Phase transition and thermodynamic properties of a two-component weakly interacting Bose gas in self-consistent Popov approximation

We investigate the thermodynamic properties of a homogeneous two-component weakly interacting Bose gas using the Cornwall-Jackiw-Tomboulis (CJT) effective action approach combined with the self-consistent Popov approximation. By enforcing the Hugenholtz-Pines theorem for each component, we derive gapless excitation spectra and calculate the shift in the Bose-Einstein condensation transition temperature due to inter-particle interactions. The relative shift of the critical temperature is found to be $\Delta T_c/T_c^0 = c\,(n a^3)^{1/3}$ with $c \approx 1.41$, showing good qualitative agreement with quantum Monte Carlo simulations. Numerical solutions of the coupled integral equations for the condensate densities, chemical potentials, pressure, and specific heat are presented for general two-component mixtures. Our results demonstrate significant corrections compared to the ideal gas and mean-field Hartree-Fock theory, particularly in the vicinity of the phase transition. The framework developed here provides a reliable approach for studying weakly interacting two-component Bose systems at finite temperatures.