Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0507636 · Physical Review A

Self-consistent theory for molecular instabilities in a normal degenerate Fermi gas in the BEC-BCS crossover

R. Combescot, X. Leyronas, M. Yu. Kagan

Published Feb 27, 2006• 56 citations
arXivHTMLPDF

Abstract

We investigate within a self-consistent theory the molecular instabilities arising in the normal state of a homogeneous degenerate Fermi gas, covering the whole Bose-Einstein condensate (BEC) to BCS crossover. These are the standard instability for molecular formation, the BCS instability which corresponds to the formation of Cooper pairs, and the related Bose-Einstein instability. These instabilities manifest themselves in the properties of the particle-particle vertex, which we calculate in a ladder approximation. To find the critical temperatures corresponding to these various instabilities, we handle the properties of the interacting Fermi gas on the same footing as the instabilities by making use of the same vertex. This approximate treatment is shown to be quite satisfactory in a number of limiting situations where it agrees with known exact results. The results for the BCS critical temperature and for the BE condensation are found to be in fair agreement with earlier results. The threshold for formation of molecules at rest undergoes a sizable shift toward the BEC side, due to quantum effects arising from the presence of the degenerate Fermi gas. This should make its experimental observation fairly easy. This shift remains important at least up to temperatures comparable to the Fermi energy of the gas.

DOI
10.1103/physreva.73.023618

Related papers

Condensate fraction of a Fermi gas in the BCS-BEC crossover

Luca Salasnich, Nicola Manini, Alberto Parola

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.

arXiv:cond-mat/0506074•Physical Review A•Aug 30, 2005•89 citations

Equation of state of a superfluid Fermi gas in the BCS- BEC crossover

H. Hu, Xia-Ji Liu, P. D. Drummond

PACS. 03.75.Hh – Static properties of condensates; thermodynamical, statistical and structural properties. PACS. 03.75.Ss – Degenerate Fermi gases. PACS. 05.30.Fk – Fermion systems and electron gas. Abstract. – We present a theory for a superfluid Fermi gas near the BCS-BEC crossover, including pairing fluctuation contributions to the free energy similar to that considered by Nozières and Schmitt-Rink for the normal phase. In the strong coupling limit, our theory is able to recover the Bogoliubov theory of a weakly interacting Bose gas with a molecular scattering length very close to the known exact result. We compare our results with recent Quantum Monte Carlo simulations both for the ground state and at finite temperature. Excellent agreement is found for all interaction strengths where simulation results are available. The recent experimental realization of strongly interacting Fermi gases of 6 Li and 40 K atoms near a Feshbach resonance has opened up the exciting possibility of investigating the crossover from a Bardeen-Cooper-Schrieffer (BCS) superfluid to a Bose-Einstein condensate (BEC) [1–5]. In these systems, the inter-atomic interaction strength can be varied by tuning

•Jan 1, 2006•84 citations

Self-consistent theory of atomic Fermi gases with a Feshbach resonance at the superfluid transition

Xia-Ji Liu, Hui Hu

A self-consistent theory is derived to describe the BCS--Bose-Einstein-condensate crossover for a strongly interacting Fermi gas with a Feshbach resonance. In the theory the fluctuation of the dressed molecules, consisting of both preformed Cooper pairs and ``bare'' Feshbach molecules, has been included within a self-consistent $T$-matrix approximation, beyond the Nozi\`eres and Schmitt-Rink strategy considered by Ohashi and Griffin. The resulting self-consistent equations are solved numerically to investigate the normal-state properties of the crossover at various resonance widths. It is found that the superfluid transition temperature ${T}_{c}$ increases monotonically at all widths as the effective interaction between atoms becomes more attractive. Furthermore, a residue factor ${Z}_{m}$ of the molecule's Green function and a complex effective mass have been determined to characterize the fraction and lifetime of Feshbach molecules at ${T}_{c}$. Our many-body calculations of ${Z}_{m}$ agree qualitatively well with recent measurments of the gas of $^{6}\mathrm{Li}$ atoms near the broad resonance at $834\phantom{\rule{0.3em}{0ex}}\mathrm{G}$. The crossover from narrow to broad resonances has also been studied.

arXiv:cond-mat/0505572•Physical Review A•Dec 14, 2005•59 citations

Crossover from BCS Theory to Bose–Einstein Condensation

Mohit Randeria

This article gives a detailed review of our current theoretical understanding of the crossover from cooperative Cooper pairing to independent bound-state formation and Bose–Einstein condensation in Fermi systems with increasing attractive interactions. Introduction There are two well-known paradigms for understanding the phenomena of superconductivity and superfluidity: (I) BCS theory [1], in which the normal state is a degenerate Fermi liquid that undergoes a pairing instability at a temperature T c ≪ ε f , the degeneracy scale. The formation of Cooper pairs and their condensation (macroscopic occupation of a single quantum state) both occur simultaneously at the transition temperature T c . (II) Bose–Einstein condensation (BEC) of bosons at a T c of the order of their degeneracy temperature. At a fundamental level these bosons (for example, 4 He or excitons) are invariably composite objects made up of an even number of fermions. The composite particles form at some very high temperature scale of the order of their dissociation temperature T dissoc , and these “pre-formed” bosons then condense at the BEC T c ≪ T dissoc . In most cases of experimental interest the system under consideration clearly falls into one category or the other. For instance, 3 He is a Fermi superfluid described by (I) whereas 4 He is a Bose superfluid (II). Essentially all cases of superconductivity in metals that we understand reasonably well are much closer to (I) than to (II), as evidenced by the existence of fermionic quasiparticles above T c .

Cambridge University Press eBooks•Apr 6, 1995•158 citations

Modified Fermi sphere, pairing gap, and critical temperature for the BCS-BEC crossover

Stefan Floerchinger, Michael M. Scherer, CHRISTOF WETTERICH

We investigate the phase diagram of two-component fermions in the BCS-BEC (Bose-Einstein condensate) crossover. Using functional renormalization-group equations we calculate the effect of quantum fluctuations on the fermionic self-energy parametrized by a wave-function renormalization, an effective Fermi radius, and the gap. This allows us to follow the modifications of the Fermi surface and the dispersion relation for fermionic excitations throughout the whole crossover region. We also determine the critical temperature of the second-order phase transition to superfluidity. Our results are in agreement with BCS theory including Gorkov's correction for a small negative scattering length $a$ and with an interacting Bose gas for a small positive $a$. At the unitarity point the result for the gap at zero temperature agrees well with quantum Monte Carlo simulations, while the critical temperature differs.

arXiv:0912.4050•Physical Review A•Jun 14, 2010•38 citations

Fermi excitations in a trapped atomic Fermi gas with a molecular Bose condensate

Yoji Ohashi, Adam Griffin

We discuss the effect of a molecular Bose condensate on the energy of Fermi excitations in a trapped two-component atomic Fermi gas. The single-particle Green's functions can be approximated by the well-known BCS form, in both the BCS (Cooper pairs) and BEC (Feshbach resonance molecules) domains. The composite Bose order parameter ${\tilde Δ}$ describing bound states of two atoms and the Fermi chemical potential $μ$ are calculated self-consistently. In the BEC regime characterized by $μ<0$, the Fermi quasiparticle energy gap is given by $\sqrt{μ^2+{\tilde Δ}^2}$, instead of $|{\tilde Δ}|$ in the BCS region, where $μ>0$. This shows up in the characteristic energy of atoms from dissociated molecules.

arXiv:cond-mat/0402031•Feb 2, 2004•0 citations
We investigate within a self-consistent theory the molecular instabilities arising in the normal state of a homogeneous degenerate Fermi gas, covering the whole Bose-Einstein condensate (BEC) to BCS crossover. These are the standard instability for molecular formation, the BCS instability which corresponds to the formation of Cooper pairs, and the related Bose-Einstein instability. These instabilities manifest themselves in the properties of the particle-particle vertex, which we calculate in a ladder approximation. To find the critical temperatures corresponding to these various instabilities, we handle the properties of the interacting Fermi gas on the same footing as the instabilities by making use of the same vertex. This approximate treatment is shown to be quite satisfactory in a number of limiting situations where it agrees with known exact results. The results for the BCS critical temperature and for the BE condensation are found to be in fair agreement with earlier results. The threshold for formation of molecules at rest undergoes a sizable shift toward the BEC side, due to quantum effects arising from the presence of the degenerate Fermi gas. This should make its experimental observation fairly easy. This shift remains important at least up to temperatures comparable to the Fermi energy of the gas.
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.
PACS. 03.75.Hh – Static properties of condensates; thermodynamical, statistical and structural properties. PACS. 03.75.Ss – Degenerate Fermi gases. PACS. 05.30.Fk – Fermion systems and electron gas. Abstract. – We present a theory for a superfluid Fermi gas near the BCS-BEC crossover, including pairing fluctuation contributions to the free energy similar to that considered by Nozières and Schmitt-Rink for the normal phase. In the strong coupling limit, our theory is able to recover the Bogoliubov theory of a weakly interacting Bose gas with a molecular scattering length very close to the known exact result. We compare our results with recent Quantum Monte Carlo simulations both for the ground state and at finite temperature. Excellent agreement is found for all interaction strengths where simulation results are available. The recent experimental realization of strongly interacting Fermi gases of 6 Li and 40 K atoms near a Feshbach resonance has opened up the exciting possibility of investigating the crossover from a Bardeen-Cooper-Schrieffer (BCS) superfluid to a Bose-Einstein condensate (BEC) [1–5]. In these systems, the inter-atomic interaction strength can be varied by tuning
A self-consistent theory is derived to describe the BCS--Bose-Einstein-condensate crossover for a strongly interacting Fermi gas with a Feshbach resonance. In the theory the fluctuation of the dressed molecules, consisting of both preformed Cooper pairs and ``bare'' Feshbach molecules, has been included within a self-consistent TTT-matrix approximation, beyond the Nozi\`eres and Schmitt-Rink strategy considered by Ohashi and Griffin. The resulting self-consistent equations are solved numerically to investigate the normal-state properties of the crossover at various resonance widths. It is found that the superfluid transition temperature Tc{T}_{c}Tc​ increases monotonically at all widths as the effective interaction between atoms becomes more attractive. Furthermore, a residue factor Zm{Z}_{m}Zm​ of the molecule's Green function and a complex effective mass have been determined to characterize the fraction and lifetime of Feshbach molecules at Tc{T}_{c}Tc​. Our many-body calculations of Zm{Z}_{m}Zm​ agree qualitatively well with recent measurments of the gas of 6Li^{6}\mathrm{Li}6Li atoms near the broad resonance at 834G834\phantom{\rule{0.3em}{0ex}}\mathrm{G}834G. The crossover from narrow to broad resonances has also been studied.
This article gives a detailed review of our current theoretical understanding of the crossover from cooperative Cooper pairing to independent bound-state formation and Bose–Einstein condensation in Fermi systems with increasing attractive interactions. Introduction There are two well-known paradigms for understanding the phenomena of superconductivity and superfluidity: (I) BCS theory [1], in which the normal state is a degenerate Fermi liquid that undergoes a pairing instability at a temperature T c ≪ ε f , the degeneracy scale. The formation of Cooper pairs and their condensation (macroscopic occupation of a single quantum state) both occur simultaneously at the transition temperature T c . (II) Bose–Einstein condensation (BEC) of bosons at a T c of the order of their degeneracy temperature. At a fundamental level these bosons (for example, 4 He or excitons) are invariably composite objects made up of an even number of fermions. The composite particles form at some very high temperature scale of the order of their dissociation temperature T dissoc , and these “pre-formed” bosons then condense at the BEC T c ≪ T dissoc . In most cases of experimental interest the system under consideration clearly falls into one category or the other. For instance, 3 He is a Fermi superfluid described by (I) whereas 4 He is a Bose superfluid (II). Essentially all cases of superconductivity in metals that we understand reasonably well are much closer to (I) than to (II), as evidenced by the existence of fermionic quasiparticles above T c .
We investigate the phase diagram of two-component fermions in the BCS-BEC (Bose-Einstein condensate) crossover. Using functional renormalization-group equations we calculate the effect of quantum fluctuations on the fermionic self-energy parametrized by a wave-function renormalization, an effective Fermi radius, and the gap. This allows us to follow the modifications of the Fermi surface and the dispersion relation for fermionic excitations throughout the whole crossover region. We also determine the critical temperature of the second-order phase transition to superfluidity. Our results are in agreement with BCS theory including Gorkov's correction for a small negative scattering length aaa and with an interacting Bose gas for a small positive aaa. At the unitarity point the result for the gap at zero temperature agrees well with quantum Monte Carlo simulations, while the critical temperature differs.
We discuss the effect of a molecular Bose condensate on the energy of Fermi excitations in a trapped two-component atomic Fermi gas. The single-particle Green's functions can be approximated by the well-known BCS form, in both the BCS (Cooper pairs) and BEC (Feshbach resonance molecules) domains. The composite Bose order parameter Δ~{\tilde Δ}Δ~ describing bound states of two atoms and the Fermi chemical potential μμμ are calculated self-consistently. In the BEC regime characterized by μ<0μ<0μ<0, the Fermi quasiparticle energy gap is given by μ2+Δ~2\sqrt{μ^2+{\tilde Δ}^2}μ2+Δ~2​, instead of ∣Δ~∣|{\tilde Δ}|∣Δ~∣ in the BCS region, where μ>0μ>0μ>0. This shows up in the characteristic energy of atoms from dissociated molecules.