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Advances in pure optical trapping techniques now allow the creation ofdegenerate Bose gases with internal degrees of freedom. Systems such as${}^{87}$Rb, $^{39}$K or ${}^{23}$Na in the $F=1$ hyperfine state offer anideal platform for studying the interplay of superfluidity and quantummagnetism. Motivated by the experimental developments, we study ground statephases of a two-component Bose gas loaded on an optical lattice. The system isdescribed effectively by the Bose-Hubbard Hamiltonian with onsite and nearneighbor spin-spin interactions. An important feature of our investigation isthe inclusion of interconversion (spin flip) terms between the two species,which has been observed in optical lattice experiments. Using mean-field theoryand quantum Monte Carlo simulations, we map out the phase diagram of thesystem. A rich variety of phases is identified, including antiferromagnetic(AF) Mott insulators, ferromagnetic and AF superfluids.

Competing exotic quantum phases of spin-$1/2$ ultra-cold lattice bosons with extended spin interactions