Introduction

Novel lattice geometries have been known to give rise to interesting many-body phenomena including topological states of quantum matter. In the context of ultracold quantum gases, optical lattices engineered with interfering laser beams can realize specific configurations of potentials of single or multiple periods not found in nature. For instance, double-well superlattices1,2 have matured into a powerful tool for manipulating orbital degrees of freedom3,4,5,6,7,8,9,10. Controls of atoms in the s and p orbitals of the checkerboard6 and hexagonal8 optical lattices have also been demonstrated, and correlation between these orbitals tends to give exotic quantum states6,8,11,12,13. The spatial symmetry of the orbital wavefunction dictates the complex hopping amplitudes between nearby sites. Under certain circumstances, as for the uneven double wells, the orbital hopping pattern is sufficient for producing topologically nontrivial band structures14.

Motivated by these developments, here we consider a lattice of uneven double wells where fermionic atoms are loaded up to the s and p orbital levels of the shallow and deep wells, respectively. This new configuration of orbitals beyond solid state materials realizes a topological band insulator phase of interacting fermions with edge states. Topological phase transitions to trivial band and Mott insulators are predicted as a feature of the topological state. Remarkably, the edge states persist upon dimensional crossover, which makes it promising to realize in optical lattice experiments. This route of achieving topological band insulators and superfluids is distinct from previous proposals that require rotation of the gas15,16, artificial gauge fields17, spin–orbital coupling18,19,20,21 or p-wave triplet pairing22.

Results

One-dimensional orbital ladder

We will first focus on a one-dimensional (1D) ladder system illustrated in Fig. 1b. This corresponds to the quasi-1D limit of a standard double-well optical lattice, with the optical potential given by

Figure 1: The sp orbital ladder reduced from a 2D double-well optical lattice.
figure 1

(a) An optical lattice of uneven sub-wells (light and dark blue), with parameters Vx/V1=0.3, V2/V1=1 and φ=0.6π, develops high barriers (red ridges) in the y direction, slicing the lattice into dynamically decoupled uneven two-leg ladders. (b,c) Schematic side and top views, respectively, of the ladder illustrate tunneling (t′s) of fermions prepared in the degenerate s and px levels. (d) Topological winding of Hamiltonian across the Brillouin zone.

This optical lattice has a double-well structure in the y direction. For V1,2>>Vx, there is a large tunnelling barrier between double wells in the y direction, so in low-energy physics the two-dimensional (2D) system decouples into an array of dynamically isolated two-leg ladders of A and B sub-wells (Fig. 1), with each ladder extending in the x direction. The relative well depth of the two legs is controlled by the phase φ and further by the ratio V2/V1. We will focus on a situation, similar to the setup in the experiment6, where the s orbital of leg A has roughly the same energy as the px orbital of leg B (other p orbitals have much higher energy). For example, one can choose V1=40ER, V2=20ER, Vx=4.0ER and φ=0.9π in experiments, where ER is the recoil energy , with m the mass of atom and k the wave number of the laser. Such a setup will give the A (B) wells a depth 2.7ER (8.1ER). The tunnelling rates of the various orbitals illustrated in Fig. 1c are given as ts=0.053ER, tp=0.40ER and tsp=0.064ER in the tight binding approximation. The lattice constant a=π/k will be set as the length unit in this paper. We now consider a single species of fermions occupying these orbitals, with the low-lying s orbital of leg B completely filled. Alternatively fermions can be directly loaded into the px orbital of leg B, leaving the low-lying s empty, by techniques developed in recent experiments4,6,10. With these techniques, long-lived meta-stable states of atoms in high orbitals with life time on the order of several hundred milliseconds are demonstrated achievable6.

The Hamiltonian of the sp orbital ladder is then given by

where , with and being fermion creation operators for the s and px orbitals on the A and B leg, respectively. The relative sign of the hopping amplitudes is fixed by parity symmetry of the s and px orbital wavefunctions. As depicted in Fig. 1c, the hopping pattern has a central role in producing a topological phase. With a proper global gauge choice, ts, tp and tsp are all positive. The rung index j runs from 0 to L−1 with L the system size. We consider half filling (one particle per unit cell), for which the chemical potential μ=0, and the Hamiltonian is particle-hole symmetric under transformation . Topologically nontrivial band structure of this sp orbital ladder, which shall be shown analytically next, may be heuristically speculated from the following comparison: the staggered quantum tunnelling resembles spin–orbit interaction18,19,20,21,23, when the s and p orbital states are mapped to pseudo-spin-(1/2) states. Such a staggered tunnelling can also naturally arise in the checkerboard optical lattice already engineered in the experiment6 by increasing the laser strength in one direction to reach the quasi-1D limit. The physics of the sp orbital ladder is also connected to the more familiar frustrated ladder with magnetic flux24, but the sp orbital ladder appears much easier to realize experimentally.

Topological band structure and zero energy edge states

In the momentum space the Hamiltonian takes a simple and suggestive form,

where h0(k)=(tpts)cos(k), hx=0, hy(k)=2tspsin(k) and hz(k)=−(tp+ts)cos(k). Here, is the unit matrix, σx, σy and σz are Pauli matrices in the 2D orbital space. The energy spectrum consists of two branches,

with a band gap Eg=min(2tp+2ts, 4tsp), which closes at either tsp=0 or ts+tp=0. An interesting limit that highlights the nontrivial band structure of our model is that when tp=ts=tsp, the two bands are both completely flat. To visualize the topological properties of the band structure, one notices that as k is varied from −π through 0 to +π, crossing the entire Brillouin zone, the direction of the vector winds an angle of 2π (Fig. 1d). The corresponding Berry phase is half of the angle, γ=π (see Supplementary Note 1). The orbital ladder Hamiltonian H0 belongs to the symmetry group in the notation of 25, as it has both particle-hole and time-reversal symmetry, in addition to the usual charge U(1) symmetry. Therefore, at half filling, it is a topological insulator characterized by an integer topological invariant, in this case the winding number 1, according to the general classification scheme of topological insulators and superconductors25,26.

The nontrivial topology of the ladder system also manifests in existence of edge states. It is easiest to show the edge states in the flat band limit, ts=tp=tspt, by introducing auxiliary operators, . Then the Hamiltonian only contains coupling between φ+ and φ of nearest neighbours, but not among the φ+ (or φ) modes themselves,

Immediately, one sees that the operators φ+(0) and φ(L−1) at the left and right ends are each dynamically isolated from the bulk, and do not couple to the rest of the system (Fig. 2a). These loners describe the two edge states at zero energy. They are the bonding and anti-bonding modes of s and p orbitals, that is, shared by the two rungs of the ladder. For general parameters away from the flat band limit, the wavefunctions of the edge states are found not to confine strictly at j=0 or L−1, but instead decay exponentially into the bulk with a characteristic length scale . For the lattice strength given above, the decay width is estimated to be two to three times of the lattice constant. Only for , which is potentially reachable in experiments, ξ→0, the edge states are completely confined at the edges. For tsp=0, the bulk gap closes and ξ→∞. The analytical expression for the edge state wavefunctions in the general case are discussed in Supplementary Note 2. The existence of zero energy edge states is also confirmed by numerical calculation as shown in Fig. 2b.

Figure 2: Bulk and edge eigenstates of the orbital ladder.
figure 2

(a) A pictorial representation of the simplified Hamiltonian in the flat band limit ts=tp=tsp showing the emergence of isolated edge modes. The definition of the φ operator is given in the main text. (b) The eigen energy of a ladder with finite length L=12 showing two degenerate zero energy states inside the gap. (c) The probability distribution of the in-gap states (equation (10)) for varying strengths of inter-orbital interaction Usp. The in-gap states are shown localized on the edges and survive against finite interaction. In b and c, we choose ts=tp=2tsp (taken as the energy unit).

Fractional charge and topological anti-correlations

For a finite ladder of length L with open boundary condition and populated by L fermions (half filling), L−1 fermions will occupy the valence band (bulk states) and one fermion will occupy the edge states (Fig. 2). As the two edge states are degenerate, the ground state has a double degeneracy. The edge state is a fractionalized object carrying half charge (cold atoms are charge neutral, here charge refers to the number of atoms). This becomes apparent if we break the particle-hole symmetry by going infinitesimally away from half filling, for example, tuning chemical potential μ=0+. Then, the valence band and the two edge states will be occupied. With a charge density distribution on top of the half filled background defined as , one finds , where d satisfies ξ<<d<<L (for example, take d=5ξ). A characteristic feature of the topological insulator (with the number of atoms fixed) is the topological anti-correlation of the charge at the boundaries,

In the sharp confinement limit, ξ→0, the edge states are well localized at the two ends of the ladder. The topological anti-correlation simplifies as 〈ρ(0)ρ(L−1)〉=−(1/4), and the half charge is also well localized, that is, . As the edge states are well isolated from the bulk states by an energy gap, they are stable against local Gaussian fluctuations. The coupling between the two edge states vanishes in the thermodynamic limit (L→∞), because the hybridization induced gap scales as exp(−αL) as L→∞27.

Time-reversal symmetry breaking and topological phase transition

An interesting topological phase transition to a trivial insulator can be tuned to occur when rotating the atoms on individual sites, for example, by applying the technique demonstrated in 28. Such an individual site rotation amounts to addition of an imaginary transverse (along y) tunnelling between s and px orbitals in our Hamiltonian,

This term preserves particle-hole symmetry but breaks both parity and time-reversal symmetries. The total Hamiltonian in the momentum space now reads . This Hamiltonian belongs to the symmetry group G+(U, C) and allows a Z2 classification of its topological properties25. Even though time-reversal symmetry is absent, particle-hole symmetry still ensures that Berry phase γ is quantized, with γ mod 2π=0 or π defining the trivial and topological insulator, respectively29. For our model , the topological insulating phase with γ=π is realized as long as . In another word, the Berry phase quantization is robust against the time-reversal symmetry breaking term Δy, and this topological phase is protected by particle-hole symmetry. For Δy greater than , Berry phase vanishes and the system becomes a trivial band insulator. At the critical point the band gap closes. Apart from the Berry phase, the topological distinction between and can also be seen from a gapped interpolation29 as discussed in Supplementary Note 3. Besides probing the half charges on the boundaries, another signature for the critical point of the topological phase transition is the local density fluctuation, . δρ is 1/ when Δy=0, independent of other parameters ts, tp and tsp, and decreases monotonically with increasing Δy (see Supplementary Note 4). The peaks of dδρ2/dΔy reveal the critical points (Fig. 3b) and provide a reliable tool of detecting the topological phase transition in experiments.

Figure 3: Phase transition between topological and trivial band insulators.
figure 3

(a) A domain wall between a topological insulator (ts=tp=tsp, Δy=0, left) and a trivial insulator (ts=tp=tsp=0, right). The circle represents the delocalized fermion shared by two neighbouring rungs as depicted in Fig. 2a, whereas the ellipse represents localized fermion without hopping. The additional charge 1/2 in the middle is the fractional charge carried on the domain wall. (b) The derivative of density fluctuation, −(dδρ2/dΔy). It develops sharp peaks, measurable in experiments, along the line of topological critical points.

It is feasible to prepare the ladder with phase separation: for example, a topological insulator on the left half but a trivial insulator on the right half. This can be achieved by rotating the lattice sites on half of the ladder only. The system is now described by:

with a field configuration η(j), which satisfies the boundary conditions η(j=−∞)=0 and η(j=+∞)=π, and . The charge distribution induced by the domain wall (the phase boundary) is calculated both numerically and from effective field theory shown in Supplementary Note 5. Both approaches cross-verify that the domain wall carries half charge (Fig. 3a). The half charge can be detected30 by the single-site imaging technique in experiments31,32.

Stability against interactions and transition to a Mott insulator

We further examine the stability of the topological phase and its quantum phase transitions in the presence of interaction using exact diagonalization. For single-species fermions on the sp orbital ladder, the leading interaction term is the on-site repulsion between different orbitals,

We compute the fidelity metric g as function of the interaction strength Usp (see the Methods section). A peak in the fidelity metric indicates a quantum phase transition33. Our numerical results (Fig. 4) show that the topological phase is stable for , with robust in-gap (zero energy) states33 localized on the edges (Fig. 2c). For stronger interaction the ladder undergoes a quantum phase transition to a Mott insulator phase. With ts=tp the Mott state exhibits ferro-orbital order with order parameter defined as (see discussions in Supplementary Note 6). Such ferro-orbital order gets weaker as ts gets smaller. A rich phase diagram of orbital ordering is expected and will be investigated in the future.

Figure 4: Phase transition between topological and ferro-orbital (FO) Mott insulators.
figure 4

Top panel shows the fidelity metric g and the ferro-orbital order parameter λsp, across the transition from the topological insulator (TI) to the Mott insulator. Bottom panel shows the particle/hole chemical potential (μp/μh). The finite charge gap μpμh in the bulk calculated with periodic boundary condition (dashed lines) comparing with the vanishing gap with open boundary condition (solid lines) indicates in-gap states on the edge. The length L is 12, and ts=tp=2tsp (taken as the energy unit) in this plot.

Coupled ladders and flat band in two dimensions

Remarkably, the zero-energy edge states of the sp orbital ladder survive even when the system is extended to two dimensions with finite inter-ladder coupling (for example, by reducing V1,2 relative to Vx in the setup of Fig. 1a). The zero modes of individual ladder morph into a flat band with double degeneracy (Fig. 5). The lack of dispersion in the y direction is related to the inter-ladder hopping pattern, which does not directly couple the edge states but only s and p orbitals on different rungs (Supplementary Fig. 1). The unexpected flat band in 2D is an exact consequence of the p orbital parity and hence is protected by symmetry. The flat dispersion can be rigorously proved using an unitary transformation, and arguments based on the quantization of Berry phase in Supplementary Note 7. The flat band makes the edge states in this 2D optical lattice distinct from that of quantum Hall effect previously proposed with lattice rotation16,34, artificial gauge field35 or optical flux36. Such a flat band is reminiscent of that at the zigzag edge of graphene, but with the difference that the present flat band is protected by the parity of the orbital wavefunctions. The diverging density of states associated with the flat band provides a fertile ground for interaction-driven many body instabilities. Future work will tell whether strongly correlated topological states exist in such 2D interacting systems.

Figure 5: Energy spectra of coupled ladders.
figure 5

In this plot we choose ts=tp=2tsp (taken as the energy unit here), and length L=200. An open (periodic) boundary condition is applied in the x (y) direction. a and b show the spectra with the small and large inter-ladder coupling, tsp=tsp/5 and tsp=tsp, respectively. A flat band (red line) at zero energy with double degeneracy generically appear for 0<tsp<tsp.

Discussion

The mathematical description of the sp orbital ladder H0 is similar to the celebrated Su-Schrieffer-Heeger (SSH) model37. Although the two systems belong to the same symmetry class of 1D topological insulators25, the orbital ladder contains new physics beyond the SSH model. First, the edge states of the sp orbital ladder have quite different spatial structures. For example, the sharp confinement of the edge states only requires , rather than the energy spectrum being dispersionless. In contrast, sharp confinement coincides with the flat band limit in the SSH model, when one of its hoppings vanishes. Moreover, the edge modes form flat bands in the presence of inter-ladder coupling. The original SSH model does not have such a nice property. Second and more importantly, tunnelling in the orbital ladder can form π flux loops and allow time-reversal symmetry breaking, as shown in discussed above. This gives rise to an interesting Z2 topological insulating state25. In contrast, breaking time-reversal symmetry within the SSH model of spinless fermions is impossible because the tunnelling is strictly 1D and thus cannot form flux loops.

Finally, we discuss a potential connection of the orbital ladder to Majorana physics38,39. The orbital ladder at half filling maps to two decoupled Majorana chains (see Supplementary Note 8)22. Note that the Majorana number is 1 for the double Majorana chains22. Topologically protected Majorana fermions with Majorana number −1 (ref. 22) can be realized on the orbital ladder using schemes similar to those proposed in 18,19,20,21, for example, by inducing weak pairing of the form The staggered quantum tunnelling tsp mimics the spin–orbit coupling. For the sp orbital ladder with 2ts<|μ|<2tp, we find that the Majorana number is −1 and the resulting Majorana zero modes are topologically protected. The topologically protected Majorana state is a promising candidate for topological quantum computing40,41.

Methods

Imaginary transverse tunnelling

By rotating individual lattice sites the induced bare coupling term is , where the angular momentum operator is . This term couples the px to py orbitals of the B leg. One can tune the rotating frequency to match Ω with the transverse tunnelling tsy from the py orbital of the B leg to the nearby s orbital on the A leg. Despite the large energy band gap () which separates the py orbitals from the degenerate s and px orbitals (bear in mind that the s and px orbitals are from different legs of the ladder), the low-energy effective Hamiltonian receives a standard second order effect from virtual processes, in which a particle jumps from a px orbital to the on-site py orbital and then to the nearby s orbital. The correction is given by , which makes an imaginary transverse tunnelling between nearby s and px orbitals.

Interaction effects

To characterize the stability of the topological phase against the inter-orbital interaction Hint (see equation (9)), we use the exact diagonalization method to calculate the fidelity metric , where is the ground state wavefunction of the Hamiltonian H=H0+Hint for a finite chain of length L with N=L fermions. In presence of interaction, the edge states survive as in-gap states (zero energy single particle/hole excitations)33. The energy of single particle (hole) excitation is defined as μp (μh) by μp=EL+1EL (μh=ELEL−1), where EN is the ground state energy of the ladder loaded with N fermions. The spatial distribution of the in-gap states is defined as the density profile (Δnj) of a hole created out of the ground state, which is

where is the ground state with N fermions. The density profile Δnj is found to be localized on the edges when , and to delocalize when approaching the critical point and finally disappear (Fig. 2c). The Mott state appearing at the strong coupling regime has a ferro-orbital order , with (Fig. 4). In our numerical calculation of finite system size, the correlation matrix is calculated and the strength of the ferro-orbital order λsp is defined as the maximum eigenvalue of [C]/L, extrapolated to the thermodynamic limit.

Additional information

How to cite this article: Li, X. et al. Topological states in a ladder-like optical lattice containing ultracold atoms in higher orbital bands. Nat. Commun. 4:1523 doi: 10.1038/ncomms2523 (2013).