The special role of the first Matsubara frequency for superconductivity near a quantum-critical point -- the non-linear gap equation below $T_c$ and spectral properties in real frequencies

Understand

Near a quantum-critical point in a metal a strong fermion-fermion interaction, mediated by a soft boson, destroys fermionic coherence and also gives rise to superconductivity.

• In a class of large $N$ models, one would naively expect an incoherent (non-Fermi liquid) normal state behavior to persist down to $T=0$.
• However, this is not the case for quantum-critical systems described by Eliashberg theory, where non-Fermi liquid part of the self-energy $\Sigma (\omega_m)$ is large for a generic Matsubara frequency $\omega_m = \pi T(2m+1)$, but vanishes at $\omega_m = \pm \pi T$, while the pairing interaction between fermions with these two frequencies remains strong.
• This peculiarity gives rise to a non-zero $T_c$ even at large $N$[Y.