Multifractal wave functions of simple quantum maps

TL;DR

This study numerically analyzes the multifractal properties of wave functions in two one-dimensional quantum maps, revealing similarities and differences with the Anderson transition and highlighting the unique features of the intermediate map.

Contribution

It provides a detailed numerical comparison of multifractal exponents in quantum maps, introducing new insights into their properties and differences from the Anderson transition.

Findings

Wave functions of the Anderson map show multifractality similar to 3D Anderson transition but weaker.

Intermediate map wave functions share some multifractal features with Anderson transition eigenfunctions.

Distinct properties observed in the intermediate map, such as a large linear regime for exponents and different moment distributions.

Abstract

We study numerically multifractal properties of two models of one-dimensional quantum maps, a map with pseudointegrable dynamics and intermediate spectral statistics, and a map with an Anderson-like transition recently implemented with cold atoms. Using extensive numerical simulations, we compute the multifractal exponents of quantum wave functions and study their properties, with the help of two different numerical methods used for classical multifractal systems (box-counting method and wavelet method). We compare the results of the two methods over a wide range of values. We show that the wave functions of the Anderson map display a multifractal behavior similar to eigenfunctions of the three-dimensional Anderson transition but of a weaker type. Wave functions of the intermediate map share some common properties with eigenfunctions at the Anderson transition (two sets of multifractal exponents, with similar asymptotic behavior), but other properties are markedly different (large linear regime for multifractal exponents even for strong multifractality, different distributions of moments of wave functions, absence of symmetry of the exponents). Our results thus indicate that the intermediate map presents original properties, different from certain characteristics of the Anderson transition derived from the nonlinear sigma model. We also discuss the importance of finite-size effects.