  Motivated by recent experiments [Y.J. Lin {\it et al.}, Nature {\bf 471}, 83 (2011)], we study Mott phases and superfluid-insulator (SI) transitions of two-species ultracold bosonic atoms in a two-dimensional square optical lattice with nearest neighbor hopping amplitude $t$ in the presence of a spin-orbit coupling characterized by a tunable strength $\gamma$. Using both strong-coupling expansion and Gutzwiller mean-field theory, we chart out the phase diagrams of the bosons in the presence of such spin-orbit interaction. We compute the momentum distribution of the bosons in the Mott phase near the SI transition point and show that it displays precursor peaks whose position in the Brillouin zone can be varied by tuning $\gamma$. Our analysis of the critical theory of the transition unravels the presence of unconventional quantum critical points at $t/\gamma=0$ which are accompanied by emergence of an additional gapless mode in the critical region. We also study the superfluid phases of the bosons near the SI transition using a Gutzwiller mean-field theory which reveals the existence of a twisted superfluid phase with an anisotropic twist angle which depends on $\gamma$. Finally, we compute the collective modes of the bosons and point out the presence of reentrant SI transitions as a function of $\gamma$ for non-zero $t$. We propose experiments to test our theory. 
Superfluid-Insulator transition of two-species bosons with spin-orbit   coupling