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

TL;DR

This paper develops a Ginzburg-Landau theoretical framework for a trapped Fermi gas across the BEC-BCS crossover, incorporating density equations and analyzing fluctuations, valid near the transition temperature and connecting to the Gross-Pitaevskii equation.

Contribution

It derives a coupled Ginzburg-Landau and density equation system for inhomogeneous Fermi gases across the BEC-BCS crossover, extending the theory's applicability.

Findings

The theory is valid near the transition temperature on both sides of the crossover.

The Ginzburg-Landau equation maps onto the Gross-Pitaevskii equation at zero temperature in the BEC regime.

Order parameter fluctuations and molecule coupling renormalization are analyzed on the BEC side.

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.