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arXiv:2402.13664 · Physical Review A

Quantum Monte Carlo and perturbative study of two-dimensional Bose-Fermi mixtures

Jacopo D'Alberto, Lorenzo Cardarelli, Davide Emilio Galli, Gianluca Bertaina, P. Pieri

Published May 2, 2024• 5 citations
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Abstract

The authors investigate the thermodynamic properties of a two-dimensional dilute Bose-Fermi mixture of ultracold atoms at zero temperature through two complementary methods: perturbation theory to the second order in the interactions and quantum Monte Carlo. They find good agreement between analytic expressions and numerical results for weak interactions, while significant discrepancies appear in the regime close to mechanical instability, indicating phase separation of the bosonic component.

DOI
10.1103/physreva.109.053302

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The authors investigate the thermodynamic properties of a two-dimensional dilute Bose-Fermi mixture of ultracold atoms at zero temperature through two complementary methods: perturbation theory to the second order in the interactions and quantum Monte Carlo. They find good agreement between analytic expressions and numerical results for weak interactions, while significant discrepancies appear in the regime close to mechanical instability, indicating phase separation of the bosonic component.
We study a resonant Bose-Fermi mixture at zero temperature by using the fixed-node diffusion Monte Carlo method. We explore the system from weak to strong boson-fermion interaction, for different concentrations of the bosons relative to the fermion component. We focus on the case where the boson density n(B) is smaller than the fermion density n(F), for which a first-order quantum phase transition is found from a state with condensed bosons immersed in a Fermi sea, to a Fermi-Fermi mixture of composite fermions and unpaired fermions. We obtain the equation of state and the phase diagram, and we find that the region of phase separation shrinks to zero for vanishing n(B).
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 nonperturbative 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.
The equation of state of a weakly interacting two-dimensional Bose gas is studied at zero temperature by means of quantum Monte Carlo methods. Going down to as low densities as $n{a}^{2}\ensuremath{\propto}{10}^{\ensuremath{-}100}$ permits us to obtain agreement on beyond mean-field level between predictions of perturbative methods and direct many-body numerical simulation, thus providing an answer to the fundamental question of the equation of state of a two-dimensional dilute Bose gas in the universal regime (i.e., entirely described by the gas parameter na2n{a}^{2}na2). We also show that the measure of the frequency of a breathing collective oscillation in a trap at very low densities can be used to test the universal equation of state of a two-dimensional Bose gas.
We study thermal properties of a trapped Bose-Bose mixture in a dilute regime using quantum Monte Carlo methods. Our main aim is to investigate the dependence of the superfluid density and the condensate fraction on temperature for the mixed and separated phases. To this end we use the diffusion Monte Carlo method in the zero-temperature limit and the path-integral Monte Carlo method for finite temperatures. The results obtained are compared with solutions of the coupled Gross-Pitaevskii equations for the mixture at zero temperature. We notice the existence of an anisotropic superfluid density in some phase-separated mixtures. Our results also show that the temperature evolution of the superfluid density and condensate fraction is slightly different, showing noteworthy situations where the superfluid fraction is smaller than the condensate fraction.
The properties of interacting ultracold Bose systems were investigated using quantum Monte Carlo simulations. The superfluidity and Bose-Einstein condensation of very dilute two species trapped bosons mixtures as a function of temperature were investigated. Depending on the interactions, the system can be in three different modes: mixed, separated into two blobs, and separated in a shell regime. Anisotropy of superfluidity was observed in the case of two blobs. A transition was also observed in which the superfluidity fraction becomes smaller than the condensate fraction for each regime. We further investigated the properties of one dimensional strongly interacting bosonic systems in periodic potentials. The transition of the superfluid to the Mott insulator phase in the robust and fragile superfluid regimes is shown in detail. We investigated transition from a one-dimensional system to a two-dimensional one of helium-4 in the nanopore. We analyzed the scaling of the helium-4 superfluidity with the product of the temperature and the length of the simulation cell and the agreement of the results with the theoretical predictions. The transition of the helium-4 system from one-dimensional to two-dimensional with an increase in the linear density within the nanopores of radius R = 4 A was observed, which is visible in the behavior of the correlation function and suprafluidity.
In this thesis, we investigate ultracold Bose-Fermi mixtures in both two and three dimensions using a non-perturbative approach based on the Lowest Order Constrained Variational (LOCV) method. This variational technique, originally developed in the context of nuclear matter, is applied here to dilute quantum gases at zero temperature, allowing for an accurate treatment of two-body correlations beyond perturbative limits. We begin by analysing Bose-Fermi mixtures near a broad Fano-Feshbach resonance, which enables precise tuning of the interspecies interaction strength. Particular attention is devoted to the low-dimensional case, where quantum fluctuations are enhanced and experimental realizations have become increasingly accessible. While the three-dimensional problem serves as a useful benchmark, the two-dimensional Bose-Fermi mixture presents unique challenges. Notably, no prior non-perturbative studies exist on its mechanical stability. A central goal of this work is to determine the conditions under which the 2D Bose-Fermi mixture remains mechanically stable as the boson-fermion attraction is increased. Our results demonstrate that the LOCV approach captures the relevant physics from weak to moderate coupling, offering a complementary perspective to perturbative methods or alternative non-perturbative approaches such as the T-matrix formalism. In this coupling range, we identify the critical strength of boson-boson repulsion necessary to prevent phase separation or collapse, thereby providing the first theoretical variational prediction of a stability phase diagram in this two dimensional regime. Overall, this work fills a gap in the literature and provides a useful reference point for future theoretical and experimental studies of strongly interacting Bose-Fermi systems in reduced dimensionality.