Conductance fluctuations in disordered superconductors with broken time-reversal symmetry near two dimensions

TL;DR

This paper investigates conductance fluctuations in disordered superconductors with broken time-reversal symmetry near two dimensions, extending previous metallic system analyses to new symmetry classes using renormalization group methods.

Contribution

It applies a one-loop renormalization group analysis to disordered superconductors in symmetry classes C and D, revealing the relevance of high-gradient operators near two dimensions.

Findings

Infinite high-gradient operators become relevant near two dimensions.

The analysis extends to symmetry class D and compares with one-dimensional exact results.

Method applies broadly to sigma models on Riemannian symmetric spaces.

Abstract

We extend the analysis of the conductance fluctuations in disordered metals by Altshuler, Kravtsov, and Lerner (AKL) to disordered superconductors with broken time-reversal symmetry in $d=(2+\epsilon)$ dimensions (symmetry classes C and D of Altland and Zirnbauer). Using a perturbative renormalization group analysis of the corresponding non-linear sigma model (NL$\sigma$M) we compute the anomalous scaling dimensions of the dominant scalar operators with $2s$ gradients to one-loop order. We show that, in analogy with the result of AKL for ordinary, metallic systems (Wigner-Dyson classes), an infinite number of high-gradient operators would become relevant (in the renormalization group sense) near two dimensions if contributions beyond one-loop order are ignored. We explore the possibility to compare, in symmetry class D, the $\epsilon=(2-d)$ expansion in $d<2$ with exact results in one dimension. The method we use to perform the one-loop renormalization analysis is valid for general symmetric spaces of K\"ahler type, and suggests that this is a generic property of the perturbative treatment of NL$\sigma$Ms defined on Riemannian symmetric target spaces.