in the 2 cˆ magnetic Brillouin zone. For noninteracting particles, there is ! a macroscopic degeneracy associated with the occupation of ε Δ HBdG ¼ ; ð88Þ these q degenerate single-particle states: a BEC could form in −Δ −ε any linear superposition of these states; or indeed a “frag- mented” condensate could form (Mueller et al., 2006). The † T where ðc;ˆ cˆ Þ is a 2Ns-component column vector formed inclusion of interactions U ≠ 0 is required to resolve this ˆ by listing all fermionic destruction cα¼1;…;Ns and creation macroscopic degeneracy. The simplest way to include inter- † cˆα 1 … operators. The quasiparticle excitations are deter- actions is via Gross-Pitaevskii mean-field theory. This is valid ¼ ; ;Ns 2 2 HBdG for sufficiently weak interactions U ≪ J and at high mean mined by the spectrum of the Ns × Ns matrix particle density n¯ ≫ 1. Studies of the mean-field ground state ! ! ðλÞ ðλÞ (in regimes of both weak and strong interactions, Un¯ ≪ J and u BdG u Eλ ¼ H ð89Þ Un¯ ≫ J) show that repulsive interactions stabilize a simple vðλÞ vðλÞ BEC with the form of a vortex lattice (Straley and Barnett, λ 1993; Zhang et al., 2010; Powell et al., 2011). This state in terms of the 2N -component vector of amplitudes uð Þ s α¼1;…;Ns breaks the underlying translational symmetry of the lattice, so ðλÞ and vα 1 … , where λ labels the 2N eigenvalues. The matrix there are several discrete (symmetry-related) states that are ¼ ; ;Ns s ϕ 2π 3 degenerate. For example, for flux AB ¼ = it has the form BdG Hamiltonian (88) has a special symmetry: of vortex lines along diagonals of the lattice; see Fig. 25. ! ! 0 0 Numerical studies of the vortex lattice ground states for I I HBdG ¼ − ½HBdG ; ð90Þ Un¯ ≫ J have been conducted for a wide range of flux I 0 I 0 ϕ AB: these show complex behavior with many competing vortex lattice structures (Straley and Barnett, 1993). where 0 and I are the Ns × Ns null and identity matrices, Similar vortex lattice states appear in ladder systems. In respectively. This has the consequence that for any eigenstate experiments on weakly interacting bosons on a two-leg ladder ðλÞ ðλÞ T ðu ; v Þ with eigenvalue Eλ there is another eigenstate with flux, Atala et al. (2014) demonstrated a transition ! ! ¯ between a uniform superfluid phase with Meissner-like chiral uðλÞ vðλÞ currents and a vortex phase (with broken translational sym- ¼ ð91Þ λ¯ ðλÞ metry along the ladder) as a function of the tunneling strength vð Þ u across the rungs of the ladder, akin to the Meissner and vortex − lattice phases of a type-II superconductor. with eigenvalue Eλ¯ ¼ Eλ: i.e., the spectrum is symmetric in energy around E ¼ 0. (Note that energies have been defined relative to the chemical potential μ.) This intrinsic particle- 2. Topological superfluids hole symmetry represents an inherent redundancy in the In both the tight-binding lattices and the Raman-dressed flux theory, by which the eigenstates of HBdG must include both lattices, situations can arise in which there are nonlocal the destruction and the creation operator of any quasiparticle interactions between fermions in a single band. This allows state (Bernevig and Hughes, 2013). Specifically, defining the attractive p-wave pairing and can lead to interesting forms of operator associated with each eigenvector λ ¼ 1; …; 2Ns topological superfluidity with “Majorana” excitations. Here we X ˆ ðλÞ ðλÞ † discuss the general features of these topological superfluids. Cλ ¼ uα cˆα þ vα cˆα ð92Þ Their properties are understood within mean-field theory, as α described by the BdG Hamiltonian. This takes the form the Hamiltonian may be expressed in the diagonalized form X 1 2 ˆ ˆ ˆ† ˆ ˆ† ˆ† ˆ ˆ 1 XNs H − μN ¼ εαβcαcβ þ ðΔαβcαcβ þ ΔαβcβcαÞ ; ð86Þ ˆ ˆ ˆ † ˆ 2 H − μN ¼ EλC Cλ: ð93Þ α;β 2 λ λ¼1

ð†Þ where cˆα are fermionic creation or annihilation operators The symmetry (91) allows this to be written as for the single-particle states labeled by α ¼ 1; …;Ns, which XNs encodes both positional and internal (spin) degrees of free- ˆ ˆ 1 ˆ † ˆ ˆ † ˆ H − μN EλC Cλ E¯ C C¯ 94 dom. They obey the fermionic anticommutation relations ¼ 2 λ þ λ λ¯ λ ð Þ † † † λ¼1 fcˆα; cˆβg¼fcˆα; cˆβg¼0, fcˆα; cˆβg¼δαβ. The Ns × Ns matrix ε XNs describes the conventional particle motion and potentials 1 ˆ † ˆ ˆ ˆ † ¼ EλðCλCλ − CλCλÞð95Þ and must be Hermitian εαβ ¼ εβα; the Ns × Ns matrix Δ 2 λ¼1 represents the superconducting pairing and is antisymmet- Δ −Δ ric αβ ¼ βα. XNs 1 – ˆ † ˆ − It is convenient to express the Bogoliubov de Gennes ¼ Eλ CλCλ 2 ; ð96Þ Hamiltonian as (up to a constant shift) λ¼1

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where the sum is now over just Ns eigenstates. These are topological superconductors there is a clear physical reason to conventionally chosen to be states for which Eλ ≥ 0, such that prefer the Majorana operators: the Majoranas are spatially ˆ ˆ † the ground state j0i is defined by Cλj0i¼0, and Cλ are the localized, each one tied to a single boundary or defect. Thus, quasiparticle creation operators. local probes couple directly to the Majorana operators. In † For spinless fermions in translationally invariant settings we Cˆ ð Þ P contrast, the quasiparticle operators λ0 are nonlocal. The ˆ −iq·rα can use the plane-wave operators c˜q ∝ αe cˆα to write locality of the Majorana operators is shown explicitly in Appendix A for the Kitaev model, where one Majorana 1 X ε Δ ˜ˆ operator acts on the right boundary and one acts on the left ˆ ˆ ˆ† ˆ q q cq H − μN ¼ ðc˜qc˜−qÞ ð97Þ 2 ˆ† boundary. q Δq −εq c˜−q That the Majorana modes are spatially localized causes the Δ −Δ system to have properties that are robust to external pertur- with the superconducting gap required to satisfy −q ¼ q. bations, including disorder potentials: the exact particle-hole The BdG matrix (88) reduces to a momentum-dependent symmetry enforces the mode to have E ¼ 0. Departures from 2 2 HBdG × matrix q . An important example is the Kitaev model E ¼ 0 can arise only from mixing with other Majorana modes; of a p-wave superfluid in 1D, described in detail in since the intervening superconducting state is gapped these Appendix A, for which corrections are suppressed exponentially in the distance between the local Majorana modes, e.g., the length of the −2J cosðqaÞ − μ −2iΔ sinðqaÞ — BdG 1D Kitaev chain. Note that the two Majorana modes one at Hq ¼ ð98Þ 2iΔ sinðqaÞ μ þ 2J cosðqaÞ each end of the 1D Kitaev chain—describe a single fermionic excitation, such that one should view the Majorana as a ≡ − hðqÞ · σ: ð99Þ fractionalized quasiparticle. In contrast, the zero-energy edge states of the SSH model correspond to two separate fermionic The energy eigenvalues show the above particle-hole sym- excitations, one at each end of the chain. As discussed in Sec. II.E.1 the edge states of the SSH model are sensitive to metry Eq ¼jhðqÞj. Furthermore, as shown in Appendix A, the bands are characterized by a winding number relating to local perturbations which break the chiral symmetry. how hðqÞ encircles the origin, analogous to the winding The above example for the 1D topological superconductor number of the SSH model. This winding number can be used has a very natural generalization to 2D. There, the bulk to identify a topological superconducting phase. spectrum is of the form On a finite geometry this topological phase hosts an exact ϵq − μ Δq Eλ 0 BdG zero-energy state on its boundary, i.e., with 0 ¼ in Hq ≡ −h q · σ; 102 ¼ ð Þ ð Þ Eq. (96). The prescription that we used to define the form Δq μ − ϵq (96) is ambiguous for such zero modes. Consider the operator ˆ q Cλ constructed via Eq. (92) from the eigenvector of such a where now runs over a 2D Brillouin zone. Topological 0 superconducting states can appear in situations where the zero mode λ0. The particle-hole symmetry implies that there is ¯ hðqÞ acquires all three components, such as in the continuum another zero-energy state with label λ0, which, using Eq. (91), † px þ ipy superfluid, for which ˆ ¯ ˆ would lead to an operator Cλ0 ¼ Cλ0. Together, these two ˆ † ˆ operators C and C¯ describe the destruction or creation of a ℏ2 q 2 2 − μ Δ λ0 λ0 hz ¼ j j = m ;hx þ ihy ¼ 0ðqx þ iqyÞ: ð103Þ ¯ 0 fermionic quasiparticle at Eλ0 ¼ Eλ0 ¼ . Since this excitation has zero energy, it describes a ground state that is twofold For μ > 0, the quasiparticle spectrum is fully gapped, and the degenerate depending on whether this zero-energy mode is unit vector hðqÞ=jhðqÞj wraps the sphere as q runs over all occupied or filled. It is therefore immaterial which of these values, indicating that this is a topological phase (Read and operators we choose to view as a particle creation or Green, 2000). At μ ¼ 0 there is a gap-closing transition to a destruction. Indeed, we could also choose to work in terms superconducting phase at μ < 0 which is nontopological. of operators that are linear superpositions. One particular The 1D surface of this topological 2D superconductor has choice is to define the Majorana operators an edge mode that has Majorana character, albeit in a setting in which there is a continuum of edge modes. In this 2D setting ˆ ˆ † localized Majorana modes arise as bound states on the cores of γˆ1 ≡ ðCλ þ Cλ Þ; ð100Þ 0 0 quantized vortices, i.e., point defects localized in the bulk of ˆ ˆ † the system (introduced by rotation or other external means). γˆ2 ≡ iðCλ − C Þ; ð101Þ 0 λ0 It is natural to search for topological superfluids using cold atoms. Many routes to p-wave pairing of single-component γˆ ; γˆ 2δ which obey anticommutation relations f i jg¼ ij. Since fermions have been suggested. These include methods involv- γˆ† γˆ the Majorana operators are self-adjoint i ¼ i, they can be ing a p-wave Feshbach resonance (Gurarie and Radzihovsky, viewed as describing particles that are their own antiparticles. 2007), long-range dipolar interactions (Baranov et al., 2012), The above transformation to Majorana operators appears or induced interactions via a background BEC (Wu and rather arbitrary: from a mathematical perspective one can Bruun, 2016). In the context of this review, the connections always choose to work either in terms of the two Majorana arise through the use of spin-orbit coupling (optical dressing) γˆ γˆ ˆ ˆ † operators 1, 2 or in terms of Cλ0 and its adjoint Cλ0 . For to allow contact interactions (e.g., s-wave pairing) to lead to

Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019 015005-33 N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms effective p-wave interactions between single-component fer- b. Lattice effects mions. Proposals of this kind have been presented in 3D, 2D The use of optical lattices to generate topological energy (Zhang et al., 2008), and 1D (Nascimb`ene, 2013; Yan, Wan, bands naturally causes cold atomic gases to explore new and Wang, 2015). aspects of FQH physics. The underlying lattice makes the single-particle states differ from those of the continuum 3. Fractional quantum Hall states Landau level and can influence the nature of the many-body The conditions for realizing fractional quantum Hall (FQH) ground states. For the Harper-Hofstadter model, the FQH states in 2D semiconductor systems are well understood states found at low flux density, where the bands resemble the (Prange and Girvin, 1990). The application of a strong continuum Landau level, are replaced by other strongly magnetic field breaks the single-particle energy spectrum into correlated phases at high flux densities where lattice effects degenerate Landau levels. When a Landau level is partially become relevant. filled with electrons, the large number of ways in which the It has been shown that states of the same form as FQH states of the continuum Landau level, the so-called “fractional electrons can occupy the single-particle states gives a very ” high degeneracy. This degeneracy is lifted by repulsive Chern insulator (FCI) states, can be formed, for bosons and fermions, in a variety of topological energy bands starting interparticle interactions, leading to strongly correlated from models (such as the Haldane model) which are far FQH ground states at certain ratios of particle density to flux removed from the continuum Landau level (Parameswaran, density ν ¼ n=nϕ. FQH states are characterized by a nonzero Roy, and Sondhi, 2013). Typically such models require the energy gap to making density excitations in the bulk. For introduction of further-neighbor tunneling terms to flatten the temperatures below this gap they behave as incompressible lowest energy band, such that one can enter a regime of strong liquids: with a bulk energy gap, but carrying gapless edge correlations (mean interaction energy larger than bandwidth) modes. In this sense they resemble integer quantum Hall without mixing with higher bands. Indeed, it has been shown states. However, the edge modes are not simply described as that by tailoring the tunneling Hamiltonian, one can construct single-particle states but involve fractionalized quasiparticles models for which the exact many-body ground states are FQH (Wen, 1995). Similarly, the gapped particlelike excitations in states (Kapit and Mueller, 2010) or FCI states (Behrmann, the bulk of the system have fractional charge and are predicted Liu, and Bergholtz, 2016). No general theorem exists con- to have fractional quantum exchange statistics (Stern, 2008). cerning the nature of the many-body ground state in a given The achievement of similar regimes with ultracold atoms Chern band. However, beyond flatness of the energy band, it is would allow the exploration of several novel variations of believed that flatness of the geometry of the states, as FQH physics. measured by the Berry curvature and by a quantity known as the Fubini-Study metric, is advantageous in stabilizing a. Bosons topological many-body phases (Parameswaran, Roy, and Cold atom experiments have the potential to allow the first Sondhi, 2013). exploration of FQH states for bosons. Theory shows that A particularly interesting aspect of the topological energy contact-interacting bosons in the lowest Landau level exhibit bands in lattices is that these can differ qualitatively from a FQH states provided the filling factor is not too large. These continuum Landau level, specifically if their Chern number states include robust variants of interesting phases—the Moore- differs from unity. Indeed, for the Harper-Hofstadter model at ϕ π ϵ ϵ Read and Read-Rezayi phases—which are expected to exhibit flux AB ¼ þ , with small, the lowest energy band has non-Abelian particle exchange statistics (Cooper, 2008). The Chern number of 2, and so is topologically distinct from the stabilization, and exploration, of non-Abelian phases is a much lowest Landau level. Numerical calculations show the appear- sought after goal, in part to find the first evidence that nature ance of FQH phases for particles occupying this energy band. does exhibit this exotic possibility of many-body quantum These are examples of FQH states that have no counterpart in theory and in part in connection with the possible relevance for the continuum Landau level but that are stabilized by the quantum information processing (Nayak et al., 2008). lattice itself (Möller and Cooper, 2009, 2015; He et al., 2017). The physics of interacting bosons in the lowest Landau level can be accessed in the Harper-Hofstadter model at relatively c. Symmetry-protected topological phases ϕ ≲ 2π 3 low flux density j ABj = for which the lowest band is The topological states of noninteracting Fermi systems similar to the continuum Landau level and where similar FQH (topological insulators and superconductors) are now viewed states appear for bosons (Sørensen, Demler, and Lukin, 2005; as examples of a more general class of a symmetry-protected Palmer and Jaksch, 2006; Hafezi et al., 2007; Möller and topological (SPT) phase, which allows for interparticle inter- Cooper, 2009). Optical flux lattices, involving spin-orbit actions and bosonic statistics. SPT phases are defined by the coupling, can lead to 2D energy bands that are very similar conditions that they are gapped phases, with gapless edge to the lowest Landau level: topological bands with unit Chern modes, but unlike the FQH effect they do not involve particle number and with very narrow energy dispersion. An advantage fractionalization. Hence these states have a unique ground of this approach is that the flux density nϕ can be high, so FQH state on a periodic geometry, and the many-body ground state states are expected at high particle density n where interactions has only short-range entanglement (Senthil, 2015). The phases are strong. Exact diagonalization studies have established of strongly interacting bosons in Chern bands include cases of stable FQH states of bosons including exotic non-Abelian “integer” quantum Hall states of the bosons, which provide phases (Cooper and Dalibard, 2013; Sterdyniak et al., 2015). one of the cleanest realizations of SPT phases (Möller and

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Cooper, 2009, 2015; He et al., 2017). Here integer refers to the Cooper, 2010). Thus, the FQH (or FCI) phases of these fact that the Hall conductance is an integer as opposed to hard-core bosons can be viewed as quantum spin liquids fractional. Such phases still arise from strong interparticle (Kalmeyer and Laughlin, 1987). Other predicted quantum interactions, albeit without fractionalized quasiparticles. spin liquids may also find realization with these Bose- Hubbard models (L¨auchli and Moessner, 2015). The phases d. Ladders of interacting fermions in Hubbard-like models with tunneling phases have also been studied theoretically. Much of the focus Cold atomic gases provide ways in which to study ladder- has been on the Haldane model for spin-1=2 fermions with on- like systems which are quasi-1D variants of FQH systems. site repulsion U. For systems close to half filling there is a These arise both in Harper-Hofstadter models with super- competition between the band insulator, the Mott insulator, lattices to create ladder geometries, and in systems involving a and various spin-symmetry-broken phases (Zheng et al., synthetic dimension for which there is naturally tight confine- 2015; Imriška, Wang, and Troyer, 2016; Vanhala et al., ment. Both situations have been shown theoretically to 2016). The Mott insulating state naturally leads to frustrated support strongly correlated states that are closely related to spin models, with possible unconventional forms of ordering FQH states (Cornfeld and Sela, 2015; Petrescu and Le Hur, at very low temperatures, at the superexchange energy scale 2015; Calvanese Strinati et al., 2017). A precise connection J2=U or below, including chiral spin states (Arun et al., 2016) between FQH states on an infinite 2D system and the states in and chiral spin liquids (Hickey et al., 2016). these quasi-1D settings can be made by considering the quantum Hall wave functions on a cylindrical geometry which is infinite in one direction but has finite circumference L in the c. Number-conserving topological superfluids other. Studies of the evolution of the ground state from 2D The theory of topological superfluids, Sec. V.B.2, is based (L ≫ a¯, with a¯ the mean interparticle spacing) into the on the mean-field treatment of pairing described by the “squeezed geometry” (small L ≲ a¯) show that many FQH Bogoliubov–de Gennes theory. In this theory, the number states of the 2D systems evolve smoothly into charge-density- of pairs of fermions is not conserved. For a 1D system, such a wave (CDW) states of the quasi-1D geometry. For example, mean-field theory can be appropriate in settings in which the the bosonic Laughlin state, at filling factor ν ¼ 1=2,evolves superfluid pairing is proximity induced, e.g., by coupling to a into a CDW state which breaks translational symmetry to bulk superfluid of fermion pairs with which pairs of particles double the unit cell to give two degenerate ground states of the can be exchanged and which can impose a fixed super- CDW. The fractionally charged quasiparticles of the FQH conducting order parameter. However, in the absence of such a states map onto domain walls between the different sym- proximitizing medium the quantum fluctuations in 1D sys- metry-related CDW states. tems preclude the existence of any long-range order in the superconducting pairing and the structure of the mean-field 4. Other strongly correlated phases theory should fail. Theoretical work has shown how 1D a. Chiral Mott insulator topological superfluids can still arise in such number-conserving settings. Models with microscopic symmetry can give For the Harper-Hofstadter-Hubbard model at strong inter- rise to zero-energy modes, arising as exact degeneracies in the actions U ≳ J the Gross-Pitaevskii approach fails. Just as in spectrum of an open chain in the topological phase (Fidkowski the case of vanishing magnetic field, there can be incom- et al., 2011; Sau et al., 2011; Iemini et al., 2015, 2017; Lang pressible Mott phases, with ni ¼ integer. However, other and B¨uchler, 2015). This degeneracy is the many-body strongly correlated phases are predicted to appear. One counterpart of the Majorana modes of the mean-field 1D striking example is the chiral Mott insulator: an incompress- topological superconductor. In settings without such a sym- ible phase at integer filling (like the Mott insulator), but which metry, topological degeneracies can arise in geometries in carries a nonzero local current in the ground state (as does a which modulation of the parameters along the chain lead to vortex lattice). Numerical calculations indicate that, within a multiple interfaces between topological and nontopological region of stability between Mott insulator and superfluid, the phases (Ruhman, Berg, and Altman, 2015; Ruhman and chiral Mott insulator can form the ground state of the Bose- Altman, 2017). Hubbard model with external flux on a variety of 2D lattices ć (Dhar et al., 2013; Zaletel et al., 2014; Vasi et al., 2015). In C. Experimental perspectives ladder systems the Mott-vortex phase found in numerical calculations (Greschner et al., 2015; Petrescu and Le Hur, Many of the experimental consequences of topological 2015; Piraud et al., 2015) can be viewed as the 1D analog of bands described in Sec. IV rely on the interactions between the the chiral Mott insulator. particles being sufficiently weak that they can be treated as noninteracting. It is therefore important to consider what b. Chiral spin states experimental observables can be used to characterize the properties of atoms in topological bands when interactions For hard-core interactions U → ∞, interesting many-body cannot be neglected. states can still arise in the Harper-Hofstadter-Hubbard model for bosons provided ni ≠ integer so the hard-core bosons are mobile. These bosons can be viewed as a spin-1=2 quantum 1. Equilibrium observables magnet, with the phases of the tunneling matrix elements Some of the most important observables for characterizing introducing frustrated magnetic interactions (Möller and the properties of the atoms are well established from studies of

Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019 015005-35 N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms atomic gases in other settings without topological character. 4. rf excitation Measurements of the equation of state and observations of Another natural probe of the edge states is to measure the density-density correlations in time-of-flight imaging will be spectrum for the removal of particles via rf excitation, ideally crucial for establishing the existence and forms of strongly performed with single-site resolution (Bakr et al., 2009; correlated phases such as fractional quantum Hall states. Sherson et al., 2010) to focus on the boundary. Such spectra Similarly, the possibility to image at the single-site level in have been proposed as a means to detect localized Majorana quantum gas microscopes (Bakr et al., 2009; Sherson et al., modes in topological superfluids, appearing as a near zero- 2010) will allow precise characterization of microscopic energy contribution inside the spectral gap (Grosfeld et al., structure in these phases, including of the local currents, 2007; Nascimb`ene, 2013). A definition of fractional statistics for example, in the chiral Mott insulator phase. A method for is provided by Haldane’s exclusion statistics (Haldane, 1991): detecting local currents was presented in experiments from the a generalized version of Pauli blocking, by which the frac- Munich group (Atala et al., 2014). tional quasiparticles reduce the number of available states for other quasiparticles in a well-defined, but fractional, manner. 2. Collective modes An observation of this effect requires counting many-body Observations of the collective mode frequencies provide a states and may be possible in precision spectroscopy of small sensitive way to detect properties of many-body systems FQH clusters (Cooper and Simon, 2015) in which the (Dalfovo et al., 1999). For BECs formed in bands with exclusion statistics reveal themselves in the count of spectral geometrical character, the collective mode frequencies are lines. sensitive to the Berry curvature at the band minimum (Price and Cooper, 2013). Consider a weakly interacting BEC in a 5. Adiabatic pumping single band minimum. We take the minimum to have isotropic For topologically ordered systems, the existence of frac- Ωe effective mass M and a local Berry curvature z. The effect tional low-energy particlelike excitations allows for new of Berry curvature on the collective modes can be readily features in Thouless pumping. Specifically, under one full determined by adapting the standard hydrodynamics approach cycle of the adiabatic evolution of the pump, it is possible to (Pethick and Smith, 2002) to include the anomalous velocity transfer a fractional particle number across the system. (In the from the Berry curvature. For a spherical harmonic trapping examples discussed above, the number of particles transferred 2 potential VðrÞ¼ð1=2ÞΛjrj , the collective modes have the was constrained to be an integer, set by the Chern number.) same spatial structure as for Ω ¼ 0 (Stringari, 1996), but the This fractional pumping is related to the existence of a ground frequencies depend on the Berry curvature. For example, for state degeneracy: one cycle of the adiabatic pump converts small Ω the three angular-momentum components of the one ground state into another degenerate ground state, and dipole mode are split by Δω ¼ ΛΩ=2ℏ, leading to a preces- multiple cycles (transferring multiple fractional particles) are sional motion of the center-of-mass oscillation of the cloud at required before the system returns to its starting state. This this frequency. picture forms the basis for understanding of the quantization of the Hall conductance in the fractional quantum Hall effect 3. Edge states of 2D systems (Laughlin, 1981; Halperin, 1982). For narrow strips of quantum Hall systems, this physics smoothly evolves For incompressible fluids that are topological, e.g., a into the pumping of fractional charges in commensurate topological insulator of noninteracting fermions, or a FQH charge-density waves. The manifestation of such pumping fluid, there exists a special class of low-energy collective for 1D fermionic systems with synthetic dimension was modes, which are the gapless edge states of the fluid. For described by Mazza et al. (2015), Zeng, Wang, and Zhai noninteracting particles in Chern bands, these are the edge (2015), and Taddia et al. (2017). states discussed in Sec. II.D. For interacting systems these may not be easily described as single-particle excitations, “ ” but can still appear as long-lived surface excitations. 6. Hall conductivity from heating Measurements of the propagation of surface waves can An interesting way in which to measure the Hall conduc- directly probe the edge-state structure, for example, allowing tivity σxy of a system, i.e., the transverse current density in one to detect the number of edge channels and their response to a uniform force, is to measure the heating rate, as ω respective velocities (Cazalilla, Barberán, and Cooper, set by the rate of power absorption Pð Þ caused by the F 2005). The edge states also lead to highly characteristic application of a circularly polarized force ðtÞ¼ dynamics in far-from-equilibrium situations in which the F0ðcos ωt; sin ωtÞ. The dc Hall conductivity is found to confining potential is removed (Goldman et al., 2013). Edge be set by states appear also at interfaces between bulk regions of Z 2 ∞ 1 differing topologies, prepared, for example, by spatial σ ω ω − ω xy ¼ π 2 d ω ½Pþð Þ P−ð Þ; ð104Þ modulation of the lattice potential (Goldman et al., 2016) F0Asyst 0 and have been probed experimentally for the SSH model (Leder et al., 2016). Experiments using a quantum gas i.e., the difference of the power absorption rates divided by microscope have shown the influence of strong interparticle frequency and integrated over all drive frequencies (Tran interactions on the chiral edge states in Harper-Hofstadter et al., 2017; Asteria et al., 2019). For noninteracting particles ladders (Tai et al., 2017). the power absorption can be found from measurements of the

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Γ rates of depletion of initially occupied states , via eliminated by suitable selection of different transitions or ℏω Γ P ¼ð Þ , allowing the Hall conductivity to be deter- working with different atoms; however, fundamentally differ- mined via measurements of this depletion rate. For a filled ent approaches are also possible. Chern band, the integrated difference of depletion rates (104) is Of particular interest is the possibility to replace the laser therefore expected to recover the quantized Hall conductivity fields with rf or microwave magnetic fields generated by a of the filled bands. Equation (104) is however valid even for microfabricated atom chip. Unlike optical fields, these fields interacting particles, arising from a general sum rule for the have practically no off-resonant emission, solving the in- linear response functions, and so provides a possible way to principle atomic physics limitation, in exchange for the measure the Hall conductivity also in strongly correlated technical complexity of working in the vicinity of an atom chip. phases. In one category of proposals, the atom chip serves simply as the source of a large time-modulated gradient magnetic field VI. OUTLOOK (Anderson, Spielman, and Juzeliūnas, 2013). This produces in effect a series of pulses that generates spin-dependent gauge A. Turning to atomic species from the lanthanide family fields; the most simple implementation of this technique was realized in the lab (Luo et al., 2016), and although spontaneous In this review we explored several classes of lattice schemes scattering was eliminated, the fairly low frequency of their for which a nontrivial topology originates from a two-photon drive led to significant micromotion induced heating effects. Raman coupling between various sublevels of the electronic In a second category of proposals (Goldman et al., 2010), ground state manifold. So far most experiments of this type the atom chip consists of a large array of microfabricated were performed with atoms from the alkali-metal family, parallel wires which give a near-field radio-frequency mag- which are relatively easy to manipulate and cool down to netic field that drives transitions in the same way as Raman quantum degeneracy. However, for such atoms, recoil heating lasers, as described here in the context of synthetic dimensions due to spontaneous emission of photons may cause severe or intrinsic spin-dependent lattices. Because these fields are problems. Indeed the desired Raman coupling is significant structured at the micrometer scale, far below the free-space only when the laser is detuned from the resonance by less than wavelength of rf fields, these are near-field structures and the the fine-structure splitting Δ of the resonance line (see f:s: atomic ensemble must be on the micrometer scale from the Appendix C). Because of this relatively small detuning, chip’s surface. spontaneous emission of photons occurs with a non-negligible rate γ. More precisely, one can define the merit factor C. Other topological insulators and topological metals M ¼ κ=γ, where κ is the desired Raman matrix element. Taking as an example the case of the fermionic alkali-metal We have focused on specific recent experimental realiza- 40 M ∼ Δ Γ ∼ 105 atom K, one finds after optimization f:s:= , tions of topological energy bands using cold gases: in 2D where Γ stands for the natural width of the electronic excited systems and in 1D systems with chiral symmetry. Routes to state [see, e.g., Dalibard (2016) for details]. If one takes as a achieving topological superfluid phases in cold gases, arising typical value ℏκ equal to the recoil energy, the photon from BCS pairing of fermions, have also been described in −1 scattering rate is γ ∼ 0.3 s , leading to the heating rate Sec. V.B.2. _ ∼ 100 E kB × nK=s. This may be too large for a reliable As discussed in Sec. II.E other forms of topological energy production of strongly correlated topological states. bands can arise depending on dimensionality and the global A more favorable class of atoms is the lanthanide family, symmetries that are imposed, according to the tenfold way with species such as erbium or dysprosium. These atoms have (Chiu et al., 2016). Important cases in solid state systems are 2 two outer electrons and an incomplete inner shell (6s and the Z2 topological insulators that arise in spin-orbit coupled 4f10 for Dy). Because of this inner shell, the electronic systems with time-reversal symmetry, in 2D and 3D. There ground state has a nonzero orbital angular momentum are proposals for how to realize bands with this topology for (L ¼ 6 for Dy). The lower part of the atomic spectrum cold gases. The required TRS can be implemented by fine- contains lines corresponding to the excitation either of one tuned engineering of the relevant terms in the Hamiltonian of the outer electrons or of one electron of the inner shell. By (Goldman et al., 2010). TRS can also be established as an choosing a laser excitation close to a narrow line resonance intrinsic property for cold gases: in the absence of Zeeman and thus with a large detuning Δb from the closest broad splittings and of any circularly polarized light fields (B´eri and line, one reaches after optimization M ∼ Δb=Γb. Because Cooper, 2011). In 2D, the insulating state formed by filling a Z Δb is now of the order of an optical frequency, the merit 2 topological band exhibits a quantized Hall effect for the factor is M ∼ 107, leading to γ ∼ 10−3 s−1 andtothe spin current. A similar quantized spin Hall response also arises _ ∼ 0 1 in a related setting without spin-orbit coupling, in which spin residual heating E kB × . nK=s. up and spin down fill Chern bands with equal and opposite B. Topological lattices without light Chern number: such energy bands have been realized for bosons in cold atoms (Aidelsburger et al., 2013) using laser- Laser fields constitute a common element for all of the assisted tunneling to generate a Harper-Hofstadter model with techniques described herein, and in virtually all of these cases equal and opposite fluxes for the two spin states (Kennedy these fields lead to unwanted off-resonant scattering leading et al., 2013). to heating, atom loss, or both. As discussed in the previous Cold atomic gases provide a natural setting with which to section, these scattering processes might be mitigated or explore certain forms of topological band that are more

Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019 015005-37 N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms difficult to implement in solid state materials. The sublattice D. Far-from-equilibrium dynamics (chiral) symmetry is very fragile in solid state systems, as it is typically broken by any disorder potential. However, it can Cold atomic gases readily allow the study of coherent readily arise for optical lattice potentials in cold atom gases for quantum dynamics in far-from-equilibrium settings. Starting which disorder can be negligible (Essin and Gurarie, 2012; with a cold gas at thermal equilibrium, a sudden change of the “ ” Wang, Deng, and Duan, 2014). Similarly, there are predicted Hamiltonian, a so-called quantum quench, typically leaves to be topological insulators beyond those classified in the the system in a far-from-equilibrium state. Here we discuss tenfold way, arising from lattice symmetries which rely on a some of the consequences of quantum quenches between high spatial regularity that can arise in cold gases. These Hamiltonians for which the ground states have different include topological invariants stabilized by crystalline lattice topological character. We focus on cases of noninteracting symmetries (Fu, 2011), and the Hopf insulator (Moore, Ran, particles in topological energy bands. and Wen, 2008; Deng et al., 2013), which also relies on a form Consider first an optical lattice potential which is varied of translational symmetry for stability (Liu, Vafa, and Xu, slowly in time, such that the topological invariant of the lowest 2017). Note also that cold gases allow topology in dimensions energy band differs between initial and final Hamiltonians. higher than d ¼ 3 to be explored, through the use of synthetic If the relevant topological invariant is symmetry protected, dimensions provided by internal degrees of freedom or by it is possible to have a smooth evolution between these two viewing a phase degree of freedom as an additional quasi- cases provided that the Hamiltonian breaks the symmetry at momentum. Recent experimental work (Lohse et al., 2018) intervening times. An example of this is the pumping used pumping to demonstrate the topological response of an sequence of the RM model, which breaks the chiral symmetry effective 4D quantum Hall system (Zhang and Hu, 2001) of the SSH model and therefore allows a smooth evolution based on the theoretical proposal by Price et al. (2015). between the topologically distinct phases of the SSH model. An area of growing interest in solid state settings concerns When there is no symmetry protection (e.g., Chern bands in so-called topological metals, or semimetals (Chiu et al., 2D), then the change in band topology requires the band gap 2016). (There exist analogous topologically stable forms of to close at some intermediate time, as illustrated in Fig. 26. For gapless superconductors.) As for topological insulators, the a BEC formed close to the band minimum, such a change in classification of topological metals depends also on the band topology need not induce any phase transition: the Bloch dimensionality and the existence (or absence) of symmetries. wave functions of those states which the bosons occupy can However, since metals involve bands that are only partially evolve smoothly, allowing adiabatic evolution of the BEC, filled by fermions, they cannot be characterized by the albeit into a very different local wave function. This allows, topological invariants used for insulators, which involve for example, the adiabatic formation of a dense vortex lattice integrals over the filled energy bands. Instead, topological (Baur and Cooper, 2013). However, for noninteracting fer- metals can be characterized by topological invariants defined mions that fill the lowest energy band, the ground states of the in terms of integrals over the Fermi surface which separates initial and final Hamiltonians have different topological filled from empty states (Volovik, 2003). One example of a characters, so these two states must be separated by a phase topological metal is provided by the 2D honeycomb lattice transition. with nearest-neighbor hopping. This realizes the band struc- The far-from-equilibrium dynamics following a quantum ture of graphene in which there are two Dirac points in BZ, quench between Hamiltonians whose ground states have each of which leads to a Fermi surface when the Fermi energy different topologies has been explored for noninteracting lies close to the Dirac point. Each of these Fermi surfaces is a fermions which fill a band. One striking result is that the topological metal, characterized by the Berry phase of Bloch topological invariant of the many-body state is often preserved states around the Fermi surface, which is π. This value is a under unitary time evolution. This has been shown for topological invariant, i.e., it is robust to continuous changes topological superfluids (Foster et al., 2013, 2014; of the underlying parameters, provided the system retains both Sacramento, 2016), and for Chern bands (Caio, Cooper, time-reversal symmetry and inversion symmetry. (These and Bhaseen, 2015; D’Alessio and Rigol, 2015). [Different symmetries ensure that the Berry curvature of the bands behavior can arise for symmetry-protected topological invar- vanishes, except for singular delta-function contributions at iants (McGinley and Cooper, 2018).] In interpreting this result the Dirac nodes which lead to the π Berry phases.) The Dirac it is crucial to distinguish between the topology of the node is topologically stable and can only be created or Hamiltonian and the topology of the many-body state. The annihilated if it merges within another Dirac point. A related former is defined as the topological invariant constructed for example that arises in 3D is the “Weyl point.” This can be the lowest energy band of the Hamiltonian, with Bloch wave ð0Þ viewed as a point source of Berry curvature in reciprocal functions juq i, while the latter is defined as the topological space. The integral of the flux of Berry curvature through any closed 2D surface in reciprocal space is 2πC, where C is EEE required to be an integer. If this surface encloses a Weyl point, jCj¼1. The Weyl point is topologically stable to deforma- kkk tions of the Hamiltonian, without any symmetry requirements, ν=0 ν=? ν=1 unless it annihilates with a second Weyl point of opposite charge. Optical lattice with Weyl points can be constructed FIG. 26. The change of band Hamiltonian that causes the using the laser-induced tunneling methods described in topological index of the lowest band ν to change, via the closing Sec. III.D (Dubček et al., 2015). of a band gap.

Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019 015005-38 N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms invariant constructed from the wave functions which the hðqÞ, then this is an energy eigenstate of the Hamiltonian. s fermions occupy juqi. These two sets of wave functions need In the quench experiment, the system is prepared with only coincide if the many-body wave function is the ground einitialðqÞ¼ð0; 0; −1Þ for all wave vectors [the ground state state of the Hamiltonian. Out of equilibrium, for example, of an initial Hamiltonian with hinitialðqÞ¼ð0; 0; 1Þ] and then following a quantum quench, the Bloch wave function the Hamiltonian is changed to its final value hðqÞ. For general occupied by the particle at momentum q need not be an q this is not aligned with einitialðqÞ so the wave function eigenstate of the Hamiltonian, so it becomes time dependent evolves in time, precessing around the local hðqÞ. This leads to s juqðtÞi. We consider situations in which the periodicity of the the appearance of nonzero components of the vector ðex;eyÞ lattice is preserved, such that wave vector q remains a good which indicate interband coherences. The evolution can lead quantum number. That the topological invariant of the state is to the creation of vortex-antivortex pairs in these components preserved is guaranteed provided the unitary evolution of the ðe ;e Þ. The appearance of each vortex-antivortex pair was ˆ x y s −iHqt=ℏ s Bloch states juqðtÞi ¼ e juqð0Þi is smooth in momen- shown to be associated with a cusp in the Loschmidt echo, tum space q, which is true for short-range hopping. [For hence giving rise to the characteristic feature of a “dynamical topological invariants that are symmetry protected, it is also phase transition” (Heyl, 2018). Such singular features can required that the symmetry is not broken either explicitly or arise even for quenches within a single topological phase, and dynamically (McGinley and Cooper, 2018).] That said, at long so are not connected to the topological phase transition itself. times, the Bloch wave functions of the many-body state Instead, the change in the topology of the Hamiltonian can be s juqðtÞi will become rapidly varying as a function of q. When found by tracing the time evolution of eðq;tÞ and constructing the variation in q is so fast that this cannot be viewed as the linking number of the trajectories (in q and t) of any two smooth on the scale of 2π=L, with L the typical sample values of e [e.g., e ¼ð1; 0; 0Þ and (0,0,1)] (Wang et al., 2017). dimension, then the bulk topological invariant becomes ill This linking number is the Hopf index of the map eðq;tÞ. This defined. Thus for any finite system, there is an upper time procedure has been successfully carried out in experiments scale after the quench for which it is meaningful to expect the (Tarnowski et al., 2017). A related approach has recently been Chern number to be preserved. Simple estimates lead to the used to demonstrate the topological character of the effective conclusion that this time is of order L=v, with v a character- Hamiltonian in a spin-orbit coupled BEC (Sun et al., 2018). istic group velocity of the final Hamiltonian (Caio, Cooper, The above considerations rely on the assumption that the and Bhaseen, 2016). Such systems can still be characterized fermions are noninteracting. It will be of interest to explore the by the Bott index (Loring and Hastings, 2010), which extent to which these, or similar, approaches can be applied in provides a real-space formulation of the Chern index appli- the presence of interparticle interactions. Being gapped phases cable also to finite-sized systems. This has been used to find of matter, associated with filled bands, the ground states of the protocols for preparing nonequilibrium (Floquet) systems by Hamiltonian are expected to be robust to weak interactions. which the Bott index undergoes transitions between topo- However, even weak interactions can lead to the generation of logical values (D’Alessio and Polkovnikov, 2013; Ge and entanglement between single-particle states at different wave Rigol, 2017). vectors under far-from-equilibrium dynamics, which may be Although the topological invariant of the wave function is viewed as a form of decoherence. unchanged following the quench to a new Hamiltonian with Moving beyond the phases of noninteracting fermions in different topology, this does not mean that there are no topological bands, it will be of interest to explore similar observable consequences of the new Hamiltonian. Local quench dynamics in topological phases that arise only because physical observables can be strongly influenced by the new of strong interparticle interactions. This is an area where Hamiltonian, despite the fact that the nonlocal topological results remain limited. Recent work on the Haldane phase of a invariant of the state is unchanged. Indeed, theory shows that spin-1 chain (a symmetry-protected topological phase of this under a quantum quench of the Haldane model the edge interacting quantum spin system) has shown that the “string current quickly adapts to become close to that of the ground order” that characterizes the topological phase is lost follow- state of the final Hamiltonian (Caio, Cooper, and Bhaseen, ing the quench (Calvanese Strinati et al., 2016), suggesting a 2015). Furthermore, by following the dynamics of the Bloch difference from the noninteracting fermion cases described s wave functions juqðtÞi detailed information on the final above. One important difference concerns the role of sym- Hamiltonian can be recovered (Gong and Ueda, 2017; metry protection under dynamical evolution (McGinley and Wang et al., 2017). This has been demonstrated in experi- Cooper, 2018). ments by the Hamburg group (Tarnowski et al., 2017; Fl¨aschner et al., 2018), using the band-mapping techniques E. Invariants in Floquet-Bloch systems described in Sec. IV.A.2 to reconstruct the dynamical evolu- s tion of the occupied wave functions juqðtÞi. The experiments The topological invariants we focused on are those of static use a two-band model, for which the Hamiltonian at q may be single-particle Hamiltonians, in which the spatial periodicity written as leads to the existence of a Bloch Hamiltonian that depends on a quasimomentum q within a BZ. If the Hamiltonian is time ˆ ˆ Hq ¼ h0ðqÞ1 − hðqÞ · σˆ: ð105Þ varying, but periodic with period T ≡ 2π=ω, then energy is replaced by a Floquet quasienergy that is defined up to the The wave function of the fermion with q can be represented by addition of integer multiples of ℏω. The combination of both a three-component unit vector eðqÞ.IfeðqÞ is aligned with temporal and spatial periodicities causes Floquet-Bloch

Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019 015005-39 N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms systems to have topological invariants that are distinct from For a band that winds in quasienergy s times, the current is those of static Hamiltonians. quantized at sa in each cycle T. The quantized current from a (1) The periodicity of the Floquet spectrum allows the filled Floquet-Bloch band that winds s times in quasienergy Floquet bands to wind in quasienergy, by an integer multiple precisely matches the current expected from an adiabatic of h=T as q runs over the BZ. This winding in quasienergy pump, as described in Sec. II.F, operated cyclically with gives rise to topological invariants of Floquet-Bloch bands period T and with Chern number s. Indeed, settings in which (Kitagawa et al., 2010) that are absent in static settings. adiabatic Thouless pumping occurs give rise to Floquet-Bloch Figure 27 shows an example of a Floquet-Bloch band in a 1D spectra in which a band winds in quasienergy. (The adiaba- system that winds in quasienergy once across the BZ. ticity condition corresponds to the relevant Floquet-Bloch To understand the physical significance of such situations, band crossing with all Floquet-Bloch bands that wind in the it is instructive to consider the band of Fig. 27 (top) to be filled opposite sense.) However, the general structure of the Floquet- with noninteracting fermions. [How one might prepare such a Bloch states and the associated topological invariants state is itself an interesting question (Dauphin et al., 2017; (Kitagawa et al., 2010) are not restricted to such adiabatic Lindner, Berg, and Rudner, 2017). Indeed, nonadiabatic settings. effects associated with the switch-on of a pump can lead to (2) New features also arise when one considers the edge deviation from quantization (Privitera et al., 2018).] The net states on systems with a boundary. One finds that finite-size current carried by the filled band is systems can have protected edge states, at quasienergies Z between the bulk bands, even if the topological invariants π ˆ F L =a 1 dϵq L constructed from the Floquet Hamiltonian Hq are trivial for all I ≡ dq ¼ ; ð106Þ 2π −π=a ℏ dq T of the bulk energy bands (Kitagawa et al., 2010). A simple example of a 2D model which exhibits such “anomalous edge where a is the lattice constant and L is the total length of the states” is illustrated in Fig. 28. One period T is broken into system. The number of particles in the filled band is N ¼ L=a, four subperiods T=4, during which the tunneling amplitude J so the current per particle is is turned on only for a subset of the bonds. This tunneling is chosen to satisfy ðJ=ℏÞðT=4Þ¼π such that a particle that I a starts on one side of the active bond is transferred to the other ¼ : ð107Þ 4 N T side of the bond during the time T= . For particles in the bulk, the net action after all four parts of the cycle is to return the Thus, the current is equivalent to a displacement of each particle by a lattice constant a in each period of the cycle T.

FIG. 27. Floquet energy spectrum in units of ℏω for the driven Rice-Mele model in the topological (top) and nontopological (bottom) cases. The parameters ðJ0=J; Δ=JÞ are varied along a FIG. 28. A simple model that shows anomalous edge states. circle of radius 0.5 with angular frequency ω ¼ 0.23J=ℏ. The (a) A cycle consists of four steps in which specific sets of bonds circle is centered on the point (2,0) in the nontopological case and are active and one step in which there is only a sublattice energy on the point (1,0), i.e., the vortex in Fig. 7, in the topological one. offset δAB. (b) Trajectories of particles initially in the bulk (blue In the topological case a particle starting in state A at momentum closed loop) or on the edges [green (red) lines on top (bottom) q ¼ 0 arrives in B after an adiabatic motion across the full BZ, edges]. (c) Floquet spectrum showing the dispersionless bulk leading to the net current (107) for a filled band. The gray areas band and the two anomalous edge states with equal and opposite highlight given “temporal Brillouin zones,” of energy width ℏω. nonzero velocities. From Rudner et al., 2013.

Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019 015005-40 N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms particle to its starting point (up to some overall phase). Thus the Recent works have constructed a general theory of the Floquet operator for time evolution over one cycle [see topological classification of time-varying single-particle Eq. (B2)] is proportional to the identity and has a bulk Hamiltonians (Nathan and Rudner, 2015; Fruchart, 2016; Floquet-Bloch band structure with constant quasienergy and Roy and Harper, 2016). Extensions of these ideas to interact- whose wave functions have trivial topological character. ing, many-body phases is an active area of theoretical However, a particle that starts on the edge of the system is research. transported along the edge over one period; this motion appears as dispersive (chiral) edge states in the Floquet spectrum of the F. Open systems finite-size system; see Fig. 28(c).Theunderstandingofsuch So far we focused our studies on the topological properties anomalous edge states is that the topological invariant of a band of the ground state of an isolated system. The possibility to constructed from the Floquet operator determines the change in introduce a coupling between this system and an environment the number of edge states as the quasienergy passes through the opens new possibilities and raises new questions that we band (Rudner et al., 2013). An edge state can pass through a set now discuss. We restrict our presentation to studies in direct of topologically trivial bands and satisfy periodicity in quasie- relation with atomic gas implementations. nergy and quasimomentum; see Fig. 29. To compute the Consider first the case where the environment is at a number of anomalous edge states requires one to know not nonzero temperature T. If the coupling is sufficiently weak, just the stroboscopic evolution defined by the Floquet operator, the energy levels of the system remain relevant and its steady but also the full time-evolution operator at intermediate times, state is now a statistical mixture of these levels. Since bands from which an additional invariant can be constructed (Rudner with various topologies that were empty at T ¼ 0 now acquire et al.,2013). Anomalous edge states have not yet been seen in a finite population, the Chern number calculated via a thermal atomic systems, but have been observed in experiments on light average will not be an integer anymore. One could naively propagation in photonic structures (Maczewsky et al.,2017; conclude that the system loses all its topological properties Mukherjee et al., 2017). when T becomes non-negligible with respect to the gap (3) The concept of particle-hole symmetry needs be protecting the bands that are populated at zero temperature. generalized for Floquet-Bloch systems. For static However, one may look for more subtle topological Hamiltonians, particle-hole symmetry (E → −E) stabilizes invariants that can be associated with the density matrix (edge) modes at energies E ¼ 0 (SSH model or Majorana describing the system at nonzero T. A possible direction mode in the Kitaev model). The periodicity of the Floquet consists of generalizing the notion of geometric phase, using the concept of parallel transport for density matrices intro- quasienergy ϵ under ϵ → ϵ þ ℏω means that symmetry ϵ → duced by Uhlmann (1986). This line was investigated by −ϵ arises for ϵ ¼ 0 or ϵ ¼ ℏω=2. Indeed, topologically stable Huang and Arovas (2014) and Viyuela, Rivas, and Martin- edge modes at quasienergies ϵ ¼ ℏω=2 have been theoreti- Delgado (2014), who could derive in this way a classification cally demonstrated in periodically driven lattice models with of topological phases at nonzero T. It was subsequently sublattice (chiral) symmetry analogous to the SSH model revisited with a somewhat different perspective by Budich (Asbóth, Tarasinski, and Delplace, 2014), and in Floquet and Diehl (2015), who pointed out possible ambiguities of the superfluid systems where they appear as Majorana modes previous approaches for a model 2D system. Another direc- (Jiang et al., 2011). tion consists of establishing an equivalence between classes of density matrices using local unitary operations (Chen, Gu, and Wen, 2010), and deducing from this equivalence the desired classification of topological phases. It was developed for the particular case of mixed Gaussian states of free fermions by Diehl et al. (2011) and Bardyn et al. (2013), and recently generalized by Grusdt (2017) [see also van Nieuwenburg and Huber (2014) for the specific case of 1D systems]. Independently of the tool that is used to define the topological class of density matrices, it is worth emphasizing that on the one hand these topological features are in principle observable, but on the other hand their relation with usual physical quantities associated with the response of the system is still the subject of ongoing research (Budich, Zoller, and Diehl, 2015). Working with open systems also offers the possibility of creating novel topological states that emerge from the dissipative coupling itself. Here we take the reservoir at T ¼ 0 for FIG. 29. For Floquet-Bloch systems, the topological invariant of simplicity. In the conceptually simplest version of the scheme, the bulk band determines the change in the number of edge the coupling is engineered so that the system ends up after modes as quasienergy passes through the band. The periodicity in some relaxation time in a pure state, often called a dark state. quasienergy allows the existence of edge states even in settings “Topology by dissipation” is achieved when this dark state where the bulk bands are all topologically trivial. From Rudner possesses nontrivial topological properties. For a coupling et al., 2013. compatible with the Born-Markov approximation (Gardiner

Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019 015005-41 N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms and Zoller, 2004; Daley, 2014), the master equation describing The concept of topological pumping can also be extended the evolution of the density operator ρ of the system can be to the case of open systems, as shown by Linzner et al. (2016) written in the Lindblad form (Lindblad, 1976) ρ_ ¼ LðρÞ and Hu et al. (2017). Using a proper engineering of the where the linear operator L, the Liouvillian, acts in operator Liouvillian (108) of a fermionic 1D chain, Linzner et al. space: (2016) could generalize the result of Sec. IV.C.1 for the Rice- Mele model and prove the quantization of the variation of the 1 X L ρ ρ 2 ρ † − † ρ − ρ † many-body polarization after a closed loop in parameter ð Þ¼i½ ;Hþ2 ð Lj Lj Lj Lj Lj LjÞ: ð108Þ j space. As for the other topological features described in this section, this quantization holds even when the steady state of Here H is the Hamiltonian of the system in the absence of the master equation is a mixed state. coupling to the reservoir, and the Lj’s are the Lindblad Finally we note that dissipation also offers a new route for operators describing the various coupling channels to the revealing an existing topological order. Rudner and Levitov bath. If one neglects for simplicity the Hamiltonian evolution (2009) considered the motion of a single particle on a 1D of the system (H ¼ 0), one finds that a pure state jψi bipartite (AB) lattice similar to the one at the basis of the SSH satisfying Ljjψi¼0 for all j is an eigenstate of the model. Here dissipation corresponds to a nonzero decay rate Liouvillian with eigenvalue 0 and thus a dark state. The of the particle when it resides on the sites of sublattice A. The topology associated with this state can be readily inferred from particle is assumed to start on site B0, i.e., the B sublattice site the fact that it is the ground state of the parent Hamiltonian in the lattice cell m ¼ 0P. Here one is interested in the average P † Δ H0 ¼ L L . Conditions for the existence and uniqueness displacement h mi¼ mmPm, where Pm is the probability j j j that the particle decays from the A site of the mth lattice cell. of such dark states were discussed in the context of fermionic Quite remarkably this average displacement is quantized setups by Bardyn et al. (2012, 2013). and can take only the values 0 and 1. This binary result The steady state of the master equation will be protected corresponds to the two topological classes that we identified against small perturbations if it is an isolated point in the above for the SSH model within the Hamiltonian framework. spectrum of the Liouvillian. One defines in this case the This result holds for any values of the decay rate and on-site “damping gap” as the smallest rate at which deviations from energies, but it stops being valid if one introduces a dissipative the steady state are washed out. In this dissipative context, the component in the hopping process between the two sublat- damping gap plays a role that is formally equivalent to the tices. It was confirmed experimentally with a setup in the energy gap in the Hamiltonian context. photonic context by Zeuner et al. (2015). They used a lattice The concepts of topology by dissipation and damping gap of evanescently coupled optical waveguides, in which losses can be generalized to the case where the steady state of the and dissipation were engineered by bending the waveguides. master equation associated to Eq. (108) is a mixed state. Its The scheme of Rudner and Levitov (2009) was recently analysis is relatively simple for free fermion systems on a generalized theoretically to multipartite lattices by Rakovszky, lattice, assuming that the Lindblad operators L are linear j Asbóth, and Alberti (2017), who used a weak measurement of functions of the on-site creation and annihilation operators. the particle position as the source of dissipation. This corresponds, for example, to the case where dissipation occurs via an exchange of particles with a reservoir that is VII. SUMMARY described by a classical mean field. In this case, the system density operator is Gaussian and the already mentioned tools We have summarized the methods that have been used to developed by Bardyn et al. (2013) can be used to characterize engineer topological bands for cold atomic gases, and the its topological properties. Interestingly for such systems, a main observables that have allowed characterizations of their change of the topological properties of the system can occur geometrical and topological properties. Most experimental “ ” either when the damping gap closes or when the purity gap studies so far have been at the single-particle level. Theory closes. This notion of purity gap closure, first introduced by suggests many interesting possibilities for novel many-body Diehl et al. (2011) and later generalized by Budich and Diehl phases in regimes where interparticle interactions become (2015), corresponds to a situation where there exist bulk strong. Such systems are in regimes where theoretical under- q modes associated with a subspace (e.g., a given in standing is very limited, so experimental investigation will be momentum space) in which the state is completely mixed. particularly valuable. Accessing this regime for large gases Examples of topologically nontrivial pure or mixed states will require careful management of heating—from lattice and their corresponding edge modes were presented by Diehl modulation methods or from Raman coupling of internal et al. (2011) and Bardyn et al. (2012) in a 1D and a 2D setup, states—as well as the development of robust detection respectively. Budich, Zoller, and Diehl (2015) pointed out a schemes to uncover the underlying order. There is already remarkable property of steady states involving a mixed state: progress in this direction. It seems likely to provide a rich vein they may possess a nontrivial topological character even to explore, with much scope for experimental discoveries and when all Lindblad operators Lj are local in space. This cannot surprises. occur when the steady state is pure, an impossibility that is reminiscent from the fact that in the Hamiltonian framework, ACKNOWLEDGMENTS Wannier functions of a topologically nontrivial filled band (nonzero Chern number) must decay slowly, i.e., algebrai- We are grateful to the innumerable colleagues from cally, in space (Thouless, 1984). whom we have learned so much. We also thank J.

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Beugnon, B. Gadway, F. Gerbier, N. Goldman, Z. Hadzibabic, For the semi-infinite SSH chain (with j ≥ 1) the eigenvalue S. Nascimbene, and J. V. Porto for comments on an earlier condition is draft of this paper. N. R. C. was supported by EPSRC Grants No. EP/K030094/1 and No. EP/P009565/1 and by the Simons − 0ψ B − ψ B 1 J j J j−1 j> ; Foundation. J. D. was supported by the ERC Synergy Grant Eψ A ¼ ðA1Þ j − 0ψ B 1 UQUAM. I. B. S. was partially supported by the ARO’s J j j ¼ ; atomtronics MURI, the AFOSR’s Quantum Matter MURI, ψ B − ψ A − 0ψ B ≥ 1 NIST, and the NSF through the PFC at the JQI. E j ¼ J jþ1 J j j : ðA2Þ

APPENDIX A: TOPOLOGICAL BANDS IN ONE Assuming that there is only one edge state in this problem DIMENSION (which can be checked analytically), the chiral symmetry entails that it has a zero energy. Setting E ¼ 0 in Eqs. (A1) 1. Edges states in the SSH model and (A2) one readily finds the solution The Su-Schrieffer-Heeger (SSH) model was introduced to ψ A 0 j j ð−J =JÞ describe the electronic structure of polyacetylene (Su, ∝ : ðA3Þ ψ B 0 Schrieffer, and Heeger, 1979). This molecule has alternating j single and double bonds along the carbon chain, which are represented in a tight-binding model with one orbital per This is a localized (normalizable) state provided J0 J0 (N ¼ 1). type of site (here the A sites), in agreement with the general This classification may appear rather formal for an infinite requirement of the chiral symmetry for a state jψi at zero chain, since the labeling in terms of A and B sitesisarbitrary ˆ ˆ ˆ energy: UjΨi¼ðPA − PBÞjΨi ∝ jΨi. This solution is the 0 and one can exchange the roles of J and J without changing discrete version of the edge mode, Eq. (37), of the continuum the physical system. However, its physical relevance appears model derived in Sec. II.D.1. clearly if one considers a finite or semi-infinite chain. Then the two classes correspond to different possibilities for the edge state(s) of the chain, as an illustration of the general b. Finite chain bulk-edge correspondence. In the following we first describe One can also consider a chain with M unit cells; see Fig. 31. the case of a semi-infinite chain for which analytical For J0

FIG. 30. A semi-infinite SSH model. The left region (gray area) FIG. 31. Energy spectrum of a SSH chain of M ¼ 10 dimers. A has J ¼ 0, hence a winding number N ¼ 0. The right region may pair of edge states close to zero energy exists when J0=J is have N ¼ 0 or 1, depending on the ratio J=J0. In the latter case a notably below 1. When J0=J increases above 1, these edge states zero-energy edge state resides close to the boundary between the are gradually transformed into bulk states. Adapted from Del- two domains. place, Ullmo, and Montambaux, 2011.

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0 the quasicontinua corresponding to the two bands for J =J can be useful, notably for Z2 topological invariants (Fu and above unity. Kane, 2007). In practice, generating a boundary that preserves chiral symmetry could be delicate to achieve. Thus edge states of 4. Adiabatic pumping for the Rice-Mele model cold atom implementations of the SSH model are likely to be shifted away from zero energy. In contrast, the topological The Rice-Mele model generalizes the SSH model to the protection of the Majorana modes of the Kitaev chain case where the energies of sites A and B may differ by a 2Δ ϕ described next (Appendix A.5) does not require any such quantity . As explained in Sec. II.F, the vortex in Zak local fine-tuning. There the protecting symmetry is the around the gap-closing point (Δ ¼ 0 and J0=J ¼ 1)isa exact particle-hole symmetry of the Bogoliubov–de Gennes topological invariant which can be viewed as a Chern number Hamiltonian (90). in the periodic 2D space formed from crystal momentum −π=a < q ≤ π=a and time 0

Thus, information on the winding number N can be obtained FIG. 32. Kitaev model: A 1D chain of identical sites with ϕ 0 π by measuring q at just two points, q ¼ and =a. Since the nearest-neighbor hopping and a coherent coupling to a superfluid ϕ 2π angle q is defined only mod , this method can determine reservoir that injects and removes pairs of fermions on neighbor- only if N is even or odd. Nevertheless, this partial information ing sites.

Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019 015005-44 N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms the total population of the chain. We are therefore interested in jhðqÞj ¼ 0 at some point in the BZ, i.e., a closing of the the eigenstates and the corresponding energies of quasiparticle gap. The existence of edge states for an open chain of M sites Xn ˆ ˆ † † † can be simply revealed by taking the particular case Δ ¼ J H − μN ¼ −J cˆ cˆ 1 þ cˆ cˆ − μcˆ cˆ j jþ jþ1 j j j and μ ¼ 0 (Kitaev, 2001) (the energy spectrum as a function j o of μ is shown in Fig. 34). In this case, one can perform a Δ ˆ ˆ ˆ† ˆ† þ cjcjþ1 þ cjþ1cj : ðA8Þ canonical transformation using the new fermionic operators:

ˆ i † † C cˆ − cˆ cˆ cˆ 1 A11 For an infinite chain (or a finite chain with periodic boundary j ¼ 2 ð j j þ jþ1 þ jþ ÞðÞ conditions), translation invariance ensures that Hˆ − μNˆ takes a simple form in momentum space. It can be written in the for j ¼ 1; …;M− 1 and standard Bogoliubov–de Gennes form ! ˆ i † † CM ¼ ðcˆ − cˆM þ cˆ1 þ cˆ1ÞðA12Þ 1 X c˜ˆ 2 M ˆ − μ ˆ ˜ˆ† ˜ˆ HBdG q H N ¼ ð cq; c−q Þ q ; ðA9Þ 2 ˆ† † q c˜−q ˆ ˆ so that the fCj; Cj g satisfy canonical fermionic commutation P rules. The Hamiltonian (A8) can then be written ˜ˆ† ∝ iqja ˆ† where the operator cq je cj creates a particle with BdG MX−1 quasimomentum q and each Hamiltonian H is a 2 × 2 † 1 q Hˆ − μNˆ ¼ 2J Cˆ Cˆ − : ðA13Þ matrix: j j 2 j¼1 h ðqÞ¼−2Δ sinðqaÞ; ˆ ˆ BdG y A ground state of H − μN is obtained by solving Hq ¼ −hðqÞ · σ with hzðqÞ¼2J cosðqaÞþμ; ˆ ψ 0 1 … − 1 ðA10Þ Cjj 0i¼ ;j¼ ; ;M : ðA14Þ It is separated from the excited states by the gap 2J,as h 0 and x ¼ . Although the present physical problem is differ- expected from the above analysis for the infinite chain: For ent from the SSH and Rice-Mele models, we recover in J ¼ Δ and μ ¼ 0, jhðqÞj ¼ 2J is indeed independent of q. The Eq. (A10) a Hamiltonian in reciprocal space which has a existence of edge states in this case originates from the fact similar structure. In particular, the search for distinct topo- † that the nonlocal fermion mode Cˆ ; Cˆ , acting on both ends logical phases can be performed by analyzing the trajectory of ð M MÞ of the chain, does not contribute to the Hamiltonian (A13).It the vector hðqÞ when q travels across the BZ. h can therefore be filled or emptied at no energy cost, leading to More precisely we see from Eq. (A10) that the vector ψ always lies in a plane (here y-z), which makes the discussion two independent ground states. More precisely, if j 0i is the ˆ ψ 0 ψ ˆ † ψ formally similar to the SSH model. The quasiparticle excita- ground state satisfying CMj 0i¼ , then j 1i¼CMj 0i is ψ ψ tion spectrum is set by jhðqÞj: this is gapped for all q in the BZ also a ground state. The states j 0i and j 1i correspond to provided jμj ≠ 2J. However, there are two topologically different global parities of the total fermion number. The distinct phases; see Fig. 33.Forjμj < 2J the two-component situation is thus different from a usual superconductor, where vector hðqÞ encircles the origin, winding by 2π as q runs over the BZ: this is the topological superconducting phase, for which there exist localized (Majorana) modes on the boundaries of a finite sample. For jμj > 2J the vector hðqÞ does not encircle the origin and has a vanishing winding number: this is the nontopological superconducting phase. These two topological phases cannot be smoothly connected without causing

FIG. 33. Distinct topological phases of the Kitaev model, FIG. 34. Quasiparticle spectrum of Hˆ − μNˆ [Eq. (A8)] for an evidenced by the trajectory of hðqÞ=jhðqÞj on the unit sphere, open chain of 20 sites and Δ ¼ J. The pair of solutions ðE; −EÞ as q scans the Brillouin zone. Left (μ < −2J) and right (μ > 2J): with E ≈ 0 that appears in the topological region jμj < 2J topologically trivial phases, with a zero winding around the represents the Majorana zero energy modes at the two ends of origin. Middle (jμj < 2J): topological phase. the chain.

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the ground state is nondegenerate and represents a condensate takes account of the micromotion (related to how ti and tf of Cooper pairs, hence a superposition of states with an even fall in the period T) and the time-independent effective number of fermions. Hamiltonian Hˆ eff controls the long-time behavior. As discussed in the main text, the gapless edge modes of the In general, the determination of the effective Hamiltonian topological superconductor are better viewed in terms of the ˆ ˆ F Heff (or the closely related Ht0 ) is a difficult task. However, for Majorana operators a drive frequency ω that is large compared to other frequency scales, the effective Hamiltonian can often be approximated γˆ ≡ ˆ ˆ † ˆ† − ˆ 1 CM þ CM ¼ iðcM cMÞ; ðA15Þ by the Magnus expansion in powers of 1=ω (Eckardt, 2017). Writing the Hamiltonian in terms of its harmonics γˆ ≡ ˆ − ˆ † − ˆ† ˆ 2 iðCM CMÞ¼ ðc1 þ c1ÞðA16Þ X∞ ˆ ˆ imωt in view of the fact that these Majorana operators are spatially HðtÞ¼ Hme localized on each end of the chain. m¼−∞ X∞ ˆ ˆ imωt ˆ −imωt ¼ H0 þ Hme þ H−me ðB4Þ APPENDIX B: FLOQUET SYSTEMS AND THE MAGNUS m¼1 EXPANSION ˆ † ˆ where Hm ¼ H−m, the Magnus expansion leads to (Goldman We mention here several key results for Floquet systems of and Dalibard, 2014) use in the main text. We refer the interested reader to Eckardt (2017) for a comprehensive recent account. X∞ ˆ ˆ 1 1 ˆ ˆ Consider a Floquet system, defined by a time-varying Heff ¼ H0 þ ℏω ½Hm; H−m 1 m Hamiltonian Hˆ ðtÞ that is periodic Hˆ ðt þ TÞ¼Hˆ ðtÞ with m¼ X∞ period T ¼ 2π=ω. The time evolution over an integer number 1 1 ˆ ˆ ˆ þ 2 2 ½½Hm; H0; Hm of periods, from t0 to t0 þ NT, is described by the unitary 2 ℏω ð Þ m¼1 m operator ˆ ˆ ˆ þ½½H− ; H0; H þ: ðB5Þ Z m m i t0þNT ˆ ≡ T − ˆ 0 0 Uðt0;t0 þ NTÞ exp ℏ Hðt Þdt ; ðB1Þ The Magnus expansion underpins several results used in the t0 main text as follows: ˆ where T denotes time ordering. This can be written (i) The leading term H0 is the time-averaged Hamiltonian, as used in our discussion of inertial forces in Sec. III.C to ˆ ˆ N −ði=ℏÞHˆ F T construct nonzero Peierls phase factors for tunneling. Uðt0;t0 þ NTÞ¼½Uðt0;t0 þ TÞ ≡ e t0 ; ðB2Þ (ii) The first-order term is important in generating the next- nearest-neighbor hopping from circular shaking of the honey- which defines the effective Floquet Hamiltonian Hˆ F in t0 comb lattice, as required to simulate the Haldane model. terms of the logarithm of the time evolution operator over a Consider the three-site system shown in Fig. 35 with on-site ˆ F single period T. Clearly Ht0 is undefined up to the addition energies Eα, α ¼ A, B, C, a constant tunnel matrix element −J of integer multiples of 2πℏ=T ¼ ℏω, so its eigenvalues— along the links AB and AC, and no “bare” tunneling between defining the Floquet spectrum—have a periodicity in energy B and C: ℏω “ of . The Floquet Hamiltonian describes the strobo- X ” ˆ ˆ scopic evolution of the system, i.e., at the selected times H ¼ −JðjBihAjþjCihAjþH:c:Þþ EαPα ðB6Þ t ¼ t0;t0 þ T;…;t0 þ NT. This is relevant for understand- α ing the dynamics over time scales long compared to the ˆ drive period T. with the projectors Pα ¼jαihαj. Shaking at a frequency ω In addition to this stroboscopic evolution, the system leads to sinusoidally varying energy offsets that we model as undergoes a “micromotion” on a time scale of the period ˆ F T. This causes the Floquet Hamiltonian Ht0 to depend on the time during the cycle, via a t0-dependent unitary transformation (the Floquet spectrum is therefore invariant). A convenient way to account for this micromotion is to write the general time-evolution operator, from time ti to tf ,as(Goldman and Dalibard, 2014)

ˆ − ˆ − ℏ ˆ eff − ˆ FIG. 35. Hopping on three sites of the honeycomb lattice, with U t ;t e iKðtf Þe ði= ÞH ðtf tiÞeiKðtiÞ: B3 ð i f Þ¼ ð Þ time-varying on site energies. The unitary transformation (B7) maps this problem to time-varying tunneling matrix elements This can be achieved with a time-independent effective between nearest neighbors A ↔ B and A ↔ C. Then the first- ˆ eff Hamiltonian H and a “kick” operator that is periodic in order correction in 1=ω from the Magnus expansion leads to a ˆ ˆ 2 time KðtÞ¼Kðt þ TÞ and that has vanishing average over tunneling term of order J =ℏω between next-nearest neighbors one period (Goldman and Dalibard, 2014). The kick operator B ↔ C, with nonzero Peierls phase factor; see Eq. (B12).

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EBðtÞ¼Δ cosðωtÞ and ECðtÞ¼Δ cosðωt − ϕÞ, whereas interested reader to Eckardt and Anisimovas (2015) for a EAðtÞ is set to zero by convention. The phase offset ϕ arises comprehensive analysis of such terms. from the circular shaking and the angle between the two ϕ 2π 3 nearest-neighbor bonds (typically ¼ = ). Using the APPENDIX C: LIGHT-MATTER INTERACTION unitary transformation generalizing Eq. (52) X R Here we describe the general form that the light-matter t 0 0 ˆ i Eαðt Þdt =ℏ ˆ interaction must take for two-photon interactions and describe UðtÞ¼ e 0 Pα; ðB7Þ α the specific structure of this interaction for alkali atoms. To briefly summarize what follows, the light-matter interaction we convert these energy modulations into time-varying phase can be divided into contributions from irreducible rank-0, -1, factors on the hopping: and -2 spherical tensor operators. In alkali atoms the rank-0 and -1 contributions dominate, but in heavier atoms (or ˜ˆ sufficiently close to atomic resonance) the rank-2 contribution HðtÞ Δ ℏω ω ¼ eið = Þ sin tjBihAj can be large. In addition, a similar description of single photon −J transitions (as would be relevant to systems coupling ground iðΔ=ℏωÞ sinðωt−ϕÞ þ e jCihAjþH:c: ðB8Þ and metastable excited states) gives only the rank-0 and rank-1 contributions. Expanding in terms of the harmonics leads to Now consider a system of ultracold alkali atoms in their electronic ground state manifold illuminated by one or several ˆ Hm laser fields which nonresonantly couple the ground states with ¼ J ðΔ=ℏωÞjBihAjþJ − ðΔ=ℏωÞjAihBj −J m m the lowest electronic excited states. In the presence of an −imϕ external magnetic field only, the light-matter Hamiltonian for þ e ½J mðΔ=ℏωÞjCihAjþJ −mðΔ=ℏωÞjAihCj; the atomic ground state manifold is ðB9Þ μ ˆ Iˆ Jˆ B B Jˆ Iˆ H0 ¼ Ahf · þ ℏ · ðgJ þ gI Þ; ðC1Þ where J m are Bessel functions. Computing the first-order correction to the effective Hamiltonian (B5), one finds μ where Ahf is the magnetic dipole hyperfine coefficient, and B is the Bohr magneton. The Zeeman term includes separate 1 X∞ 1 ˆ ð1Þ ˆ ˆ Jˆ Lˆ Sˆ Lˆ H ≡ ½H ; H− ðB10Þ contributions from ¼ þ (the sum of the orbital eff ℏω m m m ˆ m¼1 and electronic spin S angular momentum) and the nuclear angular momentum Iˆ, along with their respective Land´e eff iπ=2 −iπ=2 ¼ −J ½e jBihCjþe jBihCj ðB11Þ g factors. We next consider the additional contributions to the atomic Hamiltonian resulting with off-resonant interaction 2 X∞ 2 2J sinðmϕÞ½J ðΔ=ℏωÞ with laser fields. Jeff ¼ m : ðB12Þ ℏω m As Deutsch and Jessen (1998), Dudarev et al. (2004), and m¼1 Sebby-Strabley et al. (2006) observed, conventional spin- This describes a next-nearest-neighbor tunneling term, independent (scalar Us) optical potentials acquire additional between B and C sites, which inherits a Peierls phase factor spin-dependent terms near atomic resonance: the rank-1 of π=2. A particle that encircles the plaquette A → B → C → (vector Uv) and rank-2 tensor light shifts (Deutsch and A picks up a gauge-invariant phase ψ ¼ −π=2, placing the Jessen, 1998). For the alkali atoms, adiabatic elimination of AB 1 2 1 3 2 2 system in the regime where the Haldane model has topological the excited states labeled by J ¼ = (D ) and J ¼ = (D ) bands. It is interesting to note that for this 1=ω expansion, the yields an effective atom-light coupling Hamiltonian for the 1 ground state atoms (with J ¼ 1=2): leading term in Jeff corresponds to the m ¼ contribution in 1 ω3 Eq. (B12), hence scales as = . Had we directly used the iu E × E ˆ ˆ ˆ ˆ E E vð Þ J ˆ Magnus expansion (B5) for the initial Hamiltonian (B6), HL ¼ H0 þ H1 þ H2 ¼ usð · Þþ ℏ · þ H2: this would have required us to go up to the third order of the expansion, which would have been quite involved. ðC2Þ Fortunately the unitary transformation (B7) involves an ˆ The rank-2 term H2 is negligible for the parameters of integral of the on-site energies over time, which provides a E 1 ω interest and henceforth neglected. Here is the optical gain of a factor = . The Magnus expansion at order 1 applied −2 Δ 3 ω − ω ˆ electric field, uv ¼ us FS= ð 0Þ determines the vec- to H˜ is then sufficient to obtain the relevant effective Δ ω − ω tor light shift, FS ¼ 3=2 1=2 is the fine-structure split- Hamiltonian. ting, ℏω1=2 and ℏω3=2 are the D1 and D2 transition energies, (iii) The second-order term in Eq. (B5), of order 1=ω2,is and ω0 ¼ð2ω1=2 þ ω3=2Þ=3 is a suitable average. us sets the responsible for the nonlocal interactions discussed in scale of the light shift and proportional to the atoms ac Sec. V.A.1. These arise from the contribution of the ˆ polarizability. Hubbard interaction U to H0, and from the oscillating The contributions from the scalar and vector light shifts ˆ tunneling matrix elements in Hm≠0 in the double commutator, featured in HL can be independently specified with informed leading the term of order J2U=ðℏωÞ2 of Eq. (83). We refer the choices of laser frequency ω and intensity. Evidently, the

Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019 015005-47 N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms h i u vector light shift is a contribution to the total Hamiltonian κ δ v E E E E e e ¼ þ i ð ω− × ω− þ ω × ω Þ · z z acting like an effective magnetic field ℏ þ þ uv − Im½ðEω × Eω Þ · ðe − ie Þe E E ℏ − þ x y x B iuvð × Þ eff ¼ ðC3Þ μ uv BgJ − Re½ðEω × Eω Þ · ðe − ie Þe : ðC8Þ ℏ − þ x y y that acts on Jˆ and not the nuclear spin Iˆ. Instead of using Although this effective coupling is directly derived from the full Breit-Rabi equation (Breit and Rabi, 1931) for the the initial vector light shifts, κ is composed of both static Zeeman energies, we assume that the Zeeman shifts are small and resonant couplings in a way that goes beyond the in comparison with the hyperfine splitting—the linear, or restrictive B ∝ iE × E form. This enables topological anomalous, Zeeman regime—in which case, the effective eff state-dependent lattices. Hamiltonian for a single manifold of total angular momentum Fˆ ¼ Jˆ þ Iˆ states is μ g APPENDIX D: BERRY CURVATURE AND UNIT CELL E E B F B B Fˆ H0 þ HL ¼ usð · Þþ ℏ ð þ eff Þ · : ðC4Þ GEOMETRY

Note that B acts as a true magnetic field and adds vectorially As noted in the main text, it can be convenient to perform eff a unitary transformation of the Bloch Hamiltonian with B, and since jgI=gJj ≈ 0:0005 in the alkali atoms, we ˆ 0 ˆ ˆ ˆ † ˆ −μ B Iˆ ℏ Hq ¼ UqHqUq,withUq a wave-vector-dependent unitary safely neglected a contribution BgI eff · = to the atomic Hamiltonian. We also introduced the hyperfine Land´e g factor operator. Such transformations can be used to render 87 the Hamiltonian periodic in the BZ, but also can relate gF.In Rb’s lowest energy manifold with F ¼ 1, for which different choices of unit cell without change of periodicity. J ¼ 1=2 and I ¼ 3=2, we get gF ¼ −gJ=4 ≈−1=2. In the following, we always consider a single angular momentum They must leave all physical observables unchanged. manifold labeled by F and select its energy at zero field as the The energy spectrum is invariant under this unitary 0 zero of energy. transformation Eq ¼ Eq, while the Bloch states transform 0 ˆ Bichromatic light field: Consider an ensemble of ultracold as juqi¼Uqjuqi. (In this Appendix we drop the band atoms subjected to a magnetic field B ¼ B0ez. The atoms are index for clarity.) The new Berry connection in reciprocal illuminated by several lasers with frequencies ω and ω þ δω, space is where δω ≈ jgFμBB0=ℏj differs by a small detuning δ ¼ μ ℏ − δω 0 0 0 ˆ † ˆ gF BB0= from the linear Zeeman shift between mF A ¼ ihuqj∇qjuqi¼A þ ihuqjUq½∇qUqjuqi: ðD1Þ states (where jδj ≪ δω). In this case, the complex electric field E Eω − ωt Eω − ω δω t ¼ − expð i Þþ þ exp ½ ið þ Þ contributes to It is interesting to note that, in general, this transformation the combined magnetic field, giving causes the Berry curvature to change. How can one reconcile this with the fact that the Berry curvature has iuv measurable physical consequences, e.g., within semiclass- B þ B ¼ B0e þ ½ðEω × Eω ÞþðEω × Eω Þ eff z μ − − þ þ BgJ ical dynamics (Sec. IV.B.2)? As we shall see, the answer −iδωt iδωt þðEω × Eω Þe þðEω × Eω Þe : ðC5Þ lies in noting that these unitary transformations can lead to − þ þ − changes of the positions of the orbitals within the unit B e cell, i.e., changing the internal geometry of the unit cell. The first two terms of eff add to the static bias field B0 z, and the remaining two time-dependent terms describe transitions We illustrate this for a two-band model, with a unitary ≫ B δω transformation between different mF levels. Provided B0 j eff j and are large compared to the kinetic energy scales, the Hamiltonian Uˆ 1 q ρσˆ can be simplified by time averaging to zero the time- q ¼ exp ð2i · zÞ: ðD2Þ dependent terms in the scalar light shift and making the rotating wave approximation (RWA) to eliminate the time This is the transformation used in our discussion of the dependence of the coupling fields. The resulting contribution Haldane model (33). It leads to the change to the Hamiltonian 0 1 A ðqÞ¼AðqÞ − 2 ρhσˆ ziðD3Þ ˆ r 1ˆ κ r Fˆ HRWA ¼ Uð Þ þ ð Þ · ; ðC6Þ of the Berry connection, where we defined hσˆ zi ≡ huqjσˆ zjuqi. where we identify the scalar potential Hence, the Berry curvature becomes

1 U r u Eω Eω Eω Eω Ω0 q ≡ ∇ A0 Ω q − ∇ σˆ ρ ð Þ¼ sð − · − þ þ · þ ÞðC7Þ ð Þ q × ¼ ð Þ 2½ qh zi × : ðD4Þ and the RWA effective magnetic field. This expression is valid In general, the Berry curvature changes, Ω0ðqÞ ≠ ΩðqÞ. for gF > 0 (for gF < 0 the sign of the ex and iey terms would However, since the difference is a total derivative, its integral both be positive, owing to selecting the opposite complex over the BZ vanishes, so there is no change in the Chern terms in the RWA) number of the band.

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For a consistent application of the unitary transformation, changes in cell geometry modify the effect of the forces one must consider how all relevant physical quantities trans- applied and the velocity on the scale of the unit cell. These form. Semiclassical dynamics describes the velocity of a wave modifications compensate the change in the Berry curvature packet centered on q and r in response to a uniform force. A to recover the correct semiclassical dynamics. uniform force in the original basis arises from a potential

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