The distribution of the ratios of nearest neighbor level spacings has become a popular indicator of spectral fluctuations in complex quantum systems such as the localized and thermal phases of interacting many-body systems, quantum chaotic systems, and 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 β (r) for the three ensembles indexed by β = 1, 2, 4, their k−th order spacing ratio distributions are shown to be identical to P β ′ (r), where β ′ , an integer, is a function of β 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. *

The spectral fluctuations of complex quantum systems, in an appropriate limit, are known to be consistent with those obtained from random matrices. However, this relation between the spectral fluctuations of physical systems and random matrices is valid only if the spectra are desymmetrized. This implies that the fluctuation properties of the spectra are affected by the discrete symmetries of the system. In this Rapid Communication, it is shown that in the chaotic limit the fluctuation characteristics and symmetry structure for any arbitrary sequence of measured or computed levels can be inferred from its higher-order spectral statistics without desymmetrization. In particular, we consider a spectrum composed of k > 0 independent level sequences with each sequence having the same level density. The kth order spacing ratio distribution of such a composite spectrum is identical to its nearest-neighbor counterpart with a modified Dyson index k. This is demonstrated for the spectra obtained from random matrices, quantum billiards, spin chains, and experimentally measured nuclear resonances with disparate symmetry features.

The spectra of empirical correlation matrices, constructed from multivariate data, are widely used in many areas of sciences, engineering and social sciences as a tool to understand the information contained in typically large datasets. In the last two decades, random matrix theory-based tools such as the nearest neighbour eigenvalue spacing and eigenvector distributions have been employed to extract the significant modes of variability present in such empirical correlations. In this work, we present an alternative analysis in terms of the recently introduced spacing ratios, which does not require the cumbersome unfolding process. It is shown that the higher order spacing ratio distributions for the Wishart ensemble of random matrices, characterized by the Dyson index β, is related to the first order spacing ratio distribution with a modified value of co-dimension β ′ . This scaling is demonstrated for Wishart ensemble and also for the spectra of empirical correlation matrices drawn from the observed stock market and atmospheric pressure data. Using a combination of analytical and numerics, such scalings in spacing distributions are also discussed.

Typical eigenstates of quantum systems, whose classical limit is chaotic, are well approximated as random states. Corresponding eigenvalue spectra are modeled through an appropriate ensemble described by random matrix theory. However, a small subset of states violates this principle and displays eigenstate localization, a counterintuitive 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 modify the spectral statistics. In this work, a 3×3 random matrix model is used to obtain exact results for the ratio of spacing between a generic and localized state. We consider time-reversal-invariant as well as noninvariant scenarios. These results agree with the spectra computed from realistic physical systems that display localized eigenmodes.