Ring phases of spin-orbit coupled Bose-Einstein condensate in the radial optical lattices

. We study the ground-state phases of two-dimensional spin-orbit coupled Bose-Einstein condensate loaded in the spin-dependent radial optical lattices. For strong depth of the spin-dependent radial optical lattice, three new types of the ring phases are found with isotropic spin-orbit coupling, i.e. , the ring-stripe phase with the periodic density modulation along the radial direction, the ring-supersolid phase with the periodic density modulation both the radial and azimuthal directions and the windmill-supersolid phase with four vanes along the azimuthal direction. By increasing spin-orbit coupling, the system undergoes a sequence phase transitions from the ring-stripe phase to the ring-sueprsolid phase, and to the windmill-supersolid phase for the weak depth of radial optical lattice, however, the ring-stripe phase transforms to the windmill-supersolid phase directly for the strong depth of radial optical lattice. We also discuss the ground-state phases with anisotropic spin-orbit coupling, two-component Bose-Einstein condensates are easier condensed along the direction with larger spin-orbit coupling.