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 ($\beta \rightarrow \infty$) 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 $\frac{ 1}{ \sqrt{ \omega}}$ type, van-Hove singularity (vHS) indicating non-Fermi liquid (NFL) physics. Crucially, in the strong coupling limit ($\beta u \gg 1$), 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 $-\beta F = \beta u_{ c\infty} - \gamma~\text{ ln}\left(\beta u_{ c \infty}\right)$ where $\gamma$ is a constant equal to $\frac{ 1}{ 2}$. Weak impurity scattering ($\tau \gg \beta^{ -1}$) 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.
Categories: cond-mat.str-el cond-mat.stat-mech cond-mat.supr-con hep-th