Gaussian field theories, random Cantor sets and multifractality

TL;DR

This paper introduces a novel approach linking Gaussian field theories to random Cantor sets to compute universal multifractal exponents, demonstrated on Dirac fermions with random vector potentials.

Contribution

It proposes a new method to calculate multifractal scaling exponents in Gaussian field theories using random Cantor sets, addressing a longstanding open problem.

Findings

Multifractal exponents are self-averaging.

The approach successfully characterizes the multifractal wave function.

Universal scaling exponents can be computed with this method.

Abstract

The computation of multifractal scaling properties associated with a critical field theory involves non-local operators and remains an open problem using conventional techniques of field theory. We propose a new description of Gaussian field theories in terms of random Cantor sets and show how universal multifractal scaling exponents can be calculated. We use this approach to characterize the multifractal critical wave function of Dirac fermions interacting with a random vector potential in two spatial dimensions. We show that the multifractal scaling exponents are self-averaging.