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Dimensional crossover in two-component Fermi gases with attractive interactions

Mathias Macêdo Lima

Published Mar 30, 2026• 0 citations
Publisher

Abstract

This work studies zero-temperature Fermi gases with short-ranged and attractive interactions with focus on the dimensionality aspects by confining the system from three to two dimensions. Interacting Fermi gases are superfluids characterized as being strongly correlated if the scattering length is much larger than the other scales present. In the many-body context, this category of systems requires a careful treatment since mean-field approaches are not expected to hold under the conditions which define the strong pairing. For this reason, we employ quantum Monte Carlo methods to study the dimensionality effects in two-component Fermi gases in the Bardeen-Cooper-Schrieffer to Bose-Einstein condensate (BCS-BEC) crossover. To be able to investigate such aspects, we consider a three-dimensional gas with periodic boundary conditions in the xy plane and a confining potential in the z direction capable of reducing the dimensionality of the system, which defines a dimensional BCS-BEC crossover. We calculate the equation of state as the energy per particle as a function of the dimensionality and as a function of the interaction parameter, which are characterized by the oscillator length in the z direction and the scattering length respectively. The greater confinement in the system lowers the interparticle distances, which increases the effects of the attractive interactions between antiparallel-spin fermions.We also calculate the density profile in the z direction over the crossover for different confinements and different interaction parameters to quantify their influence on the spatial distribution of the particles. This study connects the perspectives between theoretical and experimental works in low-dimensional ultracold Fermi gases by utilizing an approach that is similar to what it is realized in modern experiments.

DOI
10.11606/d.76.2026.tde-22072026-114416

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This work studies zero-temperature Fermi gases with short-ranged and attractive interactions with focus on the dimensionality aspects by confining the system from three to two dimensions. Interacting Fermi gases are superfluids characterized as being strongly correlated if the scattering length is much larger than the other scales present. In the many-body context, this category of systems requires a careful treatment since mean-field approaches are not expected to hold under the conditions which define the strong pairing. For this reason, we employ quantum Monte Carlo methods to study the dimensionality effects in two-component Fermi gases in the Bardeen-Cooper-Schrieffer to Bose-Einstein condensate (BCS-BEC) crossover. To be able to investigate such aspects, we consider a three-dimensional gas with periodic boundary conditions in the xy plane and a confining potential in the z direction capable of reducing the dimensionality of the system, which defines a dimensional BCS-BEC crossover. We calculate the equation of state as the energy per particle as a function of the dimensionality and as a function of the interaction parameter, which are characterized by the oscillator length in the z direction and the scattering length respectively. The greater confinement in the system lowers the interparticle distances, which increases the effects of the attractive interactions between antiparallel-spin fermions.We also calculate the density profile in the z direction over the crossover for different confinements and different interaction parameters to quantify their influence on the spatial distribution of the particles. This study connects the perspectives between theoretical and experimental works in low-dimensional ultracold Fermi gases by utilizing an approach that is similar to what it is realized in modern experiments.
We investigate the dimensional crossover from three to two dimensions in an ultracold Fermi gas across the whole BCS-BEC crossover. Of particular interest is the strongly interacting regime as strong correlations and pair fluctuations are more pronounced in reduced dimensions. Our results are obtained from first principles within the framework of the functional renormalization group (FRG). Here, the confinement of the transverse direction is imposed by means of periodic boundary conditions. We calculate the equation of state, the gap parameter at zero temperature, and the superfluid transition temperature across a wide range of transversal confinement length scales. Particular emphasis is put on the determination of the finite-temperature phase diagram for different confinement length scales. In the end, our results are compared with recent experimental observations and we discuss them in the context of other theoretical works.
Ultracold atomic Fermi gases have been a popular topic of research, with attention being paid recently to two-dimensional (2D) gases. In this work, we perform T=0T=0T=0 ab initio diffusion Monte Carlo calculations for a strongly interacting two-component Fermi gas confined to two dimensions. We first go over finite-size systems and the connection to the thermodynamic limit. After that, we illustrate pertinent 2D scattering physics and properties of the wave function. We then show energy results for the strong-coupling crossover, in between the Bose-Einstein condensation (BEC) and Bardeen-Cooper-Schrieffer (BCS) regimes. Our energy results for the BEC-BCS crossover are parametrized to produce an equation of state, which is used to determine Tan's contact. We carry out a detailed comparison with other microscopic results. Finally, we calculate the pairing gap for a range of interaction strengths in the strong coupling regime, following from variationally optimized many-body wave functions.
We investigate collective modes and free expansions of quasi-one- and quasi-two-dimensional (Q1D and Q2D) ultracold Fermi gases in the crossover from a Bardeen-Cooper-Schrieffer (BCS) superfluid to a Bose-Einstein condensate (BEC). We solve a superfluid order parameter equation valid for the BCS-BEC crossover by employing a time-dependent variational method. We take a trial wave function of hybrid Gaussian-parabolic type, which not only reflects the low-dimensional character of the system but also allows an essentially analytical approach for the problem. We present Q1D and Q2D criteria that are valid in various superfluid regimes and show clearly the relation between the maximum condensed particle number and the parameters of the trapping potential as well as the atom-atom interaction. We demonstrate that, due to the small particle number in Q1D and Q2D condensates, the contribution to oscillating frequencies of collective modes by the quantum pressure in the strong-confinement direction is significant and hence the Thomas-Fermi approximation cannot be used. We also show that the free expansion of Q1D and Q2D superfluid Fermi gases in the strong-confinement direction is much faster than that in the weak-confinement direction.
We investigate thermodynamics and phase structure of ultracold Fermi gases, which can be realized and measured in the laboratory with modern trapping techniques. We approach the subject from a both theoretical and experimental perspective. Central to the analysis is the systematic comparison of the BCS-BEC crossover of two-component fermions in both three and two dimensions. A dimensional reduction can be achieved in experiments by means of highly anisotropic traps. The Functional Renormalization Group (FRG) allows for a description of both cases in a unified theoretical framework. In three dimensions we discuss with the FRG the influence of high momentum particles onto the density, extend previous approaches to the Unitary Fermi Gas to reach quantitative precision, and study the breakdown of superfluidity due to an asymmetry in the population of the two fermion components. In this context we also investigate the stability of the Sarma phase. For the two-dimensional system scattering theory in reduced dimension plays an important role. We present both the theoretically as well as experimentally relevant aspects thereof. After a qualitative analysis of the phase diagram and the equation of state in two dimensions with the FRG we describe the experimental determination of the phase diagram of the two-dimensional BCS-BEC crossover in collaboration with the group of S. Jochim at PI Heidelberg.
We study the system of a dilute gas of fermions in 3-dimensions, with attractive interactions tuned to the unitarity point, using the non-perturbative Restricted Path Integral Monte Carlo (R-PIMC) method. The pairing and superfluid properties of this system are calculated at finite temperature. The total energy at very low temperature from our results agrees closely with that of previous ground-state Quantum Monte Carlo calculations. We identify the temperature T � � 0.70ǫF below which pairing correlations develop, and estimate the critical temperature for the superfluid transition Tc � 0.25ǫF from a finite size scaling analysis of the superfluid density. PACS numbers: The recent experiments, starting with generating a degenerate gas of cold atoms[3], the use of Feshbach resonance to tune the effective interaction between the fermions[4, 5], the measurements of the gap to identify a pairing scale[6] and the measurements of vortices[7] to identify the superfluid state, have all given an impetus to the study of superfluidity in the BCS-BEC crossover regime. The nature of superconductivity or superfluidity in a many-particle system with an increasing pairing attraction was shown [1, 2] to interpolate smoothly between the BCS regime for weak attraction between the fermions, and the Bose-Einstein condensation (BEC) regime for strong coupling. In the weak coupling regime, the Cooper pair size is much larger than the interparticle spacing, and the simultaneous pairing and condensation of fermions is well described by the BCS theory. In the strong coupling regime, fermions form strongly bound bosonic molecules at a pairing temperature scale T ∗ that is considerably higher than the BEC temperature Tc. In a two species system of fermions with an attractive interaction between them, the unitary point is defined by the divergence of the zero energy s-channel scattering length, as, for two particles in free space. The unitary point is interesting since it describes a strongly correlated system with universal properties[8]. In general, there are two length scales that define the system: the interparticle spacing ∝ n −1/3 , and the scattering length as which contains information about the interaction potential. At the unitary point however, since as diverges, all the properties of the system are described by a single length scale k −1
We elucidate the role of the effective range in the Bose-Einstein-condensate (BEC) to Bardeen-Cooper-Schrieffer (BCS) crossover regime of two-component fermions in three dimensions. In contrast to ultracold Fermi gases near the broad Feshbach resonance, where the interaction can be characterized by the contact-type interaction, the interaction range in general becomes important in the density-induced BEC-BCS crossover discussed in the context of condensed-matter and nuclear systems. By characterizing the nonlocal interaction in terms of the low-energy constants such as scattering length aaa and effective range rrr, we show how the crossover phenomena are affected by nonzero effective ranges. In particular, we show that the superfluid order parameter is strongly suppressed in the high-density regime and the sound velocity exhibits a nonmonotonic behavior reflecting the mechanical stability of the system. Moreover, we point out that the high-momentum tail associated with the contact parameter can be visible when the magnitude of momentum kkk is much less than 1/r1/r1/r. We also discuss the relation between the high-momentum tail and the nuclear contact formalism.