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:0904.2630 · Physical Review A

Ginzburg-Landau theory of a trapped Fermi gas with a BEC-BCS crossover

Kun Huang, Zeng-Qiang Yu, Lan Yin

Published May 4, 2009• 11 citations
arXivHTMLPDF

Abstract

The Ginzburg-Landau theory of a trapped Fermi gas with a BEC-BCS crossover is derived by the path-integral method. In addition to the standard Ginzburg-Landau equation, a second equation describing the total atom density is obtained. These two coupled equations are necessary to describe both homogeneous and inhomogeneous systems. The Ginzburg-Landau theory is valid near the transition temperature ${T}_{c}$ on both sides of the crossover. In the weakly interacting BEC region, it is also accurate at zero temperature where the Ginzburg-Landau equation can be mapped onto the Gross-Pitaevskii (GP) equation. The applicability of GP equation at finite temperature is discussed. On the BEC side, the fluctuation of the order parameter is studied and the renormalization to the molecule coupling constant is obtained.

DOI
10.1103/physreva.79.053602

Related papers

Unitarity Schrödinger Equation and Ground State Properties of the Finite Trapped Superfluid Fermi Gases in a BCS-BEC Crossover

Zhang Wen-Yuan, Wang Cheng-tao, Ma Yong-Li

On the basis of quantum hydrodynamical equations we derive a unitarity Schrödinger equation of a finite trapped superfluid Fermi gas valid in the whole interaction regime from BCS superfluid to BEC. This equation is just the Ginzburg–Laudau-type equation for the fermionic Cooper pairs in the BCS side, the Gross–Pitaevskii-type equation for the bosonic dimers in the BEC side, and a unitarity equation for a strongly interacting Fermi superfluid in the unitarity limit. By taking a modified Gauss-like trial wave function, we solve the unitarity Schrödinger equation, calculate the energy, chemical potential, sizes and profiles of the ground-state condensate, and discuss the properties of the ground state in the entire BCS-BEC crossover regimes.

Chinese Physics Letters•Apr 1, 2010•1 citation

Dynamics of Fluctuactions in the BCS-BEC Crossover in Superfluid Fermi Gases

H Vivas C

The BCS-BEC crossover phenomenon in a 3D attractive Fermi gas is revisited by using finite temperature Green functions formalism derived from the quantum path integral technique and Mean Field Approximation (MFA). Collective modes for weak and strong coupling particle-particle interaction g are obtained after recovering the dynamical coefficients in the nonlinear Time Dependent Ginzburg Landau (TDGL) equation, and numerically simulated under those particular states for the filling energy g ~ 0 and order parameter Δ( x , τ ) ~ Δ 0 . Explicit formulas for the first order average fluctuations ⟨η 2 q,ω ⟩ and the scaling law for the thermal energy associated to the non-linear collective modes are also reported.

Journal of Physics Conference Series•Dec 8, 2014•0 citations

Solutions of Ginzburg-Landau Theory for Atomic Fermi Gases Near the BCS-BEC Crossover

Shuhong, Chen, Boling +1 more

We are concerned with a time-dependent Ginzburg-Landau equations come from the superfluid atomic Fermi-gases near the Feshbach resonance from the fermion-boson model. We obtain the global existence and uniqueness of solutions to the TDGL equations near the BCS-BEC crossover.

Acta Mathematicae Applicatae Sinica English Series•Jan 1, 2015•0 citations

Dynamical behaviour of the time-dependent Ginzburg-Landau equation for atomic Fermi gases near the BCS-BEC crossover

Chunyan Xiong, Shuhong Chen

The present article studies the dynamic behaviour of the time-dependent Ginzburg-Landau equation, which describes the phenomenon of BCS-BEC crossover.First, the appropriate prior estimates are established by Gronwall's inequality and Gagliardo-Nirenberg interpolation inequality, etc.Then, we prove the existence of the global attractor in the time-dependent Ginzburg-Landau system.Finally, the evolution of the field and the influence of physical parameters on the dynamic behaviour are demonstrated by numerical experiments, thereby validating the theory in this article.

Current Science•Nov 25, 2025•0 citations

Extended Ginzburg-Landau functional for two-band superfluid Fermi gases

Serghei N. Klimin, Jacques Tempere, J. T. Devreese

The standard Ginzburg-Landau theory is valid only near the critical temperature, where the order parameter is small. As a consequence it is not straightforward to develop a Ginzburg-Landau framework for two-bandgap superconductors such as magnesium diboride, where two order parameters exist with different intrinsic critical temperatures. We derive an extended Ginzburg-Landau functional, capable of describing both single-band and two-bandgap superfluids at all temperatures below the critical temperature. In the limit of small temperatures, we retrieve the ground state energy obtained from the microscopic path-integral theory for Fermi superfluids, and for temperatures near the critical temperature, we obtain the standard Ginzburg-Landau expressions. The present formalism is applied to reveal hidden criticality and the presence of two healing length scales in two-bandgap superfluids. The developed method is focused on strong-coupling superfluid systems, in particular, the ultracold atomic Fermi gases in the BCS-BEC crossover regime.

•Sep 5, 2013•0 citations

BCS-BEC crossover of a quasi-two-dimensional Fermi gas: The significance of dressed molecules

Wei Zhang, Guin-Dar Lin, L.-M. Duan

We study the crossover of a quasi-two-dimensional Fermi gas trapped in the radial plane from the Bardeen-Cooper-Schrieffer (BCS) regime to the Bose-Einstein condensation (BEC) regime by crossing a wide Feshbach resonance. We consider two effective two-dimensional Hamiltonians within the mean-field level, and calculate the zero-temperature cloud size and number density distribution. For a model 1 Hamiltonian with renormalized atom-atom interaction, we observe a constant cloud size for arbitrary detunings. For a model 2 Hamiltonian with renormalized interactions between atoms and dressed molecules, the cloud size deceases from the BCS to BEC side, which is consistent with the picture of BCS-BEC crossover. This qualitative discrepancy between the two models indicates that the inclusion of dressed molecules is essential for a mean-field description of quasi-two-dimensional Fermi systems, especially on the BEC side of the Feshbach resonance.

arXiv:0803.2488•Physical Review A•Jun 13, 2008•36 citations
The Ginzburg-Landau theory of a trapped Fermi gas with a BEC-BCS crossover is derived by the path-integral method. In addition to the standard Ginzburg-Landau equation, a second equation describing the total atom density is obtained. These two coupled equations are necessary to describe both homogeneous and inhomogeneous systems. The Ginzburg-Landau theory is valid near the transition temperature Tc{T}_{c}Tc​ on both sides of the crossover. In the weakly interacting BEC region, it is also accurate at zero temperature where the Ginzburg-Landau equation can be mapped onto the Gross-Pitaevskii (GP) equation. The applicability of GP equation at finite temperature is discussed. On the BEC side, the fluctuation of the order parameter is studied and the renormalization to the molecule coupling constant is obtained.
On the basis of quantum hydrodynamical equations we derive a unitarity Schrödinger equation of a finite trapped superfluid Fermi gas valid in the whole interaction regime from BCS superfluid to BEC. This equation is just the Ginzburg–Laudau-type equation for the fermionic Cooper pairs in the BCS side, the Gross–Pitaevskii-type equation for the bosonic dimers in the BEC side, and a unitarity equation for a strongly interacting Fermi superfluid in the unitarity limit. By taking a modified Gauss-like trial wave function, we solve the unitarity Schrödinger equation, calculate the energy, chemical potential, sizes and profiles of the ground-state condensate, and discuss the properties of the ground state in the entire BCS-BEC crossover regimes.
The BCS-BEC crossover phenomenon in a 3D attractive Fermi gas is revisited by using finite temperature Green functions formalism derived from the quantum path integral technique and Mean Field Approximation (MFA). Collective modes for weak and strong coupling particle-particle interaction g are obtained after recovering the dynamical coefficients in the nonlinear Time Dependent Ginzburg Landau (TDGL) equation, and numerically simulated under those particular states for the filling energy g ~ 0 and order parameter Δ( x , τ ) ~ Δ 0 . Explicit formulas for the first order average fluctuations ⟨η 2 q,ω ⟩ and the scaling law for the thermal energy associated to the non-linear collective modes are also reported.
The present article studies the dynamic behaviour of the time-dependent Ginzburg-Landau equation, which describes the phenomenon of BCS-BEC crossover.First, the appropriate prior estimates are established by Gronwall's inequality and Gagliardo-Nirenberg interpolation inequality, etc.Then, we prove the existence of the global attractor in the time-dependent Ginzburg-Landau system.Finally, the evolution of the field and the influence of physical parameters on the dynamic behaviour are demonstrated by numerical experiments, thereby validating the theory in this article.
The standard Ginzburg-Landau theory is valid only near the critical temperature, where the order parameter is small. As a consequence it is not straightforward to develop a Ginzburg-Landau framework for two-bandgap superconductors such as magnesium diboride, where two order parameters exist with different intrinsic critical temperatures. We derive an extended Ginzburg-Landau functional, capable of describing both single-band and two-bandgap superfluids at all temperatures below the critical temperature. In the limit of small temperatures, we retrieve the ground state energy obtained from the microscopic path-integral theory for Fermi superfluids, and for temperatures near the critical temperature, we obtain the standard Ginzburg-Landau expressions. The present formalism is applied to reveal hidden criticality and the presence of two healing length scales in two-bandgap superfluids. The developed method is focused on strong-coupling superfluid systems, in particular, the ultracold atomic Fermi gases in the BCS-BEC crossover regime.
We study the crossover of a quasi-two-dimensional Fermi gas trapped in the radial plane from the Bardeen-Cooper-Schrieffer (BCS) regime to the Bose-Einstein condensation (BEC) regime by crossing a wide Feshbach resonance. We consider two effective two-dimensional Hamiltonians within the mean-field level, and calculate the zero-temperature cloud size and number density distribution. For a model 1 Hamiltonian with renormalized atom-atom interaction, we observe a constant cloud size for arbitrary detunings. For a model 2 Hamiltonian with renormalized interactions between atoms and dressed molecules, the cloud size deceases from the BCS to BEC side, which is consistent with the picture of BCS-BEC crossover. This qualitative discrepancy between the two models indicates that the inclusion of dressed molecules is essential for a mean-field description of quasi-two-dimensional Fermi systems, especially on the BEC side of the Feshbach resonance.