Dzyaloshinskii-Moriya Interaction and Spiral Order in Spin-orbit Coupled Optical Lattices

We show that the recent experimental realization of spin-orbit coupling in ultracold atomic gases can be used to study different types of spin spiral order and resulting multiferroic effects. Spin-orbit coupling in optical lattices can give rise to the Dzyaloshinskii-Moriya (DM) spin interaction which is essential for spin spiral order. By taking into account spin-orbit coupling and an external Zeeman field, we derive an effective spin model in the Mott insulator regime at half filling and demonstrate that the DM interaction in optical lattices can be made extremely strong with realistic experimental parameters. The rich finite temperature phase diagrams of the effective spin models for fermions and bosons are obtained via classical Monte Carlo simulations.

is the SO coupling strength. In the deep Mott insulator regime, the degeneracy in spin configurations is lifted by second order virtual processes. The effective Hamiltonian H eff can be obtained using perturbation theory. We take the Mott insulator as the unperturbed state and derive the corrections of the effective Hamiltonian by the standard Schrieffer-Wolf transformation 17 The first two terms are Heisenberg exchange and Zeeman terms, respectively, while the last two terms arise from SO coupling. In solid state systems the third term is called the DM interaction 33,34 , which is believed to drive multiferroic behavior. The definition of the D vector and the Γ tensor will be presented below. The structure of these terms can be derived from basic symmetry analyses but the coefficients must be computed microscopically. In the following we derive the coefficients in Eq. 3 by considering the coupling between four internal degenerate ground states { ; ; ; ; } α ∈ ↑ ↑ , ↑ ↓ , ↓ ↑ , ↓ ↓ through the spin independent and dependent tunnelings t σ and λ. The couplings are different for fermions and bosons, as illustrated in Fig. 1. Fermionic atoms. For fermionic atoms, there are only two possible excited states ex = ;0 ↑ ↓ and 0; ↑ ↓ , as shown schematically in Fig. 1(a). We find J J t t U 2 4 x y ;0 ↓ ↓ , 0; ↑ ↑ , 0; ↑ ↓ , 0; ↓ ↓ , as shown in Fig. 1

The last term in Eq. 3 reads as
, where

for fermions (bosons). This term arises from the coupling between states 1 and 4 ,
Here the real part contributes asymmetric terms to the Heisenberg model, while the imaginary part contributes to Γ ij . In a square lattice with d d 0 x y = , this term vanishes. However, for tilted lattices, such as triangular and honeycomb, this term should be significant.
Lattice parameters. We estimate the possible parameters that can be achieved in a square optical where k L is the wavevector of the laser. The SO coupling coefficient is given by k m R γ ∼ / , k R is the wavevector of the external Raman lasers, and k k R L in most cases. The Raman lasers are pure plane waves, and serve as a perturbation to the hopping between adjacent sites.
We use the Wannier functions of the lowest band without SO coupling to calculate the tight binding parameters t and λ. In a square lattice, coordinates decouple and the Bloch functions are Mathieu Our numerical results are presented in Fig. 2 is in general much larger than t and can be controlled through a Feshbach resonance independently.
In Fig. 2 / reaches the maximum value of 1.0 at t λ = . This is in sharp contrast to models of weak multiferroic effects in solids with D J D J 0 001 0 1 / = / . − . , which is generally induced by small atomic displacements 35 . Optical lattices, by contrast, can be tuned to exhibit either weak or strong DM terms. This enhanced tunability enables optical lattice systems to single out the effects of strong DM interactions and study the impact of the DM term.
There are notable differences between our model and corresponding models in solids (i) In solids the SO coupling arises from intrinsic (atomic) SO coupling and D is generally along the z direction (out of plane). However, in our model D is in the plane and the out of plane component is zero.
where N s is the number of sites. However, these two order parameters do not fully characterize the phase diagrams because in some cases there are still local magnetic or spiral orders although both M and P P = are vanishingly small. In these cases, we also take into account the spin structure factor: S k ( ) shows peaks at different positions in momentum space for different phases. For instance, the peak of the spin structure factor is at k 0 0 = ( , ) for ferromagnetic phases, k π π = ( , ) for antiferromagnetic phases, and 0 π ( , ) (or 0 π ( , )) for the flux spiral phase (P 0 = but with nontrivial local spin structure). General spiral orders correspond to other k. We obtain the phase diagrams by analyzing both the order parameters and spin structure factors. We have not checked for long range order in the spin structure factor. We expect quasi-long range order to accompany magnetized phases at low h, e.g., a ferromagnetic phase for 1 ξ  . The phase diagrams of an 8 8 × lattice in Fig. 3 show a rich interplay between magnetic orders and spin spiral orders. For instance, for fermions with small SO coupling ( 0 25 ξ < . ), the ground states are anti-ferromagnetic states with zero (non-zero) magnetization for a Zeeman field h J , therefore the pure flux phase with zero spiral order can be observed. Similarly, the increasing SO coupling for bosonic atoms gives rise to a series of transitions from simply magnetic (ferromagnetic at small h) order to simply magnetic spiral order (with zero total spiral order but local spiral structure), then to magnetic spiral orders (or non-magnetic spiral orders) and finally to flux spiral orders. The emergence of the spiral order and flux order with increasing SO coupling can be clearly seen from the change of the spin structure factors in Fig. 4, which shift from k 0 0 = ( , ) or π π ( , ) to 0 π ( , ) and 0 π ( , ). The spin spiral order phase transition temperature is comparable to the magnetic phase transition temperature, J 0 . In Fig. 5(a), we plot the spin configuration of fermions at T J 0 05 0 = .
, 1 0 ξ = . and h 1 5 = . (MS phase), which shows clear spiral ordering. The corresponding order parameters P and M are plotted in Fig. 5 , which is comparable to the magnetic critical temperature 17 (In 2D, the Heisenberg model has a critical temperature T J c 0 = in mean-field theory). Note that spiral order can also exist in the frustrated model without SO coupling, however, the critical temperature is generally much smaller than the magnetic phase transition temperature 11,41 . Our results therefore show that SO coupling in the absence of frustration provides an excellent platform to search for spiral order and multiferroics-based states in optical lattices.

Discussion
Finally we note that different spiral orders may be observed using optical Bragg scattering methods 42 , which probe different spin structure factors for different spiral orders. Similar methods have been widely used in solid state systems. Furthermore, in optical lattices, the local spin magnetization at each lattice site (thus the magnetic order M) as well as the local spin-spin correlations (thus the spiral order P) can be measured directly 43,44 , which provides a powerful new tool for understanding the physics of spiral orders and multiferroic effects in optical lattices.
Note added. During the preparation of this manuscript (the initial version is available at arXiv:1205.6211) we became aware of work 45-47 on similar topics.

Methods
The phase diagrams of an 8 8 × lattice are computed by classical MC methods for both fermions and bosons. The results are obtained after 10 6 thermalization steps followed by 10 6 sampling steps in each MC run at low temperature (T J 0 05 0 = . ). We have checked that for lower temperatures the phase diagrams do not change quantitatively. We also verify that similar phase diagrams can be obtained for larger system sizes, however, the spiral orders in a larger optical lattice become more complicated, and the boundary between different quantum phases is shifted.  . Similar features can also been found for bosons with the same parameters.