Superconductivity Near a Quantum Critical Point: Bounds on the Transition Temperature in the $\gamma$-Model

TL;DR

This paper derives rigorous, tight bounds on the superconducting transition temperature near a quantum critical point within the gamma-model, advancing understanding of non-Fermi liquid behavior and unconventional superconductivity.

Contribution

It introduces a simplified linear algebra approach to obtain closed-form upper and lower bounds on transition temperatures in gamma-models, improving upon previous estimates.

Findings

Lower bounds match existing results, confirming their accuracy.

Upper bounds are significantly tighter and align closely with numerical data.

The method simplifies the derivation process for bounds in quantum critical systems.

Abstract

Near a quantum critical point (QCP) in a metal, strong Fermion-Fermion interactions mediated by soft collective bosons give rise to two competing phenomena: non-Fermi liquid behavior and superconductivity that deviates from conventional BCS and Migdal-Eliashberg theories. We consider the problem of obtaining closed-form analytical lower and upper bounds on transition temperatures for such systems. We focus mainly on a class of models known as the gamma-model, which generalizes the Eliashberg theory of Superconductivity where the effective interaction potential scales as V(Omega) ~ 1/|Omega|^gamma. Building on a recent reformulation of Migdal-Eliashberg theory, expressed as a classical infinite spin chain with nonlocal interactions, and employing a simple linear algebra framework, we derive rigorous closed-form expressions for upper and lower bounds on the superconducting transition temperature for any gamma > 0. While our lower bounds coincide precisely with those previously in the literature, our derivation offers a more streamlined and accessible method. Moreover, our upper bounds are substantially tighter than any existing estimates and converge rapidly enough to the numerical results from various prior studies.