Exotic Quantum Spin Models in Spin-Orbit-Coupled Mott Insulators

We study cold atoms in an optical lattice with synthetic spin-orbit coupling in the Mott-insulator regime. We calculate the parameters of the corresponding tight-binding model using Peierls substitution and “localized Wannier states method” and derive the low-energy spin Hamiltonian for bosons and fermions. The spin Hamiltonian is a combination of Heisenberg model, quantum compass model and Dzyaloshinskii-Moriya interactions and it has a rich classical phase diagram with collinear, spiral and vortex phases. We discuss the state of the art of experiments to realize and detect magnetic orderings in strongly correlated optical lattices.

Figure 1

Tunneling amplitudes $T_y^{(1,1)}$ and $T_y^{(1,2)}$ obtained by Peierls substitution (solid red and dash-dotted blue line) and the localized Wannier states method (full dots and empty circles) for different strengths of Rashba SOC ($\alpha =\beta$) and $V_x=V_y=10E_R$.

Figure 2

Spin textures: (a) ferromagnet (${\theta }_x={\theta }_y=0$); (b) spiral wave (${\theta }_x=0.5$, ${\theta }_y=0.2$); (c) vortex phase (${\theta }_x={\theta }_y=1$); (d) antivortex phase (${\theta }_x=\pi -1$, ${\theta }_y=1$); (e) stripes (${\theta }_x=1.6$, ${\theta }_y=0.7$). Phases (a),(b) and (e) are coplanar and shown in a two-dimensional representation.

Figure 3

Classical phase diagram of the Hamiltonian $H_{\mathrm{B}}$ (see text for details): ferromagnet (black corner dots); spiral waves [dark orange (commensurate with four-sites periodicity), light orange (commensurate with three-sites periodicity), red (others)]; stripes (yellow); vortex phase (dark blue) and antivortex phase (light blue).