Multifractality in a broad class of disordered systems

TL;DR

This paper investigates the multifractal properties of various disordered systems at criticality, revealing infinitely many independent critical exponents and calculating them to two-loop order using renormalized field theory.

Contribution

It introduces a comprehensive analysis of multifractality in disordered systems, including the derivation of multiple critical exponents and their connection to the random resistor network.

Findings

Each cumulant has its own independent critical exponent.

Multifractal exponents are determined to two-loop order.

Higher cumulant amplitudes can vanish depending on the model.

Abstract

We study multifractality in a broad class of disordered systems which includes, e.g., the diluted x-y model. Using renormalized field theory we analyze the scaling behavior of cumulant averaged dynamical variables (in case of the x-y model the angles specifying the directions of the spins) at the percolation threshold. Each of the cumulants has its own independent critical exponent, i.e., there are infinitely many critical exponents involved in the problem. Working out the connection to the random resistor network, we determine these multifractal exponents to two-loop order. Depending on the specifics of the Hamiltonian of each individual model, the amplitudes of the higher cumulants can vanish and in this case, effectively, only some of the multifractal exponents are required.