Finite Temperature Phases of Two Dimensional Spin-Orbit Coupled Bosons

We determine the finite temperature phase diagram of two dimensional bosons with two hyperfine (pseudo-spin) states coupled via Rashba-Dresselhaus spin-orbit interaction using classical field Monte Carlo calculations. For anisotropic spin-orbit coupling, we find a transition to a Berenzinskii-Kosterlitz-Thouless superfluid phase with quasi-long range order. We show that the spin-order of the quasi-condensate is driven by the anisotropy of interparticle interaction, favoring either a homogeneous plane wave state or stripe phase with broken translational symmetry. Both phases show characteristic behavior in the algebraically decaying spin density correlation function. For fully isotropic interparticle interaction, our calculations indicate a fractionalized quasi-condensate where the mean-field degeneracy of plane wave and stripe phase remains robust against critical fluctuations. In the case of fully isotropic spin-orbit coupling, the circular degeneracy of the single particle ground state destroys the algebraic ordered phase in the thermodynamic limit, but a cross-over remains for finite size systems.