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 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-beta (r) for the three ensembles indexed by beta = 1, 2, 4, their kth order spacing ratio distributions are shown to be identical to P-beta' (r), where beta', 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.