Magnetic Phase Transition in a Mixture of Two Interacting Superfluid Bose Gases at Finite Temperature

The miscibility condition for a binary mixture of two interacting Bose-Einstein condensates is shown to be deeply affected by interaction driven thermal fluctuations. These give rise to a first order phase transition to a demixed phase with full spatial separation of the two condensates, even if the mixture is miscible at zero temperature. Explicit predictions for the isothermal compressibility, the spin susceptibility, and the phase transition temperature $T_M$ are obtained in the framework of Popov theory, which properly includes beyond mean-field quantum and thermal fluctuations in both the spin and density channels. For a mixture of two sodium condensates occupying the hyperfine states $|F=1⟩,|m_F=1⟩$ and $|F=1,m_F=-1⟩$, respectively, $T_M$ is predicted to occur at about 0.7 times the usual BEC critical temperature.

Figure 1

Isothermal compressibility (a) and spin susceptibility (b) Eq. (1) for binary mixtures of Bose gases, with interaction parameters $gn/(k_B{T}_{\mathrm{BEC}})=0.1$ and $\delta g/g=0.07$. The blue solid and the red dashed lines are the predictions of HF theory [Eq. (2)] and Popov theory [Eq. (6)], respectively. Both quantities are normalized to the mean-field $T=0$ values, ${\kappa }_{T,M}(T=0)=2/(g\pm g_{12})$.

Figure 2

Difference of free energies between the miscible state ($m=0$) and the phase-separated state described in the main text, for $gn/(k_B{T}_{\mathrm{BEC}})=0.1$ and $\delta g/g=0.07$. Blue solid line: $T<T^{*}$, red dashed line: $T^{*}<T<T_M$, green dotted line $T>T_M$. The vertical lines indicate the critical magnetization $m=n-2\zeta (3/2)/{\lambda }_T^3$ above which the minority component is purely thermal.

Figure 3

Phase diagram for binary condensates with $gn/(k_B{T}_{\mathrm{BEC}})=0.1$. The blue solid and the red dashed lines are the phase transition temperature $T_M$, and characteristic temperature $T^{*}$, respectively. The gray area corresponds to the regime of phase separation.