Fermi liquid

A Fermi liquid is an interacting fermion phase whose low-energy excitations are long-lived quasiparticles near a Fermi surface. Its effective interactions are classified by scaling only the frequency and momentum normal to that surface.

A Fermi surface is the codimension-one surface in momentum space separating occupied and unoccupied zero-temperature states of a fermionic system.

Fermi-surface renormalization-group scaling sends frequency and normal displacement from the Fermi surface to zero while leaving tangential momentum fixed. Generic interactions are irrelevant, while forward-scattering and opposite-momentum Cooper channels can remain marginal.

BCS theory describes a superconducting instability caused by attractive interactions between opposite-momentum fermions near a Fermi surface.

The Cooper instability is the logarithmic growth of an attractive BCS coupling at low energy. If ${\nu }_F$ is the density of states and $V<0$, perturbation theory breaks down at a scale proportional to $\Lambda e^{-1/({\nu }_F|V|)}$.

At zero temperature and weak coupling, the BCS gap equation gives $\Delta \simeq 2\Lambda e^{-1/({\nu }_F|V|)}$. Its exponential scale agrees with the renormalization-group scale of the Cooper instability.