Symmetry classification of spin-orbit-coupled spinor Bose-Einstein condensates

We develop a symmetry classification scheme to find ground states of pseudo-spin-1/2, spin-1, and spin-2 spin-orbit-coupled spinor Bose-Einstein condensates and show that as the SO(2) symmetry of simultaneous spin and space rotations is broken into discrete cyclic groups, various types of lattice structures emerge in the absence of a lattice potential. Examples include two different kagome lattices for pseudo-spin-1/2 condensates and a nematic vortex lattice in which uniaxial and biaxial spin textures align alternatively for spin-2 condensates. For the pseudo-spin-1/2 system, although mean-field states always break time-reversal symmetry, there exists a time-reversal invariantmany-body ground state, which is fragmented and expected to be observed in a microcondensate.