Higher order spacing ratios in random matrix theory and complex quantum systems

Abstract

The distribution of the ratios of nearest neighbor level spacings has become a popular indicator of spectral fluctuations in complex quantum systems like interacting many-body localized and thermalization phases, quantum chaotic systems, and also in atomic and nuclear physics. In contrast to the level spacing distribution, which requires the cumbersome and at times ambiguous unfolding procedure, the ratios of spacings do not require unfolding and are easier to compute. In this work, for the class of Wigner-Dyson random matrices with nearest neighbor spacing ratios $r$ distributed as $P_{\beta }(r)$ for the three ensembles indexed by $\beta =1,2,4$, their $k-$th order spacing ratio distributions are shown to be identical to $P_{{\beta }^{\prime }}(r)$, where ${\beta }^{\prime }$, an integer, is a function of $\beta$ and $k$. This result is shown for Gaussian and circular ensembles of random matrix theory and for several physical systems such as spin chains, chaotic billiards, Floquet systems and measured nuclear resonances.

6 pages, 5 figures, 1 table. Comments are welcome

References in corpus (10)

• Localization of interacting fermions at high temperature
• Many-body localization edge in the random-field Heisenberg chain
• The distribution of the ratio of consecutive level spacings in random matrix ensembles
• Many-body localization due to random interactions
• Statistical properties of the spectrum the extended Bose-Hubbard model
• Poisson to GOE transition in the distribution of the ratio of consecutive level spacings
• Induced Time-Reversal Symmetry Breaking Observed in Microwave Billiards
• Many-body localization and thermalization in the full probability distribution function of observables
• Exact distribution of spacing ratios for random and localized states in quantum chaotic systems
• Many-Body Localization and Level Repulsion

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• Distribution of the Ratio of Consecutive Level Spacings for Different Symmetries and Degrees of Chaos
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• Localization and fractality in disordered Russian Doll model
• Surmise for random matrices' level spacing distributions beyond nearest-neighbors