Superconductivity from Luttinger surfaces: Emergent $\mathrm{\infty }$-body SYK physics

The pairing of two electrons on a Fermi surface due to an infinitesimal attraction between them always results in a superconducting instability at zero temperature ($T=0$). The equivalent question of pairing instability on a Luttinger surface (LS) – a contour of zeros of the propagator – instead leads to a quantum critical point (QCP) that separates a non-Fermi liquid (NFL) and superconductor. A surprising and little understood aspect of pair fluctuations at this QCP is that their thermodynamics maps to that of the Sachdev-Ye-Kitaev (SYK) model in the strong coupling limit. Here, we offer a simple justification for this mapping by demonstrating that (i) LS models share the reparametrization symmetry of the $q\to \mathrm{\infty }$ SYK model with $q$-body interactions \textcolor{black}{close to the LS}, and (ii) the enforcement of gauge invariance results in a $\frac{1}{\sqrt{\tau }}$ ($\tau \sim T^{-1}$) behavior of the fluctuation propagator near the QCP, as is a feature of the fundamental SYK fermion.