Mutifractal Analysis of Broadly Distributed Observables at Criticality

TL;DR

This paper reviews the multifractal analysis of localization-delocalization transitions, extending it to general observables and relating it to critical exponents, thus providing a comprehensive framework for understanding critical phenomena in disordered systems.

Contribution

It extends multifractal analysis to broader observables, establishes relations with critical exponents, and connects it with finite size scaling and conformal mapping methods.

Findings

Multifractal spectra relate to critical exponents like $eta$ and $ u$.

Bounds for the correlation length exponent are derived from the order parameter exponent.

Multifractal analysis can determine all critical exponents needed for phase transition characterization.

Abstract

The multifractal analysis of disorder induced localization-delocalization transitions is reviewed. Scaling properties of this transition are generic for multi parameter coherent systems which show broadly distributed observables at criticality. The multifractal analysis of local measures is extended to more general observables including scaling variables such as the conductance in the localization problem. The relation of multifractal dimensions to critical exponents such as the order parameter exponent $\beta$ and the correlation length exponent $\nu$ is investigated. We discuss a number of scaling relations between spectra of critical exponents, showing that all of the critical exponents necessary to characterize the critical phenomenon can be obtained within the generalized multifractal analysis. Furthermore we show how bounds for the correlation length exponent $\nu$ are obtained by the typical order parameter exponent $\alpha_0$ and make contact between the multifractal analysis and the finite size scaling approach in 2-d by relying on conformal mapping arguments.