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

TL;DR

This paper develops a self-consistent theoretical framework to analyze molecular and pairing instabilities in a degenerate Fermi gas across the BEC-BCS crossover, providing insights into critical temperatures and molecular formation thresholds.

Contribution

It introduces a unified approach to calculate instabilities and critical temperatures in a Fermi gas, aligning well with known results and highlighting quantum effects on molecule formation.

Findings

Critical temperatures for BCS and BE condensation agree with previous studies.

Molecular formation threshold shifts toward the BEC side due to quantum effects.

Shift in formation threshold remains significant up to temperatures near the Fermi energy.

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 BEC-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 sizeable 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.