Spin-orbit-coupled mean-field Bose gas at finite temperature

We consider the spin-orbit-coupled Bose gas with repulsive mean-field interparticle interactions. We analyze the phase diagram of the system varying the temperature $T>0$, the chemical potentials, and interparticle and spin-orbit interaction couplings. Our results indicate that, for Rashba- and Weyl-type spin-orbit couplings, condensates featuring an ordering wave vector $\stackrel{P\vec}{Q}\ensuremath{\ne}\stackrel{P\vec}{0}$ are fragile with respect to thermal fluctuations and at $T>0$ the only stable thermodynamic phases involving the Bose-Einstein condensate (BEC) are those of uniform type with $\stackrel{P\vec}{Q}=\stackrel{P\vec}{0}$. On the other hand, the presence of the spin-orbit coupling stabilizes the $\stackrel{P\vec}{Q}=\stackrel{P\vec}{0}$ BEC state at any dimensionality $d>1$ and modifies either the order or the universality class of the corresponding phase transition. We emphasize the singular nature of the limit of vanishing spin-orbit interaction coupling $v$, sizable shifts of the phase boundaries upon varying $v$, and the role of the relative magnitudes of the interparticle interaction couplings for the character of the condensation transition.