Spin-orbit-coupled Bose gases with nonlocal Rydberg interactions held under a toroidal trap

We consider a spin-orbit-coupled Bose-Einstein condensate with nonlocal Rydberg interactions confined in a toroidal trap. Our results show that the interplay among nonlocal Rydberg interactions, contact interactions, spin orbit coupling, and the external toroidal trap can give rise to a rich variety of ground-state phases, such as lump necklace, chiral supersolid, standing-and plane-wave supersolid, and so on. Furthermore, we clarify the different breaking of symmetry induced by system parameters. Finally, the limiting case of a quasi-one-dimensional ring has also been investigated, and our analytical results agree well with the Gross-Pitaevskii numerical simulations. We also give an experimental protocol to observe these states in future experiments.