Statistical Signatures of Integrable and Non-Integrable Quantum Hamiltonians

TL;DR

The paper tackles the challenge of identifying quantum integrability from finite-sample spectra by proposing a probabilistic framework centered on the presence of vanishing energy gaps and a two-step protocol combining spectral decimation with higher-order gap analyses. By carefully accounting for global and dynamical symmetries and unfolding procedures, it distinguishes genuine Poisson statistics of integrable systems from mixed or mimicry spectra arising from superposed non-integrable subsectors. The authors validate the approach on permutation-group–based Hamiltonians, illustrating how block structures, locality, and boundary conditions shape level statistics and how the proposed protocol robustly separates integrable from non-integrable cases. This framework provides a practical, scalable diagnostic for spectral data and has potential to be widely applied in many-body quantum systems where the spectrum is the primary accessible object.

Abstract

Integrability is a cornerstone of classical mechanics, where it has a precise meaning. Extending this notion to quantum systems, however, remains subtle and unresolved. In particular, deciding whether a quantum Hamiltonian - viewed simply as a matrix - defines an integrable system is far from obvious, yet crucial for understanding non-equilibrium dynamics, spectral correlations, and correlation functions in many-body physics. We develop a statistical framework that approaches quantum integrability from a probabilistic standpoint. A key observation is that integrability requires a finite probability of vanishing energy gaps. Building on this, we propose a two-step protocol to distinguish integrable from non-integrable Hamiltonians. First, we apply a systematic Monte Carlo decimation of the spectrum, which exponentially compresses the Hilbert space and reveals whether level spacings approach Poisson statistics or remain mixed. The termination point of this decimation indicates the statistical character of the spectrum. Second, we analyze $k$-step gap distributions, which sharpen the distinction between Poisson and mixed statistics. Our procedure applies to Hamiltonians of any finite size, independent of whether their structure involves a few blocks or an exponentially fragmented Hilbert space. As a benchmark, we implement the protocol on quantum Hamiltonians built from the permutation group $\mathcal{S}_N$, demonstrating both its effectiveness and generality.

Figures (26)

• Figure 1: An example of level spacing distribution $P(s)$ with a non-zero probability to have zero gaps in the spectrum.
• Figure 2: Block form of an Hamiltonian with a degenerate spectrum according to the probability distribution $P(s)$ of its spacing levels and an energy resolution $\delta E$.
• Figure 4: Typical phenomenon of level repulsion in a quantum non-integrable Hamiltonian which depends on a parameter $\lambda$.
• Figure 5: 2-point correlation function $R_{2}^{(\beta)}(s)$ for the GUE and GOE. Note that both curves vanish at the origin, reflecting level repulsion, and rapidly saturate to the asymptotic value 1, which characterizes the spectral rigidity.
• Figure 6: Higher order spacing probability distribution $P_k^{(2)}(s)$ ($k=1,2,\ldots,11)$ for GUE. The histogram is an average over 250 realizations of GUE spectra, each from an $800\times800$ matrix. The $k$ distribution is sharply peaked at $s \simeq k$, reflecting level repulsion and spectral rigidity.