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.
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.
A central theme in the field of quantum many-body physics is the characterization of thermal properties, which can be accessed, for instance, through the analysis of the Hamiltonian spectrum. This thesis approaches the problem from both numerical and analytical viewpoints, treating the spectrum as a statistical object whose correlations encode predictive information about the physics of the underlying system. In the first part of this thesis, we review the relation between thermalization, quantum integrability and chaos in one-dimensional quantum many-body systems. We focus on the statistics of energy levels and establish whether quantum integrability can be assessed from a statistical viewpoint. We then consider the problem of intermediate statistics between purely integrable and non-integrable quantum many-body systems. The first contribution of this thesis is an answer to the problem of discriminating between the spectrum of a genuine integrable system and a statistical mixture of, possibly exponentially many, non-ergodic sectors. We show that the spectral decimation algorithm can capture both qualitative and quantitative features of a `characteristic symmetry sector' from an unbiased analysis of numerically available spectral data. When applied to physically relevant systems, the algorithm is able to identify traces of Hilbert space fragmentation and disorder-induced many-body localization. The second part of this thesis is dedicated to a class of Hamiltonians of particular interest to statistical spectroscopy. Such Hamiltonians involve only the permutation of sites without the presence of external interactions. Nonetheless, they generate a highly tunable and numerically implementable class of models. For instance, it is easy to engineer integrability and its breaking while preserving essential structures such as the block-diagonal decomposition. These Hamiltonians are naturally suited to generating unitary random walks on the permutation group, which can, for instance, be compared with their classical counterparts. After a detailed discussion of the numerical implementation of these Hamiltonians and their spectral properties, we depart from the analysis of the system's static properties to answer a genuine out-of-equilibrium question: do quantum coherences accelerate mixing in the computational basis? We consider an exactly solvable class of quantum stochastic walks which, in the absence of quantum coherences, admit a limit coinciding with a classical continuous-time Markov chain. While the trace distance of the associated density matrices from the maximally mixed state remains identical to that in the classical limit, we show that the probability distance in the presence of quantum coherences, which defines mixing in the computational basis, is upper bounded by the classical limit. An analysis of the slowest mode identifies a necessary condition under which quantum coherences and classical mixing become of the same order, thereby defining a slow-mode time scale. Finally, a numerical analysis of quantum stochastic walks based on random transpositions corroborates the analytical findings, in particular the accelerated mixing. We also identify a closed-form expression for the ratio of quantum to classical mixing times, which yields a collapse over numerically accessible system sizes. Overall, this thesis presents a modern approach to the theory of statistical spectroscopy by proposing analytically grounded and numerically implementable protocols to ascertain the thermal properties of a one-dimensional quantum many-body system from the analysis of the spectrum within numerically accessible volumes. It also shows that the analysis of the statistical properties of Hamiltonians can be of practical aid in the study of out-of-equilibrium properties of quantum many-body systems, such as mixing in quantum stochastic walks.
Abstract Integrability has long served as a cornerstone concept in classical mechanics, where it possesses a precise and unambiguous definition. Extending this notion to the quantum domain, however, remains a far more subtle and elusive problem. In particular, deciding whether a given quantum Hamiltonian—viewed simply as a matrix of its elements—does or does not define an integrable system is far from obvious. Yet this question is crucial: it bears directly on non-equilibrium dynamics, spectral correlations, the behaviour of correlation functions, and other fundamental properties of many-body quantum systems. In this work, we develop a statistical framework for addressing quantum integrability from a purely probabilistic standpoint. Our approach begins with the observation that a necessary signature of integrability is the finite probability of encountering vanishing energy gaps in the spectrum. On this basis, we formulate a twofold protocol capable of distinguishing between integrable and non-integrable Hamiltonians. The first step consists of a systematic Monte Carlo decimation of the spectrum, designed to reveal the emergence (or absence) of Poissonian level spacing statistics. The iterative decimation compresses the Hilbert space exponentially, and its termination point determines whether the spectrum is governed by a mixed distribution or approaches the Poisson limit. In the second step, the distinction is instead obtained by analysing the distributions of k -step gaps which can help in discriminating between Poisson and mixed statistics. This procedure applies to Hamiltonians of arbitrary finite size, regardless of whether their structure involves a finite number of blocks or an exponentially fragmented Hilbert space. As a concrete benchmark, we implement the protocol on a class of quantum Hamiltonians constructed from the permutation group S N , thereby demonstrating both its effectiveness and its broad applicability.
Investigating many-body localization (MBL) using exact numerical methods is limited by the exponential growth of the Hilbert space. However, localized eigenstates display multifractality and only extend over a vanishing fraction of the Hilbert space. Here, building on this remarkable property, we develop a simple yet efficient decimation scheme to discard the irrelevant parts of the Hilbert space of the random-field Heisenberg chain. This leads to an Hilbert space fragmentation in small clusters, allowing to access larger systems at strong disorder. The MBL transition is quantitatively predicted, together with a geometrical interpretation of MBL multifractality.
Hilbert-space fragmentation provides a mechanism for ergodicity breaking in quantum many-body systems even in the absence of disorder, leading to dynamically disconnected sectors and strong memory of initial conditions. However, identifying such structures is often challenging and typically relies on prior knowledge of conservation laws or model-specific analytical insight. Here we introduce an unbiased approach based on spectral graph theory and, within this framework, formulate the concept of nearly fragmented systems, in which perturbative processes couple otherwise fragmented sectors while preserving their dynamical imprint. By representing basis states as vertices and Hamiltonian matrix elements as edges, we map the connectivity structure of the many-body Hilbert space onto a graph and analyze it using tools such as the Laplacian spectrum, Fiedler vectors, and modularity. Exact fragmentation corresponds to disconnected graph components, while nearly fragmented systems manifest as weakly connected communities whose structure can still be resolved spectrally. Applying this framework to the one-dimensional t-J model and its perturbations, we demonstrate that graph-theoretic diagnostics reliably identify both fragmented and nearly fragmented Hilbert-space structures and capture the hierarchy of dynamical time scales that governs the system's evolution. We further show that the method extends beyond kinetically constrained models by applying it directly to the Hubbard chain, where it reveals the emergence of nearly decoupled subspaces associated with doublon dynamics and spin configurations. Our results establish the spectral graph analysis as a general and scalable tool for diagnosing fragmentation and approximate dynamical constraints in complex quantum many-body systems.
This thesis investigates the dynamics of quantum many-body systems in isolation, in three different setups. We start by considering a paradigmatic quantum chaotic model of magnetism, the Heisenberg antiferromagnet in 2 and 3 spatial dimensions. The focus is on obtaining approximations for the full counting statistics (FCS) of a subsystem order parameter, the staggered magnetization, both in and out of equilibrium (following quantum quenches). We employ mappings of spin variables to bosonic ones, together with self-consistent time-dependent mean-field schemes, to compute the FCS via Gaussian techniques. In particular, we introduce a formula for the FCS of any subsystem observable that is at most quadratic in the bosons. We obtain results both in the presence and absence of long-range magnetic order. We then turn to the first of two ergodicity-breaking scenarios, by considering Hilbert space fragmentation (HSF) in charge- and dipole-conserving quantum chains. We determine the critical charge density at which a "freezing" phase transition from weak to strong fragmentation takes place, for any on-site Hilbert space dimension. To characterize the degree of dynamical connectivity in the system, we introduce a novel framework that enables us to obtain analytic results for the strongly fragmented phase, and numerically explore the weakly fragmented one. We derive some critical exponents, and show that the entanglement entropy of typical eigenstates in the strongly fragmented phase obeys an area law. We also uncover the existence of rare eigenstates whose entropy appears to scale beyond any area law. Finally, we investigate dynamical correlators in the Lieb-Liniger gas, one of the simplest interacting integrable theories solvable by Bethe ansatz. We employ Markov chain Monte Carlo methods to sample matrix elements (form factors) that appear in the Lehmann representation of the finite-temperature single-particle bosonic Green’s function. In the physical regimes we consider, sampling is necessary because a full summation of the relevant form factors is computationally intractable. We benchmark our results against exact solutions available in specific limits, and find them to be very accurate.
We identify a hierarchical symmetry structure underlying the Hilbert space of quantum systems made of identical particles. By studying the linear map between single-body and many-body spectra, we show that the resulting spectral organization admits a natural partition into sectors induced by a symmetry governed by cyclotomic fields and their Galois groups. This approach offers a new perspective on the structure of quantum many-body spectra, defining a controlled coarse-graining of the Hilbert space in which degeneracies and spectral features can be systematically organized through a set of new quantum numbers. The resulting structure induces a hierarchy of spectral resolutions following a renormalization flow, which enables information loss control. We demonstrate that this hierarchical decomposition provides an efficient route to approximate MBDoS calculations for any identical particle system, while preserving physically relevant spectral properties. This approach provides a general symmetry-based framework for organizing and approximating many-body spectra, with potential applications to quantum thermalization, spectral statistics, and large-scale quantum simulations.