Self-consistent inclusion of disorder in the BCS-BEC crossover near the critical temperature

TL;DR

This paper presents a systematic theoretical framework to include static disorder effects in the BCS-BEC crossover near the critical temperature, bridging the gap between well-understood limits.

Contribution

It develops a controlled, functional-integral approach that accounts for Gaussian fluctuations and disorder, applicable to both continuum and lattice systems.

Findings

Recovers known BCS and BEC limits with disorder included.

Provides a consistent description of pairing fluctuations in disordered systems.

Framework applicable to multiband models and various system types.

Abstract

We develop a systematic theoretical approach to incorporate the effects of a static white-noise disorder into the BCS-BEC crossover near the critical temperature ($T_c$) of the superfluid transition. Starting from a functional-integral formulation in momentum-frequency space, we derive an effective thermodynamic potential that fully accounts for Gaussian fluctuations of the order-parameter field and its coupling to the disorder potential. The effective action, expanded to second order in both the disorder potential and the bosonic field, naturally involves third- and fourth-order terms arising from the logarithmic expansion near $T_c$. By providing a controlled description of pairing fluctuations and disorder effects, this formalism correctly recovers the well-established BCS and BEC limits. This ensures a consistent physical foundation for analyzing the entire BCS-BEC crossover, effectively anchoring the intermediate regime between these two analytically robust endpoints. The approach applies equally to continuum and lattice systems and provides a natural framework for generalizations to multiband models.