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arXiv:2211.03832 · New Journal of Physics

Universal correlations along the BEC-BCS crossover

Juan Carlos Obeso-Jureidini, G. A. Domínguez‐Castro, E Neri, R. Paredes, Vı́ctor Romero-Rochı́n

Published Oct 31, 2023• 2 citations
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Abstract

Abstract Universality of the long-distance behavior across the Bardeen–Cooper–Schrieffer (BEC)-Bose–Einstein condensate (BCS) smooth transition for two-body density correlation functions and the Cooper-pair probability density is demonstrated in a balanced mixture of a two-component Fermi gas at T = 0. It is numerically shown at the mean-field level that these two-body quantities exhibit an exponential decay in terms of the chemical potential and the low-energy behavior of the gap. A general expression is found for the two-body distributions holding for different features of finite-range potentials, such as divergences at the origin, discontinuities at a finite radius, power-law decay, and exponential decay. The correlation length characterizing the long-distance behavior unravels the dependence on the energy needed to break pairs along the BEC-BCS crossover, a quantity meaningful to the stability of the many-body state.

DOI
10.1088/1367-2630/ad0854

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Abstract Universality of the long-distance behavior across the Bardeen–Cooper–Schrieffer (BEC)-Bose–Einstein condensate (BCS) smooth transition for two-body density correlation functions and the Cooper-pair probability density is demonstrated in a balanced mixture of a two-component Fermi gas at T = 0. It is numerically shown at the mean-field level that these two-body quantities exhibit an exponential decay in terms of the chemical potential and the low-energy behavior of the gap. A general expression is found for the two-body distributions holding for different features of finite-range potentials, such as divergences at the origin, discontinuities at a finite radius, power-law decay, and exponential decay. The correlation length characterizing the long-distance behavior unravels the dependence on the energy needed to break pairs along the BEC-BCS crossover, a quantity meaningful to the stability of the many-body state.
Abstract We theoretically investigate the Bardeen–Cooper–Schrieffer (BCS) to Bose–Einstein condensation (BEC) crossover in a two-dimensional Fermi gas with the finite-range interaction by using the Hartree–Fock–Bogoliubov theory. Expanding the scattering phase shift in terms of the scattering length and effective range, we discuss the effect of the finite-range interaction on the pairing and thermodynamic properties. By solving the gap equation and the number equation self-consistently, we numerically calculate the effective-range dependence of the pairing gap, chemical potential, and pair size throughout the BCS–BEC crossover. Our results would be useful for further understanding of low-dimensional many-body problems.
We analyze the BCS-BEC crossover transition of a balanced two component mixture of fermions interacting via a finite range potential, within a mean field approach. For the analysis we consider three finite range potentials cases describing the interaction between different Fermi species: a square well, an exponential and a Yukawa potential. The T = 0 thermodynamics analysis along the BCS-BEC crossover allow us to recognize the proper variables, for finite range interactions, that capture the transition from a thermodynamic non-universal behavior at unitarity, to its universal restoration. On the other side, the determination of the pair functions along the crossover suggests that the smooth transition occurs always between the scattering resonance and the change of sign of the chemical potential. This identification follows directly from the pair wave functions behavior, which in the BCS and BEC sides become exponentially localized and oscillatory slowly decaying respectively.
The spatial structure of a two-dimensional homogeneous mixture of fermionic atoms in two hyperfine states is analyzed throughout the BEC-BCS crossover. Within the BCS-Leggett mean-field model we consider three functions: the pair wave function and the density-density correlation functions between atoms of the same and of different hyperfine states. For the correlation functions we derive analytical expressions which allow one to unveil the rich spatial structure of the superfluid. Mainly, we are able to study the large-distance behavior of the three functions, which exhibits an exponential decay and a well-defined oscillatory behavior. We report closed-form expressions for the correlation lengths and mean pair radius. Differences and similarities emerge when comparing with the three-dimensional case. Particularly, we find an expression for the large-distance correlation length, in terms of the chemical potential and the gap, valid in two and three dimensions, but whose dependence on the corresponding scattering lengths differ significantly.
Within the BCS-Leggett mean-field model of a balanced mixture of two different spin fermionic atoms, we calculate the BCS pair probability distribution and the density correlation functions both between atoms with the same and different spin states, being able to accurately extract their spatial large-distance behavior along the BEC-BCS crossover. It is found that, after different initial transients, these distributions show an exponential decay and a well-defined periodic oscillatory behavior, which we analyze throughout the whole crossover. In addition, we derive analytic expressions for the mean pair size and the correlation lengths of the same and different density correlation functions. While in general, in the BCS side, there are long-range correlations and, in the BEC region, the behavior is dominated by tight-pair formation, each distribution shows its own characteristic behavior, yielding all together a quite complete description of the spatial structure of the superfluid along the crossover.
We investigate the Bose-Einstein condensation of fermionic pairs in a uniform two-component Fermi gas, obtaining an explicit formula for the condensate density as a function of the chemical potential and the energy gap. We analyze the condensate fraction in the crossover from the Bardeen-Cooper-Schrieffer (BCS) state of weakly interacting Cooper pairs to the Bose-Einstein condensate of molecular dimers. By using the local-density approximation we study confined Fermi vapors of alkali-metal atoms for which there is experimental evidence of condensation also on the BCS side of the Feshbach resonance. Our theoretical results are in agreement with these experimental data and give the behavior of the condensate on both sides of the Feshbach resonance at zero temperature.
We study a two-dimensional paired Fermi system in which altermagnetism produces a spin splitting that depends on momentum. The interaction strength is described through the two-body binding energy, so that the chemical potential and pairing gap evolve self-consistently from the weak-coupling Bardeen-Cooper-Schrieffer (BCS) regime to the strong-coupling Bose-Einstein-condensate (BEC) regime at fixed density. The stability of the uniform paired state is examined by giving the pairs a small center-of-mass momentum and following the resulting change in free energy. We show that at zero temperature, the phase stiffness follows a universal quadratic suppression, J/J0=1−(α/α0)2J/J_0 = 1 - (α/α_0)^2J/J0​=1−(α/α0​)2, across the entire fully gapped crossover regime. Deviations from this relation emerge only on the BCS side upon the opening of gapless Bogoliubov pockets, which rapidly reduce the stiffness and can trigger an instability toward finite-momentum pairing. In the BEC regime, this universal quadratic correction corresponds to the altered effective mass of the tightly bound composite bosons. The model therefore provides a simple setting in which to compare altermagnetic pair breaking on the BCS and BEC sides of the crossover.