Superconductivity with repulsion: A variational approach

In the presence of repulsive interactions, the BCS state appears unstable, showing up as a saddle point of the mean-field free energy. This apparent paradox has sparked recent proposals to restore stability by projecting out repulsive modes, effectively restricting the problem to the attractive sector. The authors show here that a Bogoliubov variational construction yields a bounded free energy and stabilizes the BCS state, without constraining the energy landscape. Simple models highlight the role of the full BCS kernel in correctly identifying collective modes, offering a robust framework for multichannel superconductors.