gl(N|N) Super-Current Algebras for Disordered Dirac Fermions in Two Dimensions

TL;DR

The paper develops and solves a unified framework for 2D disordered Dirac fermions with particle-hole symmetry by mapping quenched disorder to current-current perturbations of gl(N|N) super-current algebras via SUSY. It derives exact beta-functions and current correlators, identifies a nearly conformal, scale-invariant subsector (PSL(N|N)) as a sigma-model description, and provides explicit N=1 solutions for all correlators. The work connects random-field XY physics to gl(N|N) currents, and shows DOS divergences at zero energy across delocalization transitions in both TRS-preserving and TRS-breaking classes, including Hatsugai et al. and Gade–Wegner universality classes. It also establishes a precise equivalence between GL(1|1)/U(1|1) sigma-models (with or without WZW terms) and current-current perturbations, suggesting a duality between sigma-model and current-algebra formulations for these disordered systems. These results offer exact, nonperturbative insights into zero-energy criticality and disorder-induced delocalization in 2D quantum systems with rich symmetry structures.

Abstract

We consider the non-hermitian 2D Dirac Hamiltonian with (A): real random mass, imaginary scalar potential and imaginary gauge field potentials, and (B) arbitrary complex random potentials of all three kinds. In both cases this Hamiltonian gives rise to a delocalization transition at zero energy with particle-hole symmetry in every realization of disorder. Case (A) is in addition time-reversal invariant, and can also be interpreted as the random-field XY Statistical Mechanics model in two dimensions. The supersymmetric approach to disorder averaging results in current-current perturbations of $gl(N|N)$ super-current algebras. Special properties of the $gl(N|N)$ algebra allow the exact computation of the beta-functions, and of the correlation functions of all currents. One of them is the Edwards-Anderson order parameter. The theory is `nearly conformal' and possesses a scale-invariant subsector which is not a current algebra. For N=1, in addition, we obtain an exact solution of all correlation functions. We also study the delocalization transition of case (B), with broken time reversal symmetry, in the Gade-Wegner (Random-Flux) universality class, using a GL(N|N;C)/U(N|N) sigma model, as well as its PSL(N|N) variant, and a corresponding generalized random XY model. For N=1 the sigma model is shown to be identical to the current-current perturbation. For the delocalization transitions (case (A) and (B)) a density of states, diverging at zero energy, is found.