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Competing interactions in population-imbalanced two-component Bose-Einstein condensates

We consider a two-component Bose-Einstein condensate with and without synthetic "spin-orbit" interactions in two dimensions. Density- and phase-fluctuations of the condensate are included, allowing us to study the impact of thermal fluctuations and density-density interactions on the physics originating with spin-orbit interactions. In the absence of spin-orbit interactions, we find that inter-component density interactions deplete the minority condensate. The thermally driven phase transition is driven by coupled density and phase-fluctuations, but is nevertheless shown to be a phase-transition in the Kosterlitz-Thouless universality class with close to universal amplitude ratios irrespective of whether both the minority- and majority 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-component density-density interactions: a phase-modulated phase with uniform amplitudes for small intercomponent interactions, a completely imbalanced, effectively single-component, condensate for intermediate spin-orbit coupling strength and suficciently large inter-component interactions, and a phase-modulated \textit{and} amplitude-modulated phase for sufficiently large values of both the spin-orbit coupling and the inter-component density-density interactions. The phase which is modulated by a single $\bv q$-vector only is observed to transition into an isoptropic liquid through a strong de-pinning transition with periodic boundary conditions, which weakens with open boundaries.