Superconductivity near a quantum critical point in the extreme retardation regime

TL;DR

This paper analyzes superconductivity near a quantum critical point with extremely retarded interactions, providing exact results for the order parameter, free energy, and specific heat in the limit of large interaction retardation.

Contribution

It offers asymptotically exact solutions for the superconducting transition and thermodynamic properties in the extreme retardation regime, extending understanding of quantum critical superconductivity.

Findings

Exact expressions for $T_c$ and the gap function near $T_c$

Demonstration of negative specific heat at low temperatures

Identification of an instability in the $ ext{gamma}$ model

Abstract

We study fermions at quantum criticality with extremely retarded interactions of the form $V(\omega_l)=(g/|\omega_l|)^\gamma$, where $\omega_l$ is the transferred Matsubara frequency. This system undergoes a normal-superconductor phase transition at a critical temperature $T=T_c$. The order parameter is the frequency-dependent gap function $\Delta(\omega_n)$ as in the Eliashberg theory. In general, the interaction is extremely retarded for $\gamma\gg 1$, except at low temperatures $\gamma>3$ is sufficient. We evaluate the normal state specific heat, $T_c$, the jump in the specific heat, $\Delta(\omega_n)$ near $T_c$, and the Landau free energy. Our answers are asymptotically exact in the limit $\gamma\to\infty$. At low temperatures, we prove that the global minimum of the free energy is nondegenerate and determine the order parameter, the free energy, and the specific heat. These answers are exact for $T\to0$ and $\gamma>3$. We also uncover and investigate an instability of the $\gamma$ model: negative specific heat at $T\to0$ and just above $T_c$.