Helical supersolid phases in spin-orbit-coupled bosons

We study the ground-state phases of a spin-orbit-coupled Bose-Einstein condensate loaded in a two-dimensional optical lattice. We find three types of helical supersolid phases, which are dependent on the anisotropy of the spin-orbit coupling with respect to the relative strength of the $y$ and $x$ directions. The period of density modulation increases with the strength of the spin-orbit coupling.

Figure 1

The density profiles of the two-component BECs with isotropic SOC ($\lambda =1$). It shows a phase transition from superfluid to striped supersolid. From upper to lower the coupling strengths are: (a) $\kappa =0$, (b) $\kappa =30$, and (c) $\kappa =60$.

Figure 2

(a) The spin texture of the two-component SOC condensates with coupling strength $\kappa =30$ and $\lambda =1$. The color of the arrows indicates the value of $S_z$. (b) and (c) are the projective views of the spin texture along the $x$ and $y$ directions, respectively. From (b) and (c) we observe that the spin vectors alternatively consist of two sets of continuous textures for even and odd lattice sites along each direction.

Figure 3

Phase transitions induced by anisotropic SOC for a given strength ${\kappa }_x=30$: (a) $\lambda =0.35$, (b) $\lambda =2.35$, and (c) $\lambda =10$. (d) The phase diagram with respect to the relative ratio of the anisotropic SOC $\lambda$. Three types of the helical SS phases, namely, the HSS phase (blue interval), the RSS phase (red interval), and the VSS phase (black interval) are identified.