We consider a two-component Bose-Einstein condensate with and withoutsynthetic "spin-orbit" interactions in two dimensions. Density- andphase-fluctuations of the condensate are included, allowing us to study theimpact of thermal fluctuations and density-density interactions on the physicsoriginating with spin-orbit interactions. In the absence of spin-orbitinteractions, we find that inter-component density interactions deplete theminority condensate. The thermally driven phase transition is driven by coupleddensity and phase-fluctuations, but is nevertheless shown to be aphase-transition in the Kosterlitz-Thouless universality class with close touniversal amplitude ratios irrespective of whether both the minority- andmajority condensates exist in the ground state, or only one condensate exists.In the presence of spin-orbit interactions we observe three separate phases,depending on the strength of the spin-orbit coupling and inter-componentdensity-density interactions: a phase-modulated phase with uniform amplitudesfor small intercomponent interactions, a completely imbalanced, effectivelysingle-component, condensate for intermediate spin-orbit coupling strength andsuficciently large inter-component interactions, and a phase-modulated\textit{and} amplitude-modulated phase for sufficiently large values of boththe spin-orbit coupling and the inter-component density-density interactions.The phase which is modulated by a single $\bv q$-vector only is observed totransition into an isoptropic liquid through a strong de-pinning transitionwith periodic boundary conditions, which weakens with open boundaries.
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