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Exotic Multi-fold Vortex Lattices of Spin-Orbit Coupled Bose-Einstein Condensates in Optical Lattices

We investigate the ground state of two-dimensional Bose-Einstein condensates with Rashba spin-orbit coupling in square optical lattices and demonstrate the existence of rich phases with different lattice structures, which is closely related to the degenerate structure of single particle energy spectrum induced by the competition of spin-orbit coupling and optical lattices. We find that the ground state is in the phase with either parity-time-reversal or parity symmetry by direct numerical simulation. We show the phase diagram of ground state in the whole regime of spin-orbit coupling strength, and particularly find that the system supports multi-fold vortex lattices, in which ground state holds half-quantum vortex lattices, vortex-antivortex pair lattices and fundamental vortex lattices, simultaneously, when single particle energy minimums touch the boundary of the first Brillouin zone.