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.
Power-law banded random matrix ensemble as a model for quantum many-body Hamiltonians
Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）；Institut für Theoretische Physik, Technische Universität Dresden（德累斯顿工业大学理论物理研究所）；Nordita, Stockholm University & KTH Royal Institute of Technology（诺迪塔，斯德哥尔摩大学与皇家理工学院）
Locality forces equal energy spacing of quantum many-body scar towers
Princeton Center for Theoretical Science, Princeton University（普林斯顿大学理论科学中心）；Princeton Quantum Initiative, Princeton University（普林斯顿大学量子倡议计划）；Walter Burke Institute for Theoretical Physics, Caltech（加州理工学院沃尔特·伯克理论物理研究所）；Department of Physics, Caltech（加州理工学院物理系）；School of Natural Sciences, Technische Universität München (TUM)（慕尼黑工业大学自然科学学院）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）
Distinct Types of Parent Hamiltonians for Quantum States: Insights from the $W$ State as a Quantum Many-Body Scar
Walter Burke Institute for Theoretical Physics, Caltech（加州理工学院沃尔特·伯克理论物理研究所）；Department of Physics, Caltech（加州理工学院物理系）；Technische Universität München (TUM)（慕尼黑工业大学）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）
