Exact calculation of multifractal exponents of the critical wave function of Dirac fermions in a random magnetic field

TL;DR

This paper derives exact multifractal exponents for the critical wave function of 2D Dirac fermions in a random magnetic field by mapping the problem to a thermodynamic model, revealing a deep connection with the Generalized Random Energy Model.

Contribution

It introduces an exact method to calculate multifractal spectra of Dirac fermions in random magnetic fields using thermodynamic mappings, linking it to the Generalized Random Energy Model.

Findings

Multifractal exponents are expressed via thermodynamic functions.

Thermodynamic functions match those of the Generalized Random Energy Model.

The approach confirms previous Gaussian field theory results.

Abstract

The multifractal scaling exponents are calculated for the critical wave function of a two-dimensional Dirac fermion in the presence of a random magnetic field. It is shown that the problem of calculating the multifractal spectrum maps into the thermodynamics of a static particle in a random potential. The multifractal exponents are simply given in terms of thermodynamic functions, such as free energy and entropy, which are argued to be self-averaging in the thermodynamic limit. These thermodynamic functions are shown to coincide exactly with those of a Generalized Random Energy Model, in agreement with previous results obtained using Gaussian field theories in an ultrametric space.