Spin-orbit-coupled Bose-Einstein condensates held under a toroidal trap

We study a quasispin-1/2 Bose-Einstein condensate with synthetically generated spin-orbit coupling in a toroidal trap and show that the system has a rich variety of ground states. As the central hole region increases, i.e., the potential changes from harmoniclike to ringlike, the condensate exhibits a variety of structures, such as a modified stripe, an alternately arranged stripe, and countercircling states. In the limit of a quasi-one-dimensional ring, the quantum many-body ground state is obtained, which is found to be the fragmented condensate.