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arXiv:2502.09711 · Physical review. B./Physical review. B

Fading ergodicity meets maximal chaos

Rafał Świętek, Patrycja Łydżba, Lev Vidmar

Published May 27, 2025• 7 citations
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Abstract

Fading ergodicity provides a theoretical framework for understanding deviations from the eigenstate thermalization hypothesis (ETH) near ergodicity-breaking transitions. In this work we demonstrate that the position of the ergodicity-breaking critical point in the quantum sun model coincides with the peak of the maximally divergent fidelity susceptibility. We further extend our analysis to the energy-resolved ergodicity-breaking transition associated with the many-body mobility edge. Specifically, we show that fidelity susceptibilities at energies away from the middle of the spectrum exhibit a divergent peak near the mobility edge. We argue that fading ergodicity provides a simple and accurate description of the ETH breakdown in the quantum sun model, which is accompanied with the emergence of a peak in fidelity susceptibility and the onset of maximal chaos at the ergodicity-breaking critical point.

DOI
10.1103/physrevb.111.184203

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Fading ergodicity provides a theoretical framework for understanding deviations from the eigenstate thermalization hypothesis (ETH) near ergodicity-breaking transitions. In this work we demonstrate that the position of the ergodicity-breaking critical point in the quantum sun model coincides with the peak of the maximally divergent fidelity susceptibility. We further extend our analysis to the energy-resolved ergodicity-breaking transition associated with the many-body mobility edge. Specifically, we show that fidelity susceptibilities at energies away from the middle of the spectrum exhibit a divergent peak near the mobility edge. We argue that fading ergodicity provides a simple and accurate description of the ETH breakdown in the quantum sun model, which is accompanied with the emergence of a peak in fidelity susceptibility and the onset of maximal chaos at the ergodicity-breaking critical point.
The ETH represents a breakthrough in many-body physics since it links thermalization of physical observables with the applicability of RMT. This framework is widely believed to hold for an overwhelming majority of physical systems, though exceptions, where conventional ETH is violated, include integrability, single-particle chaos, many-body localization, many-body scars, and Hilbert-space fragmentation. However, the mechanism of the ETH breakdown remains elusive. In this thesis, we introduce a novel scenario in many-body quantum systems, dubbed itfadingergodicity{it fading ergodicity}itfadingergodicity regime, which establishes a link between the conventional ETH and non-ergodic behavior. This framework predicts an exponent, etaetaeta, governing eigenstate fluctuations in the matrix elements of local observables. As a consequence, slow dynamics emerge as the perturbations become vanishingly small, and the Thouless energy matches the Heisenberg scale at the critical point. We conjecture this scenario to be relevant for the description of many-body systems, where the divergent relaxation time is described by the Fermi golden rule. We provide numerical and analytical arguments for its validity in the quantum sun model and related RMT models. We continue our discussion by exploring the universality of the fading ergodicity scenario. Remarkably, we show that fading ergodicity leads to a maximally divergent peak in the fidelity susceptibility, establishing the onset of itmaximalchaos{it maximal chaos}itmaximalchaos at the critical point. Furthermore, we extend this framework to finite energy density, with both fading ergodicity and maximal chaos following the many-body mobility edge to high precision. Finally, we present a scaling theory of many-body ergodicity breaking, which exhibits a critical exponent nu=1nu = 1nu=1 at the critical point, characterizing the universality class of fading ergodicity. However, in contrast to the well established Anderson localization transition, we argue that the one-parameter scaling theory is insufficient.
The eigenstate thermalization hypothesis (ETH) provides the prevailing framework for understanding quantum thermalization and ergodicity in isolated many-body systems. Yet, no general theory describes the continuous onset of ETH violation between the conventional ETH and its complete breakdown. Here, we introduce a two-parameter ansatz for the ETH violation that unifies and distinguishes two mechanisms: fading ergodicity and trapped ergodicity. While fading ergodicity captures the established route to ergodicity breaking, trapped ergodicity describes a distinct scenario in which ETH is violated in finite systems but restored in the thermodynamic limit. We test this framework in the spin-1/2 J1J_1J1​-J2J_2J2​ chains with on-site disorder and linear potential. In both cases, we find that the observed ETH violation is consistent with trapped ergodicity.
Eigenstate thermalization hypothesis (ETH) represents a breakthrough in many-body physics since it allows us to link thermalization of physical observables with the applicability of random matrix theory (RMT). Recent years were also extremely fruitful in exploring possible counterexamples to thermalization, ranging, among others, from integrability, single-particle chaos, many-body localization, many-body scars, to Hilbert-space fragmentation. In all these cases the conventional ETH is violated. However, it remains elusive how the conventional ETH breaks down when one approaches the boundaries of ergodicity, and whether the range of validity of the conventional ETH coincides with the validity of RMT-like spectral statistics. Here we bridge this gap and we introduce a scenario of the ETH breakdown in many-body quantum systems, dubbed fading ergodicity regime, which establishes a link between the conventional ETH and nonergodic behavior. We conjecture this scenario to be relevant for the description of finite many-body systems at the boundaries of ergodicity, and we provide numerical and analytical arguments for its validity in the quantum sun model of ergodicity-breaking phase transition. For the latter, we provide evidence that the breakdown of the conventional ETH is not associated with the breakdown of the RMT-like spectral statistics. Published by the American Physical Society 2024
The quantum sun model is an interacting model that exhibits sharp signatures of ergodicity breaking phase transition. Here, we show that the model exhibits a many-body mobility edge. We provide analytical arguments for its existence, complemented by the state-of-the-art numerical simulations analyzing gap ratios, Thouless times as well as entanglement entropy of eigenstates. We also introduce the quantum sun model with particle number conservation and argue that it shares many similarities with its unrestricted predecessor.
In most noninteracting quantum systems, the scaling theory of localization predicts one-parameter scaling flow in both ergodic and localized regimes. A corresponding scaling theory of many-body ergodicity breaking is still missing. Here, we introduce a scaling theory of ergodicity breaking in interacting systems, in which the divergent relaxation time follows from the Fermi golden rule, and the observable fluctuations in proximity of the ergodicity breaking critical point are described by the recently introduced fading ergodicity scenario. We argue that, in general, the one-parameter scaling is insufficient, and we show that the scaling theory predicts the critical exponent ν=1 at the ergodicity breaking critical point. Our theoretical framework may serve as a building block for two-parameter scaling theories of many-body systems.
We investigate the fidelity susceptibility, which quantifies the sensitivity of single-particle eigenstates to perturbations, in the three-dimensional Anderson model. As a function of disorder strength WWW, it exhibits two distinct peaks. The first peak signals a crossover at weak disorder strength from plane-wave states to single-particle quantum chaos, and its position shifts toward W→0W\to 0W→0 in the thermodynamic limit. The second peak emerges, to high numerical accuracy, at the critical disorder strength associated with the Anderson localization transition. We further show that the divergence of the first peak is maximal, scaling as the square of the inverse frequency cutoff, whereas the divergence of the second peak is submaximal. We relate the latter suppression to the fractal structure of single-particle eigenstates at criticality. We discuss two distinct scenarios that give rise to the peaks in the fidelity susceptibilities. Moreover, studying the scaling of typical fidelity susceptibilities above the Anderson transition, we find evidence of two distinct regimes of nonergodic behavior.