Bose-Hubbard Models with Synthetic Spin-Orbit Coupling: Mott Insulators, Spin Textures, and Superfluidity

Motivated by the experimental realization of synthetic spin-orbit coupling for ultracold atoms, we investigate the phase diagram of the Bose-Hubbard model in a non-Abelian gauge field in two dimensions. Using a strong coupling expansion in the combined presence of spin-orbit coupling and tunable interactions, we find a variety of interesting magnetic Hamiltonians in the Mott insulator (MI), which support magnetic textures such as spin spirals and vortex and Skyrmion crystals. An inhomogeneous mean-field treatment shows that the superfluid (SF) phases inherit these exotic magnetic orders from the MI and display, in addition, unusual modulated current patterns. We present a slave-boson theory which gives insight into such intertwined spin-charge orders in the SF, and discuss signatures of these orders in Bragg scattering, in situ microscopy, and dynamic quench experiments.

Figure 1

(a) Band structure of Eq. (1) for $U=0$, with $\alpha =-\beta =\pi /4$. There are four degenerate minima at ${\overrightarrow{Q}}_1,\cdots ,{\overrightarrow{Q}}_4$ in the lower band due to rotational symmetry breaking by the square lattice and a Dirac cone at the $\Gamma$ point. (b) The spin eigenstates in the lower band. The spin is locked to the momentum through the SO coupling. (c) Real-space density distribution of spin-up particles in the condensate with $\lambda =1.3$ from the GP calculation. There is a similar distribution for the spin-down particles. On the other hand, for $\lambda <1$, the spin density is uniform. The total density is uniform for all $\lambda$.

Figure 2

(a) Phase diagram in the Mott insulating regime from Monte Carlo annealing of the spin Hamiltonian Eq. (2). Spin configurations are abbreviated as described in the text. The darker area of the Spiral-2 region represents a commensurate 4-site spiral. (b) shows the $xy$-plane projection of the real-space spin configurations in the Spiral-1, 2, SkX, and VX phases. The magnetic structure factor peaks are shown in the insets.

Figure 3

Phase diagrams of the spin-orbit coupled Bose-Hubbard model in $\mu /U$ vs $t/U$ plane, showing Mott lobes and superfluid states. (a) phase diagram with $\lambda =1.5$ and $\alpha =\pi /2$ and (b) $\lambda =0.5$ and $\alpha =\pi /2$. The width of the $n=1$ lobe is given by $\lambda U$ and the critical value $(t/U{)}_c$ increases with $\lambda$. The two insets show the local spin density distribution ($\mathrm{\text{red}}=\uparrow$, $\mathrm{\text{blue}}=\downarrow$, purple in between) and bond currents for $t/U=0.08$ in the superfluid phase close to the Mott states. $\mu /U$ is tuned such that the average number of particles per site is unity. For $\lambda =1.5$, the spin density assumes $z$-antiferromagnetic order whereas for $\lambda =0.5$, the magnetic moments are in the $2\times 2$ VX phase, and restricted to the $xy$ plane. The bond currents for both $\lambda =1.5$ and $\lambda =0.5$ share the same pattern, with clockwise and anticlockwise plaquette loop currents forming Ising antiferromagnetic order.