Pairing instability on a Luttinger surface: A non-Fermi liquid to superconductor transition and its Sachdev-Ye-Kitaev dual
Superconductivity results from an instability of the Fermi surface—contour of poles of the single-particle propagator—to an infinitesimally small attraction between electrons. Here, we instead discuss the analogous problem on a model Luttinger surface, or contour of zeros of the Green function. At zero temperature ($β → \infty$) and a critical interaction strength ($u_{c∞}$) characterized by the residue of self-energy pole, we find that the pair susceptibility diverges leading to a superconducting instability. We evaluate the pair fluctuation partition function and find that the spectral density in the normal state has an interaction-driven, power-law $\frac{1}{\sqrt{ω}}$ type, van-Hove singularity (vHS), indicating non-Fermi liquid (NFL) physics. Crucially, in the strong coupling limit ($βu \gg 1$), the leading-order fluctuation free-energy terms in the normal state of this NFL-SC transition resembles the equivalent [$\textit{O}(1)$] terms of the Sachdev-Ye-Kitaev (SYK) model. This free-energy contribution takes a simple form $–βF = βu_{c∞} – γ \text{ln}(βu_{c∞}$), where $\gamma$ is a constant equal to $\frac{1}{2}$. Weak impurity scattering ($τ \gg β^{–1}$) leaves the low-energy spectral density unaffected but leads to an interaction-driven enhancement of superconductivity. Our results in this work shed light on the role played by order-parameter fluctuations in providing the key missing link between Mott physics and strongly coupled toy models exhibiting gravity duals.