Holographic superconductivity of a critical Fermi surface

Abstract

We derive a holographic formulation of triplet superconductivity in a two-dimensional metal at a ferromagnetic quantum critical point. Starting from a large-$N$ Yukawa-Sachdev-Ye-Kitaev model of compressible fermions coupled to quantum-critical Ising ferromagnetic fluctuations, we reformulate the pairing problem in terms of bilocal collective fields and analyze Gaussian fluctuations around the quantum-critical normal state. We demonstrate that the resulting pairing action can be mapped onto a scalar field theory in an emergent curved spacetime with AdS$_2 \otimes \mathbb{R}_2$ geometry. The additional holographic dimension is shown to encode the internal dynamics of Cooper pairs and is related nonlocally to the frequency dependence of the anomalous Gor'kov function via a Radon transform. Within this framework, the onset of superconductivity corresponds to a Breitenlohner-Freedman instability of the scalar field, which is shown to be equivalent to the pairing instability obtained from the linearized Eliashberg equations. The factorized AdS$_2 \otimes \mathbb{R}_2$ geometry reflects the local-in-space but critical-in-time character of fermionic excitations near a metallic quantum critical point and corresponds to what one expects in the vicinity of a Reissner-Nordstr\"om black hole. Our results provide a microscopic derivation of holographic superconductivity in a compressible quantum critical metal and clarify the geometric structure underlying quantum-critical pairing.