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 andtopological 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 thephysical 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
4. Momentum distribution and edges states
B. Wave-packet analysis of the BZ topology
1. Bloch oscillations and Zak phase in 1D
2. Measurement of the anomalous velocity
3. Interferometry in the BZ
4. Direct imaging of edge magnetoplasmons
C. Transport measurements
1. Adiabatic pumping
2. Center-of-mass dynamics in 2D
V. Interaction Effects
A. Two-body interactions
1. Beyond contact interactions
a. Continuum models
b. Tight-binding models
c. Synthetic dimensions
d. Current-density coupling
2. Floquet heating
B. Many-body phases
1. Bose-Einstein condensates
2. Topological superfluids
3. Fractional quantum Hall statesa. Bosonsb. Lattice effects
c. Symmetry-protected topological phases
d. Ladders
4. Other strongly correlated phases
a. Chiral Mott insulator
b. Chiral spin statesc. Number-conserving topological superfluids
C. Experimental perspectives

I. Introduction
II. Topology of Bloch Bands
A. Band theory
B. Geometrical phase
C. Topological invariants
1. The Zak phase
2. The Chern number
D. Edge states
1. SSH model
2. Haldane model
E. Topological insulators
1. Discrete symmetries
2. The Rice-Mele model
F. Adiabatic pumping
III. Implementations of Topological Lattices
A. Iconic models
1. Harper-Hofstadter model
2. Haldane model
B. Realization of SSH model
C. Inertial forces
D. Resonant coupling: Laser-assisted tunneling
E. Synthetic dimensions
F. Flux lattices: Intrinsic topology
1. Flux lattices
2. Fluxless lattices
IV. Experimental Consequences
A. Characterization of equilibrium properties
1. Time-of-flight measurements
2. Local measurement of the Berry curvature

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N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

1. Equilibrium observables
2. Collective modes
3. Edge states
4. rf excitation
5. Adiabatic pumping
6. Hall conductivity from “heating”
VI. Outlook
A. Turning to atomic species from the lanthanide family
B. Topological lattices without light
C. Other topological insulators and topological metals
D. Far-from-equilibrium dynamics
E. Invariants in Floquet-Bloch systems
F. Open systems
VII. Summary
Acknowledgments
Appendix A: Topological Bands in One Dimension
1. Edges states in the SSH modela. Semi-infinite chain
b. Finite chain
2. Gauge invariance and Zak phase
3. Time-reversal symmetry of the SSH model
4. Adiabatic pumping for the Rice-Mele model
5. The Kitaev model for topological superconductors
Appendix B: Floquet Systems and the Magnus Expansion
Appendix C: Light-matter Interaction
Appendix D: Berry Curvature and Unit Cell Geometry
References

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I. INTRODUCTION

Topology is a mathematical concept that refers to certain
properties that are preserved under continuous deformations.
One familiar example is the number of twists put into a belt
before its buckle is fastened. Usually we aim to fasten a belt
without any twists. But if we were to introduce a single twist
we would produce a Möbius strip. No continuous deformationof the closed belt would get rid of this uncomfortable twist.
The number of twists is said to be a “topological invariant” ofthe closed belt.
The importance of topological invariants in stabilizing
spatial deformations and defects is also well known in physics
in diverse areas ranging from cosmology to condensed matter.
For a superfluid confined to a ring, the number of times thatthe superfluid phase ϕ changes by 2π around the ringI
1
N¼
∇ϕ · dlð1Þ
2π
is a topological invariant. This winding number cannot change
under smooth deformations of the superfluid: a change would
require the superfluid density to vanish somewhere—such thatϕ is ill defined—requiring the motion of a quantized vortex
line across the ring. Here the topological stability arises fromthe interplay of the underlying space (the ring) and the form of
the local order parameter (the phase of the superfluid wave
function).
In recent years it has come to be understood that topology
enters physics in another, very fundamental way through thenature of the quantum states of particles moving throughcrystalline lattices. The energy eigenstates of electrons moving through periodic potentials are well known to form energy
Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019

bands, a result that follows from the existence of a conserved
crystal momentum via Bloch’s theorem. Remarkably, undercertain circumstances, each Bloch energy band can be
assigned a robust integer-valued topological invariant related
to how the quantum wave function of the electron “twists” as afunction of crystal momentum. This integer is invariant undercontinuous changes of material properties, “continuous”
meaning that the energy gaps to other bands should not close.
The first example of such a topological invariant for Blochenergy bands arose from a highly original analysis of theinteger quantum Hall effect in a two-dimensional (2D) lattice(Thouless et al., 1982). Recent theoretical breakthroughs haveshown that, once additional symmetries are included, topological invariants are much more widespread. These ideas
underpin a recent revolution in our understanding of band
insulators and superconductors (Hasan and Kane, 2010; Qi
and Zhang, 2011). The topological nature of these materials
endows them with physical characteristics that are insensitive
to microscopic details, a notable example being the exact
quantization of the Hall resistance in 2D irrespective of thepresence or form of a random disorder potential.
A great deal of current research focuses on understanding
the physical consequences of these new materials, andexperimental studies of topological insulators and superconductors in solid state systems continue apace. Furthermore,
there is significant activity in exploring the nature of thestrongly correlated phases of matter that arise in thesematerials, notably to construct strong-correlation variants ofthese topological states of weakly interacting electrons.
Theory suggests many interesting possibilities, which are stillseeking experimental realization and verification.
Such questions are ideally addressed using realizations withcold atomic gases. Cold atomic gases allow strongly interacting phases of matter to be explored in controlled experimental
settings. However, a prerequisite for quantum simulations ofsuch issues is the ability to generate topological energy bands
for cold atoms. This poses a significant challenge, even at thissingle-particle level. Realizing topological energy bands
typically requires either the introduction of effective orbital
magnetic fields acting on neutral atoms and/or the introduction of a spin-orbit coupling between the internal spin states ofan atom and its center-of-mass motion. This is an area of
research that has attracted significant attention over the lastyears, both theoretical and experimental. Much progress has
been made in developing techniques to generate artificial
magnetic fields and spin-orbit coupling for neutral atoms(Zhai, 2015; Dalibard, 2016; Aidelsburger, Nascimbene, andGoldman, 2017). The use of these techniques in the setting of
optical lattices has led to the realization and characterizationof topological Bloch bands for cold atoms.
In this review we describe how topological energy bands
can be generated and probed in cold atom systems. We focus
on existing experimental studies, for which the essential
behavior can be understood in terms of noninteracting
particles. We start by explaining the concepts underpinning
topological energy bands in Sec. II. We describe the key
physical effects that are required to generate these bands andhow these can be engineered in cold atom systems in Sec. III.
Our emphasis is on recent experimental developments. In
Sec. IV we describe the principal observables that have been

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N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

used to characterize the geometrical and topological characters of the resulting energy bands. In Sec. V we move beyond
single-particle physics to discuss some of the theoreticalunderstanding of the consequences of interactions in thesenovel optical lattices and to describe some of the interacting
many-body phases that can be sought in future experiments onthese systems. We conclude in Sec. VI with comments on theoutlook for future work and point out connections to broader
research areas.
Throughout this review we explain just the essential physics
underlying recent developments, so the content is necessarily
incomplete. The review should be used as a starting point fromwhich to explore the literature, rather than as a comprehensive
survey of the field. We note that we focus on topological bands
for atoms in periodic lattices. Related phenomena can appear
for photons and hybrid light-matter objects (cavity polaritons)
in novel optical materials, and we refer the interested readers
to the review by Ozawa et al. (2018). We note also that all our
discussions are based on the notion of band structure, meaningthat we ourselves restrict to periodic materials and that we do
not address the case of atoms or photons moving in quasiperiodic systems (Kunz, 1986; Kraus et al., 2012; Tanese
et al., 2014; Dareau et al., 2017).
II. TOPOLOGY OF BLOCH BANDSIn this section we provide an introduction to how topology
enters into band theory for particles moving in periodic
potentials. We focus on quantities and models that are relevant
to later sections. More comprehensive accounts are available
in review articles on topological insulators (Hasan and Kane,
2010; Qi and Zhang, 2011; Chiu et al., 2016).
A. Band theory

Here we describe the essential properties of a quantum
particle moving through a periodic lattice potential. Our aim isto understand both the resulting eigenenergies (giving theconventional band structure) along with the underlying
structure of the eigenstates (from which the band topology
is derived).
Any lattice potential is invariant under displacements
P by aset of lattice vectors R, which can be written R ¼ i mi ai interms of integer fmi g multiples of basis vectors fai g (withi ¼ 1; …; d in dimension d). From Bloch’s theorem, theenergy eigenstates take the form
ðnÞ

ðnÞ

ψ q ðrÞ ¼ eiq·r uq ðrÞ;
ðnÞ

ð2Þ

ðnÞ

with uq ðrÞ ¼ uq ðr þ RÞ having the periodicity of thelattice. These eigenstates are characterized by the band index
n and the crystal momentum q. Although we retain ℏ inequations, we refer to momentum and wave vector interchangeably. Starting from the Schrödinger equation for theðnÞenergy eigenstate ψ q ðrÞ:



1 2
ðnÞ
ðnÞ ðnÞ
−∇ þ VðrÞ ψ q ðrÞ ¼ Eq ψ q ðrÞ;2m

Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019ð3Þ

one readily finds that the Bloch states unq ðrÞ are eigenstates of
the q-dependent Hamiltonian12Ĥq ¼ −ð∇ þ iqÞ þ VðrÞ :ð4Þ
2m
If there are internal degrees of freedom, e.g., spin, then one
should replace unq ðrÞ by unq ðα; rÞ with α labeling these additional degrees of freedom.
States that differPin crystal momentum q by a reciprocallattice vector G ¼ i mi Gi are physically equivalent. (Thereciprocal lattice is constructed from basis vectors fGi gdefined by the condition Gi · aj ¼ 2πδij .) Thus, q can bechosen to be restricted to the first Brillouin zone (BZ): thelocus of points q that are closer to the origin than to any reciprocallattice vector G. For example, for a 2D lattice with basis vectors
a1 ¼ ða1 ; 0Þ and a2 ¼ ð0; a2 Þ the reciprocal lattice has basis
vectors G1 ¼ ð2π=a1 ;0Þ and G2 ¼ ð0;2π=a2 Þ, and the BZ is theregion −π < qx a1 ≤ π and −π < qy a2 ≤ π.
Many of the cases we describe will be tight-binding models,
for which there are M single-particle orbitals within each unit
cell of the lattice. These could be multiple lattice sites and/or
other internal degrees of freedom. The system Hamiltonian Ĥ
acts on the set of M × N c single-particle states jα; Ri ¼ jαi ⊗jRi with α ¼ 1; …; M labeling the states within a unit cell andR the position of each of a total of N c unit cells. For periodic
boundary conditions, the energy eigenstates are plane waves
1 X iq·R ðnÞðnÞe juq i ⊗ jRi;jψ q i ¼ pﬃﬃﬃﬃﬃﬃ
Nc R

ð5Þ
ðnÞ

where n ¼ 1; …; M is the band index and juq i are theassociated Bloch wave functions in the M-dimensional
internal space. The band energies and Bloch wave functions
follow from the spectrum of a wave-vector-dependent
Hamiltonian Ĥ q that acts in this internal space. For example,
Ĥq can be represented by the M × M matrix
H αβ
q ¼

1 X iq·ðR−R0 Þehα; R0 jĤjβ; Ri:
N c R;R0ð6Þ

Note that the Bloch Hamiltonian Ĥq and thus its matrix
representation is not uniquely defined: it can be replaced by
Ĥ0q ¼ Û q Ĥq Û †q , where Û q is a wave-vector-dependent unitarytransformation. Because the states at wave vectors q and
q þ G are physically equivalent, one can always choose the
Hamiltonian to have the same periodicity as the BZ
Ĥq ¼ ĤqþG . We make this choice throughout this section.
However, as will be explained later, some care is required indoing so when considering geometrical properties of the
bands. Other choices are discussed in Appendix A.2.
B. Geometrical phase

At its most basic level, the band structure of a periodic
ðnÞpotential is specified by the energies Eq . However, the bands
are also characterized by geometrical and topological features.
ðnÞ
These relate to how the wave functions juq i vary with wave
vector q across the BZ. No variations can occur for simple

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.,
one-band models). However, topological features arise alreadyfor two orbitals per unit cell, M ¼ 2. We describe topologicalclassifications of Bloch bands in 1D and 2D and illustratethese using two-band tight-binding models.
Our presentation relies heavily on the concept of thegeometrical phase (Berry, 1984), which we briefly review.
Consider a Hamiltonian ĤðXÞ which depends on a set ofparameters X and with nondegenerate spectrumĤðXÞjΨðnÞ ðXÞi ¼ EðnÞ ðXÞjΨðnÞ ðXÞi:ð7Þ

The system is prepared in an eigenstate jΨðnÞ ðXÞi at an initialtime ti , and the parameters X t are changed slowly in time tsuch that the state evolves adiabatically, following an instantaneous eigenstate of ĤðX t Þ. The parameters are taken arounda cycle such that X tf ¼ X ti . Since the Hamiltonian returns to
the initial form at tf , so too must any eigenstate up to an
overall phase factor (in this case of a nondegenerate spectrum). This phase has both dynamical and geometricalcontributions
ðnÞ

ðnÞ

ðnÞ

ðnÞ

jΨf i ¼ ei½γ dyn þγgeo  jΨi i;Z
1 tf ðnÞ
ðnÞ
E ðX t0 Þdt0 ;γ dyn ¼ −
ℏ ti
I
ðnÞ
γ geo ¼ ihΨðnÞ j∂ X ΨðnÞ i · dX:

ð8Þ
ð9Þ

be imposed in physical systems by the application of external
forces, inducing adiabatic dynamics of the Bloch states.
C. Topological invariants
1. The Zak phase

Owing to the periodicity of the BZ under the addition of any
reciprocal lattice vector G, a trajectory in wave vector from qi
to qf ¼ qi þ G is a closed loop. Since the Hamiltonian isperiodic, Ĥq ¼ ĤqþG , the ideas of Berry apply directly. The
integral of the Berry connection along such closed loopsðnÞ
ϕZak ¼

ð10Þ

ð11Þ

around the closed loop in parameter space X. The Berryconnection plays a role similar to that of the vector potential
for a magnetic field. It is gauge dependent, varying under local
gauge transformations jΨðnÞ i → eiΦðXÞ jΨðnÞ i. However, if X
has more than one component, one can define a gaugeinvariant Berry curvature
ðnÞ

ðnÞ

ðnÞ

Ωij ≡ ∂ Xi Aj − ∂ Xj Ai :ð12Þ

If the closed loop X can be viewed as the boundary of a 2D
surface ðX 1 ; X 2 Þ on which the Berry curvature is everywhere
well defined, then the geometrical phase (10) is the flux of the
Berry curvatureðnÞΩðnÞ ≡ ϵij ∂ Xi Aj ¼ ϵij i∂ Xi hΨðnÞ j∂ Xj ΨðnÞ i

ð13Þthrough this 2D surface. Here ϵij is the antisymmetric tensorof two indices ϵxy ¼ −ϵyx ¼ 1 and the summation overrepeated indices is assumed.
We apply these concepts to physical situations in which therole of external parameters X is played either by the crystal
momentum q or by the real-space position r. In both cases, theBerry connection and Berry curvature define local geometricproperties of the quantum states. The integrals of thesegeometric quantities over a closed manifold—the BZ for q,
or the unit cell of the lattice for r—give rise to topologicalproperties. As discussed in later sections, trajectories of q can
Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019iqi

ihuðnÞ j∂ q uðnÞ i · dq

ð14Þ

was proposed by Zak as a way to characterize the energy
bands (Zak, 1989).
The first glimpse of how the mathematics of topologicalinvariants can arise in band theory is provided by computing
the Zak phase for simple two-band tight-binding models in
one dimension (1D). We illustrate this for the Su-SchriefferHeeger (SSH) model.
The SSH model is a tight-binding model in which there
are two sites in the unit cell labeled A and B. The sites areconnected by alternating tunnel couplings J and J 0 ; see Fig. 1.
The single-particle Hamiltonian reads
XĤ ¼ − ðJ0 â†j b̂j þ J â†j b̂j−1 þ H:c:Þ;

ð15Þ

j

The geometrical phase is the integral of the Berry connectionAðnÞ ðXÞ ≡ ihΨðnÞ j∂ X ΨðnÞ i

Z q þG

where â†j and b̂†j create a particle on the A and B sites of the jthunit cell. For a bulk system with periodic boundary conditions,
we look for the energy eigenstates using the Bloch wave formjψ q i ¼

X

eijaq ðuAq jAj i þ uBq jBj iÞ;

ð16Þ

j

with a the lattice constant. The problem reduces to finding the
eigenvectors ðuAq ; uBq ÞT of the Hamiltonian in reciprocal space,which has matrix representation
Ĥ q ¼ −0

J 0 þ Je−iqa

J 0 þ Jeiqa

0


ð17Þ

within the BZ −π=a < q ≤ π=a.
The Hamiltonian (17) can be written in the general formĤq ¼ −hðqÞ · σ̂;ð18Þ

Tight-binding (SSH) model of the polyacetylene molecule. For the SSH model (17), the on-site energies for A and B
are supposed to be equal. This constraint will be relaxed for the
Rice-Mele model (40).

FIG. 1.

015005-4

N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

where σ̂ α are the Pauli operators. Throughout this review,
we use the conventional matrix representation of the Pauli
operators in which σ̂ z is diagonal. Then the SSH model has
hx ðqÞ þ ihy ðqÞ ¼ J 0 þ Jeiqa ;hz ¼ 0;ð19Þ

with hx;y ðqÞ real periodic functions of q. The energy spectrum
is composed of the two bands
ðÞEq¼ jhðqÞj ¼ ½J2 þ J 02 þ 2JJ 0 cosðqaÞ1=2 ;ð20Þwhich are separated by a gap provided jhj does not vanish at
any q. Assuming J, J0 > 0, the gap 2jJ − J 0 j closes only when
J ¼ J0 for the quasimomentum q ¼ π=a.
Provided there is a nonzero gap, i.e., jhj ≠ 0 for all q, one
can write hx þ ihy ¼ jhjeiϕq with a well-defined ϕq , and the
Bloch states are
uAq
uBq

!

1
¼ pﬃﬃﬃ21

∓ eiϕq:ð21Þ

In this pseudospin representation, the fact that hz ¼ 0 in
Eqs. (18) and (19) for the SSH model entails that these
eigenstates lie on the equator of the Bloch sphere. The
resulting Zak phases of the bands (14) are1
ðÞ
ϕZak ¼ −
2

Z

∂ϕq
dq:
BZ ∂q

ð22Þ

The winding number for the SSH model. The curves
plot the locus of ðhx ; hy Þ as q runs over the BZ. For J 0 =J > 1 thevector ðhx ; hy Þ does not encircle the origin, but for J 0 =J < 1 itencircles the origin once, indicating that these two cases havewinding numbers (23) of N ¼ 0 and 1, respectively.

FIG. 2.

A perceptive reader will notice that the topological invariant[Eq. (23)] we constructed appears to be rather unphysical. The
two parameter regimes of the SSH model which have different
winding numbers, at J 0 =J > 1 and J 0 =J < 1, could be triviallyrelated by reversing the labeling of the A and B sublattices inFig. 1, at least deep in the bulk of the system. In what sense arethese two parameter regimes topologically distinct? A key
point to note is that the winding number (23) has a basisdependent offset. In place of Eq. (16) we could just as wellhave sought an energy eigenstate of the formjψ q i ¼

Thus,

X

eijaq ðũAq jAjþN A i þ ũBq jBjþN B iÞ

ð24Þ

j

1 ðÞ
1
N ≡ − ϕZak ¼
π
2π

Z

∂ϕq
dq
BZ ∂q

X
eijaq ðũAq e−iN A aq jAj i þ ũBq e−iN B aq jBj iÞ;

ð25Þ

j

is the number of times ϕq changes by 2π as q runs over the
BZ. Since the Hamiltonian is periodic, ϕq ¼ ϕqþG modulo 2π,
N is an integer winding number, analogous to that for thephase of a superfluid around a ring (1). It measures the solidðÞangle drawn by the pseudospins uq along the equator of the
Bloch sphere when q spans the BZ.
The winding number (23) is a topological invariant of 1Dband insulators arising from Hamiltonians of the form (18): N
cannot be changed unless jhðqÞj vanishes at some q, i.e.,
unless the band gap closes. The case of the SSH model isillustrated in Fig. 2, which shows the locus of ðhx ; hy Þ as q
runs over the BZ. For J 0 =J < 1 the curve encircles the origin
once, N ¼ 1; while for J 0 =J > 1 the curve does not encircle
the origin, N ¼ 0. These two curves cannot be smoothlyinterconverted without crossing hx ¼ hy ¼ 0, i.e., without theband gap closing. One could evade this conclusion by
including terms proportional to σ̂ z in the Hamiltonian.
Then two gapped states with different N could be continuously deformed into each other. [We explore this later for the
Rice-Mele (RM) model.] However, as explained in Sec. II.E,
including σ̂ z terms would break an underlying “chiral”
symmetry of the SSH model. The winding number in 1D
is an example of a topological invariant whose existence relies
on an underlying symmetry.
Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019

¼

ð23Þ

associating site A in the cell j þ N A with site B in cell j þ N B .
Comparing Eq. (25) with (16) shows that this amounts to
replacing ðuAq ; uBq ÞT by ðũAq e−iN A aq ; ũBq e−iN B aq ÞT , i.e., to a qdependent unitary transformation Û q . In this new basisðũAq ; ũBq ÞT , the eigenstates (21) are replaced by
ũAq
ũBq

!

1
¼ pﬃﬃﬃ2

eiqaN A∓ eiϕq eiqaN B


:

ð26Þ

For N A ¼ N B this transformation may be viewed as areciprocal-space gauge transformation. A direct calculationof the Zak phase for Eq. (26) shows that the winding number
N ≡ −ϕZak =π is increased by N A þ N B as compared to thatfor Eq. (21). This example shows that the absolute value of thewinding number cannot be physically meaningful. Instead,physical consequences can involve only differences of winding numbers, which are well defined when computed within
the same reciprocal-space basis choice. In Sec. II.D we
describe how the winding number difference at a boundary
between two regions influences the spectrum of states on theedge. However, note that we already found one physicalconsequence of the winding number difference: the parameter
space of microscopic couplings becomes disconnected, in the015005-5

N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

sense that there is no way to continuously change the physicalparameters to evolve between regions of different winding
numbers without crossing a critical point at which the band
gap closes. Such gap closings between topologically distinct
regions have direct physical consequences in measurementsof the bulk excitation spectrum. In particular, for a set ofnoninteracting fermions filling one such band, the gap closing
implies a thermodynamic phase transition between two
insulating regimes, separated by a semimetallic state. We
illustrated this gap closing for the SSH model. However, thisalso holds for any 1D model in which h ¼ ðhx ; hy ; 0Þ, i.e.,with chiral symmetry. The requirement of the closing of aband gap in order to change the winding number is a defining
feature of a topological invariant of the energy band.
2. The Chern number

A topological classification of (nondegenerate) energy
bands in 2D exists without requiring any underlying symmetry and without ambiguities related to the choice of basis
in reciprocal space. The topological invariant is the Chern
number
1
C¼
2π

Z
BZ

iϵij ∂ qi huq j∂ qj juq id2 q:

ð27Þ

We suppress the band index n, but note that a Chern number
exists for each band. This topological invariant, and terminology, arises from the mathematics of fiber bundles (Stone andGoldbart, 2009). However, it can be readily interpreted more
physically in terms of the Berry curvature (13) of the BlochstatesΩðqÞ ≡ ∇q × AðqÞ · ez ¼ iϵij ∂ qi huq j∂ qj juq i:

ð28Þ

The Chern number is related to the flux of the Berry curvatureΩðqÞ through the BZ. Just as the Dirac quantization condition
requires the magnetic flux through a closed surface to bequantized (in units of h=e), the flux of the Berry curvature
through the BZ (a closed surface with the topology of a torus)
is quantized (in units of 2π). This follows by using Stokes’s
theorem to relate the integral of ΩðqÞ over the BZ to the line
integral of AðqÞ around its boundary. Since the BZ is a closed
surface this line integral must be an integer multiple of 2π.
This integer C is the number of flux quanta of Berry curvature
through the BZ. Since the Berry curvature is a gauge-invariant
quantity, so too is the Chern number.
The Chern number vanishes for systems with time-reversal
symmetry (TRS), for which uq ðrÞ ∝ u−q ðrÞ and hence ΩðqÞ ¼−Ωð−qÞ. The realization of Chern bands therefore requires ameans to break TRS. In Sec. III we describe ways in which
this can be achieved for cold atom systems. One class of
implementation involves tight-binding models, in which aspinless particle hops on a lattice with complex tunnelingmatrix elements, e.g., to represent the Peierls phase factors fora charged particle in a magnetic field. TRS is broken if, forsome closed loop on the lattice, the phases acquired on
encircling the loop in the clockwise (þϕAB ) and anticlockwise(−ϕAB ) directions differ modulo 2π, i.e., provided ϕAB is not
an integer multiple of π. Another class of implementationRev. Mod. Phys., Vol. 91, No. 1, January–March 2019

exploits internal atomic states, labeled by index α. Quite
generally, the nondissipative action of laser light on an atom is
to couple state α to state α0 with a well-defined momentum
transfer κ. An optical lattice is defined by a set of such0
couplings V κα α , which to preserve periodicity must build up a0
regular lattice in momentum space. The couplings V ακ αtherefore define a tight-binding model in momentum spacewith amplitudes and phases determined by the laser fields. For
shallow lattices, the net phase acquired on encircling a closed
loop on this momentum-space lattice determines the integrated Berry curvature through that loop, allowing lattices
with broken TRS ΩðqÞ ≠ −Ωð−qÞ to be directly constructed(Cooper and Moessner, 2012).
To illustrate how a nonzero Chern number arises in tightbinding models, we discuss here the key features of the
Haldane model (Haldane, 1988). This is a two-band model, so
the Bloch Hamiltonian is a 2 × 2 Hermitian matrix
Ĥq ¼ −hðqÞ · σ̂ð29Þwith hðqÞ a three-component vector coupled to the Paulimatrices σ̂ x;y;z . Just as in the SSH model, the energy bands
ðÞ

are Eq ¼ jhðqÞj and there is a gap provided jhq j does
not vanish at any q. However, now the Bloch state of thelower band depends on the three-component unit vectoreðqÞ ≡ hðqÞ=jhðqÞj, e.g.,
juq i ¼cosðθq =2Þsinðθq =2Þeiϕq

ð30Þ

for eðqÞ ¼ ðsin θq cos ϕq ; sin θq sin ϕq ; cos θq Þ. The Chern
number can be written in terms of this unit vector as
C ¼ N 2D ≡ −

1
ϵ
8π ij

Z
e·
BZ

∂e
∂e 2
×
d q
∂qi ∂qj

ð31Þ

N 2D counts the number of times that the unit vector eðqÞ wraps
over the unit sphere as q spans the BZ; it is the 2D analog of
the 1D winding number (23).
The Haldane model is described in detail in Sec. III. It isdefined on a honeycomb lattice, for which the unit cell
contains two sites, which we label A and B, as in Fig. 3(a).
Nearest-neighbor tunneling is off diagonal in the sublatticeindex and leads tohx ðqÞ þ ihy ðqÞ ¼ Jðeiq·a1 þ eiq·a2 þ 1Þð32Þ

in Eq. (29), where a1;2 are the lattice vectors marked on
Fig. 3(a). The Hamiltonian has been chosen to be periodic
under the addition of reciprocal lattice vectors. Note, however,that other choices can be made, related to the Hamiltonian via
unitary transformations Ĥq → Û q Ĥq Û †q . Indeed, in Sec. III wereplacehx ðqÞ þ ihy ðqÞ → Jðeiq·ρ1 þ eiq·ρ2 þ eiq·ρ3 Þ

ð33Þ

which arises for Û q ¼ expð−iσ̂ z q · ρ3 =2Þ, with ρ3 the nearestneighbor lattice vector in Fig. 3(a). This transformation

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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 ofedge states, which we now discuss.
(a)

(b)

D. Edge states

FIG. 3. (a) Real-space honeycomb lattice for the Haldane
model. (b) Illustration of the unit vector eðqÞ in the topologicalphase of the Haldane model. The dashed line shows the conventional BZ, whose corners are the points Q . The arrows show the
projection of eðqÞ on the plane ðex ; ey Þ, while the colors andcontours indicate ez . The unit vector wraps once over the sphere
within the BZ, indicating the topological character with C ¼ 1.

does not change the energy spectrum nor the topology ofthe band. However, it can be more helpful to work with
Eq. (33) when considering physical observables (Bena andMontambaux, 2009). As discussed in Appendix D, unitarytransformations of the form Û q affect the definition of the force
and current operators as well as the local Berry curvature.
The resulting band structure is well known from studies ofgraphene: there are two points at the corners of the BZ q ¼ Qwith
4π
Q ¼  pﬃﬃﬃ ð1; 0Þ;
3 3aat which the bands touch, i.e., hx þ ihy ¼ 0. Close to either
band-touching point, with q̃ ≡ q − Q , the Hamiltonian has
the 2D Dirac formĤq ≈ ℏv½q̃x σ̂ x þ q̃y σ̂ y  þ hz σ̂ zð34Þwith velocity v ¼ ð3=2ÞJa=ℏ. The terms hz arise from effects
other than nearest-neighbor tunneling and open gaps at the
Dirac points between the two bands. One such effect is anenergy splitting Δ between A and B sublattices. In this case−the coefficients hþz and hz are both equal to Δ and the gap
openings at the two Dirac points are equivalent. This causes
the resulting Bloch bands to have vanishing Chern number, so
the two resulting bands are nontopological. This is consistent
with the fact that this model has TRS. Introducing nextnearest-neighbor hopping with the Aharonov-Bohm phaseϕAB ≠ 0 mod π breaks TRS and provides a term with hþz ¼−h−z 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 firstshown by Thouless et al. (1982), a band insulator exhibits theinteger quantum Hall effect if the total Chern number of thefilled bands is nonzero, with a Hall conductivity quantized
at C times the fundamental unit of conductance e2 =h. This
result can be extended to systems subjected to disorder andRev. Mod. Phys., Vol. 91, No. 1, January–March 2019

Topological band insulators have the generic feature that
although they are bulk insulators—owing to the energy gap
between the filled and empty bands—they host gapless stateson their surfaces.
The existence of gapless edge modes for 2D systems with
nonzero Chern number is well known from studies of theinteger quantum Hall effect (Halperin, 1982). Each filled
Landau level gives rise to a chiral edge mode. This can beunderstood semiclassically in terms of the skipping orbit of
the cyclotron motion around the edge of the sample; see
Fig. 4. These semiclassical skipping orbits consist of two
features. The rapid skipping motion at the cyclotron frequency
is a feature which in a quantum description arises when the
wave function has nonzero amplitude in more than one
Landau level, such that its time dependence involves theLandau level energy spacing (cyclotron energy); it is therefore
related to the inter-Landau level excitation known as a“magnetoplasmon.” The drift of the guiding center of theorbit around the perimeter of the sample is a feature that exists
for quantum states within a single Landau level and that
represents the chiral edge mode of that Landau level. The
existence of these edge modes is required for consistency ofthe quantized bulk Hall conductance (Laughlin, 1981).
More generally, gapless edge states occur at the boundary
between two insulating regions with different values of a
topological invariant. A simple semiclassical view of thesegapless regions is provided by considering the boundary toarise from a smooth spatial variation in the parameters of the
Hamiltonian, between two phases with different topologicalindices. Since the two insulators far to the left and far to theright of the boundary are topologically distinct, at some pointin space the gap between the filled and empty bands mustclose. This gap closing motivates the existence of gapless edge
states. While this semiclassical argument applies only forsmooth variations in space, the result is a robust feature forany form of boundary, referred to as the bulk-boundary
correspondence (Hasan and Kane, 2010). The only restriction

FIG. 4. Skipping orbits for a charged particle in a uniform
magnetic field. In the bulk, the semiclassical dynamics of a wave
packet leads to a circular cyclotron orbit. The reflection of suchan orbit from the (hard wall) edge leads to a skipping motion
around the sample edge.

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N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

is that, in cases where the topological invariant relies on anunderlying symmetry, this symmetry must be preserved also
in the boundary region. This is illustrated later for the edge
state of the SSH model.
We demonstrate the emergence of edge states in the SSH
and Haldane models within continuum approximations forwhich the edge state wave functions take simple analytic
forms.

In the last line we replaced q̃ by −i∂ x to give a 1D Dirac
Hamiltonian in which MðxÞ can be viewed as a spatiallyvarying “mass.” This was a model studied by Jackiw andRebbi (1976), who showed that, provided MðxÞ changes sign,
there is a solution that is localized at the boundary with exactly
zero energy. For M þ > 0 and M − < 0, this is − R x Mðx0 Þdx0 =a iπx=a 
e
e 0
:
jΨi ¼
0

ð37Þ

1. SSH model

Consider a 1D band insulator formed by filling the lower
energy band of a Hamiltonian of the form (18), which ispurely nondiagonal in the sublattice basis and parametrized by
the two-component vector ðhx ; hy Þ. This restriction on theform of the Hamiltonian arises from the chiral symmetry of
the model as discussed further in Sec. II.E. Let the propertiesof the system depend on some quantity M such that hx þ
ihy ≡ jhðq; MÞjeiϕðq;MÞ is a function of both wave vector q and
M. We introduce a boundary between two gapped phases at
position x ¼ 0 by allowing MðxÞ to vary slowly in space, ascompared to the lattice constant a and setting Mðx ≪ 0Þ ¼
M− and Mðx ≫ 0Þ ¼ M þ . The two phases are characterizedby winding numbers N  (23) computed from ϕðq; M Þ. It is
straightforward to show that hx þ ihy must have N − − N þvortices within the boundary region; see Fig. 5. So if N þ ≠N − then jhðq; MÞj must vanish at certain points ðq; MÞ: these
are the gap-closing points, discussed semiclassically, whichlead to gapless edge states.
For the SSH model, hx þ ihy ¼ J 0 þ Jeiqa , the gap-closing
point (J 0 =J ¼ 1, qa ¼ π) hosts a single vortex. Defining
M ¼ 1 − J 0 =J and q̃ ¼ q − π=a, we expand the
Hamiltonian around M ¼ q̃ ¼ 0 to first order in q̃ (suitable
for the continuum limit, jq̃j ≪ 1=a) to give
Ĥ SSH =J ≈ q̃aσ̂ y þ MðxÞσ̂ x¼ −iaσ̂ y ∂ x þ MðxÞσ̂ x :

ð35Þ
ð36Þ

That the state is an eigenstate of σ̂ z with jσ z ¼ þ1i indicatesthat it has nonzero amplitude only on the A sublattice. For
Mþ < 0 and M − > 0, the nonzero amplitude would be on the
B sublattice. This is the gapless edge mode of the SSH model
which arises because M  describe different topologicalphases. That the state has energy E ¼ 0 is a consequenceof a chiral symmetry that protects the relevant 1D topologicalinvariant, as discussed in Sec. II.E. If the Hamiltonian were
to depart from this chiral form, for example, by including
terms hz that couple to σ̂ z , the energy of this subgap state
would depart from E ¼ 0. This could occur either by achange of the bulk Hamiltonian or due to breaking of thesymmetry near the edge: for example, arising from an on-site
potential that shifts the energy of the A site relative to the Bsite near the edge.
Although we derived this edge state for a continuum
model, the properties are robust to lattice effects provided
the chiral symmetry is preserved. A derivation of the edge
state of the SSH model for a sharp boundary is provided inAppendix A.1.
2. Haldane model

One can use this solution for the edge state of the SSH
model to construct a 1D band of edge states on a surfaceof the topological band insulator formed from the Haldane
model. Consider the low-energy theory for the Haldane model(34) in a topological phase with hz ¼∓ jHj. We impose aboundary to a nontopological phase at x > xR by adding aspatially dependent energy offset ΔðxÞσ̂ z , such that hz ðxÞ ¼
ΔðxÞ ∓ jHj. The low-energy Hamiltonians close to Q
becomeĤ q ≈ ∓ iℏvσ̂ x ∂ x þ hz ðxÞσ̂ z þ ℏvq̃y σ̂ yFIG. 5. An edge of the SSH model is represented byhx þ ihy ¼ jhq;M jeiϕðq;MÞ , which depends both on the wave
vector q and on some control parameter M that varies
smoothly between M − at x ≪ 0 and M þ at x ≫ 0. The
number of vortices of hx þ ihy within this region can becomputed by integrating ∇ϕðq; MÞ around the boundary. The
integrals along q ¼ π=a cancel (since the BZ is periodic),leaving only the integrals along M ¼ M , which give thedifference of winding numbers N − − N þ .
Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019

ð38Þ

where we replaced q̃x → −i∂ x . Translational invariance ismaintained along the y direction, such that qy (and thereforeq̃y ¼ qy − Qy ) is conserved.
For ΔðxÞ an increasing function of x at x > 0 there is aboundary between topological (x < xR ) and nontopological(x > xR ) regions where ΔðxR Þ ¼ jHj. At this point, hþz ðxR Þ ¼
0 so the gap at Qþ vanishes. The low-energy Hamiltonian
close to Qþ (38) takes the form of the Jackiw-Rebbi
Hamiltonian (36), just under a permutation of the Paulimatrices [ðσ̂ x ; σ̂ y ; σ̂ z Þ → ðσ̂ y ; σ̂ z ; σ̂ x Þ], plus a term ℏvq̃y σ̂ yfor which the zero mode (37) is also an eigenstate (noting
the permutation of the Pauli matrices), with eigenvalue
ERqy ¼ ℏvq̃x . This is the chiral edge mode, which propagatesalong the right-hand boundary xR at velocity vey. For a

015005-8

N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

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 anunderlying chiral symmetry. This chiral symmetry arises fromthe existence of a unitary operator Û that anticommutes with
the Hamiltonian Ĥ
(a)Ĥ Û ¼ −Û Ĥ :(b)

The low-energy spectrum of the Haldane model on a
finite-width strip. (a) The strip is bounded in the x direction, but
uniform along y such that the wave vector q̃y ¼ qy − Qþy isconserved. (b) The spectrum has a continuum of states in thebulk, shown shaded. The bulk bands are topological, with unit
Chern number, so a single edge state connects between these bulk
bands: the red solid line (green dashed line) shows the bandcorresponding to the edge mode on the left (right) boundary. We
show only the part of the spectrum close to the Qþ point, at which
the boundaries force a gap closing and at which the edge states
appear.
FIG. 6.

finite-width strip, with another boundary to a nontopologicalphase at x < xL , there is an edge mode with the opposite
velocity ELqy ¼ −ℏvq̃y. The dispersion is illustrated in Fig. 6.
E. Topological insulators
1. Discrete symmetries

The Chern number for 2D systems is an example of a
topological invariant for energy bands in a setting where nosymmetries constrain the form of the Hamiltonian. The fieldof topological insulators arose from the realization that, when
one constrains the Hamiltonian to have additional symmetries,
new topological invariants emerge in various spatial dimensions. The general theory of symmetry-protected topologicalinvariants for gapped fermionic systems concerns global
symmetries—time-reversal, particle-hole symmetry, chiralsymmetry—which make no reference to spatial structure ordimension (Altland and Zirnbauer, 1997). These define ten
distinct symmetry classes which, combined with the spatial
dimensionality, determine the possible topological invariants(Chiu et al., 2016). An important example is time-reversal
symmetry which leads to the existence of a Z2 topologicalinvariant in 2D and 3D for electronic systems with spin-orbit
coupling. This led to the discovery of materials which behave
as topological insulators in these dimensions (Hasan andKane, 2010; Qi and Zhang, 2011). Furthermore, this formalism also allows a unified description of topological superfluids, formed from the pairing of fermions. Within the
Bardeen-Cooper-Schrieffer (BCS) mean-field theory, the spectrum of quasiparticle excitations of such superfluids is obtained
from the Bogoliubov–de Gennes (BdG) theory, which takes theform of a noninteracting fermionic system albeit with anomalous terms that mix particle and hole excitations. The intrinsic
particle-hole symmetry of the BdG theory stabilizes gapped
topological superfluids. We discuss such topological superfluids in Sec. V.B.2.
Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019

ð39ÞThen for an energy eigenstate jΨi of eigenvalue E, the stateÛjΨi is readily shown to be an energy eigenstate with energy−E. Thus, the chiral symmetry enforces a symmetry on the
spectrum about E ¼ 0. This rather formal construction arises
in tight-binding models, such as the SSH model, in which the
Hamiltonian involves only terms that hop between two
different sublattices (labeled A and B). Defining P̂A=B as
projectors onto the A=B sublattices, then Û ≡ ðP̂A − P̂B Þsatisfies Eq. (39) if P̂A ĤP̂A ¼ P̂B Ĥ P̂B ¼ 0, i.e., provided
the Hamiltonian Ĥ 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
described in Sec. II.C.1 this form ensures that the winding
number is a topological invariant. Furthermore, the symmetryof the energy spectrum ensures that the edge mode, found in
Sec. II.D.1, is at exactly zero energy E ¼ 0 and is also an
eigenstate of Û ¼ P̂A − P̂B . This implies that there is at least
one zero energy in-gap mode, but other nonzero energy in-gap
modes are allowed by chiral symmetry provided they come inE pairs. For boundaries that break the chiral symmetry, thein-gap states will have no particular relation to zero energy. A
disordered potential that acts differently on the two sublatticesites will break chiral symmetry and also shift the energies of
in-gap states away from zero energy. However, if the disorderis such that Eq. (39) is preserved, the topology of this model isrobust and the in-gap states remain at zero energy.
2. The Rice-Mele model

Since the topological invariant of the SSH model is ensured
by a (chiral) symmetry, it is possible to smoothly evolve
between states with different topological invariants by breaking this symmetry. To illustrate this, we study the Rice-Melemodel which generalizes the SSH model by relaxing theassumption that the A and B sites have the same energy andassigning energy offsets Δ. The Hamiltonian (17) is
replaced by

ĤRM
q ¼

Δ

−J 0 − Je−iqa−J 0 − Jeiqa

−Δ


ð40Þ

within the BZ −π=a < q ≤ π=a. The two bands have energiesðÞEq¼ ½Δ2 þ jJ 0 þ Je−iqa j2 1=2 ;ð41Þ

and so are separated by a gap provided Δ ≠ 0. (For Δ ¼ 0 thegap closes at the topological transition of the SSH model
J 0 =J ¼ 1.) The eigenstate of the lower band can be written as

015005-9

N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

ð−Þ

juq i ¼sinðγ q =2Þ
−e

iϕq



cosðγ q =2Þð42Þ

with γ q and ϕq defined by −ðJ0 þ Je−iqa Þ=Δ ≡ tan γ q e−iϕq .
The resulting Zak phase (14) is shown in Fig. 7 (left) as afunction of J0 =J and Δ=J. Along the line Δ ¼ 0 the Zak phasesteps by π across J0 =J ¼ 1, consistent with the change inwinding number by one at the topological transition of the
SSH model, and Eq. (23). However, note that the gapped band
insulators at Δ ¼ 0, J 0 =J > 1 and Δ ¼ 0, J0 =J < 1 can now
be continuously connected by tracing out a path that has
Δ ≠ 0 when J0 =J ¼ 1 (Fig. 7, right).

of the loop in parameter space. Note that at this stage no
intuition has been provided on the mechanism at the basis of
the transport, nor its direction. For a physical discussion see
Sec. IV.C.1, in particular, Fig. 22.
This quantization has the same topological origin as theinteger quantum Hall effect. To see this, consider tracing out
the closed loop in parameter space as time t varies from t ¼ 0
to t ¼ T. The Zak phase at any given time t is defined by the
integral of the Berry connection along the BZ −π=a < q ≤π=a at that instant. The change in Zak phase between any twotimes t1 and t2 can be written as a line integral
I
ϕZak ðt1 Þ − ϕZak ðt2 Þ ≡F. Adiabatic pumping

Consider again the Rice-Mele model, whose Zak phase isillustrated in Fig. 7. There is a vortex in ϕZak around thegap-closing point (Δ ¼ 0 and J 0 =J ¼ 1) at which the Zakphase is undefined. The winding of ϕZak by 2π around a closed
loop encircling the gapless point, such as the loop shown in
Fig. 7, is a topological invariant of the model: this winding
is preserved under smooth variations of the Rice-Mele
Hamiltonian that do not cause the gap to close on this loop.
The existence of this invariant is at the basis of the conceptof a quantized pump (Thouless, 1983). It describes the genericsituation of a crystal with filled bands, which is characterizedby parameters that can be externally controlled. When theseparameters are varied around a closed loop, the number ofparticles that are transported is quantized. It is thus a robust
quantity that is not affected by a small change of the geometry
1.0

0.5

0.0

L

A · dqð43Þ

where we label the position in the q–t plane by a twocomponent vector q ¼ ðq1 ; q2 Þ ¼ ðq; tÞ and the associatedBerry connection Ai ≡ ihuq j∂ i juq i, with ∂ 1 ≡ ∂=∂q and∂ 2 ≡ ∂=∂t. The integration contour L is shown in Fig. 8.
The horizontal lines at fixed t ¼ t1 and t ¼ t2 recover
ϕZak ðt1 Þ − ϕZak ðt2 Þ, while the integrals on the lines at q ¼
π=a cancel as a result of the periodicity of the BZ. Applying
Stokes’s theorem, the line integral in Eq. (43) can be written as
the integral of the Berry curvature Ω ¼ ϵij ∂ i Aj over the areabounded by L. The fact that the parameters of the Hamiltonian
return to their original values as t ¼ 0 → T enforces periodicity also in t, such that the q–t plane has the topology of atorus. Thus, extending the contour to enclose the full region
from t ¼ 0 to t ¼ T, thereby computing ϕZak ð0Þ − ϕZak ðTÞ,recovers 2π times a Chern number. The relevant integral isentirely analogous to Eq. (27) with the measure d2 q replacedby dqdt.
The link between the winding of ϕZak and the transported
particle number for a filled band can be established usingarguments detailed in Appendix A.4. The transported particle number is determined by computing the change in themean 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-Melemodel, represented by the unit vector ðcos ϕZak ; sin ϕZak Þ. The
phase changes by 2π around the point Δ ¼ 0 and J 0 =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, J 0 =J < 1), obtained whenq scans the Brillouin zone. The Zak phase, given by half thesubtended solid angle, is equal to π. Green solid line: Trajectoryin the nontopological case of the SSH model (Δ ¼ 0, J 0 =J > 1)with a zero subtended solid angle. Black dotted line: Closed
trajectory obtained for the Rice-Mele model with a nonzero
energy offset Δ. A continuous variation of Δ and J 0 =J in the RiceMele model allows one to connect the topological and nontopological SSH trajectories, without going through the singularpoint J 0 =J ¼ 1, Δ ¼ 0.
Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019

The relationship of the change in the Zak phase for a 1D
band insulator, with momentum −π=a < q ≤ π=a, under adiabatic variation of parameters in time around a cycle of period T,
i.e., with time 0 < t ≤ T. The change in Zak phase between times
t1 and t2 is represented by the line integral around a contour L.
When L is extended to include the full q–t plane, the periodicity
in q → q þ 2π=a and t → t þ T ensures that the line integral
must by an integer multiple of 2π. This integer is the Chern
number defined in the q–t plane.
FIG. 8.

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N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

displacement of particles over one cycle, for example, theclosed loop in Fig. 7, is
Δx ¼

a½ϕ ðTÞ − ϕZak ð0Þ:
2π Zakð44Þ

Because ϕZak ðTÞ equals ϕZak ð0Þ plus 2π times the winding
number corresponding to the vortex of Fig. 7, the displacement is quantized in units of the lattice period a. When theband is filled with exactly one particle per state, this entails
the quantization of the number of transported particles.
III. IMPLEMENTATIONS OF TOPOLOGICAL LATTICES

To this point in this review, we have developed ourunderstanding of lattices and discussed how topology presents
additional “labels” tied to individual Bloch bands: providing a
new way to categorize band structure. This section takes thenext step and describes the currently implemented techniques
by which 1D and 2D band structures with nontrivial topology
have been created.
A. Iconic models

Cold atom experiments are often able to nearly perfectly
realize iconic topological models from condensed mattertheory. Here we briefly describe two such models—theHarper-Hofstadter model (Harper, 1955; Hofstadter, 1976)
and the Haldane model (Haldane, 1988)—as particularly
simple examples of topological lattices and explore what isessential about each of these models. This allows us to place
experimental approaches in context and to identify what typesof new terms must be added to standard optical lattices.
1. Harper-Hofstadter model

The Harper-Hofstadter model, describing charged particles
in a square lattice with a uniform of magnetic field, derives
from the simple 2D tight-binding HamiltonianXð45Þ
Ĥ ¼ −J ðâ†jþ1;m âj;m þ â†j;mþ1 âj;m þ H:c:Þ
j;mfor particles hopping in a square lattice with tunneling strength
J. Each individual site of this lattice is labeled a pair of
integers j and m. The first term in Eq. (45) denotes tunnelingalong the j direction (horizontal) and the second term marks
tunneling along the m direction (vertical). This model results

in a single, nontopological, cosinusoidal band with full
width 8J.
As shown in Fig. 9(a), our task is to imbue the tunnelingmatrix elements with nonzero Peierls phase factors so that thephase accrued by a particle encircling a single plaquette isϕAB , equivalent to the Aharonov-Bohm phase ϕAB ¼ qBA=ℏ
acquired by a particle with charge q moved around a plaquetteof area A. The resulting complex tunneling matrix elements
are required to break time-reversal symmetry and allow anonzero Chern number. As can be confirmed by the phases
depicted in Fig. 9(a), the associated Harper-Hofstadter
Hamiltonian expressed in the Landau gaugeĤ ¼ −JX
ðeimϕAB â†jþ1;m âj;m þ â†j;mþ1 âj;m þ H:c:Þ;

ð46Þ

j;m

gives a phase ϕAB for tunneling around each plaquette. For
rational ϕAB =2π ¼ p=q, expressed in reduced form, the single
band of Eq. (45) fragments into q (generally) topologicalbands, with zero aggregate Chern number.
We can focus in on the essential features of this model byconsidering the special case of one-third flux per unit cell, i.e.,ϕAB ¼ 2π=3. First take note of the tunneling phase eimϕAB for
motion along j: As shown in Fig. 9(a), this tunneling phasedepends on m and has a spatial period of three lattice sites,
implying that the lattice’s unit cell is enlarged beyond theplaquettes of the underlying square lattice (the unit cells
without magnetic flux) to three plaquettes at ϕAB ¼ 2π=3. In
Fig. 9(b), these unit cells are graphically indicated by the graydashed lines, with a representative unit cell shaded (light blue)
for clarity. Each of these unit cells is identified by integers j
and M. In order to distinguish between the three inequivalentsublattice sites within each unit cell, we introduce also the site
index s ¼ −1; 0; þ1, related to the individual plaquette index
m via m ¼ 3M þ s.
As a result of this expanded unit cell, the associated
Brillouin zone is reduced to 1=3 of its initial size along them direction, and the number of bands correspondingly
increases from 1 to 3. Following textbook techniques, we
express this Hamiltonian in the Fourier representation, i.e.,
giving states labeled by their crystal momentum q ¼ ðqj ; qm Þalong with the sublattice index s, i.e., for ϕAB ¼ 2π=3 thestates are fjq; −1i; jq; 0i; jq; þ1ig. For each crystal momentum q the Hamiltonian matrix coupling these sublattice sites
together is

1
1
exp½iqm ð3as Þ
2 cos ðqj a − 2π=3ÞCB
1
1
2 cos ðqj aÞ
H q ¼ −J @A;
exp½−iqm ð3as Þ
1
2 cos ðqj a þ 2π=3Þ0where a and as denote the nearest-neighbor lattice spacingsin the directions of increasing j and m, respectively. (Even
if m denotes a “synthetic dimension” the notion of length asremains a useful bookkeeping device, by which 3as is theside of the expanded unit cell.) The three eigenvalues of thisRev. Mod. Phys., Vol. 91, No. 1, January–March 2019

ð47Þmatrix define three separate bands. Figure 9(c) shows theresulting band structure, where each of the three bands isendowed with a nonzero Chern number. The expansion of
the unit cell to contain three sublattice sites is essential forthe formation of topological bands. Recall that Chern

015005-11

N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

(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 theconventional honeycomb geometry. We also show its deformation to a “brick-wall” geometry most relevant for its
experimental realization with cold atoms. The tunnelingmatrix elements (black solid lines) with strength J define
the underlying honeycomb lattice, along the three nearestneighbor bonds. As indicated, even the simple honeycomb
lattice describes a dimerized lattice with jAi and jBi sublatticesites (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 pairof Dirac points. Haldane’s addition of next-nearest-neighbortunneling with strength J 0 and phase ϕAB [pink dashed lines inFig. 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 theunderlying honeycomb lattice contribute the matrix

(c)

0
B
Hq;0 ¼ @

Δ
−J

P

−J

P

expð−iq · ρi Þiexpðiq · ρi Þ

−Δ

1
C
A;

ð48Þ

i

and the next-nearest-neighbor links contribute a second term0P
B i
Hq;1 ¼ −2J 0 @cosðq · Ai − ϕAB Þ
0

1

0
P

cosðq · Ai þ ϕAB Þ

C
A:

i

ð49Þ

FIG. 9. The Harper-Hofstadter model. (a) Harper-Hofstadterlattice geometry with symmetric hopping J and a flux ϕAB in eachplaquette. (b) Harper-Hofstadter lattice geometry with flux
ϕAB ¼ 2π=3 per plaquette. The individual magnetic unit cells
are delineated by gray dashed lines with a representative
magnetic unit cell set off in blue (dark shading) for clarity.
(c) Computed band structure with ϕAB ¼ 2π=3 showing the three
topological bands with Chern numbers þ1, −2, and þ1 built
from the three inequivalent sites within the magnetic unit cell.

numbers are derived from the integrated Berry curvature
over the Brillouin zone. For a tight-binding model, theBerry curvature can only be nonzero when each Blochwave function has a spin or pseudospin degree of freedom,here provided by the sublattice degree of freedom.
In Sec. III.D we describe how to experimentally imprint these
hopping phases using tailored laser fields and will commenton the limitations of different experimental approaches.
2. Haldane model

The Haldane model (Haldane, 1988), an extension of thewell-known honeycomb lattice, was an early model ofRev. Mod. Phys., Vol. 91, No. 1, January–March 2019

We have defined A1 ¼ a1 , A2 ¼ −a2 , A3 ¼ a2 − a1 , with thevectors ρi and ai labeling nearest-neighbor and next-nearestneighbor separations as indicated in Fig. 3(a). (For practicalrealizations of the brick-wall lattice, the next-nearest-neighborcoupling, along A3 , is suppressed compared to couplings
along A1;2 . However, we use equal strengths for all in Fig. 10.)Figure 10(b) plots the topological phase diagram associatedwith this model as a function of the Aharonov-Bohm tunneling phases ϕAB and tilt 2Δ. This system supports three distinct
topological regions: zones with Chern number 1, with the
majority of parameter space in the topologically trivial phasewith Chern number 0.
The Haldane model is particularly amenable to experimental study because tuning experimental parameters such as ϕAB
can directly drive topological phase transitions. While for theHarper-Hofstadter lattice, tuning ϕAB does lead to different
Chern numbers, the size of the unit cell also changes, leading
to more dramatic changes in the band structure. In Sec. III.C
we show how strongly driving the parameters of a brick-walllattice can break time-reversal symmetry and imbue thelattice’s two bands with nontrivial topology.
In these examples of topological band structure, we
identified two common elements that experimentalists needto introduce to create nontrivial topology: complex-valued

015005-12

N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

(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 qx a ¼ q · a1 andqy a 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 ϕAB ¼ π=2 andp“tilt”
ﬃﬃﬃ
Δ ¼ 0, J 0 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 0 ,showing the formation of a single Dirac point.

tunneling matrix elements and unit cells with more than one
underlying lattice site or spin degree of freedom.
B. Realization of SSH model

The 1D SSH model of polyacetylene and its generalization
the Rice-Mele model are among the most simple topologicalmodels to realize. As described in Sec. II.E.2, the Rice-Melemodel, Eq. (40), consists of a bipartite 1D lattice withtunneling strengths alternating between J and J 0 , and energiesof the two sublattice sites, staggered by Δ.
Figure 11(a) depicts a typical laser system required toapproximate this idealized model, similar to the experimental
realization of Atala et al. (2013). Here a conventional 1D
optical lattice with period λ0 =2 is generated by a pair ofcounterpropagating lasers each with wavelength λ0 . In thislattice, the tunneling is uniform with strength J0 and theenergy minima of the lattice sites are degenerate, as indicatedby the pale blue dotted curve in Fig. 11(b). A second, weaker,
lattice with period λ0 , generated by a laser with wavelengthRev. Mod. Phys., Vol. 91, No. 1, January–March 2019

2λ0 [pink dashed curve in Fig. 11(b)], gives a combined
potential [black solid curve in Fig. 11(b)] with generally
staggered energy minima and alternating tunneling. This gives
the overall potentialVðxÞ ¼ V short sin2 ðkR xÞ þ V long sin2 ½ðkR x þ ϕÞ=2;

ð50Þ

where we defined the single-photon recoil momentum ℏkR ¼2πℏ=λ0 , the associated recoil energy ER ¼ ℏ2 k2R =2m, and theatomic mass m of the atom under study.
Figure 11(c) shows the Zak phase ϕZak , Eq. (14), computedfor this physical system in terms of the experimental control
parameters. This figure depicts the singularity expected when
V long ¼ 0, which at ϕ ¼ π=2 corresponds to the location of
the topological transition in the SSH model when J ¼ J 0 .
Figure 12 shows the bottom part of the infinite band
spectrum for V short ¼ 6ER and V long ¼ 1ER , with the relative
phase set to ϕ ¼ π=2. The right part of this figure shows a

015005-13

N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

(a)

(b)

(c)

FIG. 12. Bipartite lattice band structure compared to the Rice-

FIG. 11. Implementation of the SSH and Rice-Mele models.

(a) Representative laser configuration: A pair of overlapping
lasers with wavelengths λ0 and 2λ0 subject a cloud of ultracold
atoms to 1D optical lattices, with periods λ0 =2 and λ0 ,respectively, and a spatial relative phase ϕ. (b) Energies ofthese two lattices and combined potential, showing the longperiod lattice shifted in position with respect to the short period
lattice by a controllable phase shift ϕ. The SSH model isrealized for ϕ ¼ π=2 þ nπ (for integer n), in which case allminima have the same energy and are separated by potential
barriers with staggered height. An example of band structure inthis case is shown in Fig. 12. The choices ϕ ¼ nπ (for integer n)lead to equal barrier heights between adjacent sites, andstaggered site energies, but with J0 ¼ J, Δ ≠ 0. The displayed
data are for an intermediate case of ϕ ¼ π=4 where J 0 ≠ J andΔ ≠ 0. (c) Zak phase for the lower pair of bands controlled bytuning the phase shift ϕ between the long and short period
lattices and the strength of the long-period lattice, as represented by the unit vector ðcos ϕZak ; sin ϕZak Þ.

zoom on the lowest pair of bands, which are the only relevant
ones when the temperature and the interaction energies arecomparable to or lower than ER . The result for V long ¼ 0 is
indicated for comparison with dotted lines. The BZ is reducedin size by a factor of 2 as compared to that of the short period
lattice only. This is reflected by the V long ¼ 0 bands touchingat the edge of the BZ; in 1D this marks each such linked pair as
truly being one band in a doubled BZ.
Then the additional independent control of both J, J 0 , and Δ
requires two additional experimental degrees of freedom. In
this realization these parameters are the relative phase ϕ of thelong and short period lattices (displacing one lattice withrespect to the other) and the depth of the long period lattice
V long . Figure 12 has been calculated for ϕ ¼ π=2, in whichcase all minima of VðxÞ have the same energy while thebarrier heights between adjacent minima alternate between
two values. This realizes the SSH model and the dashed lines
in the right panel of Fig. 12 show a fit of the SSH prediction(20) to the two lowest bands, providing thus the relevant
values of J and J 0 .
Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019

Mele model. Left: Band structure computed for the superlatticepotential given by Eq. (50) with V short ¼ 6ER , V long 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 (continuouslines). The energy offset between adjacent minima and a fit to theprediction (41) for the band structure of the Rice-Mele model(dashed lines) allow one to extract the practical values of Δ, J,
and J 0 . (Here Δ ¼ 0, J ¼ 0.069ER , and J 0 ¼ 0.037ER .) The
dotted lines show the (folded) lowest band for V long ¼ 0, up to a
global energy shift.

Model parameters: The V short ¼ 6ER short period lattice
depth used in these simulations, a typical laboratory scale, sets
the nominal tunneling strength of J ≈ 0.05ER . Intuitively, we
expect that the energy difference between the minima to beabout V long cosðϕÞ. Similarly, the barriers between the sites
differ in height by roughly V long sinðϕÞ. The tunneling strength
in the effective SSH model has a nontrivial, but monotonicallydecreasing, exponential-type behavior in the barrier height.
This then begs the question of obtaining the parameters of the
Rice-Mele model in Eq. (40), including the two tunnelingstrengths J and J 0 along with the energy difference Δ between
sublattice sites.
First, recall that the band structure of simple 1D opticallattice potential Vsin2 ðkR xÞ only approaches that of a tightbinding model with nearest-neighbor tunneling when V ≫ ER .
For nearest-neighbor tunneling strength J, the resulting
dispersion is simply −2J cosðπq=kR Þ. As a result the effective
nearest-neighbor tunneling strength can be directly obtained
from the lowest term in a Fourier expansion of the band
structure of the physical 1D optical lattice. [The higher terms
in the series describe longer range tunneling, which becomes
negligible for deep lattices; see Jiménez-García andSpielman (2013) for an introduction.] This approach is insufficient for the Rice-Mele band structure ½Δ2 þ δJ 2 þ
ð4J̄ 2 − δJ 2 Þcos2 ðπq=kR Þ1=2 , because fits to this dispersion
alone cannot effectively disentangle Δ from δJ ¼ J − J0 , nor
δJ from J̄ ¼ ðJ þ J 0 Þ=2. One practical resolution to thisdifficulty is to employ symmetry and evaluate the band
structure of the bipartite lattice for two cases: first computethe band structure for ϕ ¼ 0 where J and J0 are manifestly
equal, and then compute the band structure for ϕ ¼ π=2 where
Δ ¼ 0. This then allows for the independent determination ofΔ, J, and J0 . For example, for V short ¼ 6ER and V long ¼ ER this015005-14

N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms(a)procedure gives J̄ ¼ 0.053ER , almost independent of ϕ. For
ϕ ¼ 0, we further find Δ ¼ 0.43ER and δJ ¼ 0, while for
ϕ ¼ π=2, this becomes Δ ¼ 0 and δJ ¼ 0.032ER .

(b)

C. Inertial forces

The common experience of starting water in a pail spinningby moving the bucket in a circular manner, not rotating thebucket, suggests that applied inertial forces might produce
effects akin to those present in rotating systems: described bestby effective Lorentz forces. We see later how these ideas are
implemented for ultracold atoms in optical lattices and alsocome 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 importancethat the tunneling matrix elements J are real valued (morespecifically, transformations between different gauges canintroduce “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 thissection we develop a simple model illustrating how inertial
forces (linear potential gradients or equivalently spatiallyshaking 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 theunitary evolution of a time-periodic Hamiltonian which canbe cast as complex hopping amplitudes in an effective timeindependent Hamiltonian. This physics is minimally capturedby the tunnel-coupled pair of lattices sites shown in Fig. 13,
essentially comprising a single unit cell of the Rice-Melemodel. We aim for a two-site model described by the
HamiltonianĤ ¼ −JðjrihljeiϕP þ jlihrje−iϕP Þ þ Δðjrihrj − jlihljÞ
¼ −J cosðϕP Þσ̂ x þ J sinðϕP Þσ̂ y þ Δσ̂ z ;(d)FIG. 13. Inertial forces. (a) A simple double-well lattice subject

to modulation, creating (b) experimentally tunable hopping
phases. (c) Shaking or tilting in 1D gives rise to a uniform
Peierls phase factor that shifts the minima of the tight-binding
band structure (Struck et al., 2012). (d) Shaking in 2D can breaktime-reversal symmetry giving rise to topological lattices.

ð51Þ

including a laboratory-controllable tunneling phase ϕP ,alas, our lattice is born with ϕP ¼ 0. In the second line we
expressed this Hamiltonian in terms of the Pauli operatorsσ̂ x;y;z , allowing us to follow a simple analysis of a spin-1=2system (Haroche et al., 1970).
We earlier noted that gauge transformations can introduce
complex-valued tunneling phases. A gauge transformation is
simply a position-dependent unitary transformation that
adjusts the local phase of the wave function and compensates
the Hamiltonian accordingly; in our double-well model, agauge transformation in the spatial picture becomes a σ̂ z rotation in the spin picture. Evidently, the Peierls phase factor ϕP
can be fixed to a nonzero value by the choice of gauge. Sincesuch a choice is of no physical consequence, it is instead theability to change ϕP (either spatially to induce AharonovBohm fluxes or temporally to induce artificial electric fields)that is the essential content of this discussion. Here we adopt
the most straightforward gauge choice that sets ϕP ¼ 0 for
unadulterated lattices.
In the following, we show how a time-periodic linear
gradient provides control over both the tunneling amplitude
Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019(c)

J and phase ϕP . In our two-site model, this modulation (i.e.,
detuning) is described by ΔðtÞ ¼ Δðt þ TÞ, with period T,
angular frequency ω ¼ 2π=T, and with zero per-cycle average hΔðtÞiT ¼ 0.
It is straightforward to eliminate the time-dependent ΔðtÞterm in Eq. (51) (initially, with ϕP ¼ 0) by making the unitarytransformationjψ 0 ðtÞi ¼ exp


 Z
i t
Δðt0 Þσ̂ z dt0 jψðtÞi;
ℏ 0

ð52Þ

in the language of quantum optics this is akin to the transformation into the time-dependent “interaction” picture. Since
this transformation is a σ̂ z rotation it is equivalent to a timedependent gauge transformation, leading to a nonzero, timedependent Peierls phase factor
ϕP ðtÞ ¼

2
ℏ

Z t
0

Δðt0 Þdt0 :ð53ÞWhen the modulation frequency’s associated energy ℏω isgreatly in excess of the tunneling J, we make a rotating wave

015005-15

N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

approximation to replace the time-dependent terms introducedby this rotation by their time averages, giving the timeaveraged interaction picture HamiltonianĤ ¼ −Jhcos ϕP ðtÞiT σ̂ x þ Jhsin ϕP ðtÞiT σ̂ y ;ð54Þ

with a potentially nonzero dc Peierls phase factor
tanðϕP;dc Þ ¼

hsin ϕP ðtÞiT
:hcos ϕP ðtÞiT

ð55Þ

Physically, the time-dependent gauge transformation in
Eq. (52) allows the system to sample a range of Peierls phasefactors and retain a nonzero average. In effect, our task is tomake hsin ϕP ðtÞiT nonzero, and because sin is an odd function
we seek a waveform ϕP ðtÞ that takes on positive and negativevalues in an “imbalanced” manner.
The most simple example to deploy in the laboratory is amonochromatic sinusoidal modulation ΔðtÞ ¼ Δ cosðωtÞ
of the tilt. In this case, the Bessel series expansion gives
hcos ϕP ðtÞiT ¼ J 0 ð2Δ=ℏωÞ and hsin ϕP ðtÞiT ¼ 0. Because
the integrated sinusoidal waveform takes on positive andnegative values with equal frequency the average tunnelingphase is zero; however, this modulation does renormalize thetunneling strength J → J × J 0 ð2Δ=ℏωÞ as was observed
experimentally (Lignier et al., 2007). While monochromatic
sinusoidal modulation is simple to deploy, it can obscure theunderlying physics, rapidly becoming a tangle of Bessel
functions.
Instead consider a waveform consisting of two deltafunction “kicks” in each drive cycle (Sørensen, Demler,and Lukin, 2005; Anderson, Spielman, and Juzeliūnas,2013), the first with strength Δ=ω at time t ¼ 0 and thesecond with strength −Δ=ω at time f × T, a fraction f throughthe drive period T. Integrating this waveform gives the timedependent Peierls phase factor
2Δ
ϕP ðtÞ ¼
×
ℏω



1−f

for 0 ≤ t ≤ fT;−f

for fT < t ≤ T;

ð56Þa pulse-width modulated waveform with zero average, withduty cycle f ∈ ½0; 1Þ. For f ∉ f0; 1=2g, the resulting asymmetric and skewed waveform leads to a nonzero averageof hsin ϕP ðtÞiT .
For the special case Δ=ℏω ¼ π, the time-averaged Peierls
phase factor is ϕP;dc ¼ −2πðf − 1=2Þ with tunnelingstrength unchanged at J. The basic physical picture is thatafter an atom tunnels between sites it acquires a phasedifferent from what it would have acquired on its initial site,and ϕP;dc expresses the differential phase acquired upon
returning to its initial site. The time-dependent phase ϕP ðtÞ
must break time-reversal symmetry to give a nonzero
average of sin ϕP ðtÞ. Figure 13(c) depicts the first experimental realization of a nonzero Peierls phase factor
imprinted using inertial forces (Struck et al., 2013).
Rather than tilting the lattice potential, Struck et al.
(2013) found it more convenient to spatially shake thelattice potential by modulating the phase of the lasers
creating the optical standing wave, giving the potential
Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019Vðx̂; tÞ ¼ V 0 cos½2kR (x̂ − δxðtÞ)=2. Here V 0 is the lattice
depth and kR is the two-photon recoil momentum from thewavelength λ of the lasers creating the lattice potential.
Although this shaking process is physically quite different from applying a time-dependent gradient, they arefunctionally equivalent. The connection between the two
can be seen clearly in terms of a pair of time-dependenttransformations. We begin by using the spatial displacement
operator D̂x ½−δxðtÞ ¼ exp½ik̂δxðtÞ to transform to the noninertial frame comoving with the lattice, i.e., D̂x ½−δxðtÞx̂
D̂x ½−δxðtÞ† ¼ x̂ þ δxðtÞ. This exchanges the lattice’s motion
for a new time-dependent contribution to the Hamiltonian−ℏk̂∂ t δxðtÞ. The once-transformed Hamiltonian

2
ℏ2
m
V
Ĥ ðtÞ ¼k̂ − ∂ t δxðtÞ þ 0 cosð2kR x̂Þ
2m
2
ℏ
m
2
− ½∂ t δxðtÞ
2
ð1Þ

ð57Þ

contains a new time-dependent vector potential and a
global time-dependent energy shift that does not
impact the system’s dynamics. Evoking Hamilton’s equation x_ ¼ ∂ ℏk Ĥ, we see that the appearance of this vector
potential simply describes the fact that in the moving frame
the velocity of an object differs from that in the lab frame by
the instantaneous velocity of the moving frame −∂ t δx.
We complete our argument using the timedependent momentum displacement operator D̂k ½−δkðtÞ ¼
exp½−iδkðtÞx̂, a gauge transformation, that converts the timedependent vector potential into a potential gradientV̂ ¼ ½m∂ 2t δxðtÞx̂. This reminds us that inertial forces arepresent in accelerating frames and informs us that experimenters are free to use either shaken lattices or potential
gradients equivalently to produce the inertial forces required
to induce ϕP .
While this nonzero and uniform Peierls phase factor is an
essential first step for emulating Aharonov-Bohm fluxes, a
uniform Peierls phase factor in 1D can be eliminated via agauge transformation (although any temporal change can stilllead to effective electric fields). In contrast by moving to 2D
systems such as in modulated and shaken honeycombgeometry lattices, this technique has nontrivial alterations toband structure (Struck et al., 2013) and including those topologically equivalent to the Haldane model (Jotzu et al., 2014).
We now extend our discussion to 2D to understandAharonov-Bohm fluxes. Figure 13(d) depicts a minimalmodel of shaking in 2D; the top panel illustrates the two
triangular plaquettes that make up a single unit cell of atriangular lattice, while the bottom panel graphs a shaking
protocol that first accelerates parallel to the A-B link of the left
plaquette (i.e., along ex ), then accelerates parallel to B-C, andfinally accelerates parallel to C-A. For each link, this protocol
leads to the same time-dependent Peierls phase factor given in
Eq. (53), with each phase shifted in time by 2π=3, giving thesame nonzero tunneling phase ϕP;dc to each side of theplaquette. This then leads to an overall Aharonov-Bohm
phase ΦAB ¼ ϕP;dc .
It might appear that our task is complete, but have we truly
created a uniform Aharonov-Bohm phase over all plaquettes?

015005-16

N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

Unfortunately Struck et al. (2013) demonstrated that this is not
the case. Following the same argument for the second (inverted)
plaquette that completes a single unit cell shows that accumulated phases give a negative flux ΦAB ¼ −ϕP;dc , leading to a
staggered flux. Therefore on average the Aharonov-Bohm flux
through this lattice is zero. One way to understand this is that
the Peierls phase factors are created with uniform amplitude
throughout the lattice, rather than with the linear dependence on
position as for our Landau-gauge example of the HarperHofstadter Hamiltonian. Sørensen, Demler, and Lukin (2005)
proposed remedying this by applying a potential whose
gradient itself increased away from the systems center, which
they termed a quadrupole potential.
Evidently shaking of this type does not provide a straightforward route for realizing Hamiltonians such as the HarperHofstadter model with uniform fields, but it has proven a
successful route for creating a Haldane-type Hamiltonian thathas a zero average flux but still with topological bands (Jotzu
et al., 2014). Important to this realization is the fact that theeffective Hamiltonian for the shaken lattice acquires a nextnearest-neighbor hopping with a nonzero Peierls phase factor.
Beyond the time-averaged Hamiltonian discussed, perturbative corrections from the time-varying part of the nearestneighbor tunneling lead to next-nearest-neighbor tunnelingof order J 0 ∼ J2 =ℏω, arising from a second-order process
through an intermediate virtual state detuned by ℏω. Suchterms are conveniently obtained from the Magnus expansionof this tight-binding model described in Appendix B. An
analysis of the effective model for the shaken lattice that goesbeyond this Magnus expansion approach was provided byModugno and Pettini (2017).

(a)

D. Resonant coupling: Laser-assisted tunneling

The previous section outlined the broad range of engineered
tunnel couplings possible via temporal modulation of theparameters of the lasers underlying the lattice potential. Whileit was possible to create complex-valued tunneling, it was notpossible to independently control the phase and amplitude oftunneling on each lattice link: more control is required.
Following the ideas of Jaksch and Zoller (2003), we describe
how such a fine-grained control is in principle possible using
laser-assisted tunneling and how experimental implementations have approached this task. As we shall see, although thislaser-assisted tunneling is effected by temporal modulation,
the modulation results from additional lasers, rather than thelasers from which the underlying lattice is assembled.
The essential concept of this technique is straightforwardly
illustrated in the 2D square lattice depicted in Fig. 14(a): thenative tunneling along the vertical direction is first eliminatedby applying a potential gradient (i.e., tilting the lattice), thencoupling between neighboring lattices sites is reestablishedwith a traveling wave potential. Here the spatially nonuniform
phase of the traveling wave is imprinted upon atoms as they
are moved from site to site described by complex-valued
tunneling amplitudes. Because local optical phases are relatively easy to control [for example, by creating higher-order
optical modes such as Laguerre-Gauss modes, as was done byChen et al. (2018)], these techniques in principle allow for
more subtle engineering of the local Aharonov-Bohm phases
than is possible with whole-scale modulation of lattice
parameters. Still, current implementations (Aidelsburger
et al., 2013; Miyake et al., 2013) rely only on the uniformly(b)FIG. 14. Laser-induced hopping. (a) 2D square lattice (right) with a potential gradient along em (vertical) illuminated by a travelingwave 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 withdetuning 2Δ coupled by the traveling wave. (b) Technique for creating the half-flux Harper-Hofstadter Hamiltonian in tilted spindependent lattices as implemented in MIT (Miyake et al., 2013) similar to Aidelsburger et al. (2013).

Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019015005-17

N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

changing phase from plane waves to generate homogeneous
fields.
The basic principle can be understood in terms of the same
sort of two-level system discussed in Sec. III.C, but from aperspective in which the rotating wave approximation (RWA)
is valid. In the present case, we focus on two neighboring
lattice sites in Fig. 14(a), labeled by jj; mi and jj; m þ 1i
coupled by the traveling wave potential V R ðxÞ ¼ V R sinð2kR ·x − ωtÞ that locally modulates the potential intersecting em
with angle θ. Physically this is directly realized (Aidelsburger
et al., 2013; Miyake et al., 2013) by a pair of interfering lasers
giving rise to a moving standing wave with periodicity λR =2,
and recoil wave vector jkR j ¼ 2π=λR (the wavelength λR
incorporates all geometric factors present from the intersection
angle between these lasers).
Midway between these two sites, at position
x0 ¼ aðj; m þ 1=2Þ, this potential isV R ðxÞ ≈V R ið2kR ·x0 −ωtÞfe
½1 þ 2ikR · ðx − x0 Þ − c:c:g;
2i

ð58Þ

to first order in position. In this expression (1) the first term
describes a modulated, but spatially uniform shift in thepotential with no physical consequence that may therefore
be neglected; and (2) the horizontal, ej dependence drops out
at this order because the localized wave functions, both
centered at aj, are symmetric and compact in square geometry
optical lattices. The remaining terms add a modulated contribution to the detuningΔðtÞ ≈ ½kR a cosðθÞV R cosð2kR · x0 − ωtÞ∶

ð59Þthe same sort of shaking potential we studied in Sec. III.C,
now with an overall phase dependent on the center position x0 .
Here we focus on the limit in which ℏω ¼ 2Δ ≫ J that leads
to the time-independent RWA Hamiltonian
ĤRWA ¼ −JRWA ½jj; m þ 1ihj; mjeiϕRWA ðm;jÞ þ H:c;ð60Þ

with double-well tunneling strengthVRk a cosðθÞ
JRWA ¼ −J
2Δ R

ð61Þ

ϕRWA ðm; jÞ ¼ 2kR a½j sin θ þ ðm þ 12Þ cos θ:

ð62Þ



and phase

Here the local phase of the traveling wave potential 2kR · x0 at
the double well is directly imprinted onto the atoms as theytunnel in the m direction, but not when they tunnel in the jdirection. The result of this double-well analysis can beextended to the whole lattice, where the expression for
ϕRWA is unchanged, and J RWA is qualitatively the same but
quantitatively altered.
Summing the tunneling phase around any plaquette gives
an Aharonov-Bohm flux
ΦAB ¼ 2kR a sin θ;
Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019ð63Þwith no spatial dependence. As a result, the RWA Hamiltonianis gauge equivalent to the Landau-gauge Harper-Hofstadter
HamiltonianX
H ¼ − J â†jþ1;m âj;m þ JRWA eijϕAB â†j;mþ1 âj;m þ H:c:: ð64Þj;mThis Hamiltonian was realized in the manner described by both
the Munich and the Massachusetts Institute for Technology(MIT) groups (Aidelsburger et al., 2013; Miyake et al., 2013),
illustrated in Fig. 14. The MIT group used a lattice derived froma 1064 nm laser, with a traveling wave generated by beams fromthe same laser intersecting the em axis at θ ¼ π=4. Thisgeometry gives a flux ΦAB =2π ¼ 1=2 per plaquette andillustrates an important practical point of this technique. In asimilar manner, but with a different laser geometry the Munich
group realized 1=4 flux per plaquette. In both cases, the laserinduced tunneling strength is proportional to cos θ, while the
Aharonov-Bohm phase is proportional to sin θ, requiring acompromise dependent on the experimental goals. Following
this initial experiment, the Munich group retooled theirtechnique as pictured in Fig. 14(c) by using the staggered
potential inside individual four-site plaquettes and laserinduced hopping to establish tunneling along all the latticedirections, enabling the measurement of the Chern number(Aidelsburger et al., 2015); see Sec. IV.C.2.
E. Synthetic dimensions

The concept of synthetic dimensions is rooted in the fact thata lattice is no more than a set of states labeled by integers, e.g., jand m in the preceding discussions labeled the atoms at sites
described by wave functions jj; mi. This insight allows thecreation of lattices that use the atoms’ internal or “spin” degrees
of freedom as additional synthetic dimensions. Boada et al.
(2012) and Celi et al. (2014) described how the techniques
discussed in Sec. III.D can be used to create a lattice with one
spatial dimension (denoted by j) and one synthetic dimension(denoted by m to evoke the atomic mF states from which it isbuilt). Large artificial magnetic fields using synthetic dimensions were simultaneously realized at the National Institute ofStandards and Technology (NIST) and the European
Laboratory for Nonlinear Spectroscopy (LENS) (Mancini et al.,2015; Stuhl et al., 2015) using hyperfine ground states ofbosonic 87Rb and fermionic 173Yb, respectively.
Both synthetic dimension experiments then replaced photon-assisted tunneling with two-photon Raman transitions.
Physically, these transitions simultaneously change the internal atomic state and impart the two-photon recoil momentum.
For a 1D optical lattice—essentially one chain along ej
of the 1D lattice in Fig. 14—the spatially imprinted phaseis ϕsyn ¼ 2kR aj sin θ. This expression is equivalent to Eq. (62)derived for photon-assisted tunneling, but without any
dependence on m since the Raman laser’s k vector is always“perpendicular” to the synthetic m direction, renderingcos θ → 0. This thereby eliminates the geometric compromise
required to maximize the laser-assisted tunneling strength atsimultaneously large flux.
Although synthetic dimension and photon-assisted tunneling experiments can produce the same sort of magnetic
lattice geometrics, the techniques have important practical

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N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

differences. For example, spin selective measurements allow
the synthetic dimension lattice site to be measured with nearperfect “spatial” resolution. In addition the limited numberof spin states (typically 3 to 5) produce synthetic dimension
lattices with striplike geometrics with perfect hard-wallboundary conditions, rather than extended planes as forconventional 2D optical lattices. In addition synthetic dimension lattice sites with the same spatial index j but withdifferent internal index m are in reality spatially overlapping,so that the spatially local atom-atom interactions become
anisotropic: long ranged in m and short ranged in j.
Both experimental synthetic dimension realizations [87Rb
and 173Yb in Mancini et al. (2015) and Stuhl et al. (2015),
respectively] created large flux, highly elongated strips alongj, with just three sites in width along m. More recently two-leg
ladder implementations of the synthetic dimension concept
have been realized (Livi et al., 2016; Kolkowitz et al., 2017)using the optical clock transition of both 173Yb and 87Sr, andsynthetic dimensions have even been constructed using
momentum states in lieu of spin states (Meier, An, andGadway, 2016).
F. Flux lattices: Intrinsic topology

Each of the topological lattices discussed in the preceding
sections was engineered beginning with a nontopologicallattice to which modulation or light-assisted tunneling wasadded to engineer a desired topological model. In this section
we focus on a different approach for generating a topologicallattice without the need for modulation or light-assisted
tunneling. This approach applies to an atom with several
internal states and subjected to a combination of three lightmatter interactions: those that are independent of internal
atomic states, those that depend on internal atomic states, andthose that couple between the different internal atomic states.
In particular, as we show in Appendix C, the far-detuned lightmatter interaction for alkali atoms takes the form
Ĥ RWA ¼ UðrÞ1̂ þ κðrÞ · F̂;ð65Þ

where UðrÞ and κðrÞ · F̂ describe the rank-0 (scalar) and rank1 (vector) light shifts acting on an atom with internal angularmomentum operator F̂. The possible lattices formed from
Eq. (65) have a very rich range of structures, characterized by
the spatial variations of UðrÞ and the three components ofκðrÞ. Because this approach is not tied to a preexisting tightbinding lattice, it is not limited to deep optical lattices and canhave topological properties even for very shallow or weak
optical coupling strengths.
We discuss the essence of this approach in two ways: (1) we
explore “flux lattices” in which the Aharonov-Bohm flux
emulated by a Berry phase in real space leads to lattices with
nonzero effective magnetic fields, and (2) we explore theconnection between spin-dependent lattices and establishedtopological models. Indeed in our initial discussion of the
Haldane model, we identified the two sublattice sites in each
unit cell with a pseudospin degree of freedom and arrived at aspin-dependent band structure, Eq. (48); we now make thisliteral.
Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019

1. Flux lattices

Before turning to our discussion of flux lattices, we pause
to reflect on our discussion of topological lattices to thispoint. Section II introduced the concept of topologicalinvariants in terms of the Berry connection (Zak phase)or Berry curvature (Chern number) integrated over the BZ.
In the latter case, the Chern number can be cast as a
momentum-space statement of Gauss’s law, in which the
Berry curvature integrated over the toroidal BZ counts an
integer number of topological “charges” in the inside of thetorus yielding the Chern number.
A “flux lattice” is an optical lattice potential that instead isdefined as a lattice in which the integrated Berry curvature ineach spatial unit cell is nonzero, suggesting that the atoms
might behave as if large magnetic fields are present. This
provides an intuitive framework in which to link the physics ofLandau levels and lattice bands. Indeed, in general, the atom
experiences a combination of a periodic magnetic field (withnonzero average) and a periodic scalar potential. Figure 15(b)depicts a bichromatic laser configuration that gives the same κused in the initial flux lattice proposals (Cooper, 2011; Cooperand Dalibard, 2011), with four frequency degenerate in-planelasers and with a single down-going laser at a different
frequency. As shown by Juzeliūnas and Spielman (2012),
this geometry can be tuned to produce the desired effective
magnetic field with vector strength
  
  
πx
πy
κ ¼ κ⊥ cos
ex þ coseaa y   
πx
πy
sin
e;
þ κ k sin
a
a z

ð66Þ

where κ⊥ and κk are set by the intensity and polarization of thelaser fields, and a ¼ λ is equal to the laser wavelength λ. This
result follows directly from the expressions in Appendix C.
The right panel of Fig. 15(b) shows the spatial distribution of
the Berry curvature in a single unit cell with spatial extent a,with a clear non-negative mean, evaluated for κ ⊥ ¼ κk. Thus,
it achieves the goal of having a net nonzero effective magnetic
field piercing the unit cell. In both the square geometry and asimilar three-beam setup with 2π=3 intersection angle, theresulting band structure can have topological bands (Cooper,
2011; Cooper and Dalibard, 2011).
This discussion hides one subtle point: the integrated
curvature over each unit cell must be a multiple of 2π. In
the present case the integrated curvature is 8πmF over thecomplete unit cell. The circles in Fig. 15(b) mark the locationsof the minima of the adiabatic potential ℏmF jκj. For a deep
lattice, in which this potential is large compared to the recoil
energy, the atoms are strongly confined close to these minima.
Their locations define effective lattice sites of a tight-binding
description, with the unit cell containing four such sites andtherefore divided into four plaquettes. For a spin-1=2 system,with mF ¼ 1=2, each of these plaquettes will have a flux
Φ ¼ π, while for bosonic alkali atoms, mF takes on integer
values implying Φ ¼ 2πmF is a multiple of 2π. In either case,nearest-neighbor hopping on this square lattice geometry
would not break time-reversal symmetry, so the Berry

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N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

(a)

(b)

(c)

(d)

FIG. 15. Spin-dependent topological lattices. (a) Level diagram for three-level total angular-momentum f ¼ 1 case with mF stateslabeled, as is applicable for the common alkali atoms 7Li, 23Na, 39K, 41K, and 87Rb. For reference, the diagram shows the decompositionof these optical fields into σ  and π, but as discussed in the text, this is not an overly useful way of considering this problem. (b) Lasergeometry 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 topologicalband structure to emerge in a straightforward way, longer
range hopping as in the Haldane model is required. These
considerations indicate that, for this square geometry, the flux
lattice would not lead to topological bands in the deep-latticelimit. However, we reiterate that the flux lattice approach is
not restricted to deep lattices, but applies also for shallower
lattices in which the atoms can move throughout the unit celland a restriction to nearest-neighbor hopping is inappropriate.
We demonstrated that flux lattices can give rise to a net
nonzero magnetic field piercing the real-space unit cell. The
comparison with free particles in a uniform magnetic field,
which form Landau levels, suggests the appearance of
topological Chern bands. However, this is not guaranteed:such lattices may or may not be topological, as defined by theusual Chern number computed in momentum space. (Thedeep-lattice limit of the square flux lattice described above
provides an example in which the bands are not topologicaldespite the nonzero effective magnetic flux.) Moreover, as we
now discuss, there can be cases in which the net flux throughthe unit cell vanishes, yet the bands are topological (Cooperand Moessner, 2012). In general, to determine the band
topology requires a full calculation of the band structure,including the atom’s kinetic energy.
Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019

2. Fluxless lattices

A key insight from the Haldane model is that a net magnetic
field is not a prerequisite for topological band structure. It issufficient to break time-reversal symmetry, a condition thatcan readily be achieved for lattices that couple internal states.
The appearance of topological bands for zero net flux isreadily established for shallow lattices (Cooper and Moessner,2012). Here we focus on the regime of deep lattices, where aneffective (spin-dependent) tight-binding model can be developed. As we made explicit in our discussion of the Haldane
model, the two basis sites consisting of a single unit cell can be
assigned a pseudospin label, and these pseudospins are thenarrayed to form a honeycomb or brick-wall lattice. Building
from this understanding, we conclude by discussing the
topology of state-dependent lattices, without any explicit
reference to Berry phases.
Figure 15(c) displays the phase-stable lattice geometry
realized by Sun et al. (2017), which is closely related to
the flux lattice geometry, absent the vertical beam and with thein-plane beams driving Raman transitions. A predecessorof this setup described by Wu et al. (2016) generated a 2D
optical lattice with spin-orbit coupling with 87Rb atoms [a firstrealization of 2D spin-orbit coupling was reported by Huang
et al. (2016) for a bulk geometry]. The two pseudospin states

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N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

are the Zeeman levels jF ¼ 1; m ¼ −1i and jF ¼ 1; m ¼ 0iof the lowest hyperfine state of the ground state manifold.
The essential concept of this lattice is to first use the scalar
light shift from the detuned, counterpropagating beams tocreate a conventional 2D optical lattice operating in thetight-binding regime, and then to use the vector contributionof the light shift to give a combination of local effective
magnetic fields and spin-dependent tunneling. The resulting
tight-binding model is very closely related to that of the
Haldane model and shares its topological properties. The left
panel of Fig. 15(d) shows the experimentally measured band
structure in good agreement with the predictions of theory
displayed in the right panel.
IV. EXPERIMENTAL CONSEQUENCES

This section presents recent experimental investigationsof nontrivial (global or local) topological properties of energy
bands, in either 1D or 2D geometries. Interactions play anonessential role for the experiments described next; hence
phenomena addressed here correspond to single-particle (orideal gas) physics.
This section is divided into three parts. In the first one we
describe measurements that are performed on an atomic
system at equilibrium, using local probes in momentum spacethat allow one to reconstruct the topology of the occupiedband(s). In the second part we present analyses performed bylooking at the dynamics of wave packets. These wave packets
are well localized at the scale of the Brillouin zone and one canbring them close to some points of specific interest, Diracpoints, for example, using an external force. The last partis devoted to transport measurements, which are closer inspirit to the techniques that are commonly used in condensed
matter physics.
A. Characterization of equilibrium properties

discussed by Möller and Cooper (2010) and LeBlanc et al.
(2015) these differences arise because the synthetic vector
potential vanishes during TOF when the laser fields are
removed: there are physical differences in how exactly thesynthetic vector potential returns to zero when TOF begins
and physically different effective electric fields present during
this turn-off. All physical observables remain invariant to local
gauge transformations, provided these are applied consistently, i.e., changing the vector potential both before andduring TOF.
2. Local measurement of the Berry curvature

We consider here a cold atomic gas at equilibrium in a 2D
optical lattice. With the so-called band-mapping technique
one can precisely measure the distribution N ðqÞ of thequasimomentum q for each energy band. With this techniquethe laser beams forming the lattice are turned off in acontrolled manner, so that the populations of the Bloch states
forming the various energy bands in the presence of the lattice
are transferred to states with a well-defined momentum in theabsence of the lattice (Greiner et al., 2001). The measurementof N ðqÞ in combination with a suitable temporal variation of
the lattice parameters before the complete turn-off allows one
to characterize the band topology, i.e., to access not only theenergies, but also the Berry curvature ΩðqÞ.
To illustrate this point we consider a 2D lattice and use
the same two-band model as in Sec. II, assuming a unit cellwith two nonequivalent sites labeled A and B. The generic
Hamiltonian in reciprocal space is Ĥq ¼ h0 ðqÞ1̂ − hðqÞ · σ̂where ðh0 ; hÞ is a 4-vector with real components that areperiodic over the BZ. The energies of the two bands are E
q ¼
h0 ðqÞ  jhðqÞj and the Bloch states ju
icanbewrittenasqlinear combinations of jq; Ai and jq; Bi. Using the expression(30) for these Bloch states, one findsΩ ðqÞ ¼ ∇q ðhuq jÞ × ∇q ðjuq iÞ

1. Time-of-flight measurements

Before entering into a discussion of specific measurements,
we briefly comment on implications of the time-of-flight(TOF) measurements commonly used in experiment. In thevast majority of cold atom experiments, the measurement
procedure begins with the rapid removal of all applied fields(both those involved in trapping the ensemble and thoserequired to create the topological lattice of interest). After
this abrupt, projective, turn-off, the atoms then undergo aperiod of ballistic expansion followed by a measurement of
their density (often in a spin-resolved manner). In many
experiments, this procedure gives a direct measurement of
the momentum distribution as it was when the applied fields
were just removed.
While this sounds simple in principle, this procedure canappear to give measurement results that are gauge dependent.
Fortunately any supposed contradiction with general principles of local gauge invariance is illusory. For example, twodifferent experiments might well create a Harper-Hofstadterlattice with the same flux using very different laser geometries,
which naturally define the synthetic vector potential in two
different gauges. The observed momentum distribution will in
general differ in these two cases (Kennedy et al., 2015). As
Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019

¼ 12∇q ðcos θq Þ × ∇q ϕq ;ð67Þwhere ðθq ; ϕq Þ defines the direction of hðqÞ in sphericalcoordinates.
A procedure to determine θq and ϕq , hence the curvature Ω,in such a two-band situation was proposed by Hauke,
Lewenstein, and Eckardt (2014) and implemented byFläschner et al. (2016). The starting point is the momentum
distribution N q ðkÞ associated with a Bloch stateð−Þψ q ðrÞ ¼

X X
s¼A;B Rs

αq;s wðr − Rs Þeiq·Rs :ð68Þ

Here wðrÞ is the Wannier function of the band, supposed to be
identical for the two sublattices s ¼ A, B, and ðαq;A ; αq;B Þ ¼( cosðθq =2Þ; eiϕq sinðθq =2Þ). The momentum distributionð−Þ

N q ðkÞ is the square of the Fourier transform of ψ q ðrÞ.
For k inside the first BZ, it is peaked around k ¼ q and givenby jw̃ðkÞj2 jαk;A þ αk;B j2, where w̃ðkÞ is the Fourier transformof wðrÞ. When the lowest band is uniformly filled withindependent fermions, the momentum distribution of the

015005-21

N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

gas is obtained by summing the contributions N q ðkÞ over all
quasimomenta q of the BZ:
N ðkÞ ¼ jw̃ðkÞj2 ½1 − sin θk cos ϕk :

ð69ÞThis 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 tothe 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. Moreprecisely, although not sufficient to determine unambiguously
the angles θ and ϕ, this measurement already provides thevalue of the product sin θ cos ϕ. To go one step further, Hauke,
Lewenstein, and Eckardt (2014) suggested to apply an abrupt
quench to the lattice parameters so that the Hamiltonian
becomes Ĥ 0q ¼ ðℏω0 =2Þσ̂ z . One then lets the gas evolve in thelattice during a time interval t before measuring the momentum distribution. Since the evolution during this time simply
consists of adding the phase ω0 t=2 to αq;s , the momentum
distribution at time t reads
N ðk; tÞ ¼ jw̃ðkÞj2 ½1 − sin θk cosðϕk þ ω0 tÞ:ð70Þ

By repeating this procedure for various times t and measuring
the amplitude and the phase of the time-oscillating signal, one
can determine simultaneously ϕ and θ at any point in the BZ.
This procedure was implemented by Fläschner et al. (2016)using a hexagonal lattice of tubes filled with fermionic 40K
atoms. The unit cell for this graphenelike geometry contains
two sites and the lattice parameters are chosen such that there
is initially a large energy offset ℏωAB between the A and B
sites, corresponding to essentially flat bands with no tunneling. As explained in Sec. III.D, the dynamics in the lattice canbe restored by a resonant, circular shaking of the lattice at afrequency Ω ≈ ωAB. In the experiment of Fläschner et al.
(2016), the shaking was produced by a phase modulation of
the three laser beams forming the lattice, and it also resulted ina non-negligible value for the Berry curvature. Once the atoms
equilibrated in this lattice, the abrupt quench needed for theprocedure of Hauke, Lewenstein, and Eckardt (2014) was
obtained by simply switching off the modulation. Measured
amplitudes and phases of the time oscillation of N ðk; tÞ areplotted in Fig. 16, together with the Berry curvature reconstructed from Eq. (67). One then expects the integral of Ωq to
be an integer C times 2π, where C is the Chern index of the
populated band. Here the reconstructed Berry curvature leads
to a value of C compatible with 0 (Fläschner et al., 2016). This
is in agreement with the expected band topology in this case.
The method outlined by Hauke, Lewenstein, and Eckardt(2014) is reminiscent of a previous proposal by Alba et al.
(2011). There the two lattice sites A and B are supposed to beoccupied by two different internal (pseudospin) states jai andjbi. The momentum distribution measurement can be done ina spin-resolved way, which provides the local spin polarization for the lowest band [see Eq. (30)]:hz ðqÞ
N ðqÞ − N a ðqÞ:¼ cos θq ¼ bjhðqÞjN a ðqÞ þ N b ðqÞ
Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019

ð71Þ

FIG. 16. (Left column) Amplitude ∝ sin θq and (middle col-umn) phase ϕq obtained from the fits to the oscillation (70) of the
momentum distribution. From those fit results, one can reconstruct the (right column) momentum-resolved Berry curvature
given in units of the inverse reciprocal lattice vector length jbj
squared. From Fläschner et al., 2016.

The other two components of h=jhj can be obtained byinducing a coherent transition between jai and jbi (Raman
pulse) with an adjustable phase and duration in order to rotate
the pseudospin during the time of flight. Once the direction ofhðqÞ is known at all points of the Brillouin zone, the value ofthe Berry curvature follows from Eq. (67).
3. Topological bands and spin-orbit coupling

When the considered lattice has some specific geometricalsymmetries, the assessment of the topological nature of a band
can be notably simplified with respect to the procedure
outlined above. One does not need to characterize theeigenstates of the Hamiltonian at all points of the Brillouinzone to determine if the integral of the Berry curvature (67)over this zone is nonzero, and it is sufficient to concentrate onsome highly symmetric points. The basis of this simplification, which is discussed in Appendix A for the case of a 1D
lattice [see, in particular, Eq. (A6)], was outlined by Liu et al.
(2013) for a 2D square optical lattice for pseudospin 1=2
particles in the presence of spin-orbit coupling terms.
Let us briefly outline the main result of Liu et al. (2013).
Suppose that the Hamiltonian is invariant under the combined
action of the spin operator σ̂ z and the spatial operator transforming a Bravais lattice vector R into −R. Consider theenergy eigenstates (Bloch states) at the four points of the BZ:
fΛi g ¼ fð0; 0Þ; ð0; πÞ; ðπ; 0Þ; ðπ; πÞg, i ¼ 1; …; 4. The two
Bloch states jψ  ðΛi Þi at each of these locations are also
eigenstates of σ̂ z , and the corresponding eigenvalues ξi; cantake only the values þ1
Q or −1. Now one can show that the signof the product Pη ¼ i ξi;η of the four ξi;η of a given subbandη ¼  is directly related to the Chern number of this subband.
More specifically, if the sign of Pη is negative, the Chern
number of the subband η is odd, hence nonzero: thisunambiguously signals a topological band. If Pη has a positive
sign, the Chern number of the subband η is even, and this most
likely signals a nontopological character for this subband(zero Chern number).
This procedure was realized experimentally by the typeof lattice we described in Sec. III.F.1 involving two pairs ofretroreflected Raman lasers intersecting at right angles. Thisconfiguration can produce the reciprocal-space tight-binding
Hamiltonian with the following generic form (29):015005-22

N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

ð72Þ

where mz is proportional to the detuning from Raman
resonance. The two amplitudes t0 and tSO characterize thetunneling amplitudes without and with spin flip, respectively.
They can be controlled independently by varying the intensities of the two pairs of laser beams. In particular the term
proportional to tSO can be simplified in the limit of low
momenta into ∝ ky σ̂ x þ kx σ̂ y , corresponding to the usual formof the Rashba-Dresselhaus spin-orbit coupling for a bulk
material (Galitski and Spielman, 2013).
An interesting feature of this Hamiltonian is the possibility
to control the topology of the lowest band: It is nontrivial if
and only if jmz j < 4t0 . Wu et al. (2016) tested this prediction
by loading the 87Rb gas at a temperature T ∼ 100 nK such thatthe lowest band was quasiuniformly filled, whereas the
population of higher bands remains small. The polarization
defined in Eq. (71) was measured by a spin-resolved imagingof the atomic cloud after time of flight, and the product P− forthe lowest band was found to be negative in the expected
range of values of mz .
4. Momentum distribution and edges states

It is well known from integer quantum Hall physics
that the nontrivial topology of a band can give rise to aquantized Hall conductance σ xy . Applying a voltage difference V y between the two opposite edges of a rectangular 2D
sample gives rise to a global current I x ¼ σ xy V y along the x
direction. This current flows on the edges of the sample in aðþÞchiral way, with a positive value I x on the y > 0 side ofð−Þ
the sample and a negative value I x on the y < 0 side
(Hatsugai, 1993). In the absence of an applied voltage(V y ¼ 0), the edge currents are still nonzero but they exactly
ð−Þ

ðþÞ

compensate each other: I x ¼ −I x .
The possibility to engineer a 2D atomic gas with one real
dimension and one synthetic dimension that we discussed inSec. IV.B.4 offers a way to directly access these edge currents
in a cold atom experiment. Indeed when the second dimension(labeled y) is synthetic, i.e., associated with an internal degreeof freedom (pseudospin), one expects a given sign of I x for thelargest value of the pseudospin and the opposite sign for its
smallest value.
The advantage of a synthetic dimension for observing theseedge states is clear: It provides a sharp boundary to the sample,
whereas a standard 2D optical lattice would lead to edge statesthat would be smeared over several lattice sites, and hencemuch more difficult to observe. In addition, the momentum
distribution can be measured individually for each pseudospin
value. One thus obtains the value of the current on each “site”
of the synthetic direction. The transposition of such ameasurement to a real direction y implies a single-site
resolving imaging, which is much more demanding froman experimental point of view. However, the use of spin states
for synthetic dimensions greatly limits the potential extent of
the synthetic dimension: in the extreme limit of just two spinstates, the system can even behave as if it were all edge with noRev. Mod. Phys., Vol. 91, No. 1, January–March 2019

m=-5/2n(k)þ ½mz − 2t0 ( cosðaqx Þ þ cosðaqy Þ)σ̂ z ;bulk (Hügel and Paredes, 2014). For narrow systems withsynthetic extent less than q for flux p=q, the states behave
more like those in a continuum [with a single guiding center,see the supplementary material in Stuhl et al. (2015)], and as
the width further increases, the familiar topological edge
modes emerge, but remain slightly gapped at the edge of the
1D Brillouin zone from the bulk bands.
In addition, recent experiments have instead turned to using
momentum states for a synthetic dimension, in principle
allowing for more extended systems. Indeed the edge states
associated with the 1D SSH model have already been
observed using a momentum-space lattice (Meier, An, andGadway, 2016).
We show in Fig. 17 the results of an experiment performedby Mancini et al. (2015). A gas of fermionic 173Yb atoms cantunnel between the sites of an optical lattice along the x
direction, with an essentially frozen motion along the twoother (real) y and z directions. The synthetic direction consistsof three Zeeman substates m ¼ −5=2, −1=2, þ3=2 selectedamong the six Zeeman states of the ground level. The“tunneling” along this synthetic direction is provided by apair of light beams. These beams induce stimulated Ramanprocesses between the Zeeman states, hence the desired laserinduced hopping. These beams also provide an artificial gauge
field thanks to the space-dependent phase φðxÞ printed on
the atomic state in a Δm ¼ 2 transition.
The atoms are prepared in a metallic state (less than one
atom/lattice site) in the lowest band of the single-particle
Hamiltonian. Figure 17 shows the momentum distributions
nm ðkÞ along the x direction for the three values m of thepseudospin. Here the central value of the pseudospin(m ¼ −1=2) plays the role of the bulk of the material. The
momentum distribution in this internal state is thus symmetricaround 0, corresponding to a null net current. In contrast, anonzero current is associated with the side values of thepseudospin: The distribution for the largest (smallest) value ofm is displaced toward positive (respectively, negative) values.
The displacement is made even clearer in the second row of
m=-1/2m=+3/2

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0

0
-3

-2

-1

0

1

2

3

0
-3

-2

-1

k
0.2

h(k)

Ĥ q ¼ 2tSO ½sinðaqy Þσ̂ x þ sinðaqx Þσ̂ y 

0

1

2

3

-3

0.2

+0.079(6)

0.1

0

0

-0.1

-0.1

-1

0

k

1

2

3

-0.2

1

2

3

–0.062(4)

0.1

0

-2

0

0.2

+0.018(5)

0.1

-3

-1

k

-0.1
-0.2

-2

k

-3

-2

-1

0

k

1

2

3

-0.2

-3

-2

-1

0

1

2

3

k

FIG. 17. Direct visualization of an edge-state current for a two-

dimension (one real, one synthetic) lattice in the presence of an
artificial magnetic field. Upper row: Momentum distributions
nm ðkÞ along the real direction x for the three values of thepseudospin m corresponding to the synthetic direction. The valuem ¼ −1=2 can be viewed as the bulk, whereas m ¼ þ3=2 andm ¼ −5=2 corresponding to the opposite edges of the sample
along the synthetic dimension. Lower row: Function hm ðkÞ ¼
nm ðkÞ − nm ð−kÞ. From Mancini et al., 2015.

015005-23

N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

Fig. 17 where the function hm ðkÞ ¼ nm ðkÞ − nm ð−kÞ isplotted. Mancini et al. (2015) also checked that the sign ofthe edge current is reversed when the sign of the artificial
magnetic field is changed.
B. Wave-packet analysis of the BZ topology

Cold atom experiments offer the possibility to prepare theparticles in a state described by a wave packet that is well
localized in momentum space, in comparison with the size of
the Brillouin zone. In this case, the dynamics of the wave
packet directly reveals the local properties in the BZ: energy
landscape EðqÞ and Berry curvature ΩðqÞ. These properties
can be encoded either on the dynamics of the center of the
wave packet or on the interference pattern that occurs whenseveral paths can be simultaneously followed.
1. Bloch oscillations and Zak phase in 1D

Conceptually the simplest example of a wave-packet
analysis of the band topology is found in a 1D lattice of
period a, for which the BZ extends between q ¼ −π=a and
π=a. When an atom initially prepared with the quasimomentum q0 is submitted to an additional uniform force F, its
quasimomentum qðtÞ periodically spans the Brillouin zone
(Bloch oscillation) at a frequency aF=h: qðtÞ ¼ q0 þ Ft=ℏ
(mod2π=a). The phase that is accumulated in this periodic
motion contains relevant information about the topology ofthe energy bands.
Since Bloch oscillations play a key role in several instancesin the following, we briefly outline the principle of their
theoretical description. The Hamiltonian of the particle in thepresence of the periodic lattice potential VðxÞ and of the force
F reads Ĥ ¼ p̂2 =2m þ Vðx̂Þ − Fx̂. Let us write the initialðnÞðnÞstate of the particle ψðx; 0Þ ¼ eixq0 uq0 ðxÞ, where uq0 ðxÞ is theperiodic part of the Bloch function associated with the nthband and the quasimomentum q0 . One can look for a solutionof the time-dependent Schrödinger equation under the usual
Bloch form ψðx; tÞ ¼ eixqðtÞ uðx; tÞ. The evolution of u isgoverned by the periodic Hamiltonian ĤqðtÞ ¼ ½p̂ þ ℏqðtÞ2 =2m þ Vðx̂Þ; therefore the solution uðx; tÞ remains spatiallyperiodic at any time. If we add the assumption that the force
F is weak enough so that interband transitions play anegligible role, the state of the particle will adiabaticallyfollow the corresponding Bloch state of the nth band, i.e.,ðnÞuðx; tÞ ¼ eiφðtÞ uqðtÞ ðxÞ, and the relevant information is
encoded in the phase φðtÞ. Let us focus on the state of theparticle 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).
The first measurement of the Zak phase in a cold atom
context was performed by Atala et al. (2013). A 1D superlattice was made out of two standing waves with a long and ashort period [see Sebby-Strabley et al. (2006) and Trotzky
et al. (2008) for two methods for creating double-wellsuperlattice potentials], generating the potential (50) where
both the relative phase ϕ and the amplitudes V long;short arecontrol parameters. As detailed in Sec. III.B, this constitutes aRev. Mod. Phys., Vol. 91, No. 1, January–March 2019

realization of the Hamiltonian proposed by Rice and Mele(1982), reducing to the SSH model (Su, Schrieffer, andHeeger, 1979) for ϕ ¼ π=2. Here we restrict for simplicity
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
discussed in Sec. II.C.1, the Zak phase itself is not invariant
under gauge transformations in momentum space. Indeed it
depends on the choice of the relative phases between Wannier
states, i.e., the arbitrariness in deciding if the unit cell isformed by a pair fAj ; Bj g [N A ¼ N B in Eq. (24)] or by a pairfBj−1 ; Aj g [N A ¼ N B þ 1 in Eq. (24)]. However, once thisðD Þchoice is made, the gauge-invariant quantity δϕZak ≡ ϕZak1 −ðD Þ
ϕZak2 ¼ π, where D1 and D2 correspond to the two possible
dimerizations of the system, obtained by choosing either
ϕ ¼ 0 or ϕ ¼ π.
In order to measure δϕZak , Atala et al. (2013) first prepared
the atoms in a wave packet localized at the bottom of thelowest band. Then using a π=2 microwave pulse the atoms
were placed in a superposition of two spin states j↑i and j↓i,which underwent Bloch oscillations in opposite directions in
the presence of a magnetic gradient (Fig. 18). At the moment
when the two wave packets reach the edges of the BZ, the Zakphase is encoded in the relative phase between these wave
packets. In principle it could thus be read using a second π=2
microwave pulse, closing the interferometer in quasimomentum space. However, the magnetic field fluctuations in the lab
cause a random dephasing between the two arms of theinterferometer and prevent one from performing this direct
measurement. To circumvent this problem, Atala et al. (2013)used a spin echo technique. A second microwave pulse flipped
the spins as the atoms reached the edges of the BZ and at thesame moment, the dimerization was changed by switching ϕfrom 0 to π, corresponding to an exchange of states between
the lower and the upper bands. Finally the two paths wererecombined when the wave packets reached the top of theupper band, and the accumulated phase revealed the value ofδϕZak . The experimental result δϕZak =π ¼ 0.97ð2Þ was inexcellent agreement with the expected value.
2. Measurement of the anomalous velocity

We now turn to the case of a 2D lattice and weinvestigate how the semiclassical dynamics of a wave packet

FIG. 18. Two-band spectrum of the SSH Hamiltonian and Zakphase measurement. An initial wave packet is prepared at thebottom of the lowest band (left column). Each atom is in the
superposition of the ji spin states, which experience opposite
forces in the presence of a magnetic field gradient. After a
combination of exchanges of the atomic internal state and of the
dimerization (middle column), the two wave packets recombine
at the top of the highest band (right column). The measurement of
the accumulated phase provides the value of the Zak phase. From
Atala et al., 2013.

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.
The starting point is the set of equations that govern theevolution of the average quasimomentum q and average
position r when a constant force F is superimposed to thelattice potentialℏ_q ¼ F;ð73Þℏ_r ¼ ∇q EðqÞ þ ΩðqÞ × F:

ð74Þ

(a)

(b)

This set of equations is valid when the applied force F isweak enough, so that transitions to other bands can beneglected. With the first equation we recover the Blochoscillation phenomenon: The momentum drifts linearly intime in response to the applied force F. The second equation
provides the value of the velocity of the wave packet at agiven 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 aperiodic potential]. The second contribution, which is sometimes called the “anomalous velocity” (Xiao, Chang, andNiu, 2010), couples the wave-packet dynamics to the local
value of the Berry curvature ΩðqÞ. As a result of thiscontribution, the recording of a given trajectory inside theBrillouin zone allows one to reconstruct the Berry curvature
along this trajectory (Price and Cooper, 2012).
This method was implemented by Jotzu et al. (2014) inorder to analyze the topology of a 2D optical lattice in thevicinity of Dirac points. The experiment was performed with abrick-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
only nearest-neighbor couplings A → B and B → A are taken
into account, the spectrum consists of two bands touching at
two Dirac points Qþ and Q− in the BZ [Fig. 19(a)]. As
explained in Sec. III.C, an additional circular shaking of thelattice breaks the time-reversal symmetry of the system andallows one to lift the degeneracy at the Dirac points, with thetwo subbands acquiring a nontrivial topology. Another possibility to lift this degeneracy consists of simply introducing
an energy offset between sites A and B. However, in this caseeach subband is topologically trivial (Sec. II.C.2). With thissetup one can thus explore the phase diagram of the Haldane
model shown in Fig. 10(b).
Using an analysis of wave-packet dynamics close to thepoints Q , Jotzu et al. (2014) studied the transition between
the topological and trivial cases. Starting from a wave packet
at the center of the BZ, they dragged it close to Qþ or Q−
using the force F created by a magnetic gradient. The
curvature of the motion of the wave packet in the BZ
revealed the sign of the anomalous velocity, hence of the
Berry curvature. In the trivial case, ΩðQþ Þ and ΩðQ− Þ have
opposite signs, hence the Chern number which is proportional to the integral of ΩðqÞ over the BZ is zero [Fig. 19(b)].
In the topologically nontrivial case, ΩðQ Þ have the same
sign [Fig. 19(c)]. The measurements of the drift of the wave
packet performed by Jotzu et al. (2014) quantitatively
confirmed this scenario.
Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019(c)

FIG. 19. (a) Brillouin zone and Dirac points Q for the brick-wall lattice of Fig. 10. (b) When the degeneracy is lifted byintroducing an energy offset between the A and B sites, thesubbands are topologically trivial. For the lowest band, the Berrycurvature has opposite signs in the vicinity of Q . The anomalous
velocity (white arrows) thus has opposite chirality at these points.
(c) Lifting of degeneracy obtained by adding complex nextnearest-neighbor (NNN) couplings (Haldane, 1988). The Berrycurvature then keeps a constant sign over the BZ, and theanomalous velocity has the same chirality at both points. Adaptedfrom Jotzu et al., 2014.
3. Interferometry in the BZ

We now come back to the simple case of a graphenelike
lattice with only nearest-neighbor (NN) couplings, in whichcase the two subbands touch at two Dirac points Q . In this case
it is not possible to calculate the Berry curvature Ω for eachsubband because the definition (67) is singular in Q . The
situation can be viewed as an equivalent (for momentum space)of an infinitely narrow solenoid (in real space) providing a finite
magnetic flux. In the latter case it is known that the presence ofthe solenoid can be probed by interferometric means. This isindeed a paradigmatic example for the Aharonov-Bohm
effect: Using a two-path interferometer such that the solenoid
passes through its enclosed area, there exists between the twopaths a phase difference proportional to the magnetic flux. In
the case of a two-path interferometer enclosing a Dirac pointin momentum space, the phase difference between the twopaths is π (Mikitik and Sharlai, 1999).
This phase shift was measured in the cold atom context by
Duca et al. (2015). The graphenelike structure was generatedusing an optical lattice with three beams at 120° angles.
Initially the external state of the 87Rb atoms is a wave packet
located at the center of the BZ, and their internal state is agiven Zeeman state j↑i. A microwave π=2 pulse prepares acoherent superposition of j↑i and j↓i, where the magnetic
moment of j↓i is opposite to that of j↑i. Then the displacements of the two corresponding wave packets are controlledusing simultaneously: (i) a lattice acceleration along y which

015005-25

N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

FIG. 20. Aharonov-Bohm interferometer around a Dirac point. Starting from a wave packet localized at the center of the BZ of agraphenelike optical lattice, one measures the phase difference φ between the two paths of an interferometer ending at thequasimomentum (kx ¼ 0, ky ). When one of the Dirac points K or K 0 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) amagnetic gradient which creates opposite forces on themalong x; and (iii) a π microwave pulse which exchanges theatomic spins in the middle of the trajectories, in order to close
the interferometer in momentum space. The result obtainedby Duca et al. (2015), shown in Fig. 20, shows that a phasedifference of ∼π between the two paths appears if and only if
the enclosed area contains one of the two Dirac points. Using asimilar technique Li et al. (2016) could also access the Wilsonline regime, which generalizes the Berry phase concept tothe case where the state of the system belongs to a (quasi)
degenerate manifold. Li et al. (2016) investigated the casewhere the transport from one place to another in the BZ isdone in a time much shorter than the inverse of the frequency
width of the bands. In that experiment the transport wascharacterized by a single phase factor and could be analyzed
within the framework of Stückelberg interferometry (Lim,Fuchs, and Montambaux, 2014, 2015). However, in more
complex situations, interferometry can also reveal nonAbelian features of the transport (Alexandradinata, Dai, andAndrei Bernevig, 2014).
4. Direct imaging of edge magnetoplasmons

Although the synthetic-dimension systems can be describedby the same Harper-Hofstadter Hamiltonian as with real-spacelaser-assisted tunneling approaches, using the spin degree of
freedom to encode one spatial direction enables new preparation, control, and measurement opportunities. For example,
the finite number of spin states in effect generates infinitely
sharp hard-wall boundaries in the synthetic dimension; thesynthetic-dimension tunneling can be applied and removed onany experimental time scale; atoms can be initially prepared in
any initial synthetic-dimension site; and conventional time-offlight measurements, along with Stern-Gerlach techniques,
Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019

allow the near-perfect resolution of “position” in the spin
direction.
Recent synthetic-dimension experiments (Mancini et al.,2015; Stuhl et al., 2015) used these techniques to directly
image the evolution of edge magnetoplasmons. In a quantum
Hall system, an edge magnetoplasmon is the quantum analog
to the skipping cyclotron orbits which “bounce” down theedge of a system in a chiral manner, essentially following
cyclotron orbits that are interrupted by the system’s edge; see
Fig. 4. These dynamical excitations evolve with a characteristic frequency given by the cyclotron frequency and aresuperposition states between different Landau levels. Theseshould not be confused with the chiral edge modes that
underlie quantized conductance: these modes are built fromstates all within the same Landau level. These modes have notbeen observed in two-dimensional experiments; however,their analog has been observed in 1D experiments, also usingsynthetic dimensions, where the end modes of 1D systems
have been observed (Meier, An, and Gadway, 2016).
In condensed matter systems, edge magnetoplasmonscertainly have been launched and detected, both in steadystate, in magnetic focusing experiments (van Houten et al.,1989), and directly in the time domain using electrostatic gates(Ashoori et al., 1992). Cold atom experiments complete thepicture by allowing for direct space and time resolution ofthese skipping orbits.
In these experiments (Mancini et al., 2015; Stuhl et al.,2015), the system was initialized with no hopping along thesynthetic dimension, and with all atoms in either one or theother edge along the synthetic dimension. Once this initialstate was prepared, tunneling was instantly turned on. The
highly localized initial state, described by a superposition ofdifferent Landau levels, then began to evolve, skipping down
the system’s edge. Following a tunable period of evolution
time-of-flight measurements directly imaged the position

015005-26

N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

FIG. 21. Edge-magnetoplasmon trajectories, where the dis-placement 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 shownin 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
band, one can perform a transportlike experiment, measure thedisplacement of the whole cloud in the lattice after a certain
duration, and infer nontrivial topological aspects from thisdynamics (Dauphin and Goldman, 2013).
1. Adiabatic pumping

The concept of a quantized pump introduced by Thouless(1983) has been described in general terms in Sec. II.F. It
requires a lattice whose shape is controlled by at least two
effective parameters, such as Δ and J 0 − J for the Rice-Melemodel [see Eq. (40)]. One starts with a gas that uniformly fills
a band of the lattice. Then one slowly modifies the lattice
shape in a way that corresponds to a closed loop in parameter
space. As shown in Appendix A, the resulting displacement of
the center of mass of the gas is then quantized in units of thelattice spacing.
This concept can be addressed in a 1D geometry, using a
superlattice with the potential (50). As explained in Sec. III.B,
the motion of atoms in the two lowest bands of this potential
for a deep enough lattice can be described by the Rice-Mele
Hamiltonian. Suppose that the relative phase ϕ of the longperiod lattice is scanned from 0 to 2π, while the lattice depths
V long and V short are kept fixed; see Fig. 22. Initially when
ϕ ¼ 0, the energy of an A site is below that of a B site, i.e.,
Δ < 0 in the framework of the Rice-Mele model. At thismoment the superlattice is symmetric, hence the tunnel matrix
elements J 0 and J from Bj to Aj and Ajþ1 are equal. When0 < ϕ < π, the energy barrier between Aj and Bj is smaller
than that between Bj and Ajþ1 , hence J0 > J. During this time
period, Δ increases and changes sign when ϕ ¼ π=2. During
1

Because TOF measurements yield the momentum distribution,some effort is required to derive the velocity in the lattice from thisinformation.
Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019

FIG. 22. Topological pumping with a 1D superlattice describedby the potential (50). Left: The phase ϕ of the long lattice [seeEq. (50)] is varied from 0 to 2π from top to bottom. Initially thephase ϕ ¼ 0 and the particle is supposed to be localized on
the site Aj of a given lattice cell j. In the limiting case where theenergy difference between A and B sites is large compared to thetunnel matrix elements, this state would be stationary if ϕ was
kept at the value 0. When ϕ is increased up to π=2, the sites Aj
and Bj have the same energy and the particle is adiabaticallytransferred to Bj . Note that we neglect here the tunneling of theparticle from Aj to Bj−1 , assuming that it is inhibited by the large
barrier between these two sites. The particle then remains in Bj
until the phase reaches the value 3π=2, when the particle again
undergoes an adiabatic transfer, now from Bj to Ajþ1 . (Here again
we neglect tunneling across the large barrier now present between
Bj and Aj .) When the phase ϕ ¼ 2π the potential is back to its
initial value and the particle has moved by one lattice site. Notethat a motion in the opposite direction occurs if the particle starts
for the site Bj when ϕ ¼ 0. Right: In the two-band approximation
corresponding to the Rice-Mele model, the system performs aclosed loop around the origin in the parameter space ðJ 0 − J; ΔÞ.

the second half of the cycle π < ϕ < 2π, one now finds J 0 < J
and Δ again changes sign. When ϕ ¼ 2π, the system has
performed a closed loop in the parameter space ðJ 0 − J; ΔÞ,encircling the vortex localized in J 0 − J ¼ 0, Δ ¼ 0. One cantherefore conclude that the center of mass of the cloud has
moved by one lattice cell. In the limiting case of a particle
initially localized in one of the lattice sites, one can recover
this result by a simple reasoning based on the adiabatic
following of the instantaneous energy levels; see Fig. 22.
A Thouless pump was implemented in cold atom setups(Lohse et al., 2016; Lu et al., 2016; Nakajima et al., 2016).
The experiment by Nakajima et al. (2016) used a superlatticepotential similar to that represented in Fig. 22, with V long ¼
30ER and V short ¼ 20ER , respectively, with ER ¼ ℏ2 =ð8ma2 Þ.

015005-27

N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms(b)
Differential shift 2x(t)/a

(a)

(c)

1

4

2
no flux
2

0
0

(d)

50

100

150

200

BO time (ms)FIG. 24. Transport measurement in a square lattice with a flux or(e)FIG. 23. Quantization of the displacement of a cloud offermionic 171Yb atoms placed in a 1D optical superlatticedescribed by the Rice-Mele Hamiltonian. Depending on theclosed loop in parameter space ðJ 0 − J ≡ 2δ; ΔÞ, the displacement can be positive, zero, or negative. The time T represents theduration of a pump cycle. From Nakajima et al., 2016.

Using a gas of noninteracting 171Yb atoms (fermions),
Nakajima et al. (2016) verified that the displacement of thecloud when ϕ varies from 0 to 2π is equal to one lattice period,
as expected; see Fig. 23. They also checked that it is
topologically robust, i.e., it does not change if one slightly
deforms the path in parameter space by adding a time variation
modulation of V long and V short . However, if the modification issuch that the closed trajectory in parameter space ðJ 0 − J; ΔÞdoes not encircle the origin point anymore, the displacement
per pump cycle drops to zero.
2. Center-of-mass dynamics in 2D

As a last example of a cold atom probe of band topology, let
us briefly describe the analysis of the dynamics associatedwith the Harper-Hofstadter Hamiltonian (46) by Aidelsburger
et al. (2015). This Hamiltonian was implemented using thelaser-induced hopping method explained in Sec. III.D, producing the flux per plaquette ϕAB ¼ π=2. As a result of thisapplied gauge field, the lowest band of the lattice was split infour subbands, with Chern numbers ðþ1; −1; −1; þ1Þ, withthe two intermediate subbands touching at Dirac points.
Bosonic 87Rb atoms were loaded in majority in the lowest
subband of the lattice, at a temperature such that this subband
was filled quasiuniformly. Then a weak uniform force along
the y axis originating from a gradient of the light intensity ofan auxiliary beam was applied to the atoms for an adjustable
duration t. The position of the center of mass of the atom cloud
was monitored as a function of time and it revealed the
topology of the bands. At short times (typically less than
50 ms), the drift of the cloud along the x direction, similar to a
Hall current, was found to be linear with t; see Fig. 24.
Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019plaquette ϕAB ¼ π=2 (data 1). A Hall-type current is observed
through the displacement of the center of mass of the cloud along
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 theflux is zero, no Hall current is observed (data 2). Adapted from
Aidelsburger et al., 2015.

It provided a measurement of the anomalous velocity in good
agreement with the expected value for the Chern number ofthe lowest subband. For longer times t, heating originatingfrom the resonant modulation applied to the lattice induced
transitions between the various subbands, which eventually
get equally populated. Since the sum of all four Chern
numbers is zero, the drift of the center of mass then stopped.
A careful analysis of this dynamics, associated with anindependent measurement of the population of each subband,provided a measurement of the individual Chern numbers witha 1% precision.
V. INTERACTION EFFECTS

Some of the most interesting directions for future work on
cold atoms in topological optical lattices involve studies ofcollective effects that arise from interparticle interactions.
Such studies hold promise for the exploration of novel phasesof matter and to elucidate the role of topology in strongly
correlated many-body systems.
A. Two-body interactions

The methods previously described for generating topological lattices consist either of periodic modulation ofsite energies, forming a so-called “Floquet” system (seeAppendix B), or of Raman coupling of internal spin states,
leading to optically “dressed states” of the atoms. Both ofthese methods lead to effective interactions between particles
in the resulting energy bands that have some novel features.
1. Beyond contact interactions

For ultracold atoms in the continuum, the typical two-bodyinteractions are dominated by the s-wave scattering, which
can be represented by a short-range (contact) interactiongδðr1 − r2 Þ, where the Dirac δ distribution is supposed to015005-28

N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

be properly regularized. However, for particles “dressed” by alaser field as in the optical lattices described, the interactions
typically acquire nonlocal character.
a. Continuum models

For the continuum setting of optical flux lattices ofSec. III.F with laser plane waves inducing Raman couplings
between internal spin states, nonlocal interactions can arise
from the momentum dependence of the dressed-state wave
functions. For the two-state system described by the
Hamiltonian (72), an energy eigenstate in a given band canbe written as

nonadiabatic corrections to the optical dressing, allowing
two particles to coincide in real space by being in different
internal dressed states, dependent on their momentum(Cooper and Dalibard, 2011). The momentum dependenceof the dressed-state wave functions converts the contact
interactions into a momentum-dependent interaction for atoms,
thereby allowing effective p-wave scattering even at ultralow
temperatures. Similarly, dressed-state bosons can acquire amomentum-dependent interaction allowing the appearance ofd-wave and higher angular-momentum channels in the scattering (Williams et al., 2012).
b. Tight-binding models

jΨq i ¼ jψ q i ⊗ jΣq i;

ð75Þ

i.e., the product of an orbital Bloch state jψ q i of quasimomentum q and of a q-dependent spin state jΣq i written in thebasis jiz . The dependence of the spin component jΣq i on awave vector has direct consequences on the two-body scattering matrix elements within this band.
Let us first recall the simple case of spinless or fullypolarized particles. For the contact interaction V̂ ¼ gδðr̂Þ, thematrix element of V̂ for a pair of free distinguishable particles
transitioning from ðq1 ; q2 Þ → ðq1 þ Q; q2 − QÞ is equal to g
(up to a normalization factor) and it is independent of the
momentum transfer Q. Here we neglect for simplicity energydependent corrections of the s-wave scattering amplitude,
which is valid when Q−1 is larger than the s-wave scattering
length, a criterion usually satisfied in quantum gases. For two
indistinguishable particles, physical observables involve thesum over the two permutations
ðq1 ; q2 Þ → ðq1 þ Q; q2 − QÞ orðq2 − Q; q1 þ QÞĤ0 ¼ −J½eiϕP ðtÞ jrihlj þ e−iϕP ðtÞ jlihrj;ð76Þ

with a relative sign of ε ¼ 1 for bosons or fermions. The
matrix element of V̂ is doubled for polarized bosons andvanishes for polarized fermions: the latter result simply
reflects the fact that single-component fermions are insensitive
to the contact interaction, since the Pauli principle precludes
them from having the same spatial position.
Consider now the scattering of atoms in the dressed-stateband with wave functions (75) and assume for simplicity thatthe contact interaction is independent of the internal statesji. For bosons and fermions (ε ¼ 1), the transition matrix
element of V̂ for the two-path process (76) is nowhψ q1 þQ ; ψ q2 −Q jV̂jψ q1 ; ψ q2 ihΣq1 þQ jΣq1 ihΣq2 −Q jΣq2 i
þ εhψ q2 −Q ; ψ q1 þQ jV̂jψ q1 ; ψ q2 ihΣq2 −Q jΣq1 ihΣq1 þQ jΣq2 i.
For fermions, this matrix element no longer vanishes ingeneral. For example, for Q ¼ 0, it is ∝ gð1 − jhΣq2 jΣq1 ij2 Þ.
This is an important result that shows that starting fromcontact-interacting fermions occupying a nondegenerate band,the optical dressing leads to an interacting one-component
Fermi gas, more precisely to effective p-wave interactions(Zhang et al., 2008). The possibility of interaction between
two identical fermions can be viewed as arising fromRev. Mod. Phys., Vol. 91, No. 1, January–March 2019

For tight-binding lattice systems, we described in Sec. III
how periodic driving at frequency ω can be used to tailor theamplitudes and phases of the hopping matrix elements
between the sites. We demonstrated how, at rapid driving
frequencies, an effective Hamiltonian with potentially complex modified tunneling amplitudes arises. This is the effective
Floquet Hamiltonian for the modulated system, which governs
the dynamics of the particles on time scales large compared tothe period of the modulation, T ¼ 2π=ω, as described inAppendix B. In the presence of interparticle interactions, e.g.,
Hubbard interaction U between particles on the same lattice
site, new terms appear in this effective Floquet Hamiltonian,
including nonlocal interactions (Eckardt, 2017). To understand the origin of the nonlocal interactions, consider themodulated two-site system described in Sec. III.C under aharmonic drive of the energy offset between the sitesΔðtÞ ¼ ðΔ=2Þ cos ωt. The unitary transformation Eq. (52)leads to a HamiltonianϕP ðtÞ ¼

ΔsinðωtÞ:
ℏω

ð77Þ
ð78Þ

It is convenient to expand this Hamiltonian in its harmonics
0Ĥ ¼ −J

∞
X
m¼−∞


Jm


Δ imωte jrihlj þ H:c
ℏω

ð79Þ

where J m ðzÞ are Bessel functions of the first kind. The m ¼ 0
term gives the time-averaged Hamiltonian discussed inSec. III: it describes intersite tunneling with an effective
tunneling rate that is modified from the bare rate J by
J 0 ðΔ=ℏωÞ. For ℏω large compared to all other energy scales(tunneling J and any on-site-interaction energy U) the higherorder terms jmj > 0 are far off resonant, so are rapidly
oscillating and have small effects on the wave function.
Still these terms do perturb the atomic wave function, leading
to the new features we are interested in here. For a particle thatis initially in the right well jri, a rapidly oscillating perturbation V̂ m eimωt (i.e., m ≠ 0) causes a first-order correction to thewave function of

015005-29

jψ 0 i ≈ jri þ jlihljV̂ m jrieimωt=2sinðmωt=2Þ
mℏω

ð80Þ

N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms≈jri − JJ m ðΔ=ℏωÞeimωt=2sinðmωt=2Þ
jli.
mℏω

ð81Þ

Thus, these terms induce a small nonzero probability for theparticle to be in the left well,  2
J2
Δ
Jm
;
pl ¼
2
ℏω
2ðmℏωÞ
m≠0
X

UJ 2
:
ðℏωÞ2

ð84Þ

i

ð83Þ

A similar nonlocal term will arise from interactions in the rightwell. A full analysis of such effects is best achieved through aconstruction of the Floquet Hamiltonian at high drive frequency via the Magnus expansion (Goldman and Dalibard,2014) or other systematic approach (Eckardt and Anisimovas,
2015). An overview of the Magnus expansion is given in
Appendix B.
c. Synthetic dimensions

An extreme example of nonlocal interactions arises for
systems involving synthetic dimensions (Celi et al., 2014;
Mancini et al., 2015; Stuhl et al., 2015). There interactions are
typically long ranged among the set of s ¼ 1, N s internal
states which form the synthetic dimension and short ranged in
the d spatial coordinates. For a synthetic dimension formed
from an internal spin degree of freedom, the interactions are of
infinite range in the synthetic dimension and of conventional
short range in position. For example spin-exchange interactions can be viewed as a correlated tunneling, such as
ðm; m0 Þ → ðm − 1; m0 þ 1Þ. Such systems realize interesting
intermediate situations in which the single-particle physics
can be viewed as d þ 1 dimensional (for N s is large) while theinteractions remain d dimensional. For a synthetic dimension
formed from simple harmonic oscillator subband states (Price,
Ozawa, and Goldman, 2017) the interactions fall off withincreasing spacing along the synthetic dimension. Interactionsin an effective two-leg ladder, formed from spin-orbit coupledstrontium atoms, have been studied experimentally byBromley et al. (2018).
d. Current-density coupling

Finally, we note that the novel two-body interactions thatare induced by Raman coupling or Floquet modulation canalso include terms that are not just density-density interactions, but that couple the particle motion to particle density.
These effects arise naturally in photon-assisted tunnelingbetween lattice sites as a consequence of interaction-induced
Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019X
½Jðn̂i ; n̂iþ1 Þb̂†i b̂iþ1 þ H:c:;ð82Þ

after averaging over the rapid oscillations and summing overall such terms. If in addition to the tunneling (78) the static
Hamiltonian has an on-site interaction between particles in theleft well, which we consider to be a Hubbard interaction
U n̂l ðn̂l − 1Þ=2 (with n̂l the number operator in the left well),then these rapidly oscillating terms that move particles fromthe right to the left well will give rise to effective nonlocal
interactions U eff
lr n̂l n̂r , withU eff
lr ¼ Upl ∝energy shifts. The imposed energy offset between two
adjacent lattice sites is modified by the on-site interactions
U in such a way that the photon-assisted tunneling can bebrought into or out of resonance depending on the occupationsof the sites by other particles. This leads to a densitydependent tunneling term of the formshown here for spinless bosons. Such situations wereanalyzed by Aidelsburger et al. (2015), Bermudez andPorras (2015), and Račiūnas et al. (2016). The densitydependent corrections are typically a small modification of
the photon-assisted hopping in current experimental setupsof gauge fields on optical lattices [see supplementary
information in Aidelsburger et al. (2015)]. However, they
can be made to dominate in regimes where U is large, suchthat the photon drive frequency ω is not resonant with thebare detuning Δ but with Δ  U. Such situations have been
explored in experiments on two-component Fermi systems(Görg et al., 2018; Xu et al., 2018). Analogous coupling
between current and density arises also in the continuum ifinteractions lead to detunings that influence the local
dressed states, thus causing the induced vector potential
A to be density dependent (Edmonds et al., 2013). Densitymediated hopping can also arise in settings where thesingle-particle energy bands are flat (dispersionless).
Then individual particles do not move, being localized toa region of size of the Wannier orbital, and particle motion
arises only through interparticle interactions. Such settings
include the flat energy bands of frustrated lattices, such as
the Creutz ladder (Creutz, 1999; Mazza et al., 2012;
Jünemann et al., 2017) in 1D, and the kagome (Huber
and Altman, 2010) and dice lattices (Möller and Cooper,2012) in 2D, as well as for particles in dressed states ofinternal spin states in which the direct nearest-neighbortunneling can be made to vanish (Bilitewski andCooper, 2016).
2. Floquet heating

The explicit time dependence of the Hamiltonian inperiodically driven systems relaxes energy conservation andleads to forms of inelastic scattering and heating not present in
the time-independent case. For a Floquet system at frequency
ω, these correspond to the absorption (or emission) of an
integer number of “photons” of energy ℏω from the external
drive. For the “Floquet-Bloch” waves of a spatial- and timeperiodic potential, such inelastic scattering could occur evenfor a single particle that scatters from a defect in the lattice,which allows a momentum transfer. However, an importantsource of potential inelastic scattering is for pairs of particles
via the interparticle interactions, i.e., inelastic two-bodycollisions.
A general description of the inelastic scattering of FloquetBloch waves was provided by Bilitewski and Cooper (2015b).
Consider a time-periodic Hamiltonian of frequency ω anddenote the single-particle energy band, with band index ν, bythe energies ϵν ðqÞ defined as continuous functions of q over

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 beviewed as one member of a sequence of Floquet energy bandsðmÞϵν ðqÞ ¼ ϵν ðqÞ þ mℏω (Eckardt, 2017). We define inelastic
scattering to be those processes in which m or ν, or both,change under a scattering event. For weak scattering fromstate i to state f, the inelastic rate can be computed through a“Floquet Fermi golden rule” (Kitagawa et al., 2011)
γ i→f ¼

2π X
ðmÞ
ð0Þ
j⟪Φf jV̂Φi ⟫j2 δðEi − Ef − mℏωÞ;
ℏ m
ðmÞ

ð85Þ

MBL” phase is the Floquet time crystal (Moessner andSondhi, 2017).
B. Many-body phases

The topological optical lattices described support an arrayof many-body phases. Novel features arise for weaklyinteracting gases through the geometrical and topologicalcharacters of the underlying band structure. Furthermore,
there can arise interesting strongly correlated phases, driven
by strong interparticle interactions.

ð0Þ

where
jΦf;i ðtÞi ≡ e−imωt jΦf;i ðtÞiand⟪Φ1 jΦ2 ⟫≡RT
ð1=TÞ 0 hΦ1 jΦ2 idt. Here V̂ can denote a one-body potential(e.g., a lattice defect) or a two-body interaction. Processes
with nonzero Δm correspond to the exchange of Δm quanta
from the drive field, changing the energy of the atom by mℏω.
This provides a simple prescription by which to calculate theinelastic (Δm ≠ 0) contribution to the two-body scatteringof Floquet-Bloch states. Stepping beyond the two-body
calculation to a many-body setting can be achieved byanalyzing instabilities within the Gross-Pitaevskii description(Choudhury and Mueller, 2014, 2015; Lellouch et al., 2017).
Calculations for the parameters used in the experimental
studies of weakly interacting bosons in the Harper-Hofstadtermodel (Aidelsburger et al., 2015) show that the heating rates
observed in those experiments are consistent with the
expected inelastic two-body scattering processes dominatedby single-photon absorption (Bilitewski and Cooper, 2015a).
This analysis emphasizes the role played by the motion
transverse to the 2D plane along the weakly confined third
dimension. The application of an optical lattice to confine thismotion and open up gaps at the one-photon resonance is
expected to significantly reduce the heating rate in such
experiments (Bilitewski and Cooper, 2015a; Choudhury
and Mueller, 2015). The excitation of motion along tubes,
transverse to the optical lattice, has been argued to also be
responsible for heating rates in an experimental study of aperiodically modulated 1D lattice (Reitter et al., 2017). In
strongly driven systems (Weinberg et al., 2015) particle
transfer to higher bands can arise from multiphoton resonances at the single-particle level (Sträter and Eckardt, 2016).
Floquet heating in the full, many-body system presents aninteresting theoretical issue which remains an active areaof 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, andMoessner, 2014). This expectation relies on the assumptionthat energy is redistributed between all degrees of freedom
through the interparticle interactions. There can, however,arise situations in which many-body systems show steadystates that are not at infinite temperature (D’Alessio andPolkovnikov, 2013; Chandran and Sondhi, 2016), or in which
there form prethermalized states on intermediate time scales(Bukov et al., 2015; Canovi, Kollar, and Eckstein, 2016).
Furthermore, in settings involving disorder, it has been shownthat many-body localized (MBL) phases can be robust to
Floquet modulations, allowing the existence of nonthermal
steady states to arbitrarily long times (Abanin, De Roeck, andHuveneers, 2016). A striking example of such a “Floquet
Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019

1. Bose-Einstein condensates

For an optical lattice loaded with a gas of noninteracting
bosons one expects the ground state to be a Bose-Einsteincondensate (BEC), in which all particles condense at theminimum of the lowest energy band. This expectation applies
just as well to the topological optical lattices as to regular
optical lattices. However, topological optical lattices can bringseveral novel features.
(1) For systems without time-reversal symmetry (as required
to generate a Chern band), the individual Bloch wave functions
have phase variations which in general give rise to nonzero local
current density. Since the Bloch wave functions are stationary
states, these currents must be divergenceless, but this still allows
the BEC to support circulating currents in dimensions d > 1.
These currents take the form of the local current density of avortex lattice. An example is shown in Fig. 25.
(2) For topological optical lattices in d > 1 the energy
minimum in which the BEC forms will, in general, becharacterized by a nonzero Berry curvature. As describedin Sec. V.C this Berry curvature affects the collective modes of
the BEC.
(3) There can arise situations in which there are a set ofdegenerate energy minima. An important example is provided
by the Harper-Hofstadter model, for which the energy band

FIG. 25. A vortex lattice configuration for weakly interacting

bosons in the Harper-Hofstadter model at flux ϕAB ¼ 2π=3. The
gauge-invariant currents flow on the links between the sites of thesquare lattice. The pattern of current breaks the translational
invariance of the system. The strongly circulating currents(shown as solid lines and arrows) are around plaquettes thatlie along diagonal lines. These are the lattice equivalents of the“vortex cores,” now distorted from the triangular lattice expectedin the continuum models, to be pinned to the plaquettes of thesquare lattice. There are weak countercirculating currents aroundother plaquettes, such that the net particle flow vanishes, whencoarse grained on scales large compared to the lattice spacing.

015005-31

N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

typically has multiple degenerate minima. For example, atflux ϕAB ¼ 2π=q the band has q degenerate minima in themagnetic Brillouin zone. For noninteracting particles, there isa macroscopic degeneracy associated with the occupation ofthese q degenerate single-particle states: a BEC could form in
any linear superposition of these states; or indeed a “fragmented” condensate could form (Mueller et al., 2006). The
inclusion of interactions U ≠ 0 is required to resolve thismacroscopic degeneracy. The simplest way to include interactions is via Gross-Pitaevskii mean-field theory. This is valid
for sufficiently weak interactions U ≪ J and at high mean
particle density n̄ ≫ 1. Studies of the mean-field ground state(in regimes of both weak and strong interactions, U n̄ ≪ J andU n̄ ≫ J) show that repulsive interactions stabilize a simple
BEC with the form of a vortex lattice (Straley and Barnett,
1993; Zhang et al., 2010; Powell et al., 2011). This state
breaks the underlying translational symmetry of the lattice, so
there are several discrete (symmetry-related) states that aredegenerate. For example, for flux ϕAB ¼ 2π=3 it has the formof vortex lines along diagonals of the lattice; see Fig. 25.
Numerical studies of the vortex lattice ground states for
U n̄ ≫ J have been conducted for a wide range of flux
ϕAB : these show complex behavior with many competing
vortex lattice structures (Straley and Barnett, 1993).
Similar vortex lattice states appear in ladder systems. In
experiments on weakly interacting bosons on a two-leg ladderwith flux, Atala et al. (2014) demonstrated a transition
between a uniform superfluid phase with Meissner-like chiralcurrents and a vortex phase (with broken translational symmetry along the ladder) as a function of the tunneling strength
across the rungs of the ladder, akin to the Meissner and vortex
lattice phases of a type-II superconductor.
2. Topological superfluids

In both the tight-binding lattices and the Raman-dressed flux
lattices, situations can arise in which there are nonlocal
interactions between fermions in a single band. This allows
attractive p-wave pairing and can lead to interesting forms of
topological superfluidity with “Majorana” excitations. Here we
discuss the general features of these topological superfluids.
Their properties are understood within mean-field theory, as
described by the BdG Hamiltonian. This takes the formĤ − μN̂ ¼

X
α;β



1
εαβ ĉ†α ĉβ þ ðΔαβ ĉ†α ĉ†β þ Δαβ ĉβ ĉα Þ ;
2

ð†Þ

ð87Þ
ð88Þ

where ðĉ; ĉ† ÞT is a 2N s -component column vector formedby listing all fermionic destruction ĉα¼1;…;N s and creationĉ†α¼1;…;N s operators. The quasiparticle excitations are determined by the spectrum of the 2N s × 2N s matrix HBdG
!
!
ðλÞ
uðλÞ
u
¼ HBdG
ð89Þ
Eλ
vðλÞ
vðλÞ
ðλÞ

in terms of the 2N s -component vector of amplitudes uα¼1;…;N sðλÞ

and vα¼1;…;N s , where λ labels the 2N s eigenvalues. The matrix
BdG Hamiltonian (88) has a special symmetry:
!
!
0 I
0 I
BdG
BdG 
¼−
½H 
;
ð90Þ
H
I 0
I 0
where 0 and I are the N s × N s null and identity matrices,
respectively. This has the consequence that for any eigenstateðuðλÞ ; vðλÞ ÞT with eigenvalue Eλ there is another eigenstate
!
!
vðλÞ
uðλ̄Þ
¼
ð91Þ
uðλÞ
vðλ̄Þ
with eigenvalue Eλ̄ ¼ −Eλ : i.e., the spectrum is symmetric inenergy around E ¼ 0. (Note that energies have been defined
relative to the chemical potential μ.) This intrinsic particlehole symmetry represents an inherent redundancy in thetheory, by which the eigenstates of HBdG must include both
the destruction and the creation operator of any quasiparticle
state (Bernevig and Hughes, 2013). Specifically, defining theoperator associated with each eigenvector λ ¼ 1; …; 2N s
X ðλÞ
ðλÞ
uα ĉα þ vα ĉ†α
ð92Þ
Ĉλ ¼
α

the Hamiltonian may be expressed in the diagonalized form
1 Xs
E Ĉ† Ĉ :2 λ¼1 λ λ λ
2N

ð86Þ

where ĉα are fermionic creation or annihilation operators
for the single-particle states labeled by α ¼ 1; …; N s, which
encodes both positional and internal (spin) degrees of freedom. They obey the fermionic anticommutation relations
fĉα ; ĉβ g ¼ fĉ†α ; ĉ†β g ¼ 0, fĉα ; ĉ†β g ¼ δαβ . The N s × N s matrix
ε describes the conventional particle motion and potentials
and must be Hermitian εαβ ¼ εβα ; the N s × N s matrix Δ
represents the superconducting pairing and is antisymmetric Δαβ ¼ −Δβα .
It is convenient to express the Bogoliubov–de Gennes
Hamiltonian as (up to a constant shift)Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019

 
ĉ
1 †
BdG
Ĥ − μN̂ ¼ ðĉ ĉÞH
;
2
ĉ†
!
ε
Δ
BdG
;
¼
H
−Δ −εĤ − μN̂ ¼

ð93Þ

The symmetry (91) allows this to be written as

015005-32s
1X
E Ĉ† Ĉ þ Eλ̄ Ĉ†λ̄ Ĉλ̄
2 λ¼1 λ λ λ

ð94Þ

s
1X
E ðĈ† Ĉ − Ĉλ Ĉ†λ Þ
2 λ¼1 λ λ λ

ð95Þ



1
Eλ Ĉ†λ Ĉλ − ;
2
λ¼1

ð96Þ

N

Ĥ − μN̂ ¼

N

¼

¼

Ns
X

N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

where the sum is now over just N s eigenstates. These areconventionally chosen to be states for which Eλ ≥ 0, such thatthe ground state j0i is defined by Ĉλ j0i ¼ 0, and Ĉ†λ are thequasiparticle creation operators.
For spinless fermions in translationally invariant settings weP
can use the plane-wave operators c̃ˆ q ∝ α e−iq·rα ĉα to writeεq
1 X ˆ† ˆðc̃q c̃−q ÞĤ − μN̂ ¼
2 q
Δq

Δq

 ˆ c̃q−εqc̃ˆ †−qð97Þ

with the superconducting gap required to satisfy Δ−q ¼ −Δq.
The BdG matrix (88) reduces to a momentum-dependent
2 × 2 matrix HBdG
q . An important example is the Kitaev modelof a p-wave superfluid in 1D, described in detail in
Appendix A, for which

HBdG
¼
q

−2J cosðqaÞ − μ−2iΔ sinðqaÞ

2iΔ sinðqaÞ

μ þ 2J cosðqaÞ

≡ − hðqÞ · σ:


ð98Þ
ð99Þ

The energy eigenvalues show the above particle-hole symmetry E
q ¼ jhðqÞj. Furthermore, as shown in Appendix A,
the bands are characterized by a winding number relating to
how hðqÞ encircles the origin, analogous to the winding
number of the SSH model. This winding number can be used
to identify a topological superconducting phase.
On a finite geometry this topological phase hosts an exact
zero-energy state on its boundary, i.e., with Eλ0 ¼ 0 in
Eq. (96). The prescription that we used to define the form(96) is ambiguous for such zero modes. Consider the operatorĈλ0 constructed via Eq. (92) from the eigenvector of such azero mode λ0 . The particle-hole symmetry implies that there isanother zero-energy state with label λ̄0 , which, using Eq. (91),
would lead to an operator Ĉλ̄0 ¼ Ĉ†λ0 . Together, these two
operators Ĉ†λ0 and Ĉλ̄0 describe the destruction or creation of afermionic quasiparticle at Eλ0 ¼ Eλ̄0 ¼ 0. Since this excitation
has zero energy, it describes a ground state that is twofold
degenerate depending on whether this zero-energy mode isoccupied or filled. It is therefore immaterial which of these
operators we choose to view as a particle creation or
destruction. Indeed, we could also choose to work in termsof operators that are linear superpositions. One particular
choice is to define the Majorana operatorsγ̂ 1 ≡ ðĈλ0 þ Ĉ†λ0 Þ;ð100Þ

γ̂ 2 ≡ iðĈλ0 − Ĉ†λ0 Þ;ð101Þwhich obey anticommutation relations fγ̂ i ; γ̂ j g ¼ 2δij . Since
the Majorana operators are self-adjoint γ̂ †i ¼ γ̂ i , they can beviewed as describing particles that are their own antiparticles.
The above transformation to Majorana operators appears
rather arbitrary: from a mathematical perspective one canalways choose to work either in terms of the two Majorana
operators γ̂ 1 , γ̂ 2 or in terms of Ĉλ0 and its adjoint Ĉ†λ0 . For
Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019

topological superconductors there is a clear physical reason toprefer the Majorana operators: the Majoranas are spatiallylocalized, each one tied to a single boundary or defect. Thus,
local probes couple directly to the Majorana operators. Inð†Þcontrast, the quasiparticle operators Ĉλ0 are nonlocal. The
locality of the Majorana operators is shown explicitly inAppendix A for the Kitaev model, where one Majorana
operator acts on the right boundary and one acts on the left
boundary.
That the Majorana modes are spatially localized causes thesystem to have properties that are robust to external perturbations, including disorder potentials: the exact particle-holesymmetry enforces the mode to have E ¼ 0. Departures fromE ¼ 0 can arise only from mixing with other Majorana modes;
since the intervening superconducting state is gapped these
corrections are suppressed exponentially in the distance
between the local Majorana modes, e.g., the length of the1D Kitaev chain. Note that the two Majorana modes—one ateach end of the 1D Kitaev chain—describe a single fermionicexcitation, such that one should view the Majorana as afractionalized quasiparticle. In contrast, the zero-energy edge
states of the SSH model correspond to two separate fermionicexcitations, one at each end of the chain. As discussed inSec. II.E.1 the edge states of the SSH model are sensitive to
local perturbations which break the chiral symmetry.
The above example for the 1D topological superconductor
has a very natural generalization to 2D. There, the bulk
spectrum is of the form

¼
HBdG
q

ϵq − μΔq

Δq

μ − ϵq≡ −hðqÞ · σ;

ð102Þ

where q now runs over a 2D Brillouin zone. Topologicalsuperconducting states can appear in situations where thehðqÞ acquires all three components, such as in the continuumpx þ ipy superfluid, for whichhz ¼ ℏ2 jqj2 =2m − μ;hx þ ihy ¼ Δ0 ðqx þ iqy Þ:ð103Þ

For μ > 0, the quasiparticle spectrum is fully gapped, and theunit vector hðqÞ=jhðqÞj wraps the sphere as q runs over all
values, indicating that this is a topological phase (Read andGreen, 2000). At μ ¼ 0 there is a gap-closing transition to asuperconducting phase at μ < 0 which is nontopological.
The 1D surface of this topological 2D superconductor has
an edge mode that has Majorana character, albeit in a setting inwhich there is a continuum of edge modes. In this 2D setting
localized Majorana modes arise as bound states on the cores ofquantized vortices, i.e., point defects localized in the bulk of
the system (introduced by rotation or other external means).
It is natural to search for topological superfluids using cold
atoms. Many routes to p-wave pairing of single-component
fermions have been suggested. These include methods involving a p-wave Feshbach resonance (Gurarie and Radzihovsky,2007), long-range dipolar interactions (Baranov et al., 2012),
or induced interactions via a background BEC (Wu andBruun, 2016). In the context of this review, the connections
arise through the use of spin-orbit coupling (optical dressing)to allow contact interactions (e.g., s-wave pairing) to lead to015005-33

N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

effective p-wave interactions between single-component fermions. Proposals of this kind have been presented in 3D, 2D(Zhang et al., 2008), and 1D (Nascimbène, 2013; Yan, Wan,
and Wang, 2015).
3. Fractional quantum Hall states

The conditions for realizing fractional quantum Hall (FQH)
states in 2D semiconductor systems are well understood(Prange and Girvin, 1990). The application of a strong
magnetic field breaks the single-particle energy spectrum into
degenerate Landau levels. When a Landau level is partiallyfilled with electrons, the large number of ways in which theelectrons can occupy the single-particle states gives a very
high degeneracy. This degeneracy is lifted by repulsive
interparticle interactions, leading to strongly correlated
FQH ground states at certain ratios of particle density to flux
density ν ¼ n=nϕ. FQH states are characterized by a nonzero
energy gap to making density excitations in the bulk. For
temperatures below this gap they behave as incompressible
liquids: with a bulk energy gap, but carrying gapless edge
modes. In this sense they resemble integer quantum Hall
states. However, the edge modes are not simply described as
single-particle states but involve fractionalized quasiparticles(Wen, 1995). Similarly, the gapped particlelike excitations in
the bulk of the system have fractional charge and are predicted
to have fractional quantum exchange statistics (Stern, 2008).
The achievement of similar regimes with ultracold atoms
would allow the exploration of several novel variations of
FQH physics.
a. BosonsCold atom experiments have the potential to allow the first
exploration of FQH states for bosons. Theory shows that
contact-interacting bosons in the lowest Landau level exhibit
FQH states provided the filling factor is not too large. Thesestates include robust variants of interesting phases—the MooreRead and Read-Rezayi phases—which are expected to exhibit
non-Abelian particle exchange statistics (Cooper, 2008). The
stabilization, and exploration, of non-Abelian phases is a muchsought after goal, in part to find the first evidence that naturedoes exhibit this exotic possibility of many-body quantum
theory and in part in connection with the possible relevance forquantum information processing (Nayak et al., 2008).
The physics of interacting bosons in the lowest Landau levelcan be accessed in the Harper-Hofstadter model at relatively
low flux density jϕAB j ≲ 2π=3 for which the lowest band issimilar to the continuum Landau level and where similar FQH
states appear for bosons (Sørensen, Demler, and Lukin, 2005;
Palmer and Jaksch, 2006; Hafezi et al., 2007; Möller andCooper, 2009). Optical flux lattices, involving spin-orbit
coupling, can lead to 2D energy bands that are very similar
to the lowest Landau level: topological bands with unit Chern
number and with very narrow energy dispersion. An advantageof this approach is that the flux density nϕ can be high, so FQH
states are expected at high particle density n where interactions
are strong. Exact diagonalization studies have establishedstable FQH states of bosons including exotic non-Abelianphases (Cooper and Dalibard, 2013; Sterdyniak et al., 2015).
Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019b. Lattice effects

The use of optical lattices to generate topological energy
bands naturally causes cold atomic gases to explore new
aspects of FQH physics. The underlying lattice makes the
single-particle states differ from those of the continuum
Landau level and can influence the nature of the many-bodyground states. For the Harper-Hofstadter model, the FQH
states found at low flux density, where the bands resemble thecontinuum Landau level, are replaced by other strongly
correlated phases at high flux densities where lattice effects
become relevant.
It has been shown that states of the same form as FQH
states of the continuum Landau level, the so-called “fractional
Chern insulator” (FCI) states, can be formed, for bosons andfermions, in a variety of topological energy bands startingfrom models (such as the Haldane model) which are far
removed from the continuum Landau level (Parameswaran,
Roy, and Sondhi, 2013). Typically such models require theintroduction of further-neighbor tunneling terms to flatten thelowest energy band, such that one can enter a regime of strong
correlations (mean interaction energy larger than bandwidth)
without mixing with higher bands. Indeed, it has been shownthat by tailoring the tunneling Hamiltonian, one can constructmodels for which the exact many-body ground states are FQH
states (Kapit and Mueller, 2010) or FCI states (Behrmann,Liu, and Bergholtz, 2016). No general theorem exists concerning the nature of the many-body ground state in a given
Chern band. However, beyond flatness of the energy band, it isbelieved that flatness of the geometry of the states, asmeasured by the Berry curvature and by a quantity known
as the Fubini-Study metric, is advantageous in stabilizing
topological many-body phases (Parameswaran, Roy, andSondhi, 2013).
A particularly interesting aspect of the topological energy
bands in lattices is that these can differ qualitatively from acontinuum Landau level, specifically if their Chern number
differs from unity. Indeed, for the Harper-Hofstadter model atflux ϕAB ¼ π þ ϵ, with ϵ small, the lowest energy band has
Chern number of 2, and so is topologically distinct from thelowest Landau level. Numerical calculations show the appearance of FQH phases for particles occupying this energy band.
These are examples of FQH states that have no counterpart inthe continuum Landau level but that are stabilized by thelattice itself (Möller and Cooper, 2009, 2015; He et al., 2017).
c. Symmetry-protected topological phases

The topological states of noninteracting Fermi systems(topological insulators and superconductors) are now viewed
as examples of a more general class of a symmetry-protected
topological (SPT) phase, which allows for interparticle interactions and bosonic statistics. SPT phases are defined by theconditions that they are gapped phases, with gapless edge
modes, but unlike the FQH effect they do not involve particle
fractionalization. Hence these states have a unique groundstate on a periodic geometry, and the many-body ground state
has only short-range entanglement (Senthil, 2015). The phasesof strongly interacting bosons in Chern bands include cases of“integer” quantum Hall states of the bosons, which provide
one of the cleanest realizations of SPT phases (Möller and015005-34

N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

Cooper, 2009, 2015; He et al., 2017). Here integer refers to thefact that the Hall conductance is an integer as opposed to
fractional. Such phases still arise from strong interparticle
interactions, albeit without fractionalized quasiparticles.
d. Ladders

Cold atomic gases provide ways in which to study ladderlike systems which are quasi-1D variants of FQH systems.
These arise both in Harper-Hofstadter models with superlattices to create ladder geometries, and in systems involving a
synthetic dimension for which there is naturally tight confinement. Both situations have been shown theoretically tosupport strongly correlated states that are closely related to
FQH states (Cornfeld and Sela, 2015; Petrescu and Le Hur,
2015; Calvanese Strinati et al., 2017). A precise connection
between FQH states on an infinite 2D system and the states inthese quasi-1D settings can be made by considering thequantum Hall wave functions on a cylindrical geometry whichis infinite in one direction but has finite circumference L in theother. Studies of the evolution of the ground state from 2D(L ≫ ā, with ā the mean interparticle spacing) into the“squeezed geometry” (small L ≲ ā) show that many FQH
states of the 2D systems evolve smoothly into charge-densitywave (CDW) states of the quasi-1D geometry. For example,
the bosonic Laughlin state, at filling factor ν ¼ 1=2, evolves
into a CDW state which breaks translational symmetry to
double the unit cell to give two degenerate ground states of the
CDW. The fractionally charged quasiparticles of the FQH
states map onto domain walls between the different symmetry-related CDW states.
4. Other strongly correlated phases
a. Chiral Mott insulator

For the Harper-Hofstadter-Hubbard model at strong interactions U ≳ J the Gross-Pitaevskii approach fails. Just as in
the case of vanishing magnetic field, there can be incompressible Mott phases, with ni ¼ integer. However, otherstrongly correlated phases are predicted to appear. One
striking example is the chiral Mott insulator: an incompressible phase at integer filling (like the Mott insulator), but which
carries a nonzero local current in the ground state (as does avortex lattice). Numerical calculations indicate that, within aregion of stability between Mott insulator and superfluid, thechiral Mott insulator can form the ground state of the BoseHubbard model with external flux on a variety of 2D lattices(Dhar et al., 2013; Zaletel et al., 2014; Vasić et al., 2015). In
ladder systems the Mott-vortex phase found in numericalcalculations (Greschner et al., 2015; Petrescu and Le Hur,
2015; Piraud et al., 2015) can be viewed as the 1D analog ofthe chiral Mott insulator.
b. Chiral spin states

For hard-core interactions U → ∞, interesting many-bodystates can still arise in the Harper-Hofstadter-Hubbard model
for bosons provided ni ≠ integer so the hard-core bosons aremobile. These bosons can be viewed as a spin-1=2 quantum
magnet, with the phases of the tunneling matrix elements
introducing frustrated magnetic interactions (Möller andRev. Mod. Phys., Vol. 91, No. 1, January–March 2019

Cooper, 2010). Thus, the FQH (or FCI) phases of these
hard-core bosons can be viewed as quantum spin liquids(Kalmeyer and Laughlin, 1987). Other predicted quantum
spin liquids may also find realization with these BoseHubbard models (Läuchli and Moessner, 2015). The phasesof interacting fermions in Hubbard-like models with tunnelingphases have also been studied theoretically. Much of the focus
has been on the Haldane model for spin-1=2 fermions with onsite repulsion U. For systems close to half filling there is acompetition between the band insulator, the Mott insulator,and various spin-symmetry-broken phases (Zheng et al.,2015; Imriška, Wang, and Troyer, 2016; Vanhala et al.,2016). The Mott insulating state naturally leads to frustrated
spin models, with possible unconventional forms of ordering
at very low temperatures, at the superexchange energy scale
J 2 =U or below, including chiral spin states (Arun et al., 2016)
and chiral spin liquids (Hickey et al., 2016).
c. Number-conserving topological superfluids

The theory of topological superfluids, Sec. V.B.2, is basedon the mean-field treatment of pairing described by the
Bogoliubov–de Gennes theory. In this theory, the numberof pairs of fermions is not conserved. For a 1D system, such amean-field theory can be appropriate in settings in which thesuperfluid pairing is proximity induced, e.g., by coupling to abulk superfluid of fermion pairs with which pairs of particles
can be exchanged and which can impose a fixed superconducting order parameter. However, in the absence of such aproximitizing medium the quantum fluctuations in 1D systems preclude the existence of any long-range order in the
superconducting pairing and the structure of the mean-field
theory should fail. Theoretical work has shown how 1D
topological superfluids can still arise in such number-conserving settings. Models with microscopic symmetry can giverise to zero-energy modes, arising as exact degeneracies in the
spectrum of an open chain in the topological phase (Fidkowski
et al., 2011; Sau et al., 2011; Iemini et al., 2015, 2017; Langand Büchler, 2015). This degeneracy is the many-bodycounterpart of the Majorana modes of the mean-field 1D
topological superconductor. In settings without such a symmetry, topological degeneracies can arise in geometries inwhich modulation of the parameters along the chain lead tomultiple interfaces between topological and nontopologicalphases (Ruhman, Berg, and Altman, 2015; Ruhman andAltman, 2017).
C. Experimental perspectives

Many of the experimental consequences of topologicalbands described in Sec. IV rely on the interactions between theparticles being sufficiently weak that they can be treated asnoninteracting. It is therefore important to consider what
experimental observables can be used to characterize theproperties of atoms in topological bands when interactions
cannot be neglected.
1. Equilibrium observables

Some of the most important observables for characterizing
the properties of the atoms are well established from studies of015005-35

N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

atomic gases in other settings without topological character.
Measurements of the equation of state and observations ofdensity-density correlations in time-of-flight imaging will be
crucial for establishing the existence and forms of strongly
correlated phases such as fractional quantum Hall states.
Similarly, the possibility to image at the single-site level inquantum gas microscopes (Bakr et al., 2009; Sherson et al.,2010) will allow precise characterization of microscopic
structure in these phases, including of the local currents,for example, in the chiral Mott insulator phase. A method for
detecting local currents was presented in experiments from theMunich group (Atala et al., 2014).
2. Collective modes

Observations of the collective mode frequencies provide asensitive way to detect properties of many-body systems(Dalfovo et al., 1999). For BECs formed in bands withgeometrical character, the collective mode frequencies are
sensitive to the Berry curvature at the band minimum (Priceand Cooper, 2013). Consider a weakly interacting BEC in asingle band minimum. We take the minimum to have isotropic
effective mass M  and a local Berry curvature Ωez . The effectof Berry curvature on the collective modes can be readily
determined by adapting the standard hydrodynamics approach(Pethick and Smith, 2002) to include the anomalous velocity
from the Berry curvature. For a spherical harmonic trapping
potential VðrÞ ¼ ð1=2ÞΛjrj2 , the collective modes have thesame spatial structure as for Ω ¼ 0 (Stringari, 1996), but thefrequencies depend on the Berry curvature. For example, forsmall Ω the three angular-momentum components of thedipole mode are split by Δω ¼ ΛΩ=2ℏ, leading to a precessional motion of the center-of-mass oscillation of the cloud at
this frequency.
3. Edge states

For incompressible fluids that are topological, e.g., a
topological insulator of noninteracting fermions, or a FQH
fluid, there exists a special class of low-energy collectivemodes, which are the gapless edge states of the fluid. For
noninteracting particles in Chern bands, these are the edge
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.
Measurements of the propagation of surface waves candirectly probe the edge-state structure, for example, allowing
one to detect the number of edge channels and theirrespective velocities (Cazalilla, Barberán, and Cooper,
2005). The edge states also lead to highly characteristic
dynamics in far-from-equilibrium situations in which theconfining potential is removed (Goldman et al., 2013). Edge
states appear also at interfaces between bulk regions ofdiffering topologies, prepared, for example, by spatial
modulation of the lattice potential (Goldman et al., 2016)and have been probed experimentally for the SSH model(Leder et al., 2016). Experiments using a quantum gas
microscope have shown the influence of strong interparticle
interactions on the chiral edge states in Harper-Hofstadter
ladders (Tai et al., 2017).
Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019

4. rf excitation

Another natural probe of the edge states is to measure thespectrum for the removal of particles via rf excitation, ideally
performed with single-site resolution (Bakr et al., 2009;
Sherson et al., 2010) to focus on the boundary. Such spectra
have been proposed as a means to detect localized Majorana
modes in topological superfluids, appearing as a near zeroenergy contribution inside the spectral gap (Grosfeld et al.,2007; Nascimbène, 2013). A definition of fractional statistics
is provided by Haldane’s exclusion statistics (Haldane, 1991):
a generalized version of Pauli blocking, by which the fractional quasiparticles reduce the number of available states forother quasiparticles in a well-defined, but fractional, manner.
An observation of this effect requires counting many-bodystates and may be possible in precision spectroscopy of small
FQH clusters (Cooper and Simon, 2015) in which theexclusion statistics reveal themselves in the count of spectral
lines.
5. Adiabatic pumping

For topologically ordered systems, the existence of fractional low-energy particlelike excitations allows for new
features in Thouless pumping. Specifically, under one fullcycle of the adiabatic evolution of the pump, it is possible totransfer a fractional particle number across the system. (In theexamples discussed above, the number of particles transferred
was constrained to be an integer, set by the Chern number.)This fractional pumping is related to the existence of a groundstate degeneracy: one cycle of the adiabatic pump converts
one ground state into another degenerate ground state, andmultiple cycles (transferring multiple fractional particles) are
required before the system returns to its starting state. This
picture forms the basis for understanding of the quantizationof the Hall conductance in the fractional quantum Hall effectof 2D systems (Laughlin, 1981; Halperin, 1982). For narrow
strips of quantum Hall systems, this physics smoothly evolves
into the pumping of fractional charges in commensuratecharge-density waves. The manifestation of such pumpingfor 1D fermionic systems with synthetic dimension was
described by Mazza et al. (2015), Zeng, Wang, and Zhai
(2015), and Taddia et al. (2017).
6. Hall conductivity from “heating”

An interesting way in which to measure the Hall conductivity σ xy of a system, i.e., the transverse current density inresponse to a uniform force, is to measure the heating rate, as
set by the rate of power absorption P ðωÞ caused by theapplication of a circularly polarized force F ðtÞ ¼
F0 ðcos ωt;  sin ωtÞ. The dc Hall conductivity is found to
be set byσ xy ¼

2
2
πF0 Asyst

Z ∞
0

dω

1
½P ðωÞ − P− ðωÞ;
ω þ

ð104Þ

i.e., the difference of the power absorption rates divided byfrequency and integrated over all drive frequencies (Tran
et al., 2017; Asteria et al., 2019). For noninteracting particles
the power absorption can be found from measurements of the

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N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

rates of depletion of initially occupied states Γ , viaP ¼ ðℏωÞΓ , allowing the Hall conductivity to be determined via measurements of this depletion rate. For a filled
Chern band, the integrated difference of depletion rates (104) istherefore expected to recover the quantized Hall conductivityof the filled bands. Equation (104) is however valid even forinteracting particles, arising from a general sum rule for the
linear response functions, and so provides a possible way tomeasure the Hall conductivity also in strongly correlated
phases.
VI. OUTLOOK
A. Turning to atomic species from the lanthanide family

In this review we explored several classes of lattice schemes
for which a nontrivial topology originates from a two-photon
Raman coupling between various sublevels of the electronic
ground state manifold. So far most experiments of this typewere performed with atoms from the alkali-metal family,
which are relatively easy to manipulate and cool down toquantum degeneracy. However, for such atoms, recoil heating
due to spontaneous emission of photons may cause severe
problems. Indeed the desired Raman coupling is significant
only when the laser is detuned from the resonance by less than
the fine-structure splitting Δf:s: of the resonance line (seeAppendix C). Because of this relatively small detuning,spontaneous emission of photons occurs with a non-negligible
rate γ. More precisely, one can define the merit factor
M ¼ κ=γ, where κ is the desired Raman matrix element.
Taking as an example the case of the fermionic alkali-metal
atom 40K, one finds after optimization M ∼ Δf:s: =Γ ∼ 105 ,where Γ stands for the natural width of the electronic excitedstate [see, e.g., Dalibard (2016) for details]. If one takes as a
typical value ℏκ equal to the recoil energy, the photon
scattering rate is γ ∼ 0.3 s−1 , leading to the heating rate
E_ ∼ kB × 100 nK=s. This may be too large for a reliable
production of strongly correlated topological states.
A more favorable class of atoms is the lanthanide family,
with species such as erbium or dysprosium. These atoms have
two outer electrons and an incomplete inner shell (6s2 and
4f10 for Dy). Because of this inner shell, the electronic
ground state has a nonzero orbital angular momentum(L ¼ 6 for Dy). The lower part of the atomic spectrum
contains lines corresponding to the excitation either of oneof the outer electrons or of one electron of the inner shell. By
choosing a laser excitation close to a narrow line resonanceand thus with a large detuning Δb from the closest broadline, one reaches after optimization M ∼ Δb =Γb . Because
Δb is now of the order of an optical frequency, the merit
factor is M ∼ 107 , leading to γ ∼ 10−3 s−1 and to theresidual heating E_ ∼ kB × 0.1 nK=s.
B. Topological lattices without light

Laser fields constitute a common element for all of thetechniques described herein, and in virtually all of these cases
these fields lead to unwanted off-resonant scattering leading
to heating, atom loss, or both. As discussed in the previous
section, these scattering processes might be mitigated orRev. Mod. Phys., Vol. 91, No. 1, January–March 2019eliminated by suitable selection of different transitions orworking with different atoms; however, fundamentally different approaches are also possible.
Of particular interest is the possibility to replace the laser
fields with rf or microwave magnetic fields generated by amicrofabricated atom chip. Unlike optical fields, these fields
have practically no off-resonant emission, solving the inprinciple atomic physics limitation, in exchange for thetechnical complexity of working in the vicinity of an atom chip.
In one category of proposals, the atom chip serves simply as
the source of a large time-modulated gradient magnetic field(Anderson, Spielman, and Juzeliūnas, 2013). This produces in
effect a series of pulses that generates spin-dependent gauge
fields; the most simple implementation of this technique wasrealized in the lab (Luo et al., 2016), and although spontaneous
scattering was eliminated, the fairly low frequency of theirdrive led to significant micromotion induced heating effects.
In a second category of proposals (Goldman et al., 2010),
the atom chip consists of a large array of microfabricatedparallel wires which give a near-field radio-frequency magnetic field that drives transitions in the same way as Raman
lasers, as described here in the context of synthetic dimensions
or intrinsic spin-dependent lattices. Because these fields are
structured at the micrometer scale, far below the free-space
wavelength of rf fields, these are near-field structures and theatomic ensemble must be on the micrometer scale from thechip’s surface.
C. Other topological insulators and topological metals

We have focused on specific recent experimental realizations of topological energy bands using cold gases: in 2D
systems and in 1D systems with chiral symmetry. Routes toachieving topological superfluid phases in cold gases, arisingfrom BCS pairing of fermions, have also been described inSec. V.B.2.
As discussed in Sec. II.E other forms of topological energy
bands can arise depending on dimensionality and the global
symmetries that are imposed, according to the tenfold way(Chiu et al., 2016). Important cases in solid state systems are
the Z2 topological insulators that arise in spin-orbit coupledsystems with time-reversal symmetry, in 2D and 3D. There
are proposals for how to realize bands with this topology forcold gases. The required TRS can be implemented by finetuned engineering of the relevant terms in the Hamiltonian(Goldman et al., 2010). TRS can also be established as an
intrinsic property for cold gases: in the absence of Zeeman
splittings and of any circularly polarized light fields (Béri andCooper, 2011). In 2D, the insulating state formed by filling a
Z2 topological band exhibits a quantized Hall effect for the
spin current. A similar quantized spin Hall response also arises
in a related setting without spin-orbit coupling, in which spin
up and spin down fill Chern bands with equal and opposite
Chern number: such energy bands have been realized forbosons in cold atoms (Aidelsburger et al., 2013) using laserassisted tunneling to generate a Harper-Hofstadter model withequal and opposite fluxes for the two spin states (Kennedy
et al., 2013).
Cold atomic gases provide a natural setting with which to
explore certain forms of topological band that are more

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N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

difficult to implement in solid state materials. The sublattice(chiral) symmetry is very fragile in solid state systems, as it istypically broken by any disorder potential. However, it canreadily arise for optical lattice potentials in cold atom gases forwhich disorder can be negligible (Essin and Gurarie, 2012;
Wang, Deng, and Duan, 2014). Similarly, there are predicted
to be topological insulators beyond those classified in thetenfold way, arising from lattice symmetries which rely on ahigh spatial regularity that can arise in cold gases. These
include topological invariants stabilized by crystalline latticesymmetries (Fu, 2011), and the Hopf insulator (Moore, Ran,and Wen, 2008; Deng et al., 2013), which also relies on a formof translational symmetry for stability (Liu, Vafa, and Xu,
2017). Note also that cold gases allow topology in dimensions
higher than d ¼ 3 to be explored, through the use of synthetic
dimensions provided by internal degrees of freedom or byviewing a phase degree of freedom as an additional quasimomentum. Recent experimental work (Lohse et al., 2018)used pumping to demonstrate the topological response of aneffective 4D quantum Hall system (Zhang and Hu, 2001)
based on the theoretical proposal by Price et al. (2015).
An area of growing interest in solid state settings concerns
so-called topological metals, or semimetals (Chiu et al.,2016). (There exist analogous topologically stable forms ofgapless superconductors.) As for topological insulators, theclassification of topological metals depends also on thedimensionality and the existence (or absence) of symmetries.
However, since metals involve bands that are only partiallyfilled by fermions, they cannot be characterized by the
topological invariants used for insulators, which involve
integrals over the filled energy bands. Instead, topologicalmetals can be characterized by topological invariants definedin terms of integrals over the Fermi surface which separates
filled from empty states (Volovik, 2003). One example of a
topological metal is provided by the 2D honeycomb latticewith nearest-neighbor hopping. This realizes the band structure of graphene in which there are two Dirac points in BZ,
each of which leads to a Fermi surface when the Fermi energy
lies close to the Dirac point. Each of these Fermi surfaces is a
topological metal, characterized by the Berry phase of Blochstates around the Fermi surface, which is π. This value is a
topological invariant, i.e., it is robust to continuous changesof the underlying parameters, provided the system retains bothtime-reversal symmetry and inversion symmetry. (Thesesymmetries ensure that the Berry curvature of the bands
vanishes, except for singular delta-function contributions at
the Dirac nodes which lead to the π Berry phases.) The Diracnode is topologically stable and can only be created or
annihilated if it merges within another Dirac point. A related
example that arises in 3D is the “Weyl point.” This can beviewed as a point source of Berry curvature in reciprocal
space. The integral of the flux of Berry curvature through any
closed 2D surface in reciprocal space is 2πC, where C isrequired to be an integer. If this surface encloses a Weyl point,
jCj ¼ 1. The Weyl point is topologically stable to deformations of the Hamiltonian, without any symmetry requirements,
unless it annihilates with a second Weyl point of opposite
charge. Optical lattice with Weyl points can be constructedusing the laser-induced tunneling methods described inSec. III.D (Dubček et al., 2015).
Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019

D. Far-from-equilibrium dynamics

Cold atomic gases readily allow the study of coherent
quantum dynamics in far-from-equilibrium settings. Startingwith a cold gas at thermal equilibrium, a sudden change of the
Hamiltonian, a so-called “quantum quench,” typically leaves
the system in a far-from-equilibrium state. Here we discuss
some of the consequences of quantum quenches between
Hamiltonians for which the ground states have different
topological character. We focus on cases of noninteracting
particles in topological energy bands.
Consider first an optical lattice potential which is varied
slowly in time, such that the topological invariant of the lowest
energy band differs between initial and final Hamiltonians.
If the relevant topological invariant is symmetry protected,
it is possible to have a smooth evolution between these twocases provided that the Hamiltonian breaks the symmetry atintervening times. An example of this is the pumping
sequence of the RM model, which breaks the chiral symmetryof the SSH model and therefore allows a smooth evolution
between the topologically distinct phases of the SSH model.
When there is no symmetry protection (e.g., Chern bands in2D), then the change in band topology requires the band gap
to close at some intermediate time, as illustrated in Fig. 26. For
a BEC formed close to the band minimum, such a change inband topology need not induce any phase transition: the Blochwave functions of those states which the bosons occupy canevolve smoothly, allowing adiabatic evolution of the BEC,
albeit into a very different local wave function. This allows,for example, the adiabatic formation of a dense vortex lattice(Baur and Cooper, 2013). However, for noninteracting fermions that fill the lowest energy band, the ground states of theinitial and final Hamiltonians have different topologicalcharacters, so these two states must be separated by a phasetransition.
The far-from-equilibrium dynamics following a quantum
quench between Hamiltonians whose ground states havedifferent topologies has been explored for noninteracting
fermions which fill a band. One striking result is that the
topological invariant of the many-body state is often preserved
under unitary time evolution. This has been shown for
topological superfluids (Foster et al., 2013, 2014;
Sacramento, 2016), and for Chern bands (Caio, Cooper,
and Bhaseen, 2015; D’Alessio and Rigol, 2015). [Different
behavior can arise for symmetry-protected topological invariants (McGinley and Cooper, 2018).] In interpreting this result
it is crucial to distinguish between the topology of the
Hamiltonian and the topology of the many-body state. The
former is defined as the topological invariant constructed forthe lowest energy band of the Hamiltonian, with Bloch wave
ð0Þfunctions juq i, while the latter is defined as the topological
E

E
k

ν=0

E
k

ν=?

ν=1

k

FIG. 26. The change of band Hamiltonian that causes the

topological index of the lowest band ν to change, via the closingof a band gap.

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N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

invariant constructed from the wave functions which thefermions occupy jusq i. These two sets of wave functions need
only coincide if the many-body wave function is the groundstate of the Hamiltonian. Out of equilibrium, for example,
following a quantum quench, the Bloch wave function
occupied by the particle at momentum q need not be an
eigenstate of the Hamiltonian, so it becomes time dependent
jusq ðtÞi. We consider situations in which the periodicity of thelattice is preserved, such that wave vector q remains a goodquantum number. That the topological invariant of the state ispreserved is guaranteed provided the unitary evolution of the
Bloch states jusq ðtÞi ¼ e−iĤq t=ℏ jusq ð0Þi is smooth in momentum space q, which is true for short-range hopping. [Fortopological invariants that are symmetry protected, it is also
required that the symmetry is not broken either explicitly or
dynamically (McGinley and Cooper, 2018).] That said, at longtimes, the Bloch wave functions of the many-body statejusq ðtÞi will become rapidly varying as a function of q. Whenthe variation in q is so fast that this cannot be viewed assmooth on the scale of 2π=L, with L the typical sample
dimension, then the bulk topological invariant becomes ill
defined. Thus for any finite system, there is an upper timescale after the quench for which it is meaningful to expect the
Chern number to be preserved. Simple estimates lead to theconclusion that this time is of order L=v, with v a characteristic group velocity of the final Hamiltonian (Caio, Cooper,
and Bhaseen, 2016). Such systems can still be characterized
by the Bott index (Loring and Hastings, 2010), which
provides a real-space formulation of the Chern index applicable also to finite-sized systems. This has been used to findprotocols for preparing nonequilibrium (Floquet) systems bywhich the Bott index undergoes transitions between topological values (D’Alessio and Polkovnikov, 2013; Ge andRigol, 2017).
Although the topological invariant of the wave function isunchanged following the quench to a new Hamiltonian withdifferent topology, this does not mean that there are noobservable consequences of the new Hamiltonian. Local
physical observables can be strongly influenced by the new
Hamiltonian, despite the fact that the nonlocal topologicalinvariant of the state is unchanged. Indeed, theory shows that
under a quantum quench of the Haldane model the edge
current quickly adapts to become close to that of the groundstate of the final Hamiltonian (Caio, Cooper, and Bhaseen,2015). Furthermore, by following the dynamics of the Blochwave functions jusq ðtÞi detailed information on the final
Hamiltonian can be recovered (Gong and Ueda, 2017;
Wang et al., 2017). This has been demonstrated in experiments by the Hamburg group (Tarnowski et al., 2017;
Fläschner et al., 2018), using the band-mapping techniques
described in Sec. IV.A.2 to reconstruct the dynamical evolution of the occupied wave functions jusq ðtÞi. The experiments
use a two-band model, for which the Hamiltonian at q may be
written as
Ĥ q ¼ h0 ðqÞ1̂ − hðqÞ · σ̂:ð105Þ

The wave function of the fermion with q can be represented bya three-component unit vector eðqÞ. If eðqÞ is aligned with
Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019hðqÞ, then this is an energy eigenstate of the Hamiltonian.
In the quench experiment, the system is prepared witheinitial ðqÞ ¼ ð0; 0; −1Þ for all wave vectors [the ground stateof an initial Hamiltonian with hinitial ðqÞ ¼ ð0; 0; 1Þ] and thenthe Hamiltonian is changed to its final value hðqÞ. For generalq this is not aligned with einitial ðqÞ so the wave function
evolves in time, precessing around the local hðqÞ. This leads tothe appearance of nonzero components of the vector ðex ; ey Þwhich indicate interband coherences. The evolution can leadto the creation of vortex-antivortex pairs in these componentsðex ; ey Þ. The appearance of each vortex-antivortex pair was
shown to be associated with a cusp in the Loschmidt echo,
hence giving rise to the characteristic feature of a “dynamicalphase transition” (Heyl, 2018). Such singular features canarise even for quenches within a single topological phase, andso are not connected to the topological phase transition itself.
Instead, the change in the topology of the Hamiltonian can befound by tracing the time evolution of eðq; tÞ and constructing
the linking number of the trajectories (in q and t) of any twovalues of e [e.g., e ¼ ð1; 0; 0Þ and (0,0,1)] (Wang et al., 2017).
This linking number is the Hopf index of the map eðq; tÞ. This
procedure has been successfully carried out in experiments(Tarnowski et al., 2017). A related approach has recently beenused to demonstrate the topological character of the effective
Hamiltonian in a spin-orbit coupled BEC (Sun et al., 2018).
The above considerations rely on the assumption that thefermions are noninteracting. It will be of interest to explore theextent to which these, or similar, approaches can be applied in
the presence of interparticle interactions. Being gapped phasesof matter, associated with filled bands, the ground states of the
Hamiltonian are expected to be robust to weak interactions.
However, even weak interactions can lead to the generation ofentanglement between single-particle states at different wave
vectors under far-from-equilibrium dynamics, which may beviewed as a form of decoherence.
Moving beyond the phases of noninteracting fermions in
topological bands, it will be of interest to explore similar
quench dynamics in topological phases that arise only becauseof strong interparticle interactions. This is an area where
results remain limited. Recent work on the Haldane phase of aspin-1 chain (a symmetry-protected topological phase of thisinteracting quantum spin system) has shown that the “stringorder” that characterizes the topological phase is lost following the quench (Calvanese Strinati et al., 2016), suggesting a
difference from the noninteracting fermion cases describedabove. One important difference concerns the role of symmetry protection under dynamical evolution (McGinley andCooper, 2018).
E. Invariants in Floquet-Bloch systems

The topological invariants we focused on are those of static
single-particle Hamiltonians, in which the spatial periodicity
leads to the existence of a Bloch Hamiltonian that depends ona quasimomentum q within a BZ. If the Hamiltonian is timevarying, but periodic with period T ≡ 2π=ω, then energy isreplaced by a Floquet quasienergy that is defined up to theaddition of integer multiples of ℏω. The combination of both
temporal and spatial periodicities causes Floquet-Bloch

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N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

systems to have topological invariants that are distinct fromthose of static Hamiltonians.
(1) The periodicity of the Floquet spectrum allows the
Floquet bands to wind in quasienergy, by an integer multipleof h=T as q runs over the BZ. This winding in quasienergy
gives rise to topological invariants of Floquet-Bloch bands(Kitagawa et al., 2010) that are absent in static settings.
Figure 27 shows an example of a Floquet-Bloch band in a 1D
system that winds in quasienergy once across the BZ.
To understand the physical significance of such situations,
it is instructive to consider the band of Fig. 27 (top) to be filledwith noninteracting fermions. [How one might prepare such astate is itself an interesting question (Dauphin et al., 2017;
Lindner, Berg, and Rudner, 2017). Indeed, nonadiabatic
effects associated with the switch-on of a pump can lead todeviation from quantization (Privitera et al., 2018).] The net
current carried by the filled band is
L
I≡
2π

Z π=a

1 dϵq
L
dq ¼ ;
T−π=a ℏ dq

ð106Þ

where a is the lattice constant and L is the total length of thesystem. The number of particles in the filled band is N ¼ L=a,so the current per particle is
I
a
¼ :
N T

ð107Þ

For a band that winds in quasienergy s times, the current isquantized at sa in each cycle T. The quantized current from afilled Floquet-Bloch band that winds s times in quasienergy
precisely matches the current expected from an adiabatic
pump, as described in Sec. II.F, operated cyclically withperiod T and with Chern number s. Indeed, settings in which
adiabatic Thouless pumping occurs give rise to Floquet-Blochspectra in which a band winds in quasienergy. (The adiabaticity condition corresponds to the relevant Floquet-Blochband crossing with all Floquet-Bloch bands that wind in the
opposite sense.) However, the general structure of the FloquetBloch states and the associated topological invariants(Kitagawa et al., 2010) are not restricted to such adiabatic
settings.
(2) New features also arise when one considers the edge
states on systems with a boundary. One finds that finite-size
systems can have protected edge states, at quasienergies
between the bulk bands, even if the topological invariants
constructed from the Floquet Hamiltonian Ĥ Fq are trivial for allof the bulk energy bands (Kitagawa et al., 2010). A simple
example of a 2D model which exhibits such “anomalous edge
states” is illustrated in Fig. 28. One period T is broken into
four subperiods T=4, during which the tunneling amplitude J
is turned on only for a subset of the bonds. This tunneling ischosen to satisfy ðJ=ℏÞðT=4Þ ¼ π such that a particle thatstarts on one side of the active bond is transferred to the otherside of the bond during the time T=4. For particles in the bulk,
the net action after all four parts of the cycle is to return theThus, the current is equivalent to a displacement of eachparticle 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 ðJ 0 =J; Δ=JÞ are varied along acircle of radius 0.5 with angular frequency ω ¼ 0.23J=ℏ. The
circle is centered on the point (2,0) in the nontopological case andon the point (1,0), i.e., the vortex in Fig. 7, in the topological one.
In the topological case a particle starting in state A at momentum
q ¼ 0 arrives in B after an adiabatic motion across the full BZ,
leading to the net current (107) for a filled band. The gray areas
highlight given “temporal Brillouin zones,” of energy width ℏω.
Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019

FIG. 28. A simple model that shows anomalous edge states.

(a) A cycle consists of four steps in which specific sets of bonds
are active and one step in which there is only a sublattice energy
offset δAB . (b) Trajectories of particles initially in the bulk (blueclosed loop) or on the edges [green (red) lines on top (bottom)
edges]. (c) Floquet spectrum showing the dispersionless bulk
band and the two anomalous edge states with equal and opposite
nonzero velocities. From Rudner et al., 2013.

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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
Floquet operator for time evolution over one cycle [seeEq. (B2)] is proportional to the identity and has a bulk
Floquet-Bloch band structure with constant quasienergy andwhose wave functions have trivial topological character.
However, a particle that starts on the edge of the system istransported along the edge over one period; this motion appears
as dispersive (chiral) edge states in the Floquet spectrum of thefinite-size system; see Fig. 28(c). The understanding of suchanomalous edge states is that the topological invariant of a band
constructed from the Floquet operator determines the change in
the number of edge states as the quasienergy passes through theband (Rudner et al., 2013). An edge state can pass through a setof topologically trivial bands and satisfy periodicity in quasienergy and quasimomentum; see Fig. 29. To compute thenumber of anomalous edge states requires one to know notjust the stroboscopic evolution defined by the Floquet operator,
but also the full time-evolution operator at intermediate times,
from which an additional invariant can be constructed (Rudner
et al., 2013). Anomalous edge states have not yet been seen in
atomic systems, but have been observed in experiments on lightpropagation in photonic structures (Maczewsky et al., 2017;
Mukherjee et al., 2017).
(3) The concept of particle-hole symmetry needs be
generalized for Floquet-Bloch systems. For static
Hamiltonians, particle-hole symmetry (E → −E) stabilizes(edge) modes at energies E ¼ 0 (SSH model or Majorana
mode in the Kitaev model). The periodicity of the Floquet
quasienergy ϵ under ϵ → ϵ þ ℏω means that symmetry ϵ →−ϵ arises for ϵ ¼ 0 or ϵ ¼ ℏω=2. Indeed, topologically stableedge modes at quasienergies ϵ ¼ ℏω=2 have been theoretically demonstrated in periodically driven lattice models with
sublattice (chiral) symmetry analogous to the SSH model(Asbóth, Tarasinski, and Delplace, 2014), and in Floquet
superfluid systems where they appear as Majorana modes(Jiang et al., 2011).

FIG. 29. For Floquet-Bloch systems, the topological invariant of
the bulk band determines the change in the number of edge
modes as quasienergy passes through the band. The periodicity inquasienergy allows the existence of edge states even in settings
where the bulk bands are all topologically trivial. From Rudner
et al., 2013.
Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019

Recent works have constructed a general theory of the
topological classification of time-varying single-particle
Hamiltonians (Nathan and Rudner, 2015; Fruchart, 2016;
Roy and Harper, 2016). Extensions of these ideas to interacting, many-body phases is an active area of theoreticalresearch.
F. Open systems

So far we focused our studies on the topological propertiesof the ground state of an isolated system. The possibility tointroduce a coupling between this system and an environment
opens new possibilities and raises new questions that we
now discuss. We restrict our presentation to studies in direct
relation with atomic gas implementations.
Consider first the case where the environment is at anonzero temperature T. If the coupling is sufficiently weak,the energy levels of the system remain relevant and its steadystate is now a statistical mixture of these levels. Since bands
with various topologies that were empty at T ¼ 0 now acquirea finite population, the Chern number calculated via a thermal
average will not be an integer anymore. One could naively
conclude that the system loses all its topological properties
when T becomes non-negligible with respect to the gap
protecting the bands that are populated at zero temperature.
However, one may look for more subtle topologicalinvariants that can be associated with the density matrix
describing the system at nonzero T. A possible direction
consists of generalizing the notion of geometric phase, using
the concept of parallel transport for density matrices introduced by Uhlmann (1986). This line was investigated by
Huang and Arovas (2014) and Viyuela, Rivas, and MartinDelgado (2014), who could derive in this way a classificationof topological phases at nonzero T. It was subsequently
revisited with a somewhat different perspective by Budich
and Diehl (2015), who pointed out possible ambiguities of theprevious approaches for a model 2D system. Another direction consists of establishing an equivalence between classes ofdensity matrices using local unitary operations (Chen, Gu, andWen, 2010), and deducing from this equivalence the desiredclassification of topological phases. It was developed for theparticular case of mixed Gaussian states of free fermions byDiehl 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 topologicalclass 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 physicalquantities associated with the response of the system is still thesubject of ongoing research (Budich, Zoller, and Diehl, 2015).
Working with open systems also offers the possibility ofcreating novel topological states that emerge from the dissipative coupling itself. Here we take the reservoir at T ¼ 0 for
simplicity. In the conceptually simplest version of the scheme,
the coupling is engineered so that the system ends up aftersome relaxation time in a pure state, often called a dark state.
“Topology by dissipation” is achieved when this dark state
possesses nontrivial topological properties. For a coupling
compatible with the Born-Markov approximation (Gardiner

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N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

and Zoller, 2004; Daley, 2014), the master equation describing
the evolution of the density operator ρ of the system can be
written in the Lindblad form (Lindblad, 1976) ρ_ ¼ LðρÞwhere the linear operator L, the Liouvillian, acts in operator
space:
LðρÞ ¼ i½ρ; H þ

1X
ð2Lj ρL†j − L†j Lj ρ − ρL†j Lj Þ:
2 j

ð108Þ

Here H is the Hamiltonian of the system in the absence of
coupling to the reservoir, and the Lj ’s are the Lindblad
operators describing the various coupling channels to thebath. If one neglects for simplicity the Hamiltonian evolutionof the system (H ¼ 0), one finds that a pure state jψisatisfying Lj jψi ¼ 0 for all j is an eigenstate of the
Liouvillian with eigenvalue 0 and thus a dark state. The
topology associated with this state can be readily inferred fromthe fact that it is the ground state of the parent HamiltonianP
H0 ¼ j L†j Lj . Conditions for the existence and uniquenessof such dark states were discussed in the context of fermionicsetups by Bardyn et al. (2012, 2013).
The steady state of the master equation will be protectedagainst small perturbations if it is an isolated point in the
spectrum of the Liouvillian. One defines in this case the“damping gap” as the smallest rate at which deviations fromthe steady state are washed out. In this dissipative context, thedamping gap plays a role that is formally equivalent to theenergy gap in the Hamiltonian context.
The concepts of topology by dissipation and damping gap
can be generalized to the case where the steady state of themaster equation associated to Eq. (108) is a mixed state. Its
analysis is relatively simple for free fermion systems on alattice, assuming that the Lindblad operators Lj are linear
functions of the on-site creation and annihilation operators.
This corresponds, for example, to the case where dissipation
occurs via an exchange of particles with a reservoir that is
described by a classical mean field. In this case, the systemdensity operator is Gaussian and the already mentioned tools
developed by Bardyn et al. (2013) can be used to characterize
its topological properties. Interestingly for such systems, achange of the topological properties of the system can occur
either when the damping gap closes or when the “purity gap”
closes. This notion of purity gap closure, first introduced by
Diehl et al. (2011) and later generalized by Budich and Diehl(2015), corresponds to a situation where there exist bulk
modes associated with a subspace (e.g., a given q in
momentum space) in which the state is completely mixed.
Examples of topologically nontrivial pure or mixed states
and their corresponding edge modes were presented by Diehl
et al. (2011) and Bardyn et al. (2012) in a 1D and a 2D setup,
respectively. Budich, Zoller, and Diehl (2015) pointed out aremarkable property of steady states involving a mixed state:
they may possess a nontrivial topological character evenwhen all Lindblad operators Lj are local in space. This cannot
occur when the steady state is pure, an impossibility that isreminiscent from the fact that in the Hamiltonian framework,
Wannier functions of a topologically nontrivial filled band(nonzero Chern number) must decay slowly, i.e., algebraically, in space (Thouless, 1984).
Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019

The concept of topological pumping can also be extended
to the case of open systems, as shown by Linzner et al. (2016)
and Hu et al. (2017). Using a proper engineering of the
Liouvillian (108) of a fermionic 1D chain, Linzner et al.
(2016) could generalize the result of Sec. IV.C.1 for the RiceMele model and prove the quantization of the variation of the
many-body polarization after a closed loop in parameter
space. As for the other topological features described in thissection, this quantization holds even when the steady state of
the master equation is a mixed state.
Finally we note that dissipation also offers a new route forrevealing an existing topological order. Rudner and Levitov(2009) considered the motion of a single particle on a 1D
bipartite (AB) lattice similar to the one at the basis of the SSH
model. Here dissipation corresponds to a nonzero decay rateof the particle when it resides on the sites of sublattice A. The
particle is assumed to start on site B0 , i.e., the B sublattice site
in the lattice cell m ¼ 0.
PHere one is interested in the average
displacement hΔmi ¼ m mPm , where Pm is the probability
that the particle decays from the A site of the mth lattice cell.
Quite remarkably this average displacement is quantizedand can take only the values 0 and 1. This binary result
corresponds to the two topological classes that we identified
above for the SSH model within the Hamiltonian framework.
This result holds for any values of the decay rate and on-siteenergies, but it stops being valid if one introduces a dissipative
component in the hopping process between the two sublattices. It was confirmed experimentally with a setup in thephotonic context by Zeuner et al. (2015). They used a latticeof evanescently coupled optical waveguides, in which losses
and dissipation were engineered by bending the waveguides.
The scheme of Rudner and Levitov (2009) was recently
generalized theoretically to multipartite lattices by Rakovszky,
Asbóth, and Alberti (2017), who used a weak measurement of
the particle position as the source of dissipation.
VII. SUMMARY

We have summarized the methods that have been used toengineer topological bands for cold atomic gases, and themain observables that have allowed characterizations of theirgeometrical and topological properties. Most experimental
studies so far have been at the single-particle level. Theory
suggests many interesting possibilities for novel many-bodyphases in regimes where interparticle interactions become
strong. Such systems are in regimes where theoretical understanding is very limited, so experimental investigation will be
particularly valuable. Accessing this regime for large gases
will require careful management of heating—from lattice
modulation methods or from Raman coupling of internal
states—as well as the development of robust detection
schemes to uncover the underlying order. There is alreadyprogress in this direction. It seems likely to provide a rich vein
to explore, with much scope for experimental discoveries andsurprises.
ACKNOWLEDGMENTS

We are grateful to the innumerable colleagues fromwhom we have learned so much. We also thank J.

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N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

Beugnon, B. Gadway, F. Gerbier, N. Goldman, Z. Hadzibabic,
S. Nascimbene, and J. V. Porto for comments on an earlier
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
Foundation. J. D. was supported by the ERC Synergy Grant
UQUAM. I. B. S. was partially supported by the ARO’s
atomtronics MURI, the AFOSR’s Quantum Matter MURI,NIST, and the NSF through the PFC at the JQI.

For the semi-infinite SSH chain (with j ≥ 1) the eigenvalue
condition is

APPENDIX A: TOPOLOGICAL BANDS IN ONE
DIMENSION

Assuming that there is only one edge state in this problem(which can be checked analytically), the chiral symmetry
entails that it has a zero energy. Setting E ¼ 0 in Eqs. (A1)and (A2) one readily finds the solution

1. Edges states in the SSH model

The Su-Schrieffer-Heeger (SSH) model was introduced todescribe the electronic structure of polyacetylene (Su,
Schrieffer, and Heeger, 1979). This molecule has alternating
single and double bonds along the carbon chain, which are
represented in a tight-binding model with one orbital per
carbon atom (site) by alternating tunnel couplings J and J 0
between the sites; see Fig. 1. There are two sites in the unit
cell, which we label A and B and that are assigned the sameenergy in the SSH model (this constraint is relaxed in the
Rice-Mele model). The single-particle Hamiltonian is given in
Eq. (15), and we emphasized in the main text the existence of
two distinct topological classes, corresponding to J < J 0(winding number N ¼ 0) and J > J 0 (N ¼ 1).
This classification may appear rather formal for an infinite
chain, since the labeling in terms of A and B sites is arbitrary
and one can exchange the roles of J and J 0 without changing
the physical system. However, its physical relevance appears
clearly if one considers a finite or semi-infinite chain. Thenthe two classes correspond to different possibilities for theedge state(s) of the chain, as an illustration of the generalbulk-edge correspondence. In the following we first describe
the case of a semi-infinite chain for which analytical
calculations are straightforward and then discuss the caseof a finite chain.
a. Semi-infinite chainEψ Aj ¼

 −J0 ψ B − Jψ B
jj−1

j > 1;−J0 ψ Bj

j ¼ 1;

Eψ Bj ¼ −Jψ Ajþ1 − J 0 ψ Bj

j ≥ 1:



ψ Aj
ψ Bj




∝

ð−J0 =JÞj
0

ðA1Þ
ðA2Þ


:

ðA3Þ

This is a localized (normalizable) state provided J 0 < J, i.e.,
provided the insulator to the right of the boundary j ≥ 1 is in
the topological phase with N ¼ 1. Recalling that the boundary is to a region with J ¼ 0, and hence N ¼ 0, we establish
the existence of an E ¼ 0 edge state if and only if theinsulators on either side have differing topological index N.
The length ξ over which this edge state is localized isξ ∼ a lnðJ=J 0 Þ. Note that this edge state occupies only one
type of site (here the A sites), in agreement with the general
requirement of the chiral symmetry for a state jψi at zeroenergy: ÛjΨi ¼ ðP̂A − P̂B ÞjΨi ∝ jΨi. This solution is thediscrete version of the edge mode, Eq. (37), of the continuum
model derived in Sec. II.D.1.
b. Finite chain

One can also consider a chain with M unit cells; see Fig. 31.
For J 0 < J and a length Ma much larger than the extension ξof the edge state found for a semi-infinite chain, we expect
now two edge states of energy ∼0, each localized at one end of
the chain. When J 0 =J increases and approaches unity, this pairof edge states gradually hybridize and they finally merge into

We consider a semi-infinite lattice, with unit cells labeled
by j ¼ 1; 2; 3; 4; …. Such a boundary could be formed by
imposing a very large potential on the unit cells at j ≤ 0 on an
infinite lattice. However, to retain the chiral symmetry it isconvenient to view this as a boundary to a region, at j ≤ 0,
where the intercell tunneling J ¼ 0: this causes the j ¼ 1 unit
cell to be disconnected from any sites with j ≤ 0; see Fig. 30.
The region on the left j ≤ 0 has J ¼ 0 so its winding number
is N ¼ 0.

FIG. 30. A semi-infinite SSH model. The left region (gray area)
has J ¼ 0, hence a winding number N ¼ 0. The right region may
have N ¼ 0 or 1, depending on the ratio J=J 0 . In the latter case azero-energy edge state resides close to the boundary between thetwo domains.
Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019

FIG. 31. Energy spectrum of a SSH chain of M ¼ 10 dimers. A

pair of edge states close to zero energy exists when J 0 =J isnotably below 1. When J 0 =J increases above 1, these edge states
are gradually transformed into bulk states. Adapted from Delplace, Ullmo, and Montambaux, 2011.

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N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

the quasicontinua corresponding to the two bands for J 0 =J
above unity.
In practice, generating a boundary that preserves chiralsymmetry could be delicate to achieve. Thus edge states ofcold atom implementations of the SSH model are likely to be
shifted away from zero energy. In contrast, the topologicalprotection of the Majorana modes of the Kitaev chain
described next (Appendix A.5) does not require any such
local fine-tuning. There the protecting symmetry is theexact particle-hole symmetry of the Bogoliubov–de Gennes
Hamiltonian (90).
2. Gauge invariance and Zak phase

Strictly speaking the Zak phase is not a gauge invariant
quantity. Here we illustrate this point using again the SSH
model. In a local gauge transform, one can redefine thepositions of the sites Aj and Bj to be ja þ ξA and jb þ ξB ,leading to the following modified form of the Bloch wave
form:jψ q i ¼

XeiðjaþξA Þq uAq jAj i þ eiðjaþξB Þq uBq jBj i:

ðA4Þ

j

The vector hðqÞ defining the Hamiltonian in reciprocal space(18) is then changed intohx ðqÞ þ ihy ðqÞ ¼ ðJ 0 þ Jeiqa ÞeiqðξA −ξB Þ ;ðA5Þso that the Hamiltonian Ĥ q is not q periodic anymore. The
phase qðξA − ξB Þ is added to ϕq in the definition (21) of theeigenstates of Ĥq . This in turn adds the quantity πðξB − ξA Þ=ato the Zak phase [see, e.g., Atala et al. (2013)].
To decrease the ambiguity in the definition of the Zakphase, one may insist on keeping the q periodicity of ĤðqÞ,and thus hðqÞ, over the BZ. This restricts the local gaugetransform described above to ξB − ξA ¼ N 0 a, where N 0 is an
integer. The Zak phase then recovers the structure π × integer,
but the corresponding winding number is increased by N 0. In
practice this has no impact because physical consequences
involve only differences of winding numbers or phases
between regions of possibly different topology, which aregauge invariant.
3. Time-reversal symmetry of the SSH model

Additional symmetries can be exploited to simplify thecalculation (or measurement) of the topological invariant. The
SSH model has time-reversal symmetry, such that Ĥ q ¼ Ĥ−qand therefore ϕq ¼ −ϕ−q . This allows the winding number(23) to be written as an integral over half of the BZ
1
N¼
π

Z π=a
0

∂ϕq
1
dq ¼ ðϕπ=a − ϕ0 Þ:∂qπ

4. Adiabatic pumping for the Rice-Mele model

The Rice-Mele model generalizes the SSH model to thecase where the energies of sites A and B may differ by aquantity 2Δ. As explained in Sec. II.F, the vortex in ϕZakaround the gap-closing point (Δ ¼ 0 and J 0 =J ¼ 1) is a
topological invariant which can be viewed as a Chern numberin the periodic 2D space formed from crystal momentum−π=a < q ≤ π=a and time 0 < t ≤ T. This Chern number
counts the number of particles that are pumped through thesystem under this adiabatic cycle (Thouless, 1983).
To establish the link between the winding of ϕZak and thetransported mass, we start from the fact that the lattice remains
at any time periodic, so that the Bloch theorem ensures thatthe quasimomentum q is conserved during the evolution. Let
us assume a slow evolution and a nonclosing gap, so thattransitions to upper bands are negligible. The evolution
operator ÛðTÞ for a closed cycle of duration T in parameter
space ðJ 0 =J; Δ=JÞ is diagonal in the Bloch state basis jψ q i:jψ q i → ÛðTÞjψ q i ¼ eiγðqÞ jψ q i. Using the canonical commutation relation, the evolution of the position operator x̂ follows:Û † ðTÞx̂ ÛðTÞ ¼ x̂ − ∂ q γðq̂Þ, from which we can deduce theglobal displacement Δx of the atomic cloud by averaging overall states in the band. Now there are two contributions to γðqÞ:(i) The dynamical phase related to the energy EðqÞ, which has
a vanishing contribution to Δx because the average of thegroup velocity ∂ q EðqÞ over the band is zero; (ii) Zak phaseR
i 0T huq j∂ t uq idt which leads after average over the band to
Δx ¼

a
2π

Z T
0

∂ t ϕZak ðtÞdt:ðA7Þ

Because ϕZak ðTÞ equals ϕZak ð0Þ plus the winding number
corresponding to the vortex of Fig. 7, this proves the quantization of the displacement in units of the lattice period a.
5. The Kitaev model for topological superconductors

In this model one considers spinless fermions moving on a
1D chain of identical sites with the hopping coefficient J
between nearest neighbors; see Fig. 32. We assume that the
chain is coherently coupled to a superfluid reservoir that can
inject and remove pairs of fermions on neighboring sites. This
coupling is characterized by the real parameter Δ, which
stands for the p-wave superconducting gap induced by thereservoir. It also sets the chemical potential μ which controlsðA6ÞThus, information on the winding number N can be obtainedby measuring ϕq at just two points, q ¼ 0 and π=a. Since theangle ϕq is defined only mod2π, this method can determine
only if N is even or odd. Nevertheless, this partial informationRev. Mod. Phys., Vol. 91, No. 1, January–March 2019can be useful, notably for Z2 topological invariants (Fu andKane, 2007).

FIG. 32. Kitaev model: A 1D chain of identical sites withnearest-neighbor hopping and a coherent coupling to a superfluid
reservoir that injects and removes pairs of fermions on neighboring sites.

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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
the eigenstates and the corresponding energies ofĤ − μN̂ ¼

Xn−J ĉ†j ĉjþ1 þ ĉ†jþ1 ĉj − μĉ†j ĉjjoþΔ ĉj ĉjþ1 þ ĉ†jþ1 ĉ†j :ðA8Þ

For an infinite chain (or a finite chain with periodic boundary
conditions), translation invariance ensures that Ĥ − μN̂ takes asimple form in momentum space. It can be written in thestandard Bogoliubov–de Gennes form
c̃ˆ q

ðA9Þ

P
where the operator c̃ˆ †q ∝ j eiqja ĉ†j creates a particle withquasimomentum q and each Hamiltonian HBdG
is a 2 × 2
qmatrix:withiĈj ¼ ðĉ†j − ĉj þ ĉ†jþ1 þ ĉjþ1 Þ
2hy ðqÞ ¼ −2Δ sinðqaÞ;iĈM ¼ ðĉ†M − ĉM þ ĉ†1 þ ĉ1 Þ
2

ðA12Þ

so that the fĈj ; Ĉ†j g satisfy canonical fermionic commutation
rules. The Hamiltonian (A8) can then be writtenĤ − μN̂ ¼ 2J

M
−1 
X



1
Ĉ†j Ĉj −

j¼1

2

:

ðA13Þ

A ground state of Ĥ − μN̂ is obtained by solving

hz ðqÞ ¼ 2J cosðqaÞ þ μ;Ĉj jψ 0 i ¼ 0;

ðA10Þand hx ¼ 0. Although the present physical problem is different from the SSH and Rice-Mele models, we recover in
Eq. (A10) a Hamiltonian in reciprocal space which has asimilar structure. In particular, the search for distinct topological phases can be performed by analyzing the trajectory of
the vector hðqÞ when q travels across the BZ.
More precisely we see from Eq. (A10) that the vector halways lies in a plane (here y-z), which makes the discussion
formally similar to the SSH model. The quasiparticle excitation spectrum is set by jhðqÞj: this is gapped for all q in the BZ
provided jμj ≠ 2J. However, there are two topologicallydistinct phases; see Fig. 33. For jμj < 2J the two-component
vector hðqÞ encircles the origin, winding by 2π as q runs overthe BZ: this is the topological superconducting phase, forwhich there exist localized (Majorana) modes on the boundaries of a finite sample. For jμj > 2J the vector hðqÞ does notencircle 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,
evidenced by the trajectory of hðqÞ=jhðqÞj on the unit sphere,
as q scans the Brillouin zone. Left (μ < −2J) and right (μ > 2J):
topologically trivial phases, with a zero winding around theorigin. Middle (jμj < 2J): topological phase.
Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019ðA11Þfor j ¼ 1; …; M − 1 and!

1 X ˆ† ˆĤ − μN̂ ¼;ð c̃q ; c̃−q ÞHBdGq
2 qc̃ˆ †−q

HBdG¼ −hðqÞ · σqjhðqÞj ¼ 0 at some point in the BZ, i.e., a closing of thequasiparticle gap.
The existence of edge states for an open chain of M sites
can be simply revealed by taking the particular case Δ ¼ J
and μ ¼ 0 (Kitaev, 2001) (the energy spectrum as a functionof μ is shown in Fig. 34). In this case, one can perform acanonical transformation using the new fermionic operators:

j ¼ 1; …; M − 1:ðA14Þ

It is separated from the excited states by the gap 2J, as
expected from the above analysis for the infinite chain: For
J ¼ Δ and μ ¼ 0, jhðqÞj ¼ 2J is indeed independent of q. The
existence of edge states in this case originates from the factthat the nonlocal fermion mode ðĈM ; Ĉ†M Þ, acting on both endsof the chain, does not contribute to the Hamiltonian (A13). It
can therefore be filled or emptied at no energy cost, leading to
two independent ground states. More precisely, if jψ 0 i is theground state satisfying ĈM jψ 0 i ¼ 0, then jψ 1 i ¼ Ĉ†M jψ 0 i isalso a ground state. The states jψ 0 i and jψ 1 i correspond todifferent global parities of the total fermion number. The
situation is thus different from a usual superconductor, where

FIG. 34. Quasiparticle spectrum of Ĥ − μN̂ [Eq. (A8)] for an

open chain of 20 sites and Δ ¼ J. The pair of solutions ðE; −EÞwith E ≈ 0 that appears in the topological region jμj < 2J
represents the Majorana zero energy modes at the two ends ofthe chain.

015005-45

N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

the ground state is nondegenerate and represents a condensate
of Cooper pairs, hence a superposition of states with an evennumber of fermions.
As discussed in the main text, the gapless edge modes of the
topological superconductor are better viewed in terms of the
Majorana operatorsγ̂ 1 ≡ ĈM þ Ĉ†M ¼ iðĉ†M − ĉM Þ;

ðA15Þ

γ̂ 2 ≡ iðĈM − Ĉ†M Þ ¼ −ðĉ†1 þ ĉ1 Þ

ðA16Þ

takes account of the micromotion (related to how ti and tf
fall in the period T) and the time-independent effective
Hamiltonian Ĥeff controls the long-time behavior.
In general, the determination of the effective HamiltonianĤeff (or the closely related ĤFt0 ) is a difficult task. However, fora drive frequency ω that is large compared to other frequency
scales, the effective Hamiltonian can often be approximatedby the Magnus expansion in powers of 1=ω (Eckardt, 2017).
Writing the Hamiltonian in terms of its harmonics

in view of the fact that these Majorana operators are spatiallylocalized on each end of the chain.
APPENDIX B: FLOQUET SYSTEMS AND THE MAGNUS
EXPANSION

We mention here several key results for Floquet systems ofuse in the main text. We refer the interested reader to Eckardt(2017) for a comprehensive recent account.
Consider a Floquet system, defined by a time-varying
Hamiltonian ĤðtÞ that is periodic Ĥðt þ TÞ ¼ ĤðtÞ withperiod T ¼ 2π=ω. The time evolution over an integer numberof periods, from t0 to t0 þ NT, is described by the unitary
operatoriÛðt0 ; t0 þ NTÞ ≡ T exp −ℏZ t þNT0

Ĥðt0 Þdt0 ;ðB1Þ

where T denotes time ordering. This can be writtenÛðt0 ; t0 þ NTÞ ¼ ½Ûðt0 ; t0 þ TÞN ≡ e−ði=ℏÞĤt0 T ;FðB2Þwhich defines the effective Floquet Hamiltonian ĤFt0 in
terms of the logarithm of the time evolution operator over a
single period T. Clearly ĤFt0 is undefined up to the additionof integer multiples of 2πℏ=T ¼ ℏω, so its eigenvalues—
defining the Floquet spectrum—have a periodicity in energyof ℏω. The Floquet Hamiltonian describes the “stroboscopic” evolution of the system, i.e., at the selected times
t ¼ t0 ; t0 þ T; …; t0 þ NT. This is relevant for understanding the dynamics over time scales long compared to thedrive period T.
In addition to this stroboscopic evolution, the system
undergoes a “micromotion” on a time scale of the period
T. This causes the Floquet Hamiltonian ĤFt0 to depend on thetime 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 generaltime-evolution operator, from time ti to tf , as (Goldman and
Dalibard, 2014)Ûðti ; tf Þ ¼ e−iK̂ðtf Þ e−ði=ℏÞĤ ðtf −ti Þ eiK̂ðti Þ :
eff

ðB3Þ

This can be achieved with a time-independent effective
Hamiltonian Ĥ eff and a “kick” operator that is periodic intime K̂ðtÞ ¼ K̂ðt þ TÞ and that has vanishing average over
one period (Goldman and Dalibard, 2014). The kick operator
Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019Ĥm eimωtm¼−∞¼ Ĥ0 þ∞X

Ĥm eimωt þ Ĥ−m e−imωt

ðB4Þ

m¼1

where Ĥ†m ¼ Ĥ−m , the Magnus expansion leads to (Goldman
and Dalibard, 2014)
Ĥeff ¼ Ĥ0 þ
þ

∞
1 X
1
½Ĥ ; Ĥ ℏω m¼1 m m −m∞
1 X
1½½Ĥm ; Ĥ 0 ; Ĥ m 22ðℏωÞ m¼1 m2

þ ½½Ĥ−m ; Ĥ 0 ; Ĥ m  þ    :



t0

∞
X

ĤðtÞ ¼

ðB5Þ

The Magnus expansion underpins several results used in themain text as follows:(i) The leading term Ĥ 0 is the time-averaged Hamiltonian,as used in our discussion of inertial forces in Sec. III.C toconstruct nonzero Peierls phase factors for tunneling.
(ii) The first-order term is important in generating the nextnearest-neighbor hopping from circular shaking of the honeycomb lattice, as required to simulate the Haldane model.
Consider the three-site system shown in Fig. 35 with on-siteenergies Eα , α ¼ A, B, C, a constant tunnel matrix element −Jalong the links AB and AC, and no “bare” tunneling between
B and C:Ĥ ¼ −JðjBihAj þ jCihAj þ H:c:Þ þ

X
Eα P̂αðB6Þαwith the projectors P̂α ¼ jαihαj. Shaking at a frequency ωleads to sinusoidally varying energy offsets that we model as

FIG. 35. Hopping on three sites of the honeycomb lattice, with

time-varying on site energies. The unitary transformation (B7)maps this problem to time-varying tunneling matrix elements
between nearest neighbors A ↔ B and A ↔ C. Then the firstorder correction in 1=ω from the Magnus expansion leads to atunneling term of order J 2 =ℏω between next-nearest neighbors
B ↔ C, with nonzero Peierls phase factor; see Eq. (B12).

015005-46

N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

EB ðtÞ ¼ Δ cosðωtÞ and EC ðtÞ ¼ Δ cosðωt − ϕÞ, whereas
EA ðtÞ is set to zero by convention. The phase offset ϕ arises
from the circular shaking and the angle between the twonearest-neighbor bonds (typically ϕ ¼ 2π=3). Using theunitary transformation generalizing Eq. (52)
X i R t E ðt0 Þdt0 =ℏP̂α ;ÛðtÞ ¼
e 0 α

ðB7Þ

α

we convert these energy modulations into time-varying phasefactors on the hopping:ˆH̃ðtÞ
¼ eiðΔ=ℏωÞ sin ωt jBihAj−Jþ eiðΔ=ℏωÞ sinðωt−ϕÞ jCihAj þ H:c:ðB8Þ

Expanding in terms of the harmonics leads to
Ĥm¼ J m ðΔ=ℏωÞjBihAj þ J −m ðΔ=ℏωÞjAihBj−Jþ e−imϕ ½J m ðΔ=ℏωÞjCihAj þ J −m ðΔ=ℏωÞjAihCj;ðB9Þ
where J m are Bessel functions. Computing the first-ordercorrection to the effective Hamiltonian (B5), one finds
ð1ÞĤeff ≡∞1 X
1
½Ĥ ; Ĥ ℏω m¼1 m m −m

ðB10Þ

¼ −J eff ½eiπ=2 jBihCj þ e−iπ=2 jBihCj

ðB11Þ

∞
2J 2 XsinðmϕÞ½J m ðΔ=ℏωÞ2:ℏω m¼1m

ðB12Þ

J eff ¼

This describes a next-nearest-neighbor tunneling term,
between B and C sites, which inherits a Peierls phase factorof π=2. A particle that encircles the plaquette A → B → C →A picks up a gauge-invariant phase ψ AB ¼ −π=2, placing thesystem in the regime where the Haldane model has topologicalbands. It is interesting to note that for this 1=ω expansion, theleading term in Jeff corresponds to the m ¼ 1 contribution in
Eq. (B12), hence scales as 1=ω3 . Had we directly used theMagnus expansion (B5) for the initial Hamiltonian (B6),this would have required us to go up to the third order ofthe expansion, which would have been quite involved.
Fortunately the unitary transformation (B7) involves anintegral of the on-site energies over time, which provides again of a factor 1=ω. The Magnus expansion at order 1 appliedˆ is then sufficient to obtain the relevant effective
to H̃
Hamiltonian.
(iii) The second-order term in Eq. (B5), of order 1=ω2, is
responsible for the nonlocal interactions discussed inSec. V.A.1. These arise from the contribution of the
Hubbard interaction U to Ĥ0 , and from the oscillating
tunneling matrix elements in Ĥm≠0 in the double commutator,
leading the term of order J2 U=ðℏωÞ2 of Eq. (83). We refer theRev. Mod. Phys., Vol. 91, No. 1, January–March 2019

interested reader to Eckardt and Anisimovas (2015) for acomprehensive analysis of such terms.
APPENDIX C: LIGHT-MATTER INTERACTION

Here we describe the general form that the light-matterinteraction must take for two-photon interactions and describe
the specific structure of this interaction for alkali atoms. To
briefly summarize what follows, the light-matter interaction
can be divided into contributions from irreducible rank-0, -1,
and -2 spherical tensor operators. In alkali atoms the rank-0
and -1 contributions dominate, but in heavier atoms (orsufficiently close to atomic resonance) the rank-2 contribution
can be large. In addition, a similar description of single photontransitions (as would be relevant to systems coupling groundand metastable excited states) gives only the rank-0 and rank-1
contributions.
Now consider a system of ultracold alkali atoms in theirelectronic ground state manifold illuminated by one or several
laser fields which nonresonantly couple the ground states withthe lowest electronic excited states. In the presence of an
external magnetic field only, the light-matter Hamiltonian for
the atomic ground state manifold isĤ 0 ¼ Ahf Î · Ĵ þ

μB
B · ðgJ Ĵ þ gI ÎÞ;ℏ

ðC1Þ

where Ahf is the magnetic dipole hyperfine coefficient, and μB
is the Bohr magneton. The Zeeman term includes separate
contributions from Ĵ ¼ L̂ þ Ŝ (the sum of the orbital L̂
and electronic spin Ŝ angular momentum) and the nuclear
angular momentum Î, along with their respective Landég factors. We next consider the additional contributions to
the atomic Hamiltonian resulting with off-resonant interaction
with laser fields.
As Deutsch and Jessen (1998), Dudarev et al. (2004), andSebby-Strabley et al. (2006) observed, conventional spinindependent (scalar U s ) optical potentials acquire additional
spin-dependent terms near atomic resonance: the rank-1(vector Uv ) and rank-2 tensor light shifts (Deutsch andJessen, 1998). For the alkali atoms, adiabatic elimination of
the excited states labeled by J ¼ 1=2 (D1) and J ¼ 3=2 (D2)yields an effective atom-light coupling Hamiltonian for the
ground state atoms (with J ¼ 1=2):
ĤL ¼ Ĥ0 þ Ĥ1 þ Ĥ 2 ¼ us ðE · EÞ þ

iuv ðE × EÞ
· J þ Ĥ2 :ℏðC2Þ

The rank-2 term Ĥ 2 is negligible for the parameters ofinterest and henceforth neglected. Here E is the opticalelectric field, uv ¼ −2us ΔFS =3ðω − ω0 Þ determines the vector light shift, ΔFS ¼ ω3=2 − ω1=2 is the fine-structure splitting, ℏω1=2 and ℏω3=2 are the D1 and D2 transition energies,and ω0 ¼ ð2ω1=2 þ ω3=2 Þ=3 is a suitable average. us sets thescale of the light shift and proportional to the atoms acpolarizability.
The contributions from the scalar and vector light shifts
featured in H L can be independently specified with informed
choices of laser frequency ω and intensity. Evidently, the015005-47

N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atomshiuκ ¼ δ þ i v ðEω− × Eω− þ Eωþ × Eωþ Þ · ez ezℏ
uv− Im½ðEω− × Eωþ Þ · ðex − iey Þexℏ
uv
− Re½ðEω− × Eωþ Þ · ðex − iey Þey :ℏvector light shift is a contribution to the total Hamiltonian
acting like an effective magnetic field
Beff ¼

iuv ðE × EÞ
μB gJ

ðC3Þ

that acts on Ĵ and not the nuclear spin Î. Instead of using
the full Breit-Rabi equation (Breit and Rabi, 1931) for the
Zeeman energies, we assume that the Zeeman shifts are smallin comparison with the hyperfine splitting—the linear, or
anomalous, Zeeman regime—in which case, the effective
Hamiltonian for a single manifold of total angular momentumF̂ ¼ Ĵ þ Î states is
H 0 þ HL ¼ us ðE · EÞ þ

μB g FðB þ Beff Þ · F̂:ℏðC4Þ

Note that Beff acts as a true magnetic field and adds vectoriallywith B, and since jgI =gJ j ≈ 0:0005 in the alkali atoms, wesafely neglected a contribution −μB gI Beff · Î=ℏ to the atomic
Hamiltonian. We also introduced the hyperfine Landé g factor
gF . In 87Rb’s lowest energy manifold with F ¼ 1, for which
J ¼ 1=2 and I ¼ 3=2, we get gF ¼ −gJ =4 ≈ −1=2. In thefollowing, we always consider a single angular momentum
manifold labeled by F and select its energy at zero field as thezero of energy.
Bichromatic light field: Consider an ensemble of ultracold
atoms subjected to a magnetic field B ¼ B0 ez . The atoms are
illuminated by several lasers with frequencies ω and ω þ δω,where δω ≈ jgF μB B0 =ℏj differs by a small detuning δ ¼
gF μB B0 =ℏ − δω from the linear Zeeman shift between mF
states (where jδj ≪ δω). In this case, the complex electric field
E ¼ Eω− expð−iωtÞ þ Eωþ exp ½−iðω þ δωÞt contributes tothe combined magnetic field, giving
B þ Beff ¼ B0 ez þ

iuv
½ðEω− × Eω− Þ þ ðEωþ × Eωþ Þ
μB g Jþ ðEω− × Eωþ Þe−iδωt þ ðEωþ × Eω− Þeiδωt :

ðC5Þ

The first two terms of Beff add to the static bias field B0 ez , andthe remaining two time-dependent terms describe transitions
between different mF levels. Provided B0 ≫ jBeff j and δω arelarge compared to the kinetic energy scales, the Hamiltonian
can be simplified by time averaging to zero the timedependent terms in the scalar light shift and making therotating wave approximation (RWA) to eliminate the timedependence of the coupling fields. The resulting contribution
to the Hamiltonian
Ĥ RWA ¼ UðrÞ1̂ þ κðrÞ · F̂;ðC6Þ

where we identify the scalar potentialUðrÞ ¼ us ðEω− · Eω− þ Eωþ · Eωþ Þ

Rev. Mod. Phys., Vol. 91, No. 1, January–March 2019

Although this effective coupling is directly derived fromthe initial vector light shifts, κ is composed of both static
and resonant couplings in a way that goes beyond therestrictive Beff ∝ iE × E form. This enables topologicalstate-dependent lattices.

APPENDIX D: BERRY CURVATURE AND UNIT CELL
GEOMETRY

As noted in the main text, it can be convenient to perform
a unitary transformation of the Bloch HamiltonianĤq0 ¼ Û q Ĥ q Û †q , with Û q a wave-vector-dependent unitary
operator. Such transformations can be used to renderthe Hamiltonian periodic in the BZ, but also can relatedifferent choices of unit cell without change of periodicity.
They must leave all physical observables unchanged.
The energy spectrum is invariant under this unitarytransformation E0q ¼ Eq , while the Bloch states transformas ju0q i ¼ Û q juq i. (In this Appendix we drop the band
index for clarity.) The new Berry connection in reciprocal
space isA0 ¼ ihu0q j∇q ju0q i ¼ A þ ihuq jÛ †q ½∇q Û q juq i:

ðD1Þ

It is interesting to note that, in general, this transformation
causes the Berry curvature to change. How can one
reconcile this with the fact that the Berry curvature has
measurable physical consequences, e.g., within semiclassical dynamics (Sec. IV.B.2)? As we shall see, the answer
lies in noting that these unitary transformations can lead tochanges of the positions of the orbitals within the unit
cell, i.e., changing the internal geometry of the unit cell.
We illustrate this for a two-band model, with a unitarytransformation
Û q ¼ exp ð12iq · ρσ̂ z Þ:ðD2ÞThis is the transformation used in our discussion of the
Haldane model (33). It leads to the change
A0 ðqÞ ¼ AðqÞ − 12 ρhσ̂ z i

ðD3Þ

of the Berry connection, where we defined hσ̂ z i ≡ huq jσ̂ z juq i.
Hence, the Berry curvature becomesΩ0 ðqÞ ≡ ∇q × A0 ¼ ΩðqÞ − 12½∇q hσ̂ z i × ρ:ðC7Þand the RWA effective magnetic field. This expression is valid
for gF > 0 (for gF < 0 the sign of the ex and iey terms would
both be positive, owing to selecting the opposite complex
terms in the RWA)

ðC8Þ

ðD4Þ

In general, the Berry curvature changes, Ω0 ðqÞ ≠ ΩðqÞ.
However, since the difference is a total derivative, its integral
over the BZ vanishes, so there is no change in the Chern
number of the band.

015005-48

N. R. Cooper, J. Dalibard, and I. B. Spielman: Topological bands for ultracold atoms

For a consistent application of the unitary transformation,
one must consider how all relevant physical quantities transform. Semiclassical dynamics describes the velocity of a wave
packet centered on q and r in response to a uniform force. A
uniform force in the original basis arises from a potentialV̂ ¼ −F · r̂:ðD5Þ

Under the unitary transformation this becomesÛ q V̂ Û †q ¼ −F · r̂ − 12 F · ρσ̂ z :ðD6Þ

We used the fact that, with q replaced by the momentum
operator, the unitary operator (D2) effects translations in real
space, albeit in a spin-dependent manner, r̂ → r̂ þ ð1=2Þρσ̂ z .
Indeed, this transformation encodes the important physicaleffect of the unitary transformation: it amounts to spatial
displacement of the orbitals within the unit cell, here separatingσ z ¼ 1 by ρ. In the new basis the force F has two effects.
First, it provides a uniform potential gradient which leads toℏ_q ¼ F as usual. Second, it applies a state-dependent potentialthat shifts the energy of the wave packet by −ð1=2ÞF · ρhσ̂ z iwith the expectation value taken for the wave packet’s
momentum q. This second contribution can be incorporated
as an F-dependent change in the dispersion relation
E0q ¼ Eq − 12 F · ρhσ̂ z i:

ðD7Þ

The displacement of orbitals within the unit cell also changes
the velocity operator. In the new basis the velocity is
11v̂0 ≡ ∇q Ĥ0q ¼ ∇q ½Û q Ĥq Û †q 
ℏ
ℏ
¼

1
i
Û q ½∇q Ĥ q Û †q − ρ½Ĥ0q ; σ̂ z ℏ2ℏ

1¼ Û q v̂Û †q − ρσˆ_ z :
2

ðD8Þ
ðD9Þ
ðD10Þ

There is a correction to the velocity v̂ ≡ ð1=ℏÞ∇q Ĥq beyond
the mere unitary transformation Û q v̂Û †q . In the last line thiscorrection is written in terms of the rate of change of σ̂ z (inthe Heisenberg picture) to indicate its physical origin: themotion of a particle between orbitals within the unit cellcorresponds to motion through space with nonzero velocity.
In the semiclassical dynamics of the wave packet, this gives
the correction
v0 ¼ v − 12 ρhσˆ_ z i ¼ v − 12 ρ_q · ∇q hσ̂ z i:

ðD11ÞStarting from the standard semiclassical dynamics (74) in theoriginal (unprimed) basis, and using the transformations(D4), (D7), and (D11), one recovers the correct semiclassicaldynamics in the new (primed) basis
v0 ¼

1∇ E0 þ Ω0 × q_ :
ℏ q q

ðD12Þ

Thus, unitary transformations of this form lead to changes in
the positions of the orbitals within the unit cell: theseRev. Mod. Phys., Vol. 91, No. 1, January–March 2019changes in cell geometry modify the effect of the forces
applied and the velocity on the scale of the unit cell. These
modifications compensate the change in the Berry curvature
to recover the correct semiclassical dynamics.

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