Multifractal dimensions for orthogonal-to-unitary crossover ensemble

TL;DR

This paper derives semi-analytical formulas for multifractal dimensions in the orthogonal-to-unitary crossover ensemble, revealing finite-size effects and applying these to quantum systems like kicked rotors and spin chains.

Contribution

It provides new semi-analytical expressions for ensemble-averaged multifractal dimensions in the orthogonal-to-unitary crossover, including shifted and scaled variants for better crossover analysis.

Findings

Finite-size multifractal dimensions distinguish ergodic classes.

Derived formulas match Monte Carlo simulations.

Applied to quantum systems, capturing crossover behavior.

Abstract

Multifractal analysis is a powerful approach for characterizing ergodic or localized nature of eigenstates in complex quantum systems. In this context, the eigenvectors of random matrices belonging to invariant ensembles naturally serve as models for ergodic states. However, it has been found that the finite-size versions of multifractal dimensions for these eigenvectors converge to unity logarithmically slowly with increase in the system size $N$. In fact, this strong finite-size effect is capable of distinguishing the ergodicity behavior of orthogonal and unitary invariant classes. Motivated by this observation, in this work, we provide semi-analytical expressions for the ensemble-averaged multifractal dimensions associated with eigenvectors in the orthogonal-to-unitary crossover ensemble. Additionally, we explore shifted and scaled variants of multifractal dimensions, which, in contrast to the multifractal dimensions themselves, yield distinct values in the orthogonal and unitary limits as $N\to\infty$ and therefore may serve as a convenient measure for studying the crossover. We substantiate our results using Monte Carlo simulations of the underlying crossover random matrix model. We then apply our results to analyze the multifractal dimensions in a quantum kicked rotor, a Sinai billiard system, and a correlated spin chain model in a random field. The orthogonal-to-unitary crossover in these systems is realized by tuning relevant system parameters, and we find that in the crossover regime, the observed finite-dimension multifractal dimensions can be captured very well with our results.