Multifractality and intermediate statistics in quantum maps

TL;DR

This paper investigates the multifractal nature of wave functions in quantum maps with intermediate spectral statistics, linking fractal dimensions to classical map parameters and spectral properties, with potential implications for quantum transitions.

Contribution

It provides analytical and numerical analysis connecting multifractality of wave functions to classical parameters in quantum maps, bridging spectral statistics and wave function structure.

Findings

Generalized fractal dimensions relate to classical map parameters

Wave functions exhibit multifractality across spectral regimes

Results applicable to Anderson and quantum Hall transitions

Abstract

We study multifractal properties of wave functions for a one-parameter family of quantum maps displaying the whole range of spectral statistics intermediate between integrable and chaotic statistics. We perform extensive numerical computations and provide analytical arguments showing that the generalized fractal dimensions are directly related to the parameter of the underlying classical map, and thus to other properties such as spectral statistics. Our results could be relevant for Anderson and quantum Hall transitions, where wave functions also show multifractality.