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arXiv:2603.20540

Quantum Chaos in Many-Body Systems Without a Classical Analogue

Fotis I. Giasemis

cond-mat.stat-mechnlin.CDquant-phSubmitted Mar 20, 2026• Announced Mar 24, 2026• 0 citations
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Abstract

In classical systems, chaos is clearly defined via the behavior of trajectories. In quantum systems with a classical analogue one finds that the transition from regular to chaotic dynamics is signified by a change in the spectral statistics. This has been found to remain true for quantum systems with no classical analogue, including many-body systems. Furthermore, quantum chaotic systems explore all the allowed configurations in the Hilbert space, i.e. they are ergodic, while integrable systems, and systems in the many-body localized phase, are restricted to a certain subspace of the available phase space, and hence strongly break ergodicity. In this dissertation, we study the intermediate behavior between ergodicity and localization, i.e. the weak breaking of ergodicity. The model examined is the PXP spin chain model, where spins are allowed to flip only under certain kinetic constraints. We start by reproducing some already established results. First, we explore the eigenstate thermalization hypothesis (ETH) for this model and demonstrate the existence of a small number of states, throughout the PXP spectrum, that violate the ETH. Then we study the level-spacing statistics of the model, a well-known quantum chaos diagnostic, which turns out to be close to semi-Poisson and approach Wigner--Dyson statistics for large system sizes. Moreover, we examine various aspects of the model that have not been studied before. For example, the eigenvector component statistics, another quantum chaos diagnostic, for the PXP model turn out to be non-Gaussian. Finally, we perform a quench, in order to study how the energy spreads throughout the system, and observe ballistic fronts.

DOI
10.26240/heal.ntua.23630
Comments
Master thesis

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In classical systems, chaos is clearly defined via the behavior of trajectories. In quantum systems with a classical analogue one finds that the transition from regular to chaotic dynamics is signified by a change in the spectral statistics. This has been found to remain true for quantum systems with no classical analogue, including many-body systems. Furthermore, quantum chaotic systems explore all the allowed configurations in the Hilbert space, i.e. they are ergodic, while integrable systems, and systems in the many-body localized phase, are restricted to a certain subspace of the available phase space, and hence strongly break ergodicity. In this dissertation, we study the intermediate behavior between ergodicity and localization, i.e. the weak breaking of ergodicity. The model examined is the PXP spin chain model, where spins are allowed to flip only under certain kinetic constraints. We start by reproducing some already established results. First, we explore the eigenstate thermalization hypothesis (ETH) for this model and demonstrate the existence of a small number of states, throughout the PXP spectrum, that violate the ETH. Then we study the level-spacing statistics of the model, a well-known quantum chaos diagnostic, which turns out to be close to semi-Poisson and approach Wigner--Dyson statistics for large system sizes. Moreover, we examine various aspects of the model that have not been studied before. For example, the eigenvector component statistics, another quantum chaos diagnostic, for the PXP model turn out to be non-Gaussian. Finally, we perform a quench, in order to study how the energy spreads throughout the system, and observe ballistic fronts.
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One of the central problems in many-body physics, both classical and quantum, is the relations between different notions of chaos. Ergodicity, mixing, operator growth, the eigenstate thermalization hypothesis, and spectral chaos are defined in terms of completely different objects in different contexts, don't necessarily co-occur, but still seem to be manifestations of closely related phenomena. In this dissertation, we study the relation between two notions of chaos: thermalization and spectral chaos. We define a quantity called the Total Return Probability (TRP) which measures how a system forgets its initial state after time TTT, and show that it is closely connected to the Spectral Form Factor (SFF), a measure of chaos deriving from the energy level spectrum of a quantum system. The main thrust of this work concerns hydrodynamic systems -- systems where locality prevents charge or energy from spreading quickly, this putting a throttle on thermalization. We show that the detailed spacings of energy levels closely capture the dynamics of these locally conserved charges. We also study spin glasses, a phase of matter where the obstacle to thermalization comes not from locality but from the presence of too many neighbors. Changing one region requires changing nearby regions which requires changing nearby-to-nearby regions, until only catastrophic realignments of the whole system can fully explore phase space. In spin glasses we find our clearest analytic link between thermalization and spectral statistics. We analytically calculate the spectral form factor in the limit of large system size and show it is equal to the TRP. Finally, in the conclusion, we discuss some ideas for the future of both the SFF and the TRP.
The Eigenstate Thermalization Hypothesis explains thermalization in isolated quantum systems through the statistical properties of observables in the energy eigenbasis. We investigate the crossover from integrability to chaos in the spin-1/21/21/2 XXZ chain, establishing a direct correspondence between the spectral correlations of the Hamiltonian and local observables expressed in the energy eigenbasis as a signature of ergodicity breaking. By introducing a local perturbation that drives the system from integrability to chaos, we track the standard ETH indicators and the eigenstate entanglement entropy. We introduce a submatrix-based framework for analyzing local observables in the energy eigenbasis. By extracting real-symmetric blocks along the diagonal of the local observables represented in eigenbasis, we show that these submatrices exhibit both the short-range and long-range spectral features of the Hamiltonian. Remarkably, this correspondence persists even in a partially ergodic regime, indicating that the emergence of chaos is already encoded locally within the observables' matrix structure and that small blocks are sufficient to capture the underlying spectral correlations.
We investigate the quantum-classical correspondence in open quantum many-body systems using the SU(3) Bose-Hubbard trimer as a minimal model. Combining exact diagonalization with semiclassical Langevin dynamics, we establish a direct connection between classical trajectories characterized by fixed-point attractors, limit cycles, or chaos and the spectral and structural properties of the quantum steady state. We show that classical dynamical behavior, as quantified by the sign of the Lyapunov exponent, governs the level statistics of the steady-state density matrix: non-positive exponents associated with regular dynamics yield Poissonian statistics, while positive exponents arising from chaotic dynamics lead to Wigner-Dyson statistics. Strong symmetries constrain the system to lower-dimensional manifolds, suppressing chaos and enforcing localization, while weak symmetries preserve the global structure of the phase space and allow chaotic behavior to persist. To characterize phase-space localization, we introduce the phase-space inverse participation ratio IPR, which defines an effective dimension D of the Husimi distribution's support. We find that the entropy scales as S∝ln⁡NDS \propto \ln N^DS∝lnND, consistently capturing the classical nature of the underlying dynamics. This semiclassical framework, based on stochastic mixtures of coherent states, successfully reproduces not only observable averages but also finer features such as spectral correlations and localization properties. Our results demonstrate that dissipative quantum chaos is imprinted in the steady-state density matrix, much like in closed systems, and that the interplay between dynamical regimes and symmetry constraints can be systematically probed using spectral and phase-space diagnostics. These tools offer a robust foundation for studying ergodicity, localization, and non-equilibrium phases of open quantum systems.
In the past decades, it was recognized that quantum chaos, which is essential for the emergence of statistical mechanics and thermodynamics, manifests itself in the effective description of the eigenstates of chaotic Hamiltonians through random matrix ensembles and the eigenstate thermalization hypothesis.Standard measures of chaos in quantum many-body systems are level statistics and the spectral form factor.In this work, we show that the norm of the adiabatic gauge potential, the generator of adiabatic deformations between eigenstates, serves as a much more sensitive measure of quantum chaos.We are able to detect transitions from integrable to chaotic behavior at perturbation strengths orders of magnitude smaller than those required for standard measures.Using this alternative probe in two generic classes of spin chains, we show that the chaotic threshold decreases exponentially with system size and that one can immediately detect integrability-breaking (chaotic) perturbations by analyzing infinitesimal perturbations even at the integrable point.In some cases, small integrability breaking is shown to lead to anomalously slow relaxation of the system, exponentially long in system size.
Characterizing states of matter through the lens of their ergodic properties is a fascinating new direction of research. In the quantum realm, the many-body localization (MBL) was proposed to be the paradigmatic ergodicity breaking phenomenon, which extends the concept of Anderson localization to interacting systems. At the same time, random matrix theory has established a powerful framework for characterizing the onset of quantum chaos and ergodicity (or the absence thereof) in quantum many-body systems. Here we numerically study the spectral statistics of disordered interacting spin chains, which represent prototype models expected to exhibit MBL. We study the ergodicity indicator g=log_{10}(t_{H}/t_{Th}), which is defined through the ratio of two characteristic many-body time scales, the Thouless time t_{Th} and the Heisenberg time t_{H}, and hence resembles the logarithm of the dimensionless conductance introduced in the context of Anderson localization. We argue that the ergodicity breaking transition in interacting spin chains occurs when both time scales are of the same order, t_{Th}≈t_{H}, and g becomes a system-size independent constant. Hence, the ergodicity breaking transition in many-body systems carries certain analogies with the Anderson localization transition. Intriguingly, using a Berezinskii-Kosterlitz-Thouless correlation length we observe a scaling solution of g across the transition, which allows for detection of the crossing point in finite systems. We discuss the observation that scaled results in finite systems by increasing the system size exhibit a flow towards the quantum chaotic regime.