# Fermi gas throughout the BCS-BEC crossover: Comparative study of t-matrix approaches with various degrees of self-consistency

## TL;DR

This study systematically compares different levels of self-consistency in t-matrix approximations to understand their impact on thermodynamic and dynamical properties of a Fermi gas across the BCS-BEC crossover.

## Contribution

It provides a comprehensive, unbiased comparison of all variants of the t-matrix approximation with varying self-consistency degrees in an attractive Fermi gas.

## Key findings

- Different self-consistency levels significantly affect thermodynamic predictions.
- The study clarifies which t-matrix variants are more appropriate for describing the normal phase.
- Results inform the choice of theoretical approaches for Fermi gases in the BCS-BEC crossover.

## Abstract

The diagrammatic t-matrix approximation has often been adopted to describe a dilute Fermi gas. This approximation, originally considered by Galitskii for a repulsive inter-particle interaction (Galitskii-1958), has later been widely utilized for an attractive Fermi gas to describe the BCS-BEC crossover from strongly overlapping Cooper pairs in weak coupling to non-overlapping composite bosons in strong coupling. Several variants of the $t$-matrix approximation have been considered in the literature, which are distinguished by the degree of self-consistency allowed in the building blocks of the diagrammatic structure. Here, we perform a systematic and comparative study of all possible variants on the degree of self-consistency for the $t$-matrix approximation in an attractive Fermi gas, which enables us to confront their outcomes for thermodynamic and dynamical quantities on the same footing in an unbiased way. For definiteness, only the normal phase above the superfluid critical temperature is considered. The dispute that can be raised in this context, about the adequateness of introducing progressive degrees of self-consistency over and above the non-self-consistent $t$-matrix approximation for an attractive Fermi gas, parallels the recent interest in the literature on assessing the importance of various degrees of self-consistency in the context of semiconductors and insulators.