The BCS gap equation for spin-polarized fermions

TL;DR

This paper analyzes the BCS gap equation for spin-polarized fermions, establishing conditions for solution existence, deriving bounds for critical temperature, and exploring phase diagram complexity with population imbalance.

Contribution

It extends the understanding of the BCS gap equation to spin-polarized systems, linking solution existence to spectral properties and providing bounds for critical temperature.

Findings

Solution existence linked to spectral properties for certain parameters

Derived bounds for critical temperature in small coupling limit

Identified complexity in phase diagram for high polarization

Abstract

We study the BCS gap equation for a Fermi gas with unequal population of spin-up and spin-down states. For $\cosh(\delta_\mu/T) \leq 2$, with $T$ the temperature and $\delta_\mu$ the chemical potential difference, the question of existence of non-trivial solutions can be reduced to spectral properties of a linear operator, similar to the unpolarized case studied previously in \cite{FHNS,HHSS,HS}. For $\cosh(\delta_\mu/T) > 2$ the phase diagram is more complicated, however. We derive upper and lower bounds for the critical temperature, and study their behavior in the small coupling limit.