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

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+ϵ)d=(2+ϵ) dimensions (symmetry classes C and D of Altland and Zirnbauer). Using a perturbative renormalization group analysis of the corresponding non-linear sigma model (NLσM) 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 ϵ=(2−d)ϵ=(2−d) expansion in d<2d<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 Kähler type, and suggests that this is a generic property of the perturbative treatment of NLσMs defined on Riemannian symmetric target spaces.