Spinor F=1 Bose-Einstein condensates loaded in two types of radially-periodic potentials with spin-orbit coupling

We consider two-dimensional spinor F=1 Bose-Einstein condensates in two types of radially-periodic potentials with spin-orbit coupling, i.e., spin-independent and spin-dependent radially-periodic potentials. For the Bose-Einstein condensates in a spin-independent radially-periodic potential, the density of each component exhibits the periodic density modulation along the azimuthal direction, which realizes the necklacelike state in the ferromagnetic Bose-Einstein condensates. As the spin-exchange interaction increases, the necklacelike state gradually transition to the plane wave phase for the antiferromagnetic Bose-Einstein condensates with larger spin-orbit coupling. The competition of the spin-dependent radially-periodic potential, spin-orbit coupling, and spin-exchange interaction gives rise to the exotic ground-state phases when the Bose-Einstein condensates in a spin-dependent radially-periodic potential.