When Order Fades: Unraveling the Limits of Quantum Chaos

Author: Denis Avetisyan

New research reveals a shared mechanism governing the breakdown of predictable behavior in both complex quantum systems and carefully constructed mathematical models.

Fading ergodicity emerges within specific parameter regimes in both the Rosenzweig-Porter model-where it is observed for <span class="katex-eq" data-katex-display="false"> 1 < \gamma < 2 </span>-and the ultrametric model-where it is observed for <span class="katex-eq" data-katex-display="false"> 1/\sqrt{2} < \alpha < 1 </span>-indicating that, in both systems, the observable relaxation time, or Thouless time <span class="katex-eq" data-katex-display="false"> t_{Th} </span>, consistently falls below the Heisenberg time <span class="katex-eq" data-katex-display="false"> t_{H} </span>. — Fading ergodicity emerges within specific parameter regimes in both the Rosenzweig-Porter model-where it is observed for $1 < \gamma < 2$ -and the ultrametric model-where it is observed for $1/\sqrt{2} < \alpha < 1$ -indicating that, in both systems, the observable relaxation time, or Thouless time $t_{Th}$ , consistently falls below the Heisenberg time $t_{H}$ .

This review explores the phenomenon of fading ergodicity in random matrix ensembles and many-body systems, connecting it to the Thouless energy and the transition to non-thermalizing dynamics.

The breakdown of ergodicity-the loss of equal time spent in all accessible states-remains a central challenge in understanding many-body quantum systems. This work, ‘Fading ergodicity and quantum dynamics in random matrix ensembles’, investigates the subtle transition to ergodicity breaking through a comparative analysis of the Rosenzweig-Porter and ultrametric random matrix models. By calibrating these models via their Thouless times, we demonstrate a unifying framework where local observables exhibit similar statistical properties and thermalize on timescales shorter than the Heisenberg time, identifying a shared fading-ergodicity regime. Does this correspondence suggest a universal mechanism governing the onset of non-ergodic behavior in a broader range of complex quantum systems?

The Illusion of Thermal Equilibrium

Statistical mechanics, a cornerstone of modern physics, traditionally posits that an isolated system, given enough time, will reach a state of thermal equilibrium. This implies the system’s macroscopic properties – temperature, pressure, and so on – become stable and predictable, describable by statistical ensembles like the MicrocanonicalEnsemble. This ensemble defines probabilities based on energy, assuming all accessible microstates within a narrow energy range are equally likely. The power of this approach lies in its ability to predict average behavior without needing to track the detailed evolution of every particle. However, this foundational assumption of thermalization isn’t universally valid, particularly when dealing with the intricate dynamics of many-body quantum systems where interactions and correlations can significantly hinder the path to equilibrium, prompting a need for more nuanced theoretical frameworks.

Many-body quantum systems, unlike classical systems, aren’t always destined for simple equilibrium. Their inherent complexities, arising from the interconnectedness of numerous particles, allow for dynamics that actively resist reaching a steady state. This is particularly evident following a QuantumQuench – an abrupt change to a system’s parameters. Such a sudden shift doesn’t allow the system to adjust gradually; instead, it’s jolted into a period of non-equilibrium evolution. The resulting behavior can be far from the predictions of traditional statistical mechanics, exhibiting persistent oscillations, memory of the initial state, or even the emergence of novel phases of matter. Investigating these deviations is crucial, as it reveals the limitations of applying equilibrium concepts to truly isolated quantum systems and demands a deeper understanding of their underlying dynamical rules.

Predicting the future states of isolated quantum systems demands a precise understanding of how these systems evolve, and deviations from traditional equilibrium assumptions fundamentally alter those predictions. While statistical mechanics often relies on the concept of thermalization – a system settling into a predictable state – many real-world quantum systems exhibit dynamics that resist this simplification. These non-equilibrium behaviors can manifest as persistent oscillations, slow relaxation to equilibrium, or even the complete breakdown of statistical predictability. Consequently, accurately modeling these systems requires going beyond standard techniques and incorporating a detailed understanding of the underlying quantum dynamics; failing to do so can lead to significant errors in forecasting their long-term behavior, particularly in areas like quantum computing and materials science where precise control and predictability are paramount.

The Eigenstate Thermalization Hypothesis (ETH) proposes that the long-time behavior of isolated quantum systems can be understood by analyzing the properties of their individual energy eigenstates, suggesting that these states already encode the thermal distribution expected at equilibrium. While remarkably successful in describing a broad range of systems, the ETH is not a universally valid principle; certain classes of quantum systems demonstrably violate its assumptions. These violations often arise in systems with many-body localization, strong disorder, or specific integrability constraints, leading to the emergence of non-thermal phases and persistent oscillations in observables even at late times. Consequently, researchers are actively exploring the boundaries of the ETH’s applicability and developing theoretical frameworks to describe quantum dynamics beyond its reach, recognizing that a complete understanding of isolated quantum systems requires acknowledging scenarios where energy eigenstates fail to fully capture the system’s complex behavior.

Analysis of the autocorrelation function <span class="katex-eq" data-katex-display="false">C(t)</span> reveals a Thouless energy Γ-dependent decay and scaling of <span class="katex-eq" data-katex-display="false">\Delta Q</span> with system size <span class="katex-eq" data-katex-display="false">L</span>, demonstrating fading ergodicity for both the Unique Metastate (UM) and Replica-Symmetric Parametric (RP) models across varying α and γ parameters. — Analysis of the autocorrelation function $C(t)$ reveals a Thouless energy Γ-dependent decay and scaling of $\Delta Q$ with system size $L$ , demonstrating fading ergodicity for both the Unique Metastate (UM) and Replica-Symmetric Parametric (RP) models across varying α and γ parameters.

Spectral Fingerprints of Dynamical Breakdown

The Spectral Function, denoted as $A(E)$ , provides a direct mapping of the energy distribution within a quantum system and details its dynamic evolution. Specifically, $A(E)$ represents the probability of finding the system with a particular energy $E$ . Analyzing the shape and features of the Spectral Function allows for the identification of energy levels, their corresponding lifetimes, and the strength of interactions within the system. Broadening of spectral peaks indicates shorter lifetimes and stronger interactions, while sharp peaks signify long-lived states and weaker coupling. Furthermore, the Spectral Function is crucial for understanding relaxation processes, as it directly relates to the density of states and the transition rates between energy levels, effectively quantifying how energy is distributed and dissipated within the quantum system.

Analysis of time-dependent fluctuations and the level spacing ratio provides quantitative insight into the relaxation process of a quantum system. Time-dependent fluctuations characterize the rate at which the system loses memory of its initial state, while the level spacing ratio, $r = \frac{min(\delta_i)}{max(\delta_i)}$ , where $\delta_i$ represents the spacing between adjacent energy levels, provides statistical information about the local density of states. Specifically, the distribution of the level spacing ratio is sensitive to the type of underlying quantum chaos; systems exhibiting regular behavior display a distribution differing significantly from those demonstrating chaotic dynamics. Quantifying these distributions allows determination of relaxation times and provides evidence for, or against, ergodic behavior within the system.

Random matrix theory (RMT) provides statistically robust predictions for the spectral properties of complex quantum systems exhibiting ergodic behavior; therefore, measurable deviations from these predictions indicate a departure from ergodicity. Specifically, the level spacing ratio distribution, typically characterized by a scaling exponent of approximately 2 for systems described by Gaussian orthogonal/unitary ensembles, will exhibit modified exponents when ergodicity breaks down. These modifications manifest as altered spectral features, including the appearance of localized states or resonances not predicted by RMT, and are directly related to the system’s inability to explore all accessible phase space. Analysis of these deviations allows for the identification of underlying mechanisms causing the breakdown of ergodicity, such as the presence of symmetries, integrability, or many-body localization.

The Thouless time, τ, quantifies the longest relaxation time within a quantum system and serves as a critical parameter for identifying non-ergodic behavior. Analysis demonstrates a correlation between the scaling exponent of the level spacing ratio, denoted as γ, and the fluctuations of matrix elements influencing the system’s dynamics. Specifically, the observed scaling exponent is approximately 2-γ, indicating that as the Thouless energy decreases – and relaxation times lengthen – the level spacing ratio deviates from the predictions of random matrix theory. This deviation signals a breakdown in ergodicity, where the system no longer explores all accessible phase space, and the spectral properties reflect this restricted exploration.

The local density of states, <span class="katex-eq" data-katex-display="false">{\mathcal{P}(\omega)}</span>, exhibits a power-law decay proportional to <span class="katex-eq" data-katex-display="false">1/\omega^{2}</span> in the random phase approximation (RP) model and a more complex dependence, <span class="katex-eq" data-katex-display="false">1/\omega^{1-d\_{2}^{(0)}}</span> with fractal dimension <span class="katex-eq" data-katex-display="false">d\_{2}^{(0)}\approx 0.57</span>, near the critical point in the unitary model, as determined by the Thouless energy extracted from the spectral function. — The local density of states, ${\mathcal{P}(\omega)}$ , exhibits a power-law decay proportional to $1/\omega^{2}$ in the random phase approximation (RP) model and a more complex dependence, $1/\omega^{1-d\_{2}^{(0)}}$ with fractal dimension $d\_{2}^{(0)}\approx 0.57$ , near the critical point in the unitary model, as determined by the Thouless energy extracted from the spectral function.

The Emergence of Fading Ergodicity

Fading Ergodicity is characterized by a divergence of the Thouless time, $\tau_{\text{th}}$ , indicating a loss of quantum ergodicity and a breakdown of the assumption that a quantum system explores all accessible phase space equally. This divergence is accompanied by observable modifications to the statistical properties of energy levels; specifically, level repulsion transitions from Wigner-Dyson statistics to Poissonian behavior, and the emergence of spectral gaps. The Thouless time, representing the timescale for dephasing due to interactions, becomes increasingly large as the system deviates from ergodicity, signifying a localization effect where quantum states remain confined to smaller regions of phase space. These changes in spectral statistics and the extended Thouless time are key indicators of the transition towards a non-ergodic regime.

Structured random matrix ensembles, such as the Rosenzweig-Porter model and the Ultrametric model, provide a framework for simulating quantum systems exhibiting a transition away from traditional ergodic behavior. The Rosenzweig-Porter model introduces long-range hopping terms to a random matrix, creating correlations between matrix elements and a degree of structure. The Ultrametric model, conversely, enforces a hierarchical structure on the matrix elements based on ultrametric distance, resulting in a fragmented phase space. Both models deviate from the Gaussian Orthogonal Ensemble (GOE) or Gaussian Unitary Ensemble (GUE) by introducing these correlations, allowing for investigation of systems where the standard assumptions of ergodicity-namely, equal time spent in all accessible phase space regions-no longer hold. These ensembles serve as tractable theoretical tools for understanding systems demonstrating fading ergodicity and related phenomena like fractal eigenstates.

Structured random matrix ensembles, such as the Rosenzweig-Porter and Ultrametric models, deviate from traditional random matrix theory by incorporating non-random correlations into the matrix elements. This blend of randomness and structure results in eigenstates that are not extended throughout the system, but instead exhibit fractal characteristics – possessing a complex, self-similar geometry at different scales. Consequently, the ergodic hypothesis, which assumes that a quantum system explores all accessible states equally over time, breaks down in these systems. The localization induced by this interplay leads to a departure from the standard predictions of ergodic theory, necessitating alternative descriptions of quantum dynamics and spectral properties.

The QuantumSunModel, a physical system mimicking the behavior of structured random matrices, provides empirical validation of theoretical predictions regarding fading ergodicity. Analysis of temporal fluctuations within this model demonstrates a scaling behavior of $𝒟⁻²/η$ , where 𝒟 represents the system’s disorder and η characterizes the degree of localization. This observed scaling directly aligns with analytical results derived from calculations of matrix element fluctuations, confirming the connection between theoretical models-such as the Rosenzweig-Porter and Ultrametric models-and experimentally accessible quantum systems. These findings establish the QuantumSunModel as a valuable platform for investigating the breakdown of standard ergodic assumptions in systems exhibiting both randomness and structure.

For the UM [RP] model, the exponent of temporal fluctuations <span class="katex-eq" data-katex-display="false">2/\eta_{t}</span> and the fractal dimension <span class="katex-eq" data-katex-display="false">d_{2}^{(0)}</span> both exhibit a transition at the critical point indicated by the dash-dotted line, signaling ergodicity breaking and a loss of equilibration, and are related to the function <span class="katex-eq" data-katex-display="false">d_{2}^{\rm(eig)}=2-\gamma</span> as calculated by Eqs. (31) and (36). — For the UM [RP] model, the exponent of temporal fluctuations $2/\eta_{t}$ and the fractal dimension $d_{2}^{(0)}$ both exhibit a transition at the critical point indicated by the dash-dotted line, signaling ergodicity breaking and a loss of equilibration, and are related to the function $d_{2}^{\rm(eig)}=2-\gamma$ as calculated by Eqs. (31) and (36).

The Persistence of Fractal Eigenstates

The complexity of quantum states in many-body systems isn’t always captured by simple descriptions of localization or extended behavior; instead, eigenstates can exhibit fractal characteristics. The $\text{FractalDimension}$ serves as a precise measure of this complexity, quantifying how effectively an eigenstate fills space. Unlike a fully localized state confined to a small region, or an extended state spread uniformly, a fractal eigenstate occupies space in a self-similar, intricate pattern – a complexity that increases its effective volume. A higher $\text{FractalDimension}$ indicates a greater space-filling capacity and a stronger departure from traditional localization, suggesting that the quantum state is neither entirely confined nor completely delocalized, but occupies an intermediate, complex regime. This property isn’t merely a geometric curiosity; it fundamentally alters the system’s behavior, influencing how quantum information propagates and how long a state persists before decaying.

The unusual geometry of fractal eigenstates profoundly impacts how a quantum system evolves over time. Unlike typical quantum systems that settle into a predictable state-decaying exponentially-systems governed by these eigenstates exhibit persistent oscillations and a reluctance to reach equilibrium. This behavior stems from the eigenstates’ ability to occupy a complex, space-filling pattern, effectively trapping the quantum state and preventing its rapid dissipation. Consequently, the system doesn’t simply lose energy; instead, it cycles through different configurations, creating sustained, albeit decaying, oscillations in measurable properties. This deviation from exponential decay is not a flaw, but a fundamental characteristic of these non-ergodic systems, directly linked to the intricate, fractal structure of the underlying quantum states and influencing predictions of long-term quantum survival $\propto 𝒟⁻κ$ .

The Diagonal Ensemble offers a powerful simplification for modeling the protracted, non-exponential decay characteristic of quantum systems exhibiting fractal eigenstates. This approach circumvents the need to fully diagonalize the complex Hamiltonian by focusing solely on its diagonal matrix elements – those representing the energy of individual states. By treating these diagonal elements as random variables drawn from a specific probability distribution, the ensemble effectively captures the essential features governing the system’s long-time evolution. This allows researchers to predict key dynamical properties, such as the survival probability of a quantum state, with significantly reduced computational cost. Essentially, the Diagonal Ensemble posits that the intricate off-diagonal interactions, while important for determining the specific eigenstates, play a less critical role in dictating the overall, emergent behavior over extended timescales, offering a valuable tool for understanding quantum dynamics in complex, non-ergodic systems.

Predicting the longevity of a quantum state-its survival probability-within these non-ergodic systems hinges on a thorough understanding of their unique dynamics, which deviate sharply from typical quantum behavior. Research reveals a compelling relationship between the system’s complexity, quantified by its fractal dimension $𝒟$ , and its energy scale, specifically the Thouless energy. The minimum fluctuation in quantum properties, denoted as $ΔQ$ , scales inversely with the fractal dimension raised to a power κ, expressed as $\propto 𝒟⁻κ$ , where κ is approximately equal to $γ⁻¹$ . This correlation suggests that the intricate, space-filling nature of fractal eigenstates-and their resulting persistent oscillations-directly impacts how long a quantum state remains distinguishable, offering a pathway to characterize and ultimately control quantum coherence in these complex systems.

Analysis of fluctuations in the diagonal matrix elements for both the Random Phase (RP) and Unitary Matrix (UM) models reveals that the scaling of these fluctuations with system size <span class="katex-eq" data-katex-display="false">L</span> aligns with theoretical predictions <span class="katex-eq" data-katex-display="false"> \eta = 2/(2-\gamma)</span> for the RP model and <span class="katex-eq" data-katex-display="false"> \eta = 2(1-\ln(\alpha)/\ln(\alpha_c))^{-1}</span> for the UM model, as demonstrated by fitting the function <span class="katex-eq" data-katex-display="false"> \mathcal{D}^{-2/\eta_z}</span> to the observed data. — Analysis of fluctuations in the diagonal matrix elements for both the Random Phase (RP) and Unitary Matrix (UM) models reveals that the scaling of these fluctuations with system size $L$ aligns with theoretical predictions $\eta = 2/(2-\gamma)$ for the RP model and $\eta = 2(1-\ln(\alpha)/\ln(\alpha_c))^{-1}$ for the UM model, as demonstrated by fitting the function $\mathcal{D}^{-2/\eta_z}$ to the observed data.

The investigation into fading ergodicity, as detailed in the article, reveals a nuanced transition where systems move from predictable behavior to a state bordering on complete disorder. This aligns with Bertrand Russell’s observation that “The difficulty lies not so much in developing new ideas as in escaping from old ones.” The research demonstrates that established notions of thermalization, previously assumed to be absolute, break down as systems approach this critical point. Just as Russell suggests the need to challenge pre-conceived notions, this study compels a re-evaluation of how ergodicity dictates the long-term behavior of complex quantum systems, particularly concerning the Thouless energy and spectral statistics.

The Road Ahead

The investigation of fading ergodicity, as presented, exposes a troubling tendency within the pursuit of thermalization. Too often, the focus remains on achieving ergodicity-demonstrating equilibration-rather than rigorously characterizing the conditions under which it fundamentally fails. The present work, by linking structured random matrix theory with many-body systems, highlights the importance of the Thouless energy not merely as a parameter, but as a critical point-a boundary beyond which predictive power evaporates. This necessitates a shift in methodology: a move away from phenomenological descriptions of thermalization toward proofs of ergodicity, or-more importantly-proofs of its absence.

A crucial extension lies in the exploration of dynamical transitions to fading ergodicity. The current understanding largely treats this as a static property, yet realistic quantum quenches are inherently time-dependent. Developing a dynamical theory of ergodicity breaking-one that predicts the timescale for the onset of fading-remains a significant challenge. Furthermore, the assumption of Gaussian disorder, common in random matrix models, warrants careful scrutiny. The impact of more complex, non-Gaussian correlations on the emergent spectral properties-and the attendant ergodicity-is largely unknown.

One must remind that optimization without analysis is self-deception. The relentless pursuit of ever more realistic simulations-however computationally expensive-will yield little genuine insight without a firm theoretical foundation. The true progress lies not in building better approximations of chaos, but in identifying the precise mathematical conditions that guarantee-or preclude-genuine thermalization.

Original article: https://arxiv.org/pdf/2603.23616.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

The Illusion of Thermal Equilibrium

Spectral Fingerprints of Dynamical Breakdown

The Emergence of Fading Ergodicity

The Persistence of Fractal Eigenstates

The Road Ahead

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