# Attractive interaction in BCS theory

The common way to discuss BCS superconductivity, as I have seen in several books, is to assume that we have a Fermi liquid, on top of which we add an interaction of the form $$H_{int}=\sum_{p,p^\prime,q} V(q)\psi^\dagger_\uparrow(-p) \psi^\dagger_\downarrow(-p^\prime) \psi_\downarrow(p^\prime+q)\psi_\uparrow(p-q)$$ The assumption is that as long as $$V$$ is negative we get a Cooper instability and a condensate will form at low enough temperatures. This doesn't make sense to me as if $$V\propto 1/q^2$$ (in 3D) we just get a correction to the Coulomb interaction and remain with a Fermi liquid.

In that case, what do we actually need to demand on $$V(q)$$ to get superconductivity? Do we explicitly need a constant term?

• You mean $V$ is negative, since the interaction is attractive. Dec 18 '20 at 14:12
• Sure. I corrected it, thank you. Dec 18 '20 at 14:13