Topological bands for ultracold atoms

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

Figure 1

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).

Figure 2

The winding number for the SSH model. The curves plot the locus of $(h_x,h_y)$ as $q$ runs over the BZ. For $J^{\prime }/J>1$ the vector $(h_x,h_y)$ does not encircle the origin, but for $J^{\prime }/J<1$ it encircles the origin once, indicating that these two cases have winding numbers (23) of $N=0$ and 1, respectively.

Figure 3

(a) Real-space honeycomb lattice for the Haldane model. (b) Illustration of the unit vector $\mathit{e}(\mathit{q})$ in the topological phase of the Haldane model. The dashed line shows the conventional BZ, whose corners are the points ${\mathit{Q}}_{\pm }$. The arrows show the projection of $\mathit{e}(\mathit{q})$ on the plane $(e_x,e_y)$, while the colors and contours indicate $e_z$. The unit vector wraps once over the sphere within the BZ, indicating the topological character with $\mathcal{C}=1$.

Figure 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 such an orbit from the (hard wall) edge leads to a skipping motion around the sample edge.

Figure 5

An edge of the SSH model is represented by $h_x+\mathrm{i}{h}_y=|h_{q,M}|{\mathrm{e}}^{\mathrm{i}\varphi (q,M)}$, which depends both on the wave vector $q$ and on some control parameter $M$ that varies smoothly between $M_{-}$ at $x\ll 0$ and $M_{+}$ at $x\gg 0$. The number of vortices of $h_x+\mathrm{i}{h}_y$ within this region can be computed by integrating $\mathbf{\nabla }\varphi (q,M)$ around the boundary. The integrals along $q=\pm \pi /a$ cancel (since the BZ is periodic), leaving only the integrals along $M=M_{\pm }$, which give the difference of winding numbers $N_{-}-N_{+}$.

Figure 6

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 ${\tilde{q}}_y=q_y-Q_y^{+}$ is conserved. (b) The spectrum has a continuum of states in the bulk, 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 band corresponding to the edge mode on the left (right) boundary. We show only the part of the spectrum close to the ${\mathit{Q}}^{+}$ point, at which the boundaries force a gap closing and at which the edge states appear.

Figure 7

Left: Zak phase of the lower band of the Rice-Mele model, represented by the unit vector $(\cos {\varphi }_{\mathrm{Zak}},\sin {\varphi }_{\mathrm{Zak}})$. The phase changes by $2\pi$ around the point $\mathrm{\Delta }=0$ and $J^{\prime }/J=1$ 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 ($\mathrm{\Delta }=0$, $J^{\prime }/J<1$), obtained when $q$ scans the Brillouin zone. The Zak phase, given by half the subtended solid angle, is equal to $\pi$. Green solid line: Trajectory in the nontopological case of the SSH model ($\mathrm{\Delta }=0$, $J^{\prime }/J>1$) with a zero subtended solid angle. Black dotted line: Closed trajectory obtained for the Rice-Mele model with a nonzero energy offset $\mathrm{\Delta }$. A continuous variation of $\mathrm{\Delta }$ and $J^{\prime }/J$ in the Rice-Mele model allows one to connect the topological and nontopological SSH trajectories, without going through the singular point $J^{\prime }/J=1$, $\mathrm{\Delta }=0$.

Figure 8

The relationship of the change in the Zak phase for a 1D band insulator, with momentum $-\pi /a<q\le \pi /a$, under adiabatic variation of parameters in time around a cycle of period $T$, i.e., with time $0<t\le T$. The change in Zak phase between times $t_1$ and $t_2$ is represented by the line integral around a contour $\mathcal{L}$. When $\mathcal{L}$ is extended to include the full $q–t$ plane, the periodicity in $q\to q+2\pi /a$ and $t\to t+T$ ensures that the line integral must by an integer multiple of $2\pi$. This integer is the Chern number defined in the $q–t$ plane.

Figure 9

The Harper-Hofstadter model. (a) Harper-Hofstadter lattice geometry with symmetric hopping $J$ and a flux ${\varphi }_{\mathrm{AB}}$ in each plaquette. (b) Harper-Hofstadter lattice geometry with flux ${\varphi }_{\mathrm{AB}}=2\pi /3$ 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 ${\varphi }_{\mathrm{AB}}=2\pi /3$ showing the three topological bands with Chern numbers $+1$, $-2$, and $+1$ built from the three inequivalent sites within the magnetic unit cell.

Figure 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 ${\varphi }_{\mathrm{AB}}$. 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 $q_{x}a=\mathit{q}·{\mathit{a}}_1$ and $q_{y}a=\mathit{q}·{\mathit{a}}_2$. (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 ${\varphi }_{\mathrm{AB}}=\pi /2$ and “tilt” $\mathrm{\Delta }=0$, $J^{\prime }=J/10$. (e) Band structure at the topological transition, marked by the blue circle in (b), with ${\varphi }_{\mathrm{AB}}=\pi /2$ and $\mathrm{\Delta }=-3\sqrt{3}{J}^{\prime }$, showing the formation of a single Dirac point.

Figure 11

Implementation of the SSH and Rice-Mele models. (a) Representative laser configuration: A pair of overlapping lasers with wavelengths ${\lambda }_0$ and $2{\lambda }_0$ subject a cloud of ultracold atoms to 1D optical lattices, with periods ${\lambda }_0/2$ and ${\lambda }_0$, respectively, and a spatial relative phase $\varphi$. (b) Energies of these two lattices and combined potential, showing the long-period lattice shifted in position with respect to the short period lattice by a controllable phase shift $\varphi$. The SSH model is realized for $\varphi =\pi /2+n\pi$ (for integer $n$), in which case all minima have the same energy and are separated by potential barriers with staggered height. An example of band structure in this case is shown in Fig. 12. The choices $\varphi =n\pi$ (for integer $n$) lead to equal barrier heights between adjacent sites, and staggered site energies, but with $J^{\prime }=J$, $\mathrm{\Delta }\ne 0$. The displayed data are for an intermediate case of $\varphi =\pi /4$ where $J^{\prime }\ne J$ and $\mathrm{\Delta }\ne 0$. (c) Zak phase for the lower pair of bands controlled by tuning the phase shift $\varphi$ between the long and short period lattices and the strength of the long-period lattice, as represented by the unit vector $(\cos {\varphi }_{\mathrm{Zak}},\sin {\varphi }_{\mathrm{Zak}})$.

Figure 12

Bipartite lattice band structure compared to the Rice-Mele model. Left: Band structure computed for the superlattice potential given by Eq. (50) with $V_{\text{short}}=6E_{\mathrm{R}}$, $V_{\mathrm{long}}=E_{\mathrm{R}}$, and a relative phase $\varphi =\pi /2$ corresponding to the particular case of the SSH model. Right: Zoom on the lower pair of bands (continuous lines). The energy offset between adjacent minima and a fit to the prediction (41) for the band structure of the Rice-Mele model (dashed lines) allow one to extract the practical values of $\mathrm{\Delta }$, $J$, and $J^{\prime }$. (Here $\mathrm{\Delta }=0$, $J=0.069E_{\mathrm{R}}$, and $J^{\prime }=0.037E_{\mathrm{R}}$.) The dotted lines show the (folded) lowest band for $V_{\mathrm{long}}=0$, up to a global energy shift.

Figure 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 ([218]). (d) Shaking in 2D can break time-reversal symmetry giving rise to topological lattices.

Figure 14

Laser-induced hopping. (a) 2D square lattice (right) with a potential gradient along ${\mathit{e}}_m$ (vertical) illuminated by a traveling wave potential. The coupling of any pair sites of this lattice $|j,m⟩$ and $|j,m+1⟩$ is qualitatively described as a two-level system with detuning $2\mathrm{\Delta }$ coupled by the traveling wave. (b) Technique for creating the half-flux Harper-Hofstadter Hamiltonian in tilted spin-dependent lattices as implemented in MIT ([167]) similar to [2].

Figure 15

Spin-dependent topological lattices. (a) Level diagram for three-level total angular-momentum $f=1$ case with $m_F$ states labeled, as is applicable for the common alkali atoms ${}^7\mathrm{Li}$, ${}^{23}\mathrm{Na}$, ${}^{39}\mathrm{K}$, ${}^{41}\mathrm{K}$, and ${}^{87}\mathrm{Rb}$. For reference, the diagram shows the decomposition of these optical fields into ${\sigma }_{\pm }$ and $\pi$, but as discussed in the text, this is not an overly useful way of considering this problem. (b) Laser geometry for a typical optical flux lattice (left) producing a real-space Berry curvature with nonzero average (right). (c) Experimental geometry for spin-dependent topological lattice ([222]). (d) Directly observed topological band structure (left) and computed (right). From [222].

Figure 16

(Left column) Amplitude $\propto \sin {\theta }_{\mathit{q}}$ and (middle column) phase ${\varphi }_{\mathit{q}}$ 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 $|\mathit{b}|$ squared. From [79].

Figure 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 $n_m(k)$ along the real direction $x$ for the three values of the pseudospin $m$ corresponding to the synthetic direction. The value $m=-1/2$ can be viewed as the bulk, whereas $m=+3/2$ and $m=-5/2$ corresponding to the opposite edges of the sample along the synthetic dimension. Lower row: Function $h_m(k)=n_m(k)-n_m(-k)$. From [161].

Figure 18

Two-band spectrum of the SSH Hamiltonian and Zak phase measurement. An initial wave packet is prepared at the bottom of the lowest band (left column). Each atom is in the superposition of the $|\pm ⟩$ 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 [14].

Figure 19

(a) Brillouin zone and Dirac points ${\mathit{Q}}_{\pm }$ for the brick-wall lattice of Fig. 10. (b) When the degeneracy is lifted by introducing an energy offset between the $A$ and $B$ sites, the subbands are topologically trivial. For the lowest band, the Berry curvature has opposite signs in the vicinity of ${\mathit{Q}}_{\pm }$. The anomalous velocity (white arrows) thus has opposite chirality at these points. (c) Lifting of degeneracy obtained by adding complex next-nearest-neighbor (NNN) couplings ([101]). The Berry curvature then keeps a constant sign over the BZ, and the anomalous velocity has the same chirality at both points. Adapted from [125].

Figure 20

Aharonov-Bohm interferometer around a Dirac point. Starting from a wave packet localized at the center of the BZ of a graphenelike optical lattice, one measures the phase difference $\phi$ between the two paths of an interferometer ending at the quasimomentum ($k_x=0$, $k_y$). When one of the Dirac points $K$ or $K^{\prime }$ is inside the enclosed area of the interferometer, the measured value for $\phi$ is in good agreement with the prediction $\phi =\pi$. From [72].

Figure 21

Edge-magnetoplasmon trajectories, where the displacement was obtained by integrating the velocity. From [220].

Figure 22

Topological pumping with a 1D superlattice described by the potential (50). Left: The phase $\varphi$ of the long lattice [see Eq. (50)] is varied from 0 to $2\pi$ from top to bottom. Initially the phase $\varphi =0$ and the particle is supposed to be localized on the site $A_j$ of a given lattice cell $j$. In the limiting case where the energy difference between $A$ and $B$ sites is large compared to the tunnel matrix elements, this state would be stationary if $\varphi$ was kept at the value 0. When $\varphi$ is increased up to $\pi /2$, the sites $A_j$ and $B_j$ have the same energy and the particle is adiabatically transferred to $B_j$. Note that we neglect here the tunneling of the particle from $A_j$ to $B_{j-1}$, assuming that it is inhibited by the large barrier between these two sites. The particle then remains in $B_j$ until the phase reaches the value $3\pi /2$, when the particle again undergoes an adiabatic transfer, now from $B_j$ to $A_{j+1}$. (Here again we neglect tunneling across the large barrier now present between $B_j$ and $A_j$.) When the phase $\varphi =2\pi$ the potential is back to its initial value and the particle has moved by one lattice site. Note that a motion in the opposite direction occurs if the particle starts for the site $B_j$ when $\varphi =0$. Right: In the two-band approximation corresponding to the Rice-Mele model, the system performs a closed loop around the origin in the parameter space $(J^{\prime }-J,\mathrm{\Delta })$.

Figure 23

Quantization of the displacement of a cloud of fermionic ${}^{171}\mathrm{Yb}$ atoms placed in a 1D optical superlattice described by the Rice-Mele Hamiltonian. Depending on the closed loop in parameter space $(J^{\prime }-J\equiv 2\delta ,\mathrm{\Delta })$, the displacement can be positive, zero, or negative. The time $T$ represents the duration of a pump cycle. From [177].

Figure 24

Transport measurement in a square lattice with a flux or plaquette ${\varphi }_{\mathrm{AB}}=\pi /2$ (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 $x(t)$ with time, in agreement with the expected Chern index of this subband. At longer times, heating processes equalize the populations of the subbands and the Hall drift stops. When the flux is zero, no Hall current is observed (data 2). Adapted from [3].

Figure 25

A vortex lattice configuration for weakly interacting bosons in the Harper-Hofstadter model at flux ${\varphi }_{\mathrm{AB}}=2\pi /3$. The gauge-invariant currents flow on the links between the sites of the square 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 that lie along diagonal lines. These are the lattice equivalents of the “vortex cores,” now distorted from the triangular lattice expected in the continuum models, to be pinned to the plaquettes of the square lattice. There are weak countercirculating currents around other plaquettes, such that the net particle flow vanishes, when coarse grained on scales large compared to the lattice spacing.

Figure 26

The change of band Hamiltonian that causes the topological index of the lowest band $\nu$ to change, via the closing of a band gap.

Figure 27

Floquet energy spectrum in units of $\hslash \omega$ for the driven Rice-Mele model in the topological (top) and nontopological (bottom) cases. The parameters $(J^{\prime }/J,\mathrm{\Delta }/J)$ are varied along a circle of radius 0.5 with angular frequency $\omega =0.23J/\hslash$. The circle is centered on the point (2,0) in the nontopological case and on 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 $q=0$ 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 $\hslash \omega$.

Figure 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 ${\delta }_{AB}$. (b) Trajectories of particles initially in the bulk (blue closed 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 [203].

Figure 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 in quasienergy allows the existence of edge states even in settings where the bulk bands are all topologically trivial. From [203].

Figure 30

A semi-infinite SSH model. The left region (gray area) has $J=0$, hence a winding number $N=0$. The right region may have $N=0$ or 1, depending on the ratio $J/J^{\prime }$. In the latter case a zero-energy edge state resides close to the boundary between the two domains.

Figure 31

Energy spectrum of a SSH chain of $M=10$ dimers. A pair of edge states close to zero energy exists when $J^{\prime }/J$ is notably below 1. When $J^{\prime }/J$ increases above 1, these edge states are gradually transformed into bulk states. Adapted from [66].

Figure 32

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

Figure 33

Distinct topological phases of the Kitaev model, evidenced by the trajectory of $\mathit{h}(q)/|\mathit{h}(q)|$ on the unit sphere, as $q$ scans the Brillouin zone. Left ($\mu <-2J$) and right ($\mu >2J$): topologically trivial phases, with a zero winding around the origin. Middle ($|\mu |<2J$): topological phase.

Figure 34

Quasiparticle spectrum of $\widehat{H}-\mu \widehat{N}$ [Eq. (a8)] for an open chain of 20 sites and $\mathrm{\Delta }=J$. The pair of solutions $(E,-E)$ with $E\approx 0$ that appears in the topological region $|\mu |<2J$ represents the Majorana zero energy modes at the two ends of the chain.

Figure 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 $A\leftrightarrow B$ and $A\leftrightarrow C$. Then the first-order correction in $1/\omega$ from the Magnus expansion leads to a tunneling term of order $J^2/\hslash \omega$ between next-nearest neighbors $B\leftrightarrow C$, with nonzero Peierls phase factor; see Eq. (b12).