Phase diagram of two-component mean-field Bose mixtures

We revisit the structure of the phase diagram of the two-component mean-field Bose mixture at finite temperatures, considering both the cases of attractive and repulsive interspecies interactions. In particular, we analyze the evolution of the phase diagram upon driving the system towards collapse and point out its distinctive features in this limit. We provide analytical insights into the global structure of the phase diagram and the properties of the phase transitions between the normal phase and the phases involving Bose-Einstein condensates. \emph{Inter alia} we analytically demonstrate that for sufficiently weak interspecies interactions $a_{12}$ the system generically exhibits a line of quadruple points but has no triple nor tricritical points in the phase diagram spanned by the chemical potentials $μ_1$, $μ_2$ and temperature $T$. In contrast, for sufficiently large, positive values of $a_{12}$, the system displays both triple and tricritical points but no quadruple points. As pointed out in recent studies, in addition to the phase transitions involving condensation, the mixture may be driven through a liquid-gas type transition, and we clarify the conditions for its occurrence. We finally discuss the impact of interaction- and mass-imbalance on the phase diagram of the mixture.

💡 Research Summary

The paper presents a comprehensive analytical study of the phase diagram of a two‑component Bose mixture within a mean‑field framework. The authors consider a Hamiltonian consisting of kinetic energy terms for each species and a long‑range, weak interaction term proportional to the product of the total particle numbers, with intra‑species couplings a₁, a₂ > 0 and an inter‑species coupling a₁₂ that may be either attractive or repulsive. By employing a Kac‑scaling argument, the mean‑field interaction is shown to be equivalent to the Hartree‑Fock treatment of dilute weakly interacting gases, ensuring that the results are relevant beyond the strictly long‑range limit.

The grand‑canonical free‑energy density ω is expressed in terms of a function Φ(t₁,t₂). In the thermodynamic limit V → ∞ the saddle‑point evaluation of the functional integral becomes exact, and the stationary conditions ∂Φ/∂t_i = 0 lead to coupled algebraic equations for the densities n₁, n₂ (Eqs. 9‑10). The 1/V terms in these equations represent condensate densities; when they remain finite as V → ∞ a Bose‑Einstein condensate (BEC) is present. The authors systematically solve these self‑consistency equations and compare the values of Φ for all possible solutions to determine the globally stable phase.

A central result concerns the role of the discriminant D = a₁a₂ − a₁₂². For D > 0 (including weakly attractive a₁₂) the system is stable and supports four distinct phases: a normal (non‑condensed) phase, BEC₁ (condensation of species 1 only), BEC₂ (condensation of species 2 only), and BEC₁₂ (both species condensed). All phase boundaries are continuous, and a line of quadruple points exists where the four phases coexist. The location of these quadruple points is given analytically by Eq. (18), which depends on temperature, masses, and the combination a₂ + a₁₂ (m₂/m₁)^{3/2}. As a₁₂ is tuned from small positive values toward negative values, the BEC₁₂ wedge widens; at the limit a₁₂ → −√(a₁a₂)⁺ (D → 0⁺) the wedge opens to 180°, and the quadruple‑point line moves toward the origin in the (μ₁, μ₂) plane.

When D < 0 (strongly repulsive inter‑species interaction) the mixed BEC₁₂ phase disappears. Instead, the BEC₁ and BEC₂ phases are separated by a first‑order transition accompanied by phase separation. In this regime the phase diagram contains triple points (coexistence of normal, BEC₁, and BEC₂) and tricritical points where two continuous transition lines meet a first‑order line. The authors provide exact criteria for the existence of these multicritical points, showing that they arise from the competition between multiple minima of Φ.

In addition to condensation transitions, the paper identifies a liquid‑gas‑type transition occurring entirely within the normal phase. This transition is signaled by a region where the second derivative of Φ with respect to density becomes negative, leading to a van der Waals‑like loop in the pressure‑density isotherm. The authors prove that such a loop inevitably appears at sufficiently high temperature or for sufficiently large positive a₁₂, and they give explicit mean‑field conditions for its occurrence.

The impact of mass and interaction imbalance is also explored. Different masses modify the thermal de Broglie wavelengths λ_i, shifting the individual BEC critical temperatures, while unequal intra‑species couplings change the shape of the stability region defined by D. The analysis shows that imbalance can move the positions of quadruple, triple, and tricritical points, and in extreme cases can generate new multicritical structures where several of these points merge.

Overall, the work delivers a unified analytical picture of the full three‑dimensional phase diagram (T, μ₁, μ₂) of a two‑component Bose mixture. It clarifies under which conditions the system exhibits continuous versus first‑order condensation, when liquid‑gas coexistence appears, and how multicritical points depend on the sign and magnitude of the inter‑species interaction as well as on mass and interaction asymmetries. The results are directly testable in current ultracold‑atom experiments, where density jumps associated with first‑order transitions and the location of BEC critical lines can be measured with high precision, providing a stringent benchmark for mean‑field theories of multicomponent quantum gases.