Exact distribution of spacing ratios for random and localized states in quantum chaotic systems

TL;DR

This paper derives exact formulas for the distribution of spacing ratios between localized and generic states in quantum chaotic systems, accounting for time-reversal symmetry, and confirms results with physical system spectra.

Contribution

It introduces a 3x3 random matrix model to exactly compute spacing ratios involving localized states in quantum chaos, extending understanding of spectral statistics.

Findings

Exact distribution formulas for spacing ratios involving localized states.

Results agree with spectra from physical systems with localized eigenmodes.

Applicable to both time-reversal-invariant and non-invariant systems.

Abstract

Typical eigenstates of quantum systems, whose classical limit is chaotic, are well approximated as random states. Corresponding eigenvalue spectra is modeled through appropriate ensemble of random matrix theory. However, a small subset of states violate this principle and display eigenstate localization, a counter-intuitive feature known to arise due to purely quantum or semiclassical effects. In the spectrum of chaotic systems, the localized and random states interact with one another and modifies the spectral statistics. In this work, a $3 \times 3$ random matrix model is used to obtain exact result for the ratio of spacing between a generic and localized state. We consider time-reversal-invariant as well as non-invariant scenarios. These results agree with the spectra computed from realistic physical systems that display localized eigenmodes.