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We investigate the thermal physics of a Bose-Hubbard model with Rashbaspin-orbit coupling starting from a strong coupling mean-field ground state.The essential role of the spin-orbit coupling $\left(\gamma\right)$ is topromote condensation of the bosons at a finite wavevector ${k}_{0}$. We findthat the bosons display either homogeneous or phase-twisted or orbital orderedsuperfluid phases, depending on $\gamma$ and the inter-species interactionstrength $\lambda$. We show that an increase of $\gamma$ leads to suppressionof the critical interaction $U_c$ for the superfluid to Mott insulatortransition in the ground state, and a reduction of the $T_c$ for superfluid toBose-liquid transition at a fixed interaction. We capture the thermalbroadening in the momentum distribution function, and the real space profilesof the thermally disordered magnetic textures, including their homogenizationfor $T \gtrsim T_{c}$. We provide a Landau theory based description of theground state phase boundaries and thermal transition scales, and discussexperiments which can test our theory.

Thermal transitions of the modulated superfluid for spin-orbit coupled correlated bosons in an optical lattice