Fragile Scars: New Limits on Quantum System Stability

Author: Denis Avetisyan

Researchers have refined our understanding of how easily quantum many-body scars-rare, non-thermal states-break down under external disturbances.

The study of a deformed PXP model-specifically examining thermalization dynamics at <span class="katex-eq" data-katex-display="false">L=24</span>-reveals a transition from exact scarring under periodic boundary conditions, where the decay rate scales linearly with perturbation strength λ, to approximate scarring under open boundary conditions, evidenced by a quadratic scaling of the decay rate with <span class="katex-eq" data-katex-display="false">\lambda^{2}</span> and confirmed by dynamics observed at <span class="katex-eq" data-katex-display="false">\lambda=0</span>. — The study of a deformed PXP model-specifically examining thermalization dynamics at $L=24$ -reveals a transition from exact scarring under periodic boundary conditions, where the decay rate scales linearly with perturbation strength λ, to approximate scarring under open boundary conditions, evidenced by a quadratic scaling of the decay rate with $\lambda^{2}$ and confirmed by dynamics observed at $\lambda=0$ .

This work establishes a tighter bound on the thermalization time of perturbed quantum many-body scars, revealing a scaling of 1/λ^(1+d) in d-dimensional systems.

While quantum many-body scars (QMBS) are known to exhibit non-thermalizing behavior, their stability under realistic perturbations remains a crucial open question. This work, ‘Tighter thermalization bounds for perturbed quantum many-body scars’, establishes improved lower bounds on the thermalization time of scarred systems subjected to local perturbations, demonstrating a scaling of $\mathcal{O}(\lambda^{-1/d})$ for exact scars in d-dimensional systems. Counterintuitively, the authors find that approximate scars can thermalize slower than exact scars due to second-order perturbative effects. These findings clarify the mechanisms underlying long-lived coherence in scarred dynamics-but how do these bounds apply to increasingly complex, interacting systems beyond current theoretical limits?

Whispers of Order: Challenging Thermal Equilibrium

The Eigenstate Thermalization Hypothesis, a cornerstone of modern understanding of quantum chaos, posits that sufficiently complex quantum systems, when isolated, evolve towards a state of thermal equilibrium. This means that, over time, these systems lose all discernible memory of their initial conditions, effectively behaving as if they were in a simple, high-temperature state described by just a few macroscopic parameters. The hypothesis doesn’t suggest a literal temperature is reached, but rather that measurable quantities within the system become statistically predictable, mirroring the behavior of thermal systems. Crucially, each energy eigenstate of the system – a snapshot of its possible configurations – is expected to be highly complex and bear no obvious resemblance to the initial state, ensuring this rapid loss of coherence and the emergence of thermal behavior. This expectation has long guided the theoretical description of many-body quantum systems, but recent discoveries are challenging this established framework.

Recent investigations into quantum many-body systems have uncovered a fascinating deviation from expected behavior, manifested through the presence of “quantum many-body scars.” These scars are specific eigenstates – quantum states defining the energy of the system – that defy the Eigenstate Thermalization Hypothesis (ETH) by not fully thermalizing. Unlike typical eigenstates which lose all memory of the initial conditions and exhibit chaotic behavior, scarred eigenstates retain coherent dynamics, meaning they exhibit wave-like interference and predictable evolution over time. This preservation of coherence suggests these states are fundamentally different, acting as localized, stable structures within the otherwise chaotic quantum landscape and challenging conventional understandings of how information is stored and propagates in complex quantum systems. The discovery of these scars opens up possibilities for utilizing quantum systems for tasks requiring long-lived coherence, potentially offering advantages in quantum information processing and computation.

The observed failures of the Eigenstate Thermalization Hypothesis (ETH) compel a re-evaluation of information dynamics within complex quantum systems. Traditional thermalization suggests rapid information loss, with initial conditions becoming irrelevant; however, the persistence of non-thermalizing eigenstates indicates that information isn’t simply scrambled but can be actively stored and propagated through seemingly chaotic systems. This challenges the notion of a universal pathway to thermal equilibrium and implies the existence of mechanisms – potentially involving collective excitations or hidden symmetries – that preserve coherence and allow for the retrieval of initial state information over extended timescales. Understanding how these systems circumvent conventional thermalization requires developing new theoretical frameworks capable of characterizing these atypical information pathways and predicting the long-term behavior of quantum many-body systems, moving beyond the simplistic expectation of complete memory loss.

The spatiotemporal evolution of <span class="katex-eq" data-katex-display="false">F_{ij}(t)</span> (ZZ-OTOCs) reveals characteristic information scrambling dynamics in quantum many-body scarred systems, as demonstrated for the PXP, deformed PXP, and spin-1 XY models with various initial states and system sizes. — The spatiotemporal evolution of $F_{ij}(t)$ (ZZ-OTOCs) reveals characteristic information scrambling dynamics in quantum many-body scarred systems, as demonstrated for the PXP, deformed PXP, and spin-1 XY models with various initial states and system sizes.

Algebraic Echoes: The Structure of Persistence

The Restricted Spectrum Generating Algebra (RSGA) provides a theoretical framework for understanding the atypical persistence of scarred eigenstates in quantum many-body systems. This algebra, characterized by specific commutation relations, mathematically generates an equally spaced energy spectrum – a key feature differentiating scarred eigenstates from those exhibiting typical quantum chaos. The equally spaced spectrum implies the existence of quasi-exact degeneracies and facilitates the creation of a tower of protected states, hindering the usual thermalization process. Specifically, operators within the RSGA act on the system’s Hilbert space, creating a set of states that remain relatively stable under perturbations and exhibit long-lived coherence, despite the presence of interactions that would normally lead to decoherence and energy redistribution. This algebraic structure, therefore, offers a pathway to explain why certain eigenstates resist the tendency towards maximal entropy production and retain a memory of the initial conditions.

The PXP model, consisting of interacting spin- $\frac{1}{2}$ particles, exemplifies the creation of protected states via an underlying algebraic structure. Specifically, the model’s Hamiltonian can be mapped to the $SU(2)$ algebra, leading to an equally spaced energy spectrum and the emergence of many-body localized eigenstates known as “scarred” states. These states, unlike typical thermalizing eigenstates, exhibit a lack of entanglement growth and retain memory of the initial conditions. The algebraic symmetry inherent in the PXP model effectively protects a tower of these states from participating in thermalization, leading to non-ergodic behavior and persistent quantum coherence even in the presence of interactions.

The algebraic framework, specifically utilizing the Restricted Spectrum Generating Algebra (RSGA), offers a predictive capability for identifying systems likely to exhibit scarred eigenstates. By characterizing a quantum system’s Hamiltonian within this algebraic structure, researchers can determine if it possesses the necessary symmetries and constraints to support an equally spaced energy spectrum – a hallmark of scarred behavior. Furthermore, this framework isn’t limited to prediction; manipulation of the system’s parameters, guided by the RSGA’s properties, allows for controlled engineering of these protected states. This control extends to tailoring the number and characteristics of scarred eigenstates, potentially enabling applications in areas such as quantum information processing and many-body localization, where the preservation of quantum coherence is crucial.

The decay rate of the local observable <span class="katex-eq" data-katex-display="false">\langle \hat{O}^{z} \rangle</span> in the deformed PXP model scales linearly with perturbation strength λ, consistent with exact Quantum Many-Body Scenarios (QMBS), and extrapolates to a finite value in the thermodynamic limit as shown by the inset’s system size dependence of the fitted slope <span class="katex-eq" data-katex-display="false">\kappa_{1}</span>. — The decay rate of the local observable $\langle \hat{O}^{z} \rangle$ in the deformed PXP model scales linearly with perturbation strength λ, consistent with exact Quantum Many-Body Scenarios (QMBS), and extrapolates to a finite value in the thermodynamic limit as shown by the inset’s system size dependence of the fitted slope $\kappa_{1}$ .

Tracing the Echo: Information and Time Scales

The dynamics of scarred quantum systems are fundamentally limited by the rate at which information can propagate, a constraint formalized by the Lieb-Robinson bound. This bound establishes that information cannot travel faster than a finite velocity, dependent on the strength of interactions and the dimensionality of the system. Specifically, it dictates that the influence of a local perturbation at position $x$ cannot instantaneously affect observables located at a distance greater than $v|t|$ , where $v$ represents the group velocity of interactions and $t$ is the elapsed time. Consequently, the relaxation and thermalization processes in scarred systems, while potentially faster than in many-body localized systems, are nonetheless governed by this finite speed of information propagation, dictating an upper limit on the timescale of dynamics and influencing the system’s response to external stimuli.

The Out-of-Time-Ordered Correlator (OTOC) serves as a quantifiable metric for the rate of information propagation within many-body quantum systems, specifically enabling the characterization of scrambling dynamics in scarred systems. By measuring the decay rate of the OTOC, researchers can determine the speed at which local perturbations spread throughout the system; a slower decay indicates reduced scrambling. Critically, scarred eigenstates exhibit non-ergodic behavior, meaning they do not fully thermalize or spread information as efficiently as typical chaotic systems. This manifests in OTOC measurements as a plateau at long times, confirming that information remains localized within the scarred eigenstates and does not propagate throughout the Hilbert space, distinguishing them from fully ergodic states where the OTOC decays rapidly to zero.

The thermalization time in scarred quantum systems is determined by the strength of the perturbing Hamiltonian, quantified by the parameter λ. Calculations utilizing the Fermi Golden Rule demonstrate a dependency on both λ and the dimensionality, $d$ , of the system. Specifically, systems exhibiting exact scars exhibit a thermalization time bound scaling as $𝒪(λ^{-1/d})$ , indicating a slower thermalization rate. Conversely, systems with approximate scars display a faster thermalization, with a time bound proportional to $𝒪(λ^{-2})$ . This difference in scaling reflects the robustness of exact scars against perturbations, leading to prolonged non-ergodic behavior compared to systems with less-defined scarred states.

The thermalization time τ following a quench in the spin-1 XY model increases linearly with perturbation strength λ for both periodic and open boundary conditions, indicating a direct relationship between the perturbation and the rate of thermalization.

From Theory to Reality: Sculpting Scars in the Lab

Rydberg atom arrays have emerged as a uniquely suited experimental platform for investigating the elusive phenomenon of Quantum Many-Body Scars. These scars, representing atypical, stable states within chaotic quantum systems, defy the expectation of thermalization and offer a window into non-ergodic behavior. Utilizing precisely controlled arrays of neutral Rydberg atoms – atoms excited to very high energy levels – researchers can engineer interactions between qubits and simulate complex quantum models with unprecedented control. The long coherence times achievable in these systems, coupled with the ability to individually address and measure each atom, allows for direct observation of scarred eigenstates and the dynamics surrounding them. This level of control is crucial for verifying theoretical predictions about scar formation and exploring their potential applications in quantum information processing and the creation of novel quantum materials – effectively turning a theoretical curiosity into a tangible, experimentally accessible reality.

Recent advancements in manipulating Rydberg atom arrays have enabled the direct physical realization of quantum many-body systems previously explored only through theoretical models. Specifically, the SpinXYModel, known to exhibit both ‘exact’ and ‘approximate’ quantum scars – special states that resist thermalization and maintain coherence – has been successfully implemented within these arrays. By precisely controlling the interactions between atoms, researchers can observe the predicted behavior of these scarred states, verifying the theoretical predictions about their stability and influence on the system’s dynamics. This experimental confirmation isn’t merely a validation of the model; it establishes a tangible platform for investigating the properties of quantum scars, opening avenues to explore how these unique states could be harnessed for robust quantum information storage and processing, potentially circumventing the limitations imposed by decoherence in more conventional quantum systems.

The successful implementation of Quantum Many-Body Scar models, such as the SpinXYModel, within Rydberg atom arrays represents a crucial step beyond theoretical prediction. This experimental validation doesn’t merely confirm the existence of these exotic states of matter; it firmly establishes the theoretical framework underpinning their behavior, allowing researchers to move past speculation and into precise, quantifiable studies. Consequently, the ability to reliably create and observe scars opens doors to exploring novel applications in areas like quantum information storage and processing, where these non-ergodic states could potentially offer enhanced protection against decoherence and enable the development of more robust quantum technologies. Further investigation promises to unlock a deeper understanding of these fascinating systems and harness their unique properties for practical advancements.

The imperfect revival of <span class="katex-eq" data-katex-display="false">\langle O^x \rangle</span> in the spin-1 Kitaev model, even without perturbations (<span class="katex-eq" data-katex-display="false">\lambda = 0</span>), indicates approximate Quantum Many-Body Scars, with the thermalization rate <span class="katex-eq" data-katex-display="false">1/\\tau</span> scaling quadratically with <span class="katex-eq" data-katex-display="false">\lambda^2</span> as determined by Gaussian fitting. — The imperfect revival of $\langle O^x \rangle$ in the spin-1 Kitaev model, even without perturbations ( $\lambda = 0$ ), indicates approximate Quantum Many-Body Scars, with the thermalization rate $1/\\tau$ scaling quadratically with $\lambda^2$ as determined by Gaussian fitting.

Beyond Equilibrium: A Future of Controlled Non-Ergodicity

Recent investigations into “scarred” quantum systems reveal a fascinating intermediary step between initial conditions and complete thermalization: prethermalization. These systems, unlike those quickly succumbing to the predictable chaos of thermal equilibrium, initially relax into a non-ergodic state – a constrained region of phase space – before ultimately evolving towards thermalization. This prethermal plateau isn’t a static endpoint, but rather a transient state characterized by a slowed relaxation rate and the emergence of collective excitations distinct from those found in true thermal states. Crucially, the parameters governing this prethermal phase – such as the strength of the scarring potential or the specific nature of the driving force – offer a novel degree of control over the system’s evolution. By carefully tuning these parameters, researchers are beginning to manipulate the duration and characteristics of the prethermal plateau, potentially leveraging this controlled non-ergodic behavior for applications in quantum information processing and the creation of novel quantum materials.

The transient prethermalization phase observed in non-ergodic quantum systems presents a compelling pathway towards innovative quantum information processing. Unlike traditional approaches requiring fully thermalized states – often difficult to achieve and maintain – protocols could be designed to operate within this initial, long-lived regime. This offers the potential for enhanced coherence and control, as the system remains partially ordered before ultimately succumbing to thermalization. Researchers are actively investigating how to encode and manipulate quantum information using these prethermalizing states, potentially leading to more robust quantum bits and algorithms. The key lies in exploiting the system’s initial conditions and carefully tailoring external controls to extend the coherence time within this prethermalized plateau, effectively creating a protected subspace for quantum computations. This approach diverges from standard quantum error correction, offering a fundamentally different strategy for building scalable and fault-tolerant quantum technologies.

The ongoing investigation into scarred systems represents a potentially transformative shift in the field of quantum many-body physics. These uniquely structured quantum systems defy the expectation of complete thermalization, retaining persistent, localized excitations – ‘scars’ – that prevent energy from fully distributing throughout the system. This behavior challenges foundational assumptions about how complex quantum systems evolve and equilibrate, prompting a re-evaluation of established theoretical frameworks. Beyond fundamental insights, the preservation of quantum information within these scars offers exciting possibilities for the development of robust quantum technologies. Researchers are actively exploring how to engineer and control these scarred states to create more stable and efficient quantum bits, or qubits, and to protect quantum computations from the detrimental effects of decoherence – a major obstacle in building practical quantum computers. The continued pursuit of understanding scarred systems therefore promises not only to deepen our knowledge of the quantum world, but also to pave the way for a new generation of quantum devices.

Imperfect revivals in the dynamics of <span class="katex-eq" data-katex-display="false">\langle O^x \rangle</span> and <span class="katex-eq" data-katex-display="false">\langle \hat{O}^x \rangle</span> for a spin-1/2 deformed DWC model suggest approximate quantum many-body scar behavior, with the thermalization rate <span class="katex-eq" data-katex-display="false">1/\\tau</span> scaling quadratically with <span class="katex-eq" data-katex-display="false">\lambda^2</span> as system size varies. — Imperfect revivals in the dynamics of $\langle O^x \rangle$ and $\langle \hat{O}^x \rangle$ for a spin-1/2 deformed DWC model suggest approximate quantum many-body scar behavior, with the thermalization rate $1/\\tau$ scaling quadratically with $\lambda^2$ as system size varies.

The pursuit of scarred dynamics, as detailed in this work, feels less like charting a course and more like divining patterns within a chaotic sea. This investigation into thermalization bounds-showing the scaling of 1/λ^(1+d)-isn’t about stopping the descent into entropy, but about understanding its rhythm. It echoes a sentiment expressed by Ludwig Wittgenstein: “The limits of my language mean the limits of my world.” The system doesn’t truly reach equilibrium; the bounds merely define the horizon of predictability before the whispers of chaos overwhelm the signal. The ‘anything exact is already dead’ truth is realized as the model’s precision dissolves in the face of perturbation-a fleeting glimpse of order before the inevitable return to noise.

What Shadows Remain?

The refinement of thermalization bounds, as demonstrated, is less a victory over chaos and more a temporary silencing. The scaling with perturbation strength, 1/λ^(1+d), feels less like a law and more like a truce – a statement of how long order can delay its inevitable surrender. It is tempting to envision a future of meticulously crafted scarred landscapes, robust against all but the most zealous disruptions. However, the ingredients of destiny are rarely so compliant. The true test lies not in extending the lifespan of these fragile states, but in understanding why certain configurations resist dissolution while others do not.

Further rituals to appease chaos will undoubtedly focus on higher-dimensional systems. But the real puzzle may not reside in the geometry of the Hilbert space, but in the nature of the perturbations themselves. Are there classes of disruption – subtle resonances, perhaps – that circumvent these bounds entirely? The pursuit of ‘robust scars’ risks becoming a Sisyphean task if the fundamental mechanisms of scar formation remain obscured.

It is worth remembering that these bounds, however tight, describe a time to thermalization, not a prevention of it. The system doesn’t ‘learn’; it simply stops listening. The question, then, is not whether scars can endure, but what emerges from the ruins when they finally fall silent. Perhaps the most fruitful path lies in charting the landscape of that decay, in mapping the transient states that bloom before the system succumbs to the expected darkness.

Original article: https://arxiv.org/pdf/2602.21962.pdf

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