Superfluid–Mott-insulator transition in the spin-orbit-coupled Bose-Hubbard model

We consider a square optical lattice in two dimensions and study the effects of both the strength and symmetry of spin-orbit coupling and Zeeman field on the ground-state, i.e., Mott-insulator (MI) and superfluid (SF), phases and phase diagram, i.e., MI-SF phase-transition boundary, of the two-component Bose-Hubbard model. In particular, based on a variational Gutzwiller ansatz, our numerical calculations show that the spin-orbit-coupled SF phase is a nonuniform (twisted) one, with its phase (but not the magnitude) of the order parameter modulating from site to site. Fully analytical insights into the numerical results are also given.

Figure 1

The atomic-limit ($t={\gamma }_x={\gamma }_y=0$) phase diagrams are shown as functions of (a) $\mu$ and $h_z$ for $h_y=0$, and (b) $\mu$ and $h_y$ for $h_z=0$ (solid line) and $h_z=0.1U$ (dashed line). The MI lobes are labeled by (a) $(N_{\uparrow },N_{\downarrow })$ and (b) $N=N_{\uparrow }+N_{\downarrow }$. While we set $U_{\uparrow \downarrow }=0.3U$ in these figures, they are schematically correct as long as $0<U_{\uparrow \downarrow }<U=U_{\uparrow \uparrow }=U_{\downarrow \downarrow }$.

Figure 2

The phase ${\theta }_{0\downarrow }$ of the order parameter ${\Delta }_{0\downarrow }$ of the reference site 0 may depend on the choice of coordinate system and is not a gauge-independent quantity within the mean-field theory. This can be seen by comparing the order parameters using the coordinate systems shown in (a) and (b) as discussed in the text.

Figure 3

Ground-state phase diagrams with out-of-plane $h_z$ Zeeman fields. The MI-SF phase-transition boundaries are shown as functions of $\mu$ and $t$ for the first two MI lobes, i.e., $N=1$ and 2, where we consider (a)–(c) Rashba SOC and (d)–(f) ERD SOC. Here, we set $h_y=0$, $U_{\uparrow \uparrow }=U_{\downarrow \downarrow }=U$, and $U_{\uparrow \downarrow }=0.3U$ in all figures. In addition, the black solid lines are guides to the eye, which are obtained from Eq. (15) (see the text for details).

Figure 4

Ground-state phase diagrams with ERD${}_x$ SOCs (${\gamma }_x={\gamma }_E,{\gamma }_y=0$) and general Zeeman fields. The rest of the parameters are specified in Fig. 3.

Figure 5

Ground-state phase diagrams with ERD${}_y$ SOCs (${\gamma }_x=0,{\gamma }_y={\gamma }_E$) and general Zeeman fields. The rest of the parameters are specified in Fig. 3.

Figure 6

Ground-state phase diagrams with Rashba SOCs (${\gamma }_x={\gamma }_y={\gamma }_R$) and general Zeeman fields. The rest of the parameters are specified in Fig. 3.