Beyond Nearest-neighbour Universality of Spectral Fluctuations in Quantum Chaotic and Complex Many-body Systems

TL;DR

This paper extends the understanding of spectral fluctuations in quantum chaotic and complex many-body systems by analyzing intermediate-range statistics, specifically the next-nearest-neighbor spacing distribution, to test universality beyond nearest-neighbor measures.

Contribution

It derives effective level-repulsion parameters and Wigner-surmise-like results for nNNSD across different regimes, supported by analytical, numerical, and model comparisons, extending RMT universality.

Findings

nNNSD matches RMT predictions across regimes

Universality extends to longer-range spectral statistics

Orthogonal-to-Unitary and diluted-Symplectic-to-Unitary crossovers are equivalent

Abstract

Discerning chaos in quantum systems is an important problem as the usual route of Lyapunov exponents in classical systems is not straightforward in quantum systems. A standard route is the comparison of statistics derived from model physical systems to those from random matrix theory (RMT) ensembles, of which the most popular is the nearest-neighbour-spacings distribution (NNSD), which almost always shows good agreement with chaotic quantum systems. However, even in these cases, the long-range statistics (like number variance, spectral rigidity etc.), which are also more difficult to calculate, often show disagreements with RMT. As such, a more stringent test for chaos in quantum systems, via an analysis of intermediate-range statistics is needed, which will additionally assess the extent of agreement with RMT universality. In this paper, we deduce the effective level-repulsion parameters and the corresponding Wigner-surmise-like results of the next-nearest-neighbor spacing distribution (nNNSD) for integrable systems (semi-Poissonian statistics) as well as the three classical quantum-chaotic Wigner-Dyson regimes, by stringent comparisons to numerical RMT models and benchmarking against our exact analytical results for $3\times 3$ Gaussian matrix models, along with a semi-analytical form for the nNNSD in the Orthogonal-to-Unitary symmetry crossover. To illustrate the robustness of these RMT based results, we test these predictions against the nNNSD obtained from quantum chaotic models as well as disordered lattice spin models. This reinforces the Bohigas-Giannoni-Schmit and the Berry-Tabor conjectures, extending the associated universality to longer range statistics. In passing, we also highlight the equivalence of nNNSD in the apparently distinct Orthogonal-to-Unitary and diluted-Symplectic-to-Unitary crossovers.