Spontaneous magnetization and anomalous Hall effect in an emergent Dice lattice

Ultracold atoms in optical lattices serve as a tool to model different physical phenomena appearing originally in condensed matter. To study magnetic phenomena one needs to engineer synthetic fields as atoms are neutral. Appropriately shaped optical potentials force atoms to mimic charged particles moving in a given field. We present the realization of artificial gauge fields for the observation of anomalous Hall effect. Two species of attractively interacting ultracold fermions are considered to be trapped in a shaken two dimensional triangular lattice. A combination of interaction induced tunneling and shaking can result in an emergent Dice lattice. In such a lattice the staggered synthetic magnetic flux appears and it can be controlled with external parameters. The obtained synthetic fields are non-Abelian. Depending on the tuning of the staggered flux we can obtain either anomalous Hall effect or its quantized version. Our results are reminiscent of Anomalous Hall conductivity in spin-orbit coupled ferromagnets.

with the contact interaction strength g 2D adjusted for a quasi-2D geometry (with a tight harmonic confinement along z) 35 . The second part gives the on-site Hamiltonian including higher band energy contribution and contact interactions. It reads 20  here U 2 denotes the energy of the composites and U sp is the additional interaction energy to occupy the p-orbital of a composite filled site. E 1 is the single-particle excitation energy of the p-band. Shaking with elliptical periodic driving force leads to 25  = − (Ω ) + (Ω + Φ). We consider the case where J sp  U 2 ,U sp ≤ Ω . That allows us to use rotating-wave approximation and Floquet theorem and to average terms fast oscillating in time (see Methods).
The sp-tunneling will be resonantly enhanced when the energy to occupy the p-bands is an integer multiple of the shaking frequency. This translates into the condition that E 1 + U sp = NΩ for integer N. The resonance order, N, can be controlled by varying either the lattice depth, interaction strength or the driving frequency.
The ground state composite structure. The composite number operator n c î commutes with the Hamiltonian (5), n H 0 c i avg    ,    =ˆ. Therefore, we can characterize a site by the presence or the absence of a composite, i.e. n 1 0 c i = , which makes the Hamiltonian, (5), quadratic in operators for a particular realization of composite configuration. For a given composite configuration, we then diagonalize the quadratic Hamiltonian and fill up the energy levels depending on the excess ↓ -fermions filling n ↓ − n ↑ ≤ 1/3. We find the ground state composite structure by comparing the energies of different composite configurations using simulated annealing on 6 × 6 up to 20 × 12 lattices with periodic boundary conditions. For details about the parameters of simulated annealing, we refer to Ref. 20. The resulting ground state self-organized structure of the composites resembles a Dice lattice and is shown in Fig. 1(a). Its basis consists of three sites denoted A, B and C. The A site consists of two orbitals p + and p − whereas the sites B and C have only s-orbitals. The basis vectors for the Dice lattice are given by a 3 2 3 2 1 = ( / , / ) and a 3 2 3 2 2 = ( / , − / ) . For any deviation from the 1/3 filling of the composites, the excess composites or vacancies will show up as impurities on top of the Dice lattice as long as the density of such impurities is small (n imp 1/3).
To understand the origin of the Dice structure, consider first a composite at some chosen site A. The energy is minimized when all the neighboring sites (forming hexagon with the site A in the center) are without composites. This facilitates the sp tunneling from a p orbital at site A to the neighboring sites. Any composite on these neighboring sites increases the energy by J sp F avg,δ . Thus, for a composite filling of n c = 1/3, the delocalization area is maximized by filling the lattice with hexagons with a composite at their center.
Assuming the ground state configuration is fixed, the effective Hamiltonian for the excess fermions is quadratic and thus easily diagonalized to yield the band structure. The behavior of the excess fermions is then that of an ideal Fermi gas with such band structure, which is easily computed.
Creation of the staggered field. For the Dice lattice considered here, one can construct two kind of plaquettes: i) The three plaquettes as shown in Fig. 1(b) where the particle traverses the closed path involving p x ↔ s↔ p x ↔ s orbitals. Such a path does not mix the p x and p y orbitals. Due to the phases of the tunneling amplitudes, a particle going through each of those plaquettes (denoted by ϕ 1 ,ϕ 2 and ϕ 3 , see Fig.1(b) and calculated along the direction of the arrow) acquire fluxes due to Aharonov-Bohm effect. We find that the induced flux is staggered in nature as the phases obey the constraint mod(ϕ 1 + ϕ 2 + ϕ 3 ,2π ) = 0. These fluxes are calculated by taking into account a single p-orbital. Figure 1(c) shows the flux strengths for N = 1. In particular, for Φ = 0, ϕ 1 = ϕ 2 = ϕ 3 = 2π/3 which is equivalent to a uniform magnetic flux of the same magnitude. With growing Φ , fluxes change with all fluxes vanishing at Φ = π/2.
ii) The other kind of plaquette involves the A sites containing p x , p y orbitals as given by the parallelogram shown in Fig. 1(d). A particle going around such plaquette picks up a non-Abelian flux. Consider the transport from site "1" to "2". The process can go either via the upper or the lower path with two consecutive sp tunnelings with strengths . The kinetic energy term around the plaquette for A sites may be written as  L 23 ( 23 ) is given by changing π/3 → − π/3 and δ + → δ − in the expression for L 12 ( 23 ). Moreover, we find that L 12 = L 34 and L 23 = L 41 (with similar relations for  ij ) and they depend on the staggered flux through the phase of the tunneling amplitudes. = .
( ) The Wilson loop parameter has (i) an intrinsic contribution (appearing at Φ = π/2), with no staggered flux through an individual plaquete in Fig. 1(b) due to the appearance of the sp-band tunneling and (ii) an extrinsic contribution due to the external staggered flux induced by shaking. As a result, the link matrices are of non-Abelian nature (W ≠ 2 ) for any shaking phase Φ .
Spontaneous magnetization. First, we study the behavior of the system in the absence of staggered flux realized for N = 1,Φ = π/2 Fig. (1). The effective non-Abelian field is intrinsic in nature and the corresponding dispersion relation for the lowest energy band is shown in Fig. 2a. The main characteristic of the dispersion relation is the appearance of two non-equivalent Dirac cones and disappearance of flat bands. This is in contrast to the dispersion relation in a normal Dice lattice where the dispersion relation contains an intersecting Dirac cone and a flat band. Moreover, above a certain Fermi energy (of the excess fermions), the first two bands are degenerate. When we introduce the staggered flux, the dispersion changes and the gap opens at the band touching points (Fig. 2b). Once the Fermi energy is higher than the gap, the two bands become degenerate again. This is in a stark contrast to other situations with nearest neighbor tunneling where staggered flux leads only to the movement of the Dirac cones 37,38 and to opening a gap one either needs long-distance tunneling 13,17 , uniform magnetic field or synthetic non-Abelian fields along with magnetic field 39 . non-zero local magnetization characterizes the breaking of time-reversal symmetry as the particles acquire local angular momentum due to the particle number difference between the p + and p − orbitals. First, we find that the presence of non-zero staggered flux immediately results in non-zero  z and in opening of the gap. Thus, appearance of non-zero  z can be used as an indirect evidence for the presence of a gap in our system. The local magnetization is shown in Fig.2c (dashed line) for a small staggered flux. It vanishes only when the first two bands are totally filled. The presence of spontaneous magnetization (spontaneous time-reversal symmetry breaking) is reminiscent of spin-orbit coupled ferromagnets 4 . Moreover, to look into the topological nature of the system, we define the intrinsic Hall conductivity,

Anomalous Hall effect.
The Berry curvature, Ω n (k), for the n-th band is given by Ω n (k) = ∇ k × 〈 u nq | ∇ k | u nq 〉 where |u nq 〉 denotes an eigenvector for the n-th band. The total Hall conductivity σ xy then depends on the Fermi energy ε F of the system as shown in [Fig.2c (solid line)]. We find that the local Berry curvature is concentrated near the Dirac points which results in a non-zero contribution to xy n σ when ε F is in the band. As ε F enters the band gap, we find that xy n σ flattens at a value > 1/2. This can be ascribed to the presence of two Dirac cones near the band gap. As we increase ε F , the contribution from the next band begins to play a role and eventually the conductivity changes sign. The second peak appears when the Fermi energy reaches the maximum of the first band. Such structures in conductivity have been predicted to arise due to the presence of magnetic monopoles in the momentum space 40 .
Quantum Anomalous Hall effect. Finally, consider the strong flux limit, e.g the case of Φ = 0 where the flux through each plaquette is 2π/3. Strong flux results in lifting the degeneracy between the first two bands (Fig. 3, top plot). The middle two bands still touch each other in the form of Dirac cones. With the degeneracy lifted, one can define Chern numbers given by ν = (2,− 4,2) resulting in the appearance of quantum Anomalous Hall effect. We have also calculated the Hall conductivity and when the Fermi energy of the excess fermions resides in the band gap, conductivity becomes integer valued (Fig. 3, bottom plot). The magnitude of the band gap is ≈ J sp . For a triangular lattice (lattice constant a = 500 nm) with lattice depth of 6E R and transverse frequency of 10E R , the sp tunneling strength in the harmonic approximation is given by J sp ~ 0.008E R assuming the scattering length of − 400 Bohr radius. This corresponds to a band gap of about ~10 nano-Kelvin which determines the temperature regime where the Hall phase can be observed. For the Dice lattice with dilute impurities, the Hall conductivity presented in this paper remain unchanged due to the topological nature of the Berry curvature for the dispersion bands 4 . The band topology discussed here can be measured in principle by using recently proposed methods of Ramsey interferometry and Bloch oscillations 41,42 , or from momentum distribution from Time-of-Flight images 43 . Moreover, the generation of local orbital angular momentum due to broken time-reversal symmetry in the chiral p-orbitals can also be detected by time-of-flight measurements 44 .

Conclusions
To summarize, we considered an unequal mixture of attractively interacting fermions in a shaken triangular lattice. Pairing produces immobile composites that gives rise to dice lattice for the excess fermions. Adjustments of shaking frequency and amplitude allow to make intra-band tunnelings negligible while resonantly enhancing interaction-induced sp-tunnelings for the excess fermions. Moreover, shaking leads to the controlled staggered magnetic field and induces (on the p-orbitals) non-Abelian character of the system. Their joint effect leads to spontaneous chiral magnetization (due to time reversal symmetry breaking) along with appearance of Anomalous Hall effect. Many fascinating question related to the findings here can be investigated further including the role of impurities, long-range interaction etc. Moreover, by using dipolar atoms, one can further study many-body effects like superconductivity 45,46 , density-waves in presence of the artificial non-Abelian gauge fields presented here. Methods The model Hamiltonian. We consider an unequal mixture of two-species ultracold fermions (denoted by ↑ , ↓ ) assuming strong attractive interactions between two species. It is energetically favorable for fermions to pair, the low energy system is then effectively composed of paired composites and the excess ↓ fermions. We denote the creation and annihilation operators for ↑ fermions as s i ↑ † and s i ↑ . For the more abundant ↓ fermions we include both s and p orbitals denoting the corresponding operators as s s p p In the main text, for simplicity, we have neglected ↑ -fermion tunneling and all the intra-band tunnelings for ↓ -fermions from the beginning. Here, let us derive the Hamiltonian without these assumptions and show that, indeed, these effects may be neglected. Here, n i

↑ (↓)
, denote number operators of ↑ ( ↓ ) fermions respectively while n i ± ↓ are number operators for the ↓ p-fermions with ± -chirality. The same amplitude, J 0 corresponds to the standard tunneling between s orbitals, the corresponding tunneling in the p-band is described by J 1 σδ . Moreover, we include density induced (bond-charge) intra-band tunneling for p-orbitals with strength J 11 δ σ . J sp is the amplitude of the hopping between s and p bands which is also induced by the interaction with ↑ fermions. The various tunneling processes in Hamiltonian (12) are shown in Fig. 4. The tunneling amplitudes are given by where  x y i 00 ( , ) is the Wannier function of the s-band and  x y i ( , ) σ with σ = ± are the Wannier functions corresponding to p + -and p − -bands in the harmonic approximation for the triangular lattice potential. The single particle Hamiltonian for the triangular lattice is denoted by H latt .
Note that the Hamiltonian (12) does not contain tunnelings of the composites themselves. Such a pair tunneling term can arise due to interaction 28 but is 3-4 orders of magnitude smaller than other tunneling terms. The composites can also tunnel via higher-order processes (discussed in 22 ). The leading term of this collective tunneling is of the second order 24 with the corresponding amplitude being proportional to J U We expand the exponential functions in (14) as: Then as U 2  Ω , after rotating-wave approximation (RWA) and projecting on our local Hilbert space, the first term of Hamiltonian (14) may be resonant only if U 2 contribution vanishes. Since this term corresponds to ↑ -fermion tunneling (which appear paired only in our subspace) this process is possible only if a paired state and a ↓ -fermion in s-orbital are neighbors ( Fig. 1(a)). Otherwise the pair (composite) is pinned. Similarly, the second term may be resonant (n n 0 ) when a ↓ -fermion in s-orbital tunnels to a neighboring empty site. The third term gives a resonant contribution via the tunneling process depicted in Fig.4(b). After RWA, all the time-independent tunneling amplitudes of the above intra-band tunnelings are changed by a factor  K 0 ( /Ω) δ . We see that to minimize the ss and pp tunnelings we have to tune the shaking amplitude such that  K 0 . Due to the type of sp coupling in Hamiltonian, (14), p-fermions may appear only in composite occupied sites. This may occur only from a site occupied by a lonely ↓ -fermion (if there were a composite at that site, an additional energy difference, U 2 , the pair energy would appear bringing the system out of the chosen resonance). After carrying RWA and in the limit of vanishing intra-band tunneling, the effective Hamiltonian reads,