The statistics of gap ratios between consecutive energy levels is a widely used tool-in particular, in the context of many-body physics-to distinguish between chaotic and integrable systems, described, respectively, by Gaussian ensembles of random matrices and Poisson statistics.In this work, we extend the study of the gap ratio distribution PðrÞ to the case where discrete symmetries are present.This is important since in certain situations it may be very impractical, or impossible, to split the model into symmetry sectors, let alone in cases where the symmetry is not known in the first place.Starting from the known expressions for surmises in the Gaussian ensembles, we derive analytical surmises for random matrices comprised of several independent blocks.We check our formulas against simulations from large random matrices, showing excellent agreement.We then present a large set of applications in many-body physics, ranging from quantum clock models and anyonic chains to periodically driven spin systems.In all these models, the existence of a (sometimes hidden) symmetry can be diagnosed through the study of the spectral gap ratios, and our approach furnishes an efficient way to characterize the number and size of independent symmetry subspaces.We finally discuss the relevance of our analysis for existing results in the literature, as well as its practical usefulness, and point out possible future applications and extensions.
The statistics of gap ratios between consecutive energy levels is a widely used tool-in particular, in the context of many-body physics-to distinguish between chaotic and integrable systems, described, respectively, by Gaussian ensembles of random matrices and Poisson statistics.In this work, we extend the study of the gap ratio distribution PðrÞ to the case where discrete symmetries are present.This is important since in certain situations it may be very impractical, or impossible, to split the model into symmetry sectors, let alone in cases where the symmetry is not known in the first place.Starting from the known expressions for surmises in the Gaussian ensembles, we derive analytical surmises for random matrices comprised of several independent blocks.We check our formulas against simulations from large random matrices, showing excellent agreement.We then present a large set of applications in many-body physics, ranging from quantum clock models and anyonic chains to periodically driven spin systems.In all these models, the existence of a (sometimes hidden) symmetry can be diagnosed through the study of the spectral gap ratios, and our approach furnishes an efficient way to characterize the number and size of independent symmetry subspaces.We finally discuss the relevance of our analysis for existing results in the literature, as well as its practical usefulness, and point out possible future applications and extensions.
In this paper we analyze the spectral level statistics of the one-dimensional ionic Hubbard model, the Hubbard model with an alternating on-site potential. In particular, we focus on the statistics of the gap ratios between consecutive energy levels. This quantity is often used in order to signal whether a many-body system is integrable or chaotic. A chaotic system has typically the statistics of a Gaussian ensemble of random matrices while the spectral properties of the integrable system follow a Poisson statistics. We find that whereas the Hubbard model without alternating potential is known to be integrable and its spectral properties follow a Poissonian statistics, the presence of an alternating potential causes a drastic change in the spectral properties, which resemble the one of a Gaussian ensemble of random matrices. However, to uncover this behavior one has to separately consider the blocks of all symmetries of the ionic Hubbard model.
The spectral fluctuations in complex quantum systems are modeled in terms of spectra from random matrices. Quantitative agreement with random matrices holds only when the quantum spectrum is a pure sequence, and not a composite spectrum that mixes levels from different symmetry sectors. It was shown earlier that discrete symmetries in composite spectra can be detected using higher-order spacing ratio statistics provided the symmetry sectors are of equal dimensions. The general case is when the symmetry sectors have unequal dimensions, and the symmetries of the Hamiltonian system are unknown or only partially known. In this work, we address how the number of symmetry sectors can be determined from a composite spectrum arising in Hermitian and non-Hermitian settings. A Best Subset Technique is proposed for determining the number of the symmetry sectors by comparing the spectral fluctuations to that of an appropriate random matrix ensemble without requiring de-symmetrization. It is demonstrated using spectra from Gaussian, Wishart and Ginibre ensembles, and also for spectra drawn from many-body quantum systems.
The study of spectrum statistics, such as the consecutive-gap ratio distribution, has revealed many interesting properties of many-body complex systems. Here we propose a two-parameter surmise expression for such distribution to describe the crossover between the Gaussian orthogonal ensemble (GOE) and Poisson statistics. This crossover is observed in the isotropic Heisenberg spin-1/2 chain with disordered local field, exhibiting the Many-Body Localization (MBL) transition. Inspired by the analysis of stability in dynamical systems, this crossover is presented as a flow pattern in the parameter space, with the Poisson statistics being the fixed point of the system, which represents the MBL phase. We also analyze an isotropic Heisenberg spin-1/2 chain with disordered local exchange coupling and a zero magnetic field. In this case, the system never achieves the MBL phase because of the spin rotation symmetry. This case is more sensitive to finite-size effects than the previous one, and thus the flow pattern resembles a two-dimensional random walk close to its fixed point. We propose a system of linearized stochastic differential equations to estimate this fixed point. We study the continuous-state Markov process that governs the probability of finding the system close to this fixed point as the disorder strength increases. In addition, we discuss the conditions under which the stationary probability distribution is given by a bivariate normal distribution.
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 Correlations between energy levels can help distinguish whether a many-body system is of integrable or chaotic nature. The study of short-range and long-range spectral correlations generally involves quantities which are very different, unless one uses the k th nearest neighbor ( k NN) level spacing distribution. For nearest-neighbor (NN) spectral spacings, the distribution in random matrices is well captured by the Wigner surmise. This well-known approximation, derived exactly for a 2 × 2 matrix, is simple and satisfactorily describes the NN spacings of larger matrices. There have been attempts in the literature to generalize Wigner’s surmise to further away neighbors. However, as we show, the current proposal in the literature fails to accurately capture numerical data. Using the known variance of the distributions from random matrix theory, we propose a corrected surmise for the k NN spectral distributions. This surmise better characterizes spectral correlations while retaining the simplicity of Wigner’s surmise. We test the predictions against numerical results and show that the corrected surmise is systematically more accurate at capturing data from random matrices. Using the XXZ spin chain with random on-site disorder, we illustrate how these results can be used as a refined probe of many-body quantum chaos for both short- and long-range spectral correlations.