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.
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.
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.
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.
Spectral statistics probe integrability versus chaos and have recently been extended to Markovian open quantum systems described by Lindbladians, whose quantum-trajectory unraveling decomposes the evolution into no-jump dynamics generated by an effective non-Hermitian Hamiltonian and recycling jumps. In this work, we perform spectrum-statistics diagnostics for Lindbladians and their effective non-Hermitian Hamiltonians. We show that recycling processes, symmetry constraints, and the Liouville-space structure crucially shape the spectral correlations. In particular, we identify a family of spectrally separable Lindbladians whose spectra exhibit robust Poisson statistics, despite the effective non-Hermitian Hamiltonian varying from Poisson to strongly correlated complex-spectrum statistics. Our work establishes a unified spectral-statistics characterization for Lindbladians and their associated effective non-Hermitian Hamiltonians, deepening our understanding of spectral properties in open 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 study generic open quantum systems with Markovian dissipation, focusing on a class of stochastic Liouvillian operators of Lindblad form with independent random dissipation channels (jump operators) and a random Hamiltonian. We perform a thorough numerical study, focusing on global spectral features, the spectral gap, and the steady-state purity and statistics. We establish that all properties follow three different regimes as a function of the dissipation strength, whose boundaries depend on the particular observable. Within each regime, we determine the scaling exponents with the dissipation strength and system size. On the analytical side, we compute the average spectral gap at arbitrary dissipation and provide simple closed-form expressions for the asymptotic values at strong and weak dissipation strength. We also consider spectral correlations in generic complex spectra, such as that of the Liouvillian. We introduce and study, both analytically and numerically, the complex ratio of the nearest-neighbor spacing by next-to-nearest-neighbor spacing. Besides the usual level repulsion, we find that complex spacing ratios have a nontrivial angular dependence which offers a distinct signature of quantum chaos. It distinguishes integrable systems, which support the flat ratio distribution characteristic of Poisson statistics, and chaotic dissipative quantum systems, which conform to Random Matrix Theory statistics. We apply this new signature of quantum chaos to random Liouvillians and boundary-driven dissipative spin-chains. The results of this thesis can help understand the long-time dynamics, steady-state properties, and spectral correlations of generic dissipative systems.
A bstract A classical dynamical system can be viewed as a probability space equipped with a measure-preserving time evolution map, admitting a purely algebraic formulation in terms of the algebra of bounded functions on the phase space. Similarly, a quantum dynamical system can be formulated using an algebra of bounded operators in a non-commutative probability space equipped with a time evolution map. Chaos, in either setting, can be characterized by statistical independence between observables at t = 0 and t → ∞, leading to the vanishing of cumulants involving these observables. In the quantum case, the notion of independence is replaced by free independence, which only emerges in the thermodynamic limit (asymptotic freeness). In this work, we propose a definition of quantum chaos based on asymptotic freeness and investigate its emergence in quantum many-body systems including the mixed-field Ising model with a random magnetic field, a higher spin version of the same model, and the SYK model. The hallmark of asymptotic freeness is the emergence of the free convolution prediction for the spectrum of operators of the form A (0) + B ( t ), implying the vanishing of all free cumulants between A (0) and B ( t ) in the thermodynamic limit for an infinite-temperature thermal state. We systematically investigate the spectral properties of A (0) + B ( t ) in the above-mentioned models, show that fluctuations on top of the free convolution prediction follow universal Wigner-Dyson statistics, and discuss the connection with quantum chaos. Finally, we argue that free probability theory provides a rigorous framework for understanding quantum chaos, offering a unifying perspective that connects many different manifestations of it.