Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 337 / 2

Under Fermi-surface renormalization-group scaling, tangential momentum $k$ is fixed whileIn frequency space the quadratic action is invariant whenequivalently, in the time representation $t\mapsto s^{-1}t$ and $\psi (t,k,\ell )\mapsto s^{-1/2}\psi (t,k,\ell )$. The free Fermi liquid action is therefore marginal.

The deformation $\mu (k){\psi }^{†}\psi$ has no factor of frequency or normal momentum, so it is relevant. It shifts the zero of the quasiparticle energy from $\ell =0$ to $\ell =\mu (k)/v_F(k)$ and hence deforms the Fermi surface.

A generic quartic interaction is irrelevant because momentum conservation fixes a normal component and leaves an additional positive power of $s$. It can be marginal only when all four momenta remain on the Fermi surface while satisfyingFor a smooth generic Fermi surface, the robust possibilities are forward or exchange scattering and the opposite-momentum BCS channel ${\mathbf{k}}_2=-{\mathbf{k}}_1$, ${\mathbf{k}}_4=-{\mathbf{k}}_3$.

The tree-level Cooper-pair scattering amplitude is simply $V$. Define the density of states at the Fermi surface, in the normalization of the question, byIn the one-loop Cooper diagram the two internal fermions have opposite momenta. After integrating over internal frequency, the remaining normal-energy integral containsConsequentlyand summing the leading geometric series gives the running BCS theory couplingFor repulsive $V>0$ it flows logarithmically toward zero. For attractive $V<0$, its denominator vanishes at the Cooper instability scaleAt this scale the normal Fermi liquid becomes unstable: opposite-momentum fermions form Cooper pairs, a superconducting or superfluid condensate develops, and a gap opens.

Put $\xi =v_F(k)\ell$ and use the same ${\nu }_F$. With a symmetric cutoff $|\xi |<\Lambda$, the BCS gap equation becomesThe frequency integral is $1/[2\sqrt{{\xi }^2+{\Delta }^2}]$, soThusThe gap and the one-loop strong-coupling scale have the same nonperturbative exponential dependence; their order-one prefactors depend on the cutoff convention and microscopic completion. The renormalization-group divergence is therefore the normal-state signal of the paired, gapped phase found from the self-consistent gap equation.