Stability of Bose-Fermi mixtures in two dimensions: a lowest-order constrained variational approach

Abstract

We investigate the problem of mechanical stability in two-dimensional Bose-Fermi mixtures at zero temperature, focusing on systems with a tunable Bose-Fermi (BF) interaction and a weak but finite boson-boson (BB) repulsion. The analysis is carried out within the framework of the lowest-order constrained variational (LOCV) approach, which allows for a non-perturbative treatment of strong interspecies correlations while retaining analytical transparency. The BF interaction is modeled by a properly regularized attractive contact potential, enabling the exploration of both the attractive and repulsive energy branches. We determine the minimal BB repulsion required to ensure mechanical stability of the mixture by evaluating the inverse compressibility matrix over the full range of BF coupling strengths, within the domain of validity of the LOCV approximation. The interaction contribution to the energy is benchmarked against available experimental data and Quantum Monte Carlo results in the single-impurity limit, showing good agreement. Our analysis reveals how the critical BB coupling depends on interaction strength, density imbalance, and mass ratio. In particular, we find that mixtures with equal boson and fermion masses exhibit enhanced stability, requiring the smallest BB repulsion to prevent mechanical instability. In this case, a relatively small BB interaction is sufficient to stabilize attractive mixtures for all values of the BF interaction. These results provide a theoretical framework for assessing stability conditions in experimentally realizable two-dimensional Bose-Fermi mixtures with tunable interactions.

Figures (7)

• Figure 1: Dimensionless healing distance ${k_\mathrm{F}} d^-$ and ${k_\mathrm{F}} d^+$ versus the coupling strength $\eta = -\ln({k_\mathrm{F}} a_\mathrm{BF})$ for the attractive and repulsive branches, respectively.
• Figure 2: (a) Correlation function $f^-(r)$ for the attractive branch, for several interaction strengths ranging from $\eta = -2$ (weak attraction) to $\eta = 2$ (strong attraction). Inset: pair correlation function $g^-(r)=f^-(r)^2$ for $\eta=-2$ (solid line) together with its small $r$ asymptotic behavior $\propto \log^2(r/a_\mathrm{BF})$ (dashed-dotted line). (b) Correlation function $f^+(r)$ for the repulsive branch, for three interaction strengths ranging from $\eta = 0$ (strong repulsion) to $\eta = 2$ (weak repulsion). Inset: pair correlation function $g^+(r)=f^+(r)^2$ for $\eta=1$.
• Figure 3: Dimensionless quantities $A^-$ and $A^+$ determining the BF interaction energy as a function of the interaction strength $\eta = -\ln({k_\mathrm{F}} a_\mathrm{BF})$ for the attractive and repulsive branches, respectively. Insets: (a) comparison between $A^-$ (line) and experimental data for the attractive polaron energy normalized by $\epsilon_{\rm F}$Koehl-2012 (squares); (b) comparison between $A^+$ (line) and QMC results for the polaron energy normalized by $\epsilon_{\rm F}$ for hard-disk BF repulsive interaction Bertaina-2021 (squares); the value obtained in Parish-2012 within a $T$-matrix approximation is also reported (asterisk). Dotted lines in both panels: weak-coupling approximation (\ref{['equ:mubbench']}); dash-dotted line in (a): strong-coupling approximation (\ref{['equ:scbench']}).
• Figure 4: Relative quantum depletion coefficients $D^-$ (a) and $D^+$ (b) as a function of the interaction strength $\eta = -\ln({k_\mathrm{F}} a_\mathrm{BF})$ for the attractive and repulsive branches, respectively.
• Figure 5: Critical value of the BB repulsion strength $\zeta_c$ as a function of the BF interaction strength $\eta = -\ln({k_\mathrm{F}} a_\mathrm{BF})$ for $m_{\rm B}=m_{\rm F}$ and different values of the boson concentration $x$ for the attractive (a) and repulsive (b) branches.