Spin-orbit-driven transitions between Mott insulators and finite-momentum superfluids of bosons in optical lattices

Synthetic spin-orbit coupling in ultracold atomic gases can be taken to extremes rarely found in solids. We study a two-dimensional Hubbard model of bosons in an optical lattice in the presence of spin-orbit coupling strong enough to drive direct transitions from Mott insulators to superfluids. Here we find phase-modulated superfluids with finite momentum that are generated entirely by spin-orbit coupling. We investigate the rich phase patterns of the superfluids, which may be directly probed using time-of-flight imaging of the spin-dependent momentum distribution.

Figure 1

Comparison of SOC strengths in solids and cold atoms. $h, \alpha$, and $E_F$ denote the Zeeman energy, SOC coefficient, and Fermi energy, respectively. For GaAs, the effective mass is $m^{*}=0.067m_0$ [29], where $m_0$ is the electron mass, the Rashba SOC strength is $\alpha =(0.04–0.06)\times {10}^{-11}\mathrm{eV}\mathrm{m}$ [30], and the $g$ factor is $g^{*}=-0.45$ [31]. For InAs the parameters are $m^{*}=0.026m_0$ [29], $\alpha =(0.28–1.4)\times {10}^{-11}\mathrm{eV}\mathrm{m}$ [32], and $g^{*}=-15.1$ [33]. For InSb the parameters are $m^{*}=0.0135$ [29], $\alpha K_F=(1.0–1.2)\times {10}^{-11}\mathrm{eV}\mathrm{m}$ [34], and $g^{*}=-51$ [33]. And for the metallic surfaces $m^{*}\sim 0.255m_0$ [2, 3], where the $g$ factor is assumed to be $g^{*}=2$. The parameters for these four different examples are plotted at an external magnetic field of 5 Tesla. A high carrier density, $n={10}^{11}{\mathrm{cm}}^{-2}$, is used for the semiconductors. For ${\mathrm{SrTiO}}_3/{\mathrm{LaAlO}}_3$ oxide interfaces, the data are taken from Ref. [35]. Additional feasible parameter regimes are plotted as horizontal and vertical bars.

Figure 2

Schematic of spin-independent tunneling (a) and spin-dependent tunneling induced by SOC (b). In the latter case, the tunneling takes place between two neighboring sites accompanied by both spin flipping and phase variations. The phase variation during tunneling is responsible for the creation of the finite momentum superfluids.

Figure 3

Phase diagrams of of Eq. (2) obtained from Gutzwiller variational simulations for an $8\times 8$ lattice with periodic boundary condition at (a) $U_{\uparrow \downarrow }=0,\lambda =0$, (b) $U_{\uparrow \downarrow }=0,\lambda =0.04U$, (c) $U_{\uparrow \downarrow }=0.5U,\lambda =0$, and (d) $U_{\uparrow \downarrow }=0.5U,\lambda =0.04U$. The phase diagrams are determined by the amplitude of the spin-up superfluid order parameter. The spin-down superfluid order parameter produces similar results.

Figure 4

Plot of the amplitude of spin-up superfluid order parameter $|\langle b_{\uparrow }\rangle |$, the filling factor $\langle n_{\uparrow }\rangle$, and the energy density $e$ as a function of the spin-independent tunneling at $U_{\uparrow \downarrow }=0.5U, \lambda =0.04U$, and $\mu =1.33U$ for periodic (left panel) and open (right panel) boundary conditions.

Figure 5

Spin-dependent momentum distribution, Eq. (7), for different superfluids at $U_{\uparrow \downarrow }=0.5U, \lambda =0.04U, \mu =1.33U$, (a) $t=0.005U$, and (b) $t=0.05U$.

Figure 6

Correlation functions of finite momentum superfluids on a $32\times 32$ lattice with a confining potential [Eq. (8)] for $\mu =0.8U, U_{\uparrow \downarrow }=0$, and $\lambda =0.04U$. The left column shows results for $t=0.01U$, the middle column for $t=0.024U$, and the right column for $t=0.08U$. The top three panels plot the phase ${\varphi }_{\uparrow }$ of the spin-up superfluid order parameter. The middle three panels plot the magnitude and the bottom three panels plot the density. The phase patterns in the top two panels reveal a sudden change in superfluid order.

Figure 7

Plot of the kinetic-energy terms, Eq. (9), as a function of the spin-independent tunneling at $\lambda =0.04U$, for a $4\times 4, 5\times 5, 6\times 6$ lattice and an infinite-size lattice.