On the correct strong-coupling limit in the evolution from BCS superconductivity to Bose-Einstein condensation

TL;DR

This paper critically examines the fermionic T-matrix approximation in the BCS-BEC crossover, identifies its failure in the strong-coupling limit, and proposes a new diagrammatic approach that accurately describes the system across all regimes.

Contribution

The authors develop a new diagrammatic framework for the BCS-BEC crossover that correctly captures both weak and strong coupling limits, improving upon the fermionic T-matrix approximation.

Findings

The fermionic T-matrix approximation fails in the strong-coupling limit.

A new class of diagrams is introduced that accurately describes the system across all regimes.

The scattering length for composite bosons is significantly modified using the new approach.

Abstract

We consider the problem of the crossover from BCS superconductivity to Bose-Einstein condensation in three dimensions for a system of fermions with an attractive interaction, for which we adopt the simplifying assumption of a suitably regularized point-contact interaction. We examine in a critical way the fermionic (self-consistent) T-matrix approximation which has been widely utilized in the literature to describe this crossover above the superconducting critical temperature, and show that it fails to yield the correct behaviour of the system in the strong-coupling limit, where composite bosons form as tightly bound fermion pairs. We then set up the correct approximation for a ``dilute'' system of composite bosons and show that an entire new class of diagrams has to be considered in the place of the fermionic T-matrix approximation for the self-energy. This new class of diagrams correctly describes both the weak- and strong-coupling limits, and consequently results into an improved interpolation scheme for the intermediate (crossover) region. In this context, we provide also a systematic mapping between the corresponding diagrammatic theories for the composite bosons and the constituent fermions. As a preliminary result to demonstrate the numerical effect of our new class of diagrams on physical quantities, we calculate the value of the scattering length for composite bosons in the strong-coupling limit and show that it is considerably modified with respect to the result obtained within the self-consistent fermionic T-matrix approximation.