Spectral Decimation of Quantum Many-Body Hamiltonians

TL;DR

This paper introduces a spectral decimation framework to analyze quantum many-body Hamiltonians, revealing hidden symmetries, integrability, and structure in spectra, with applications to localization and fragmentation phenomena.

Contribution

It develops a systematic theory of spectral decimation that links spectral statistics to Hilbert-space structure and applies it to identify emergent symmetries and integrability in complex quantum systems.

Findings

CSS captures symmetry sectors in spectra.

Spectral decimation distinguishes chaos from integrability.

Characteristic symmetry entropy scales with system size.

Abstract

We develop a systematic theory of spectral decimation for quantum many-body Hamiltonians and show that it provides a quantitative probe of emergent symmetries in statistically mixed spectra. Building on an analytical description of statistical mixtures, we derive an explicit expression for the size of a characteristic symmetry sector (CSS), defined as the largest subsequence of levels exhibiting non-Poissonian correlations. The CSS dimension is shown to be the size-biased average of the underlying symmetry sectors, establishing a direct link between spectral statistics and Hilbert-space structure. We apply this framework to two paradigmatic settings: Hilbert-space fragmentation and disorder-induced many-body localization (MBL). In fragmented systems, the CSS reproduces the mixture prediction and isolates correlated subsectors even when the full spectrum appears nearly Poissonian. In the disordered Heisenberg chain, spectral decimation reveals the gradual emergence of integrability through a shrinking CSS, whose statistics exhibit signatures consistent with local integrals of motion. We introduce a characteristic symmetry entropy (CSE) as a finite-size scaling observable and extract, within accessible system sizes, the crossover exponents. Our results establish spectral decimation as a controlled, unbiased and computationally inexpensive diagnostic of hidden structure in many-body spectra, capable of distinguishing between chaotic dynamics, statistical mixtures, and emergent integrability.