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

10 pages, 6 figures

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• Higher-order level spacings in random matrix theory based on Wigner's conjecture
• Chaos in coupled Kerr-nonlinear parametric oscillators
• Distribution of the ratio of two consecutive level spacings in orthogonal to unitary crossover ensembles
• Superposition and higher-order spacing ratios in random matrix theory with application to complex systems
• Spectral Crossovers and Universality in Quantum Spin-chains Coupled to Random Fields
• Scaling of the Reduced Energy Spectrum of Random Matrix Ensemble
• Diagnosing First and Second Order Phase Transitions with Probes of Quantum Chaos
• Higher-order gap ratios of singular values in open quantum systems
• Universal scaling of higher-order spacing ratios in Gaussian random matrices
• Screwon spectral statistics and dispersion relation in the quantum Rajeev-Ranken model
• Imprints of $U_A(1)$ chiral anomaly and disorder in the Dirac eigenspectrum of QCD at finite temperature
• Information acquisition, scrambling, and sensitivity to errors in quantum chaos