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 \textit{poles} of the single particle propagator -- to an infinitesimally small attraction between electrons. Here, we instead discuss the analogous problem on a model \textit{Luttinger} surface, or contour of \textit{zeros} of the Green function. At zero temperature ($β\rightarrow \infty$) and a critical interaction strength ($u_{c\infty}$) 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 $\frac{1}{\sqrtω}$ 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 resemble the equivalent $\left(O(1)\right)$ terms of the Sachdev-Ye-Kitaev (SYK) model. This free energy contribution takes a simple form $-βF = βu_{c\infty} - γ~\text{ln}\left(βu_{c \infty}\right)$ where $γ$ 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 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.

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