Logarithmic Operators and Hidden Continuous Symmetry in Critical Disordered Models

TL;DR

This work analyzes (2+1)-dimensional massless Dirac fermions in a random non-Abelian gauge potential at criticality, uncovering a conserved current that signals a hidden continuous symmetry and induces logarithmic contributions in correlation functions. Through the replica approach and Knizhnik–Zamolodchikov framework, the authors derive conformal dimensions, construct four-point blocks, and reveal a logarithmic operator structure that modifies the operator product expansions. The study links these logarithmic operators to a continuous symmetry, develops the associated Ward identities, and demonstrates that deformations by the logarithmic sector produce non-power-law scaling, including a log-normal distribution for the local density of states. These results bridge disordered criticality, logarithmic CFTs, and potential connections to quantum gravity, while proving equivalence between replica and SUSY formulations for this model.

Abstract

We study the model of (2 + 1)-dimensional relativistic fermions in a random non-Abelian gauge potential at criticality. The exact solution shows that the operator expansion contains a conserved current - a generator of a continuous symmetry. The presence of this operator changes the operator product expansion and gives rise to logarithmic contributions to the correlation functions at the critical point. We calculate the distribution function of the local density of states in this model and find that it follows the famous log-normal law.