Interaction driven quantum phases in spin-orbit-coupled spin-1 bosons

We study the interplay of spin orbit coupling and strong correlations present for ultra cold spin-1 bosons on a square optical lattice. In addition to the conventional spinful Mott and superfluid phases contained in the spin-1 Bose-Hubbard model, we find new lattice symmetry breaking phases [1]. For weak interactions, the interplay between the lattice momentum and the spin-orbit wave-vector induces a phase transition from a uniform superfluid to a phase where bosons simultaneously condense at the center and edge of the Brillouin zone. This state is characterized by spin density wave order, which arises from the spin-1 nature of the system. Interactions suppress this spin density wave order, and for sufficiently strong interactions the system becomes a Mott insulator. Inside the Mott lobes with an odd-integer filling we derive the effective low energy magnetic Hamiltonian. Focusing on the quasi-one-dimensional limit we solve the strongly coupled magnetic model in three ways: in its classical limit, with a spin-wave analysis, and using the density matrix renormalization group. [1] J. H. Pixley, et. al. arXiv:1509.00005. (Accepted in PRB (R))