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arXiv:2605.00094

Graph-theory measures capture weak ergodicity breaking on large quantum systems

Heiko Georg Menzler, Rafał Świętek, Mari Carmen Bañuls, Fabian Heidrich-Meisner

quant-phcond-mat.stat-mechcond-mat.str-elSubmitted Apr 30, 2026• Announced May 4, 2026• 0 citations• Revised Oct 30, 2026 (v3)
v1 · Apr 30, 2026v2 · May 11, 2026v3 · Oct 30, 2026
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Abstract

We study the onset of weak ergodicity violations in closed quantum many-body systems and focus on cases in which they occur through a transition that is controlled by a model parameter, a scenario called a weak ergodicity-breaking transition. This situation is realized in the Rosenzweig-Porter model and the quantum sun model, which we investigate here. Our analysis is based on representing quantum systems in Fock space and examining the short-time dynamics of density autocorrelation functions. From these, we extract proxies for decay rates for each individual basis state. As a first main result, we show that the distribution of such decay rates captures known weak ergodicity-breaking transitions in both models. As a second main result, we establish a connection between the decay rates of density autocorrelations and graph-theoretical measures. Concretely, we demonstrate that moments of the distribution of the recently introduced graph-energy centrality also detect the weak ergodicity transitions, which is inherited from the behavior of the decay rates. While most numerical tools are limited to small system sizes, the quantities studied in our work can be calculated analytically for large systems of many hundreds of sites and in some cases, even in the thermodynamic limit.

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14 pages, 7 figures

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Preprint arXiv:2605.00094 Submitted on 11 May 2026

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We study the onset of weak ergodicity violations in closed quantum many-body systems and focus on cases in which they occur through a transition that is controlled by a model parameter, a scenario called a weak ergodicity-breaking transition. This situation is realized in the Rosenzweig-Porter model and the quantum sun model, which we investigate here. Our analysis is based on representing quantum systems in Fock space and examining the short-time dynamics of density autocorrelation functions. From these, we extract proxies for decay rates for each individual basis state. As a first main result, we show that the distribution of such decay rates captures known weak ergodicity-breaking transitions in both models. As a second main result, we establish a connection between the decay rates of density autocorrelations and graph-theoretical measures. Concretely, we demonstrate that moments of the distribution of the recently introduced graph-energy centrality also detect the weak ergodicity transitions, which is inherited from the behavior of the decay rates. While most numerical tools are limited to small system sizes, the quantities studied in our work can be calculated analytically for large systems of many hundreds of sites and in some cases, even in the thermodynamic limit.
We study the onset of weak ergodicity violations in closed quantum many-body systems and focus on cases in which they occur through a transition that is controlled by a model parameter. Our analysis is based on representing quantum systems in Fock space and utilizes graph-theoretical measures. As a main result, we show that the recently introduced graph-energy centrality captures known weak ergodicity-breaking transitions via characteristic changes in its distribution. While most numerical tools are limited to small system sizes, our measure can be calculated analytically for large systems of many hundreds of sites and in some cases, even in the thermodynamic limit. We conclude by demonstrating the applicability of our Fock-space based measure to a kinetically constrained quantum model, where we find evidence for a weak ergodicity-breaking transition accompanied by glassy dynamics.
The interplay between quenched disorder and interaction effects opens the possibility in a closed quantum many-body system of a phase transition at finite energy density between an ergodic phase, which is governed by the laws of statistical physics, and a localized one, in which the degrees of freedom are frozen and ergodicity breaks down. The possible existence of a quantum phase transition at finite energy density is strongly questioning our understanding of the fundamental laws of nature and has generated an active field of research called many-body localization. This thesis consists of three parts and is dedicated to the understanding and characterization of the phenomenon of many-body localization, approaching it from complementary facets. In particular, borrowing methods and tools from different fields, we analyze timely problems. The first part of the thesis is devoted to detecting the many-body localization transition and to characterize both the ergodic and the localized phase it separates. Here we provide a characterization from two different perspectives: the first one is based on the study of local entanglement properties. In the second one, using tools from quantum-chaos theory, we attempt to answer the question of understanding time-irreversibility, and thus probing the breaking of ergodicity. We analyze experimentally viable observables. Moreover, we propose two different quantities to distinguish an Anderson insulating phase from a many-body localized one, which is one of the issues in experiments. The second part focuses on understanding the existence of a putative subdiffusive multifractal phase. Analyzing the quantum dynamics of the system in this region of the phase diagram, we point out the importance of finite-size effects, questioning the existence of this multifractal phase. We speculate with a possible scenario in which the diffusivity and thus ergodicity could be restored in the thermodynamic limit. Furthermore, we find that the propagation is highly non-Gaussian, which could have an important effect on understanding the critical point of the according transition. We tackle this problem also from a different angle. A possible toy-model to understand many-body localization entails the Anderson model on a random-regular graph. Also in the latter model the possible existence of an intermediate multifractal phase has been conjectured. There, studying the survival return probability of a particle with time, we give a new characterization of multifractal phases and give indication of the possible existence of this phase. Nevertheless, we also outline possible caveats. In the last part of this thesis we study the interplay between symmetry and correlated disorder in a non-interacting fermionic system. We show another possible mechanism for breaking localization. In particular, we focus on studying information and particle transport, emphasizing how the two types of propagation can be different.
We present the study of the structure of many-body eigenfunctions in a one-dimensional disordered spin chain.We discuss the choice of an appropriate basis in the Hilbert space, where the problem can be seen as an Anderson model defined on a high-dimensional non-trivial graph, determined by the many-body Hamiltonian.The comparison with the usual behavior of wave-functions in finite dimensional Anderson localization allows us to put in light the main differences of the many-body case.At high disorder, the typical eigenfunctions do not seem to localize though they occupy a infinitesimal portion of the Hilbert space in the thermodynamic limit.We perform a detailed analysis of the distribution of the wave-function coefficients and their peculiar scaling in the small and large disorder phase.We propose a criterion to identify the position of the transition by looking at the long tails of these distributions.The results coming from exact diagonalization show signs of breaking of ergodicity when the disorder reaches a critical value that agrees with the estimation of the many-body localization transition in the same model.
We adapt tools from the theory of quantum random walks to investigate slow relaxation dynamics through the many-body state graph. Specifically, we construct a probe of heterogeneity between basis states defined using hitting times derived from the unitary time-evolution operator. We find that the state-graph geometry, encoded by the pairwise hitting time of basis states, is a highly sensitive indicator of slow relaxation dynamics in a variety of systems. We study three paradigmatic models: the Rosenzweig-Porter model, the quantum East model, and the triangular lattice gas model, exhibiting a sudden onset of slow dynamics upon tuning of a control parameter. As a global characterization of the graph geometry, we analyze the spectral radius of the hitting matrix. We find that it increases sharply at the onset of slow dynamics, spanning many orders of magnitude, with a characteristic crossing point at the transition. Our work provides a geometric framework for describing and identifying phases with slow relaxation using a unified graph-theoretic formalism.
Understanding the effects of disorder in quantum many-body systems is one of the central challenges of contemporary statistical physics. Its interplay with interactions and quantum fluctuations profoundly modifies the nature of quantum states and their dynamics, giving rise to phenomena such as localization, ergodicity breaking, and the emergence of complex collective behavior. This thesis presents new results on two classes of disordered quantum systems. The first part is devoted to the study of the many-body localization transition, a paradigmatic example of disorder-induced ergodicity breaking. The critical properties of the transition are investigated using a numerical renormalization-group approach, establishing a connection between its scaling behavior and that of the Anderson localization transition on tree-like graphs. The second part focuses on the zero-temperature quantum spin-glass phase of the two-dimensional bond-disordered Heisenberg model. Combining large-scale numerical simulations with a semiclassical expansion, evidence is provided for the existence of the spin-glass phase. The effective model governing the semiclassical fluctuations is further analyzed to characterize the low-energy excitation spectrum and its localization properties. Finally, the weak-disorder regime is investigated and its implications for the ferromagnetic phase are discussed. Overall, the results presented in this thesis contribute to the understanding of the effects of disorder in interacting quantum many-body systems, encompassing both localization phenomena and the collective behavior of disordered quantum magnets.
Recent work has proposed fading ergodicity as a mechanism for many-body ergodicity breaking. Here, we show that two paradigmatic random matrix ensembles -- the Rosenzweig-Porter model and the ultrametric model -- fall within the same universality class of ergodicity breaking when embedded in a many-body Hilbert space of spins-1/2. By calibrating the parameters of both models via their Thouless times, we demonstrate that the matrix elements of local observables display similar statistical properties, allowing us to identify the fractal phase of the Rosenzweig-Porter model with the fading-ergodicity regime. This correspondence is further supported through the analyses of quantum-quench dynamics of local observables, their temporal fluctuations and power spectra, and survival probabilities. Our findings reveal that local observables thermalize within the fading-ergodicity regime on timescales shorter than the Heisenberg time, thus providing a unified framework for understanding ergodicity breaking across these distinct models.