Generalized Haldane models on laser-coupling optical lattices

We propose two generalized Haldane models on laser-coupling optical lattices. Laser-assisted nearest neighbour tunnelings generate artificial staggered magnetic flux, facilitating the realization of topological nontrivial band structures. As generalizations of Haldane model, these models support topological insulator and semimetal phases featuring high Chern numbers. We show simple rules for computing Chern numbers of our models and display the phase diagrams. Moreover, numerical calculations of energy spectra are in perfect agreement with our theoretical expectations. Our models may serve as two new family members for generalizing Haldane model on optical lattices.

Topologically ordered phases of matter have been a hot topic in condensed matter physics since the discovery of the integer quantum Hall effect 1 . In lattice systems, the topologically nontrivial band structures are currently attracting a great deal of interest 2,3 . The band topology is characterized by a topological invariant-Chern number-taking 4 integer values and signaling topological order. Nontrivial topological order has experimental consequences, such as quantized Hall conductivity 5 and the existence of edge states 2,3 . In two-dimensional lattice systems, one of the simplest models supporting topological bands was proposed by Haldane 6 . This model features real nearest-neighbor (NN) hopping and complex next-nearest-neighbor (NNN) hopping on a honeycomb lattice. Haldane showed that this two-band model is topologically equivalent to integer quantum Hall state.
In the present paper, we propose two simple lattice models that support topological insulator and semimetal phases featuring high Chern numbers. Our tight-binding model is defined on laser-coupling optical lattice, on which complex NN tunnelings are laser assisted. The three-band model is defined on state-dependent lattice whereas the four-band model is on state-independent lattice, both pierced by effective staggered magnetic fields. As generalizations of Haldane model, these models support topological nontrivial energy bands. Chern numbers are computed in a simple way and rich phase diagrams are exhibited. Moreover, the topological edge states show time-reversal (TR) symmetry for a special case, although TR symmetry breaking in the bulk is required for a nonzero Chern number.

Results
Three-band Model. In our three-band model, fermionic ultracold atoms are trapped on honeycomb-like lattice. Atoms of three different hyperfine states are located on inequivalent sites, labeled by A, B and C (Fig. 1a). In the tight-binding regime, Hamiltonian takes the form ˆˆˆ= where ˆˆâ b c , , i i i correspond to annihilation operators on site i of three sublattices. Normal NNN tunnelings of amplitude t take place within the same triangular sublattices. Laser-assisted NN tunnelings 41-43 of amplitude λ a(b) are accompanied by recoil momentum p (Fig. 1b), generating the Peierls phases 41 p r r For simplicity, we assume recoil momentums p of the two coupling processes are the same. Within the framework of the rotating-wave approximation, resonance detunings Δ a(b) of the laser coupling processes behave like onsite potentials. To eliminate the explicit spatial dependence of our Hamiltonian, we perform a unitary transformation 36 no longer depends on spatial coordinates, playing the role of effective staggered magnetic flux (Fig. 1a). The present Hamiltonian is invariant under discrete translation. Thus it can be written in momentum Given the momentum-space Hamiltonian (4), one easily knows there are three energy bands featured by Bloch wave functions ψ n,k (n = 1, 2, 3). The topological order of each band is characterized by the Chern number , is the Berry curvature and , is the Berry connection. Berry curvature can be expressed as the z component of curl . The Chern number is nonzero only if TR symmetry is breaking, or rather, it's the artificial staggered magnetic flux φ i that gives rise to topological nontrivial band structures. Interestingly, the Chern numbers can be higher than one that is beyond the usual quantum Hall effect.
We are more interested in two symmetric cases and show their phase diagrams. For the isotropic case φ π = 2 /3 i shown in Fig. 2a, we identify different regions by Chern numbers {x, y, z} of three bands. Setting t as unit, high Chern number phases { . If just one of Δ a(b) becomes large, it reduces to an effective two-band model with Chern For the other symmetric case in Fig. 2b, parameters set as , which deviates from the sine-shaped one of Haldane model by a factor φ − (1 cos ). The magnetic flux φ affect the nontrivial scale of detuning Δ, allowing for the largest value |Δ| = 9 when it approaches ±2π/3. The coupling strength λ a(b) has no influence on Chern numbers, but it turns out to modify the energy gaps and lead to topological semimetal phases as shown in the following numerical calculations.
The defining characteristic of topological nontrivial band structure is the existence of gapless edge states. We numerically calculate the energy spectra for 1D ribbons with zigzag and armchair edges, which is periodic in x and y direction respectively. The displayed edge states within the bulk energy gaps address different topological nontrivial phases. To demonstrate previous analytical results, we show two kinds of topological phase transitions driven by resonance detuning Δ or artificial magnetic flux φ. For the isotropic case of phase diagram Fig. 2a Fig. 3a and b. One can see the edge states disappear between the upper two bands during the process of enlarging Δ b with fixed Δ a . Specially, there is a symmetry  H T is the NNN hopping term on the honeycomb lattice (Fig. 4a), where c jα is the annihilation operator of fermion with pseudospin α (internal state | 〉 e and g | 〉) on site j. Tunneling strength = t t ij a b ( ) is different in two sublattices due to the trap potential depth difference 2Δ. H L is laser-assisted NN tunneling via two-photon Raman process (Fig. 4b), and normal tunneling is suppressed by the large potential barrier 2Δ. We also introduce energy ±ε for internal states, dipole interaction d and staggered sublattice resonance detuning δ j = ±δ in H V .
Atoms ω ω − ≈ Δ and momentum p, which has no relation with internal states but with the tunneling directions. Thus we have staggered sublattice resonance detuning δ ω ω = − − Δ  2 ( ) 2 1 2 in rotating wave frame.  To eliminate the explicit spatial dependence of phase factor φ ij , unitary transformation ˆĉ c e j j ip r /2 j → ± ⋅ is performed, making the NN tunnelings real and NNN tunnelings complex. In the basis of a a b b ( , , , ) 0,1 = ± and Pauli matrices σ i and s i representing the spin and sublattice indices. There are four energy bands . It implies that half filled system is topological nontrivial when The ground state is characterized by nonzero Chern number C = ±1 if δ lies in one nontrivial interval and C = ±2 if it is within the overlap of them. Thus we expect one pair or two pairs of edge states in the two topological nontrivial phases.
The energy spectra of 1D ribbons with armchair and zigzag edges are shown in Fig. 5. As the three-band model, x y x y is also presented if 2 3 φ φ = , resulting in symmetric spectra for the armchair one. We still consider the symmetric cases φ φ φ = = . We find topological insulator phases characterized by one or two pair of edge states when resonance detuning δ = .
From the numerical results, we clearly see that the number of edge states accord with our analytical expectation. Like the previous three-band model, Peierls phases φ i and detuning δ play the key role to realize topological nontrivial phases. . Detuning δ = .

Discussion
In conclusion, we have proposed and studied two generalized Haldane models on laser-coupling optical lattice. Laser-assisted NN tunnelings generate artificial staggered magnetic flux, facilitating the realization of topological nontrivial band structures. For the three-band model, we show a simple rule for computing Chern numbers, which are obtained from the values of energy E n (k) at the two Dirac points. We also show the analytical phase diagrams of two symmetric cases, verified by numerical calculations of energy spectra. For the four-band model, the Peierls phase depend on the sublattice indices instead of internal states. Different topological nontrivial phases characterized by one pair or two pairs of edge states, can be easily transformed to each other by changing the resonance detuning δ and artificial magnetic flux φ. The effects of laser coupling on topological band structures are discussed. Compared with Haldane model, topological properties of our models are much richer, featuring high Chern number and easy control of topological insulator-semimetal phase transitions. For the staggered artificial magnetic flux, our models can be viewed as two new family members for generalizing Haldane model on optical lattice.

Methods
Here we show simple rules for computing Chern numbers of our models. For the three-band model, the Bloch wave functions of H k in equation (4) take the form with E n (k) the energy of n-th band. If wave function is analytic in the whole Brillouin zone (BZ), the contour integral in equation (6) is zero because of the periodicity of BZ. Nonzero integral comes from singular points, or rather, the two Dirac points . Actually near the Dirac points, λ a b k ( ) and δ nn k approach zero, resulting in a vanishing wave function. Here one can use = n a b c , , replacing = n 1, 2, 3 to identify energy bands. Expanding them to the second order, we obtain the normalized wave functions