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.
💡 Analysis
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.
📄 Content
Bose-Fermi (BF) mixtures constitute a paradigmatic platform for studying correlated quantum matter composed of particles with different quantum statistics. Since the early investigations of dilute 3 He- 4 He mixtures [1][2][3][4][5][6][7][8], such systems have provided insight into mediated interactions [3], impurity physics [9], phase separation [10,11], 𝑝-wave pairing [4,12], dual superfluids [12] and effects beyond the mere remit of condensed matter physics [13][14][15][16].
In the context of ultracold atomic gases [17], the advent of Feshbach resonances [18] has enabled unprecedented control over interspecies interactions, allowing the realization of BF mixtures with tunable coupling strengths in a variety of atomic combinations [19]. Several studies have appeared in both the experimental and theoretical [16, literature on BF mixture with tunable interactions.
A central issue in the realization of quantum mixtures is their mechanical stability [81,82]. Unlike purely fermionic systems, Bose-Fermi mixtures are not intrinsically stabilized by Pauli pressure alone. In particular, the presence of fermionmediated attractive interactions between bosons can render the homogeneous phase unstable unless a sufficiently strong direct boson-boson (BB) repulsion is present. Determining the minimal BB interaction required to stabilize the mixture is therefore a fundamental problem, especially when the interspecies interaction is tuned across a Feshbach resonance.
Experimental [81,82] and theoretical [83][84][85][86][87][88][89][90][91] efforts in this direction have concentrated mainly on repulsive or weaklyattractive interspecies interaction. In three dimensions, the stability of resonant BF mixtures has been studied using meanfield theory for narrow resonances [58] possibly with related Gaussian corrections [49], variational methods [61,80], and 𝑇matrix approximations [73,74,79]. These works have clarified the interplay between pairing, molecule formation, and phase stability across the resonance. However, the situation in two dimensions is qualitatively different. Two-dimensional scattering is characterized by a logarithmic energy dependence and the existence of a two-body bound state for arbitrarily weak attraction [92][93][94]. Moreover, tight confinement enables confinement-induced resonances [95,96], providing an additional experimental knob for controlling interactions in quasi-2D geometries. This situation is particularly suitable for a BF mixture with a tunable BF interaction since an additional and independent BB interaction is required to guarantee its mechanical stability [61,74,79].
Despite increasing experimental and theoretical interest in low-dimensional mixtures, a systematic investigation of the mechanical stability of a two-dimensional BF mixture with a tunable interspecies interaction is still lacking. Existing theoretical studies of 2D BF systems have focused primarily on pairing, polarons, or collective properties [75,78,[97][98][99] and the problem of stability has been addressed only in restricted weak-coupling regimes [91]. In particular, a nonperturbative analysis of stability across the full interaction crossover in two dimensions is currently missing.
In this work, we fill this gap by analyzing the mechanical stability of a homogeneous two-dimensional Bose-Fermi mixture at zero temperature with a tunable BF interaction and a weak BB repulsion. We focus on mixtures with a majority of fermions, in line with previous studies for resonant Bose-Fermi mixtures in 3D [61,70,78]. To treat the interspecies interaction nonperturbatively, we employ the lowest-order constrained variational (LOCV) approach [61,[100][101][102][103][104][105][106][107], which incorporates short-range correlations at the two-body level and naturally connects to the polaron problem in the singleimpurity limit. The variational nature of the LOCV approach allows us to describe both the attractive (lower) and repulsive (upper) branches of an attractive contact interaction modeling the effective 2D scattering length resulting after confinement to two dimensions from a Feshbach or confinement-induced resonance. These two branches effectively correspond to attractive or repulsive BF interactions, respectively [108].
By solving the LOCV equations across the full range of coupling strengths and evaluating the inverse compressibility matrix, we determine the minimal boson-boson repulsion required to stabilize the homogeneous mixture against thermodynamic instability. We analyze the dependence of this critical BB interaction on the interspecies coupling, density ratio, and mass imbalance, and discuss the regime of validity of the approach, particularly in the molecular limit where higher-order correlations become increasingly important.
The paper is organized as follows. In Sec. II we present the LOCV formalism for the two-dimensional Bose-Fermi mixture. In Sec. III we solve the LOCV equation and obtain the BF pair correlati
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