Attractive Solution of Binary Bose Mixtures: Liquid-Vapor Coexistence and Critical Point

We study the thermodynamic behavior of attractive binary Bose mixtures using exact path-integral Monte Carlo methods. Our focus is on the regime of interspecies interactions where the ground state is in a self-bound liquid phase, stabilized by beyond mean-field effects. We calculate the isothermal curves in the pressure vs density plane for different values of the attraction strength and establish the extent of the coexistence region between liquid and vapor using the Maxwell construction. Notably, within the coexistence region, Bose-Einstein condensation occurs in a discontinuous way as the density jumps from the normal gas to the superfluid liquid phase. Furthermore, we determine the critical point where the line of first-order transition ends and investigate the behavior of the density discontinuity in its vicinity. We also point out that the density discontinuity at the transition could be observed in experiments of mixtures in traps.

Figure 1

Isothermal curves of pressure as a function of density. The solid line refers to the $T=0$ result from Eq. (2) while the dotted line is the Hartree-Fock (HF) prediction from Eq. (3) holding for $n<n_{\mathrm{BEC}}$. The PIMC results correspond to $N=216$ particles. The vertical and horizontal dot-dashed lines refer respectively to the density $n_{\mathrm{BEC}}$ and pressure $p_{\mathrm{BEC}}$ at the onset of BEC. Close to each PIMC point we report the corresponding value of the reduced temperature $T/T_{\mathrm{BEC}}$, where $n{\lambda }_{T_{\mathrm{BEC}}}^3=2\zeta (3/2)$.

Figure 2

Isothermal curves of pressure as a function of density below and across the critical point. PIMC results refer to systems with $N=512$ particles and the dotted lines are a guide to the eye. Labels on each curve indicate the value of $k_{B}T$ in units of ${\hslash }^2/m{a}^2$. The inset shows the cubic fit to the points near the phase transition and the Maxwell construction used to determine the gap parameter $\mathrm{\Delta }n$.

Figure 3

Phase diagram in the temperature vs density plane for two values of $a_{12}$. The shaded area corresponds to the liquid-gas coexistence region and the cross is our estimate (with uncertainties) of the critical point. The dotted line indicates the BEC transition density $n_{\mathrm{BEC}}=2\zeta (3/2)/{\lambda }_T^3$.

Figure 4

Decay of the order parameter $\mathrm{\Delta }n$ (upper panel) and of the condensate fraction discontinuity (lower panel) with $T$ on approaching $T_c$. The vertical lines indicate our central value estimate of $T_c$.

Figure 5

Phase diagram in the temperature vs pressure plane for two values of $a_{12}$. The solid line indicates the coexistence line between liquid and gas and the cross is our estimate (with uncertainties) of the critical point. The dotted lines refer to the pressure $p_{\mathrm{BEC}}$ at the onset of the BEC transition.