Abstract.
We introduce spectral decimation as a statistical diagnostic of quantum integrability and hidden structure in many-body spectra. By progressively thinning the spectrum and monitoring level-spacing and kk-step gap statistics, the method distinguishes genuine Poisson behavior from statistical mixtures and correlated subsectors. This leads naturally to a characteristic symmetry sector (CSS), whose size quantifies the largest non-Poissonian component and directly reflects the underlying Hilbert-space organization. Applications to permutation-group Hamiltonians, fragmented systems, and the disordered Heisenberg chain show that spectral decimation can detect hidden symmetries and the emergence of integrability, providing a simple and computationally efficient probe of chaotic, mixed, and localized regimes.
(The slides will soon be available here)