The Renyi Entropy and the Multifractal Spectrum of Systems Near the Localization Transition

TL;DR

This paper demonstrates that Rènyi entropies of disordered quantum systems reveal universal multifractal properties, with potential for experimental measurement, supported by analytic and numerical evidence.

Contribution

It introduces a formalism linking Rènyi entropies to the multifractal spectrum, enabling extraction of universal multifractal information from single-particle wave functions.

Findings

Good agreement between analytic formulas and numerical simulations.

Rènyi entropies contain universal multifractal information.

Potential for experimental measurement of multifractal spectrum.

Abstract

We show that the R\'enyi entropies of single particle, extended wave functions for disordered systems contain information about the multifractal spectrum. It is shown for moments of the R\'enyi entropy, $S_{n}$, where $|n|<1$, it is possible to extract universal information about the multifractility of such systems. This is shown through a generic calculation and then illustrated through two example models. We find good agreement between our analytic formula and numerical simulations of the two test models. Our formalism is easily extendable to generic non-interacting fermion models. It is also suggested that recent experimental advances in measuring the multifractal spectrum might allow some moments of the R\'enyi entropy to be measured.