Beta-functions of non-linear $\sigma$-models for disordered and interacting electron systems

TL;DR

This paper derives and analyzes one-loop renormalization group equations for Finkel'stein non-linear sigma models, revealing that symmetry classification alone does not fully determine the scaling behavior of interacting disordered electron systems.

Contribution

It provides a comprehensive set of RG equations for various symmetry classes of non-linear sigma models, highlighting the limitations of Cartan's classification in interacting systems.

Findings

RG equations depend on symmetry class and interactions

Symmetry classification alone is insufficient for scaling predictions

The study advances understanding of disordered electron systems

Abstract

We provide and study complete sets of one-loop renormalization group equations of several Finkel'stein non-linear $\sigma$-models, the effective field theories describing the diffusive quantum fluctuations in correlated disordered systems. We consider different cases according to the presence of certain symmetries induced by the original random Hamiltonians, and we show that, for interacting systems, the Cartan's classification of symmetry classes is not enough to uniquely determine their scaling behaviors.