Superconductivity with repulsion: a variational approach

TL;DR

This paper demonstrates that the BCS superconducting solution in systems with repulsive interactions is a saddle point rather than a minimum, and shows how a variational approach restores its stability, especially in multiband models relevant to iron-based superconductors.

Contribution

The paper introduces a variational formulation that correctly stabilizes the superconducting state in the presence of repulsive interactions, correcting previous non-variational approaches.

Findings

The BCS solution appears as a saddle point under non-variational mean-field free energy.

A variational approach restores the BCS solution as a stable minimum.

In a two-band model, the s$_ ext{±}$ state is shown to be stable with the new method.

Abstract

We revisit the stability of the superconducting state within mean-field theory in the presence of repulsive pairing interactions, focusing on multiband systems where such channels naturally arise. We show that, when repulsion is present, the self-consistent BCS solution appears as a saddle point of the conventional mean-field free energy, casting doubt on its physical stability. We show that this pathology is an artifact of using a non-variational functional. Recasting the problem with Bogoliubov's variational principle restores a free energy that is bounded from below and places the BCS solution at a genuine minimum. Using a two-band toy model relevant to iron-based superconductors, we demonstrate the stability of the s$_\pm$ state and clarify how projection schemes that rely only on the interaction matrix can misidentify the attractive eigenmode that drives pairing. Our results clarify the instability issue highlighted by Aase et al. and provide a consistent foundation for analyzing fluctuations in the presence of repulsive interactions.