Echoes in Time: How Quantum Scars Drive Persistent Oscillations

Author: Denis Avetisyan

New research reveals that specific quantum states can disrupt time-translation symmetry, leading to sustained, periodic behavior in interacting spin systems.

The system’s behavior reveals a sensitivity to initial conditions, where states with smaller α-shifted gaps exhibit period-doubling in magnetization dynamics-a phenomenon absent in states with larger gaps-and correlate with significantly different normalized eigenvalues, suggesting a nuanced interplay between initial state overlap and emergent temporal patterns under parameters <span class="katex-eq" data-katex-display="false">J=1</span>, <span class="katex-eq" data-katex-display="false">h_x</span> and <span class="katex-eq" data-katex-display="false">h_z</span>. — The system’s behavior reveals a sensitivity to initial conditions, where states with smaller α-shifted gaps exhibit period-doubling in magnetization dynamics-a phenomenon absent in states with larger gaps-and correlate with significantly different normalized eigenvalues, suggesting a nuanced interplay between initial state overlap and emergent temporal patterns under parameters $J=1$ , $h_x$ and $h_z$ .

The emergence of period-doubling Floquet states in kicked Ising models with long-range interactions is linked to the presence of quantum many-body scars.

The persistence of non-equilibrium behavior in interacting quantum systems remains a central challenge in many-body physics. This work, ‘Quantum many-body scars leading to time-translation symmetry breaking in kicked interacting spin models’, investigates the emergence of period-doubling oscillations in a periodically driven Ising model with long-range interactions. We demonstrate that these oscillations arise from a weak form of ergodicity breaking, linked to a minority of ‘quantum scar’ Floquet states exhibiting time-translation symmetry breaking, characterized by π-spectral pairing and long-range order. Could these findings illuminate a pathway towards realizing and controlling robust, emergent temporal order in more complex quantum systems?

The Whispers Beyond Equilibrium

For decades, condensed matter physics has primarily concerned itself with understanding systems at equilibrium – those that have settled into a stable, unchanging state. However, this approach inherently neglects the vast and complex behaviors exhibited by systems constantly subjected to external forces or ‘driven’ far from equilibrium. These driven systems, unlike their static counterparts, display dynamic phenomena – oscillations, pattern formation, and emergent behaviors – that are often more prevalent in natural and technological settings. Ignoring these non-equilibrium dynamics represents a significant gap in understanding, as many real-world materials and processes, from biological cells to electronic devices, are fundamentally driven and operate outside the realm of static stability. Investigating these systems requires new theoretical frameworks and computational techniques to capture the time-dependent, often chaotic, nature of their responses to external stimuli.

The one-dimensional Ising model, a foundational concept in statistical mechanics originally developed to describe ferromagnetism, serves as a surprisingly versatile tool for investigating systems far from thermal equilibrium. While traditionally used to predict the behavior of materials at a stable, lowest-energy state, researchers are now leveraging its simplicity to model a broader range of non-equilibrium phenomena. By adapting the model-representing interacting spins on a line-it becomes possible to simulate dynamics influenced by external drives or time-dependent fields, offering insights into systems where energy is constantly being added or removed. This approach allows for the examination of how order emerges from disorder under continuous influence, and how collective behavior deviates from predictions based on equilibrium assumptions, opening avenues for understanding complex systems in fields ranging from biology to materials science.

The conventional Ising model, a foundational tool in condensed matter physics, typically assumes interactions limited to nearest-neighbor spins. This research diverges from that established framework by incorporating long-range interactions, allowing each spin to ‘feel’ the influence of others across a considerable distance. This modification isn’t merely a technical adjustment; it fundamentally alters the system’s behavior, moving it beyond predictable patterns observed in short-range systems. Long-range interactions introduce complex correlations and collective phenomena, potentially leading to new phases of matter and responses to external stimuli. The exploration of these extended interactions unlocks a pathway to understand systems where influences aren’t localized, such as certain magnetic materials or even complex networks, and provides a richer landscape for studying non-equilibrium dynamics.

The number of long-range ordered Floquet eigenstates increases linearly with <span class="katex-eq" data-katex-display="false">NN</span> for <span class="katex-eq" data-katex-display="false"> \alpha < 5</span>, indicating a growing complexity of the system's dynamics as evidenced by the linear least-square fits and confirmed across different parameter settings (J=1, <span class="katex-eq" data-katex-display="false">h_x</span>=0.1-0.25, <span class="katex-eq" data-katex-display="false">h_z</span>=0.0-0.2, ε=0-0.1, N=19), with the total Hilbert subspace dimension shown as a dashed line. — The number of long-range ordered Floquet eigenstates increases linearly with $NN$ for $\alpha < 5$ , indicating a growing complexity of the system’s dynamics as evidenced by the linear least-square fits and confirmed across different parameter settings (J=1, $h_x$ =0.1-0.25, $h_z$ =0.0-0.2, ε=0-0.1, N=19), with the total Hilbert subspace dimension shown as a dashed line.

Orchestrating Chaos: The Floquet Solution

Periodic kicking, a driving mechanism applied to the Ising model, introduces a time-dependent perturbation that prevents the system from reaching thermal equilibrium. This technique involves applying a discrete impulse, or ‘kick’, to the system at regular time intervals $T$ . The resulting dynamics are no longer governed by static energy minimization but instead evolve into a sustained, time-periodic state where the system’s configuration repeats with the same period as the applied kicks. This non-equilibrium state is fundamentally different from the equilibrium behavior observed without external driving, as the system continually exchanges energy with the periodic force rather than relaxing to a minimum energy configuration.

Floquet Theory provides a mathematical framework for analyzing the behavior of systems periodically driven in time. When applied to the periodically kicked Ising model, it reveals that the system’s dynamics are governed by a set of time-periodic solutions known as Floquet States. These states are characterized by their associated Quasienergy levels, which are analogous to the energy levels in time-independent quantum mechanics but defined for periodic systems. Unlike energies, quasienergies are not conserved quantities; however, they remain constant in time and dictate the long-term, stable behaviors of the system. The determination of these quasienergies and their corresponding Floquet States allows for the prediction of the system’s response to the periodic drive and the identification of resonant phenomena and stability boundaries.

Floquet theory provides a means to determine the long-term dynamics of the periodically driven Ising model by transforming the time-dependent Schrödinger equation into an eigenvalue problem defined in the quasienergy domain. This approach allows for the identification of stable and unstable states, and the calculation of transition probabilities between them, effectively mapping the system’s evolution over many periods of the driving force. The resulting quasienergy spectrum reveals key signatures of non-equilibrium order, such as the presence of anomalous heating rates or the emergence of dynamical phases not found in equilibrium systems. Specifically, deviations from a simple energy conservation law, indicated by a non-zero average quasienergy change per driving cycle, signify the absorption of energy and the establishment of a steady-state, time-periodic response characteristic of a non-equilibrium ordered state.

Analysis of Floquet states reveals that varying initial conditions and parameters <span class="katex-eq" data-katex-display="false">J</span>, <span class="katex-eq" data-katex-display="false">h_x</span>, <span class="katex-eq" data-katex-display="false">h_z</span>, α, and ε modulates the system's magnetization and induces period-doubling oscillations, as demonstrated by scatter plots of reduced magnetization eigenvalues and time evolution of the zz-magnetization. — Analysis of Floquet states reveals that varying initial conditions and parameters $J$ , $h_x$ , $h_z$ , α, and ε modulates the system’s magnetization and induces period-doubling oscillations, as demonstrated by scatter plots of reduced magnetization eigenvalues and time evolution of the zz-magnetization.

Spectral Echoes of Broken Symmetry

The spectral properties of Floquet states in periodically driven systems frequently manifest as ‘spectral pairing’, wherein eigenstates organize into doublets. This pairing is characterized by two closely-spaced energy eigenvalues for each momentum value, resulting in a spectral gap separating these paired states. The presence of this gap is not merely a consequence of the periodic drive, but a direct indicator of persistent, or sustained, oscillations within the system. Specifically, the size of this spectral gap is inversely proportional to the lifetime of the oscillations; a larger gap corresponds to a longer-lived oscillatory behavior. Observation of spectral pairing, therefore, provides strong evidence for the emergence of coherent dynamics and distinguishes periodically driven systems exhibiting oscillatory behavior from those that do not.

The emergence of a spectral gap, characterized by a range of energies with no allowed states, is directly correlated with the observation of long-range order within the periodically driven system. Time-translation symmetry, which dictates that the laws of physics remain constant over time, is broken when a system exhibits persistent, coherent oscillations. The presence of both a spectral gap and long-range order confirms this breaking, as the gap arises from the system’s inability to absorb energy at certain frequencies due to the imposed periodic drive, while long-range order establishes the spatial coherence necessary for these oscillations to be sustained. This combination signifies a departure from equilibrium behavior and the establishment of a non-equilibrium state with a well-defined, time-dependent structure.

Our findings indicate a direct relationship between system size (N) and the persistence of period-doubling oscillations. Specifically, the lifetime of these oscillations scales exponentially with N, as evidenced by the linear scaling of -log(average shifted gap) with N. This means that larger systems exhibit significantly longer oscillation lifetimes; the larger the system, the slower the decay of the periodic behavior. The ‘average shifted gap’ refers to the average energy difference between paired Floquet states, and its logarithmic relationship to system size provides a quantitative measure of this exponential scaling. This behavior suggests that the observed time-translation symmetry breaking is not a finite-size effect but rather a robust property of the system in the thermodynamic limit.

Scatter plots reveal a correlation between Floquet doublet half-chain entanglement entropy and normalized magnetization eigenstates for two distinct parameter regimes (<span class="katex-eq" data-katex-display="false">J=1, h_x=0.1, h_z=0.2, \alpha=2.2, \epsilon=0, N=19</span> and <span class="katex-eq" data-katex-display="false">J=1, h_x=0.25, h_z=0.0, \alpha=0.6, \epsilon=0.1, N=19</span>), with the Page value for a fully ergodic quantum state indicated by a vertical line. — Scatter plots reveal a correlation between Floquet doublet half-chain entanglement entropy and normalized magnetization eigenstates for two distinct parameter regimes ( $J=1, h_x=0.1, h_z=0.2, \alpha=2.2, \epsilon=0, N=19$ and $J=1, h_x=0.25, h_z=0.0, \alpha=0.6, \epsilon=0.1, N=19$ ), with the Page value for a fully ergodic quantum state indicated by a vertical line.

Beyond Ergodicity: The Persistence of Scars

The breakdown of time-translation symmetry, observed in driven quantum systems, isn’t merely a fleeting instability, but a fundamental consequence of ‘Quantum Scars’ embedded within the system’s energy spectrum. These scars manifest as highly specific, non-thermal eigenstates – quantum states that stubbornly avoid the typical chaotic spreading expected of closed quantum systems. Unlike most energy levels which rapidly succumb to thermalization and lose any initial structure, scarred states retain their distinct characteristics, effectively resisting the drive towards equilibrium. This resilience stems from their unusual sensitivity to the driving force, allowing them to repeatedly absorb energy and maintain coherent oscillations even as the system evolves. Consequently, the presence of these scars provides a pathway for sustained, non-ergodic dynamics, challenging the conventional expectation that closed quantum systems will inevitably reach a state of thermal equilibrium and lose memory of their initial conditions.

The emergence of quantum scars fundamentally alters the expected behavior of closed quantum systems, leading to persistent dynamics that challenge the principle of ergodicity. Typically, ergodicity dictates that a system, given enough time, will explore all accessible states with equal probability, resulting in a decay of initial conditions and a transition towards thermal equilibrium. However, these scars-specific, non-thermal eigenstates-act as attractors in the system’s Hilbert space, concentrating the dynamics around their corresponding states and enabling sustained oscillations even in the absence of external driving forces. This persistent behavior isn’t a violation of quantum mechanics, but rather a demonstration of how specific spectral properties can dramatically shape the long-term evolution, preventing the system from fully scrambling information and offering a pathway to maintain coherence over extended periods.

Investigations reveal that the quantity of these nonthermal Floquet states doesn’t simply grow with system size, but escalates exponentially. This suggests these states are not merely a negligible quirk of the system, but a fundamental feature influencing its dynamics. While constituting a minority of the total spectral states, their exponential proliferation with increasing system size $N$ indicates a substantial and potentially dominant role in dictating long-term behavior. This challenges conventional expectations of thermalization and ergodicity, highlighting the importance of these quantum scars in sustaining persistent, non-equilibrium dynamics even within complex systems.

The Robustness of Order: Domain Walls and the Kac Factor

The emergence of stable, non-equilibrium order within the driven Ising model isn’t a consequence of uniform alignment, but rather a delicate balance maintained by the system’s inherent defects – domain walls. These boundaries, separating regions of opposing spin orientation, aren’t simply imperfections; they actively mediate the drive, absorbing and dissipating energy while simultaneously preventing the complete collapse of order. The density and arrangement of these domain walls dynamically respond to the external force, forming a network that effectively stabilizes the non-equilibrium state. Crucially, this stabilization isn’t reliant on fine-tuning parameters; the domain walls provide a robust mechanism, allowing the system to maintain order even with significant fluctuations. Their presence fundamentally alters the system’s response to the drive, shifting the focus from global coherence to the localized interactions defining these topological defects and enabling a sustained, ordered phase far from equilibrium.

The stability of any physical system hinges on its energy scaling with size; a system whose energy grows too rapidly becomes unsustainable. In this model, the inclusion of a ‘Kac Factor’ is crucial for maintaining this stability when considering long-range interactions between particles. This factor effectively renormalizes the interaction strength, ensuring that the total energy of the system scales linearly with its volume – a property known as extensivity. Without this renormalization, long-range interactions would lead to runaway energy increases, rendering the observed non-equilibrium order unstable and unrealistic. The Kac Factor, therefore, doesn’t just refine the mathematical model; it anchors it in physical reality, demonstrating a robustness that distinguishes this Floquet phase from more fragile, theoretical constructs.

Investigations are now directed towards a comprehensive charting of this newly discovered Floquet phase, achieved by systematically altering the driving parameters of the system. Researchers intend to explore how variations in the frequency, amplitude, and nature of the drive influence the stability and characteristics of the non-equilibrium order. Furthermore, extending the analysis beyond two dimensions to encompass higher dimensionality is crucial, as it’s anticipated that dimensionality will significantly affect the phase diagram and potentially reveal entirely new emergent behaviors. This detailed mapping will not only solidify the understanding of this specific system but also provide valuable insights into the broader principles governing driven, non-equilibrium phenomena and the formation of topological defects in diverse physical contexts.

The average level spacing ratio <span class="katex-eq" data-katex-display="false">\langle r \rangle</span> exhibits consistent behavior across different parameter regimes (J=1, <span class="katex-eq" data-katex-display="false">h_x</span> = 0.1, <span class="katex-eq" data-katex-display="false">h_z</span> = 0.2 for panel a and <span class="katex-eq" data-katex-display="false">h_x</span> = 0.25, <span class="katex-eq" data-katex-display="false">h_z</span> = 0.0, ε = 0.1 for panel b), closely matching theoretical predictions from the COE and Poisson distributions (red and green lines respectively), with panel (b) demonstrating consistent behavior due to Hamiltonian parity symmetry. — The average level spacing ratio $\langle r \rangle$ exhibits consistent behavior across different parameter regimes (J=1, $h_x$ = 0.1, $h_z$ = 0.2 for panel a and $h_x$ = 0.25, $h_z$ = 0.0, ε = 0.1 for panel b), closely matching theoretical predictions from the COE and Poisson distributions (red and green lines respectively), with panel (b) demonstrating consistent behavior due to Hamiltonian parity symmetry.

The pursuit of order within the kicked Ising model, as detailed in this work, feels less like discovery and more like a temporary stay of execution against chaos. The emergence of period-doubling oscillations, linked to these ‘quantum scar’ Floquet states, isn’t a revelation of underlying structure, but a localized defiance of entropy. As Michel Foucault observed, “There is no power without resistance.” These scars, these persistent oscillations, aren’t evidence of a robust, predictable system; rather, they are fleeting pockets of coherence, destined to succumb to the inevitable advance of disorder. The model whispers a promise of control, but any semblance of predictability is merely a spell cast against the darkness, functioning only until confronted by the unforgiving reality of production-or, in this case, the next kick.

The Echo of Imperfection

The persistence of these period-doubling oscillations isn’t a triumph of order, but a stubborn refusal of the system to fully succumb to chaos. The ‘quantum scars’ identified here aren’t blueprints for control, but rather privileged loci where the expected decay into thermal equilibrium is… delayed. One suspects the long-range interactions aren’t causing the symmetry breaking, but are merely widening the cracks through which it escapes. The true question isn’t how to create these scars, but to understand why most states don’t bear them.

Future investigations will likely find increasingly elaborate models attempting to map the topology of these scar states. But perhaps the more fruitful path lies in accepting the inherent limitations of such maps. Each model is, after all, a reduction – a curated lie that momentarily aligns with observation. The noise, the imperfections, are not errors to be minimized, but the very language in which the system communicates its refusal to be fully known.

The next generation of Floquet crystals won’t be about building perfect timekeepers. They will be about embracing the subtle dance between order and disorder, acknowledging that truth doesn’t reside in the frequencies themselves, but in the whispers of their decay. The system doesn’t answer questions; it holds up a mirror, reflecting back the limits of the questions asked.

Original article: https://arxiv.org/pdf/2604.20419.pdf

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