Persistent Quantum Oscillations in a Flat-Band Chain

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Author: Denis Avetisyan

New research reveals that quantum quenches in a specifically tuned XXZ chain don’t lead to thermalization, but instead exhibit sustained oscillations in entanglement and other key observables.

The study demonstrates that RĂ©nyi entanglement entropy, calculated via random Pauli measurements and analyzed with Hamming distance and classical shadow techniques-and corroborated by exact solutions-accurately characterizes the dynamical half-chain entanglement of the XXZ chain, even with varying system sizes ($N=4, 8, 12$) and focusing on subsystems of adjacent qubits with periodic boundary conditions or bulk qubits in an open chain.
The study demonstrates that RĂ©nyi entanglement entropy, calculated via random Pauli measurements and analyzed with Hamming distance and classical shadow techniques-and corroborated by exact solutions-accurately characterizes the dynamical half-chain entanglement of the XXZ chain, even with varying system sizes ($N=4, 8, 12$) and focusing on subsystems of adjacent qubits with periodic boundary conditions or bulk qubits in an open chain.

The study combines analytical solutions with digital quantum simulations to characterize the non-relaxing dynamics of the staggered XXZ chain under quantum quenches.

Understanding non-equilibrium dynamics in interacting quantum systems remains a central challenge in condensed matter physics. This is addressed in ‘Quench dynamics of the quantum XXZ chain with staggered interactions: Exact results and simulations on digital quantum computers’, where we investigate the time evolution following a sudden change in the interactions of a flat-band spin chain. Our analysis reveals persistent oscillations in entanglement and the Loschmidt echo, indicative of a lack of thermalization, and provides both analytical and numerical validations of these findings using digital quantum computers. Can these insights into quench dynamics in flat-band systems inform our understanding of more complex, disordered quantum materials?

Challenging Equilibrium: The Emergence of Persistent Quantum States

For decades, the prevailing understanding of isolated quantum systems hinged on the principle of thermal equilibrium – the idea that, given enough time, these systems would inevitably settle into a state of maximum entropy and predictable behavior. However, this expectation doesn’t universally hold true. Recent theoretical and experimental investigations reveal scenarios where quantum systems actively resist reaching equilibrium, defying the predictions of traditional quantum statistical mechanics. This breakdown occurs due to factors like strong disorder or specific interactions that hinder the system’s ability to distribute energy efficiently. Consequently, these non-equilibrium states exhibit unique properties, retaining “memory” of their initial conditions and potentially offering pathways to novel functionalities unattainable in systems governed by standard thermalization. The implications extend beyond fundamental physics, suggesting possibilities for designing quantum materials and devices with tailored and long-lived quantum properties.

Contrary to expectations rooted in traditional statistical mechanics, certain complex quantum systems demonstrably resist the pull towards thermal equilibrium. Phenomena such as many-body localization (MBL) illustrate how strong interactions between particles can effectively ‘freeze’ a system, preventing it from distributing energy and reaching a uniform temperature. Instead of thermalizing, these systems retain memory of their initial conditions and exhibit strikingly novel behaviors – from the absence of diffusion to the emergence of long-lived quantum coherence. This resistance to thermalization isn’t limited to MBL; similar effects arise in disordered systems and those driven far from equilibrium, opening pathways to explore and potentially exploit states of matter with properties fundamentally different from anything predicted by conventional thermal descriptions. The observation of these non-thermal states is not merely an academic curiosity, but a crucial step towards harnessing the full potential of quantum materials for advanced technologies.

The ability to control and harness quantum systems hinges on a deep understanding of states that deviate from traditional thermal equilibrium. Unlike systems at equilibrium, where energy is evenly distributed, these non-equilibrium states exhibit unique properties that can be exploited for advanced technologies. Specifically, maintaining quantum coherence – the delicate superposition of states crucial for quantum computation – is significantly enhanced when a system avoids thermalization. This allows for more complex and prolonged quantum operations, paving the way for more powerful quantum computers. Similarly, in quantum sensing, these non-equilibrium states can amplify subtle signals, leading to sensors with unprecedented sensitivity and precision. Research focuses on manipulating interactions and external fields to create and sustain these states, ultimately unlocking the full potential of quantum mechanics for practical applications, from materials science to medical diagnostics.

Quantum circuits were designed and implemented to generate the four Bell states: |ψ0⟩, |ψ1⟩, |ψ2⟩, and |ψ3⟩, each representing a maximally entangled state of two qubits.
Quantum circuits were designed and implemented to generate the four Bell states: |ψ0⟩, |ψ1⟩, |ψ2⟩, and |ψ3⟩, each representing a maximally entangled state of two qubits.

Probing Dynamics with Abrupt Change: The Quantum Quench

A quantum quench involves an instantaneous change to the Hamiltonian, $H$, governing a quantum system. This abrupt modification forces the system into a new ground state and, crucially, generates excitations that drive it away from its initial equilibrium. Unlike gradual changes, a quench creates a superposition of states with varying energies, resulting in time-dependent behavior that reveals information about the system’s dynamics. This technique is valuable because it allows researchers to study non-equilibrium phenomena and observe how quantum systems respond to sudden perturbations, providing insights beyond those accessible through static or slowly varying fields.

Analysis of the time evolution following a quantum quench provides detailed information regarding a system’s intrinsic dynamical properties and its reaction to external disturbances. Specifically, observing how a system relaxes or evolves after a sudden Hamiltonian change reveals characteristics such as the speed of thermalization, the presence of quasi-particle excitations, and the nature of correlations between system constituents. Measurements of quantities like correlation functions and energy transport can be directly related to the system’s response function, allowing for characterization of its fundamental properties and providing insights into its underlying physics, including the identification of relevant energy scales and conserved quantities. This approach allows for probing dynamics inaccessible through traditional equilibrium methods.

The XXZ spin chain has proven to be a valuable model system for the study of quantum quenches due to the possibility of transitioning between integrable and non-integrable regimes by varying parameters within the Hamiltonian, specifically the anisotropy parameter. Analytical solutions for the post-quench dynamics have been successfully derived for certain parameter choices, offering a benchmark for understanding the behavior in these systems. These analytical results have been computationally verified through numerical simulations, demonstrating agreement up to system sizes of $N=12$ sites, providing confidence in the methodology and enabling the exploration of finite-size effects and more complex quench protocols.

Quantum circuits were designed to implement time-evolution operators for Pauli Z, X, and Y matrices, as shown for each respective operator.
Quantum circuits were designed to implement time-evolution operators for Pauli Z, X, and Y matrices, as shown for each respective operator.

Revealing System Sensitivity: Entanglement as a Diagnostic Tool

The Loschmidt Echo (LE) quantifies a system’s sensitivity to infinitesimal perturbations of its Hamiltonian, serving as a key diagnostic for dynamical phase transitions. Mathematically, the LE is defined as $L(t) = |\langle \psi(0) | e^{-i \hat{H} t} | \psi(0) \rangle|$, where $|\psi(0)\rangle$ is the initial state and $\hat{H}$ is the Hamiltonian. A non-zero LE indicates stability against perturbations, while a vanishing LE signals a transition to a qualitatively different dynamical behavior. In this work, an analytical expression for the LE was derived based on the specific system parameters, and its validity was confirmed through comparison with numerical simulations of the time-evolved state. The rate of decay of the LE directly correlates with the system’s susceptibility to external influences and the potential for instability.

Quantification of entanglement generated during a quantum quench is achieved through calculation of the Von Neumann entropy, $S_V$, and RĂ©nyi entropies, $S_\alpha$. These metrics provide insight into the correlations between the system’s constituents following the sudden parameter change. Analysis reveals persistent, undamped oscillations in the time evolution of these entropies; unlike systems undergoing thermalization which exhibit a decay towards a stable, maximized entropy state, these sustained oscillations indicate a lack of approach to thermal equilibrium. The observed behavior suggests the system retains memory of its initial state and does not fully explore its Hilbert space following the quench.

Reconstruction of the system’s wavefunction is achieved by leveraging entanglement measures – specifically Von Neumann and RĂ©nyi Entropies – in conjunction with computational techniques such as the Hadamard Test and Classical Shadows. The Hadamard Test provides information about the overlap between the current system state and a known basis state, while Classical Shadows employs randomized measurements to efficiently estimate the wavefunction’s probability distribution. By combining data from these methods with the quantified entanglement, researchers can map the system’s state at various points in time, enabling detailed tracking of its dynamical evolution following a quantum quench. This process allows for verification of theoretical predictions regarding non-thermalization and provides insights into the system’s response to perturbations.

Figure 13:Simulation results (solid lines) for the half-chain RĂ©nyi entanglement entropy and the Loschmidt echo for a chain ofN=4N=4withΔ=2/(5+1)\Delta=2/(\sqrt{5}+1). The simulation data are based on the normalized coefficientsaka\_{k}, estimated using the the Hadamard test onibm\_fez.
Figure 13:Simulation results (solid lines) for the half-chain RĂ©nyi entanglement entropy and the Loschmidt echo for a chain ofN=4N=4withΔ=2/(5+1)\Delta=2/(\sqrt{5}+1). The simulation data are based on the normalized coefficientsaka\_{k}, estimated using the the Hadamard test onibm\_fez.

Towards Resilient Quantum Memories: Beyond Conventional Thermalization

Recent investigations utilizing the quantum quench – a sudden change in a system’s governing parameters – on a DimerizedChain have revealed a surprising phenomenon: the potential for AbsenceOfRelaxation. Typically, a disturbed quantum system rapidly degrades to thermal equilibrium. However, this specifically engineered chain, characterized by alternating strong and weak bonds, defies this expectation. Simulations demonstrate that, under certain conditions, the system doesn’t settle into a state of maximal entropy, but instead maintains coherent oscillations for extended periods. This resistance to thermalization isn’t a matter of simply slowing down the decay; it represents a fundamental departure from conventional expectations, suggesting the existence of intrinsically protected states within the chain’s unique structure. The observation highlights a promising route toward creating stable quantum memories, where information isn’t lost to environmental noise, and opens possibilities for novel quantum information processing paradigms.

The surprising resilience to thermalization observed in the dimerized chain stems from a carefully orchestrated interplay of interactions, giving rise to what are effectively protected quantum states. This protection isn’t absolute, but manifests as persistent oscillations under specific conditions-a phenomenon linked to the ratio between the interaction strength, $J$, and the energy difference, $\Delta$, between neighboring dimers. Researchers discovered that stable, non-thermalizing behavior emerges when this ratio conforms to a precise mathematical relationship: $J\Delta = p/q$, where $p$ and $q$ are integers forming an irreducible fraction. This periodicity condition effectively ‘locks’ the system into a coherent state, hindering the usual dissipation of energy and preventing the chain from reaching thermal equilibrium, thereby opening exciting possibilities for sustained quantum information storage.

The demonstrated resilience against thermalization in dimerized chains offers a promising pathway toward realizing stable quantum memories – a crucial component for scalable quantum computation. By carefully engineering the interactions within a quantum system to avoid energy relaxation, information can be stored for extended periods, overcoming a major hurdle in quantum information processing. This approach, rooted in the specific periodicity condition of $JΔ = p/q$, suggests a design principle where protected states preserve quantum information, effectively shielding it from environmental noise. Further exploration of these non-thermalizing systems holds the potential to unlock novel architectures for quantum bits and enable more complex quantum algorithms, ultimately advancing the field beyond current limitations.

The symmetric bipartition of a chain with periodic boundary conditions cuts either two post-quench couplings for even chain lengths or one coupling and one initial dimer for odd chain lengths.
The symmetric bipartition of a chain with periodic boundary conditions cuts either two post-quench couplings for even chain lengths or one coupling and one initial dimer for odd chain lengths.

The exploration of quantum dynamics, as detailed in the study of the XXZ chain, reveals a universe governed by inherent uncertainty. This aligns with the observation that, as Werner Heisenberg stated, “The more precisely the position is determined, the less precisely the momentum is known.” The persistent oscillations in entanglement entropy and the Loschmidt echo demonstrate a system resisting thermalization, a state where predictability diminishes. It underscores how the act of observation – or, in this case, measurement of quantum states – fundamentally alters the system, reinforcing the notion that algorithms, like physical laws, are not neutral descriptions but rather active constructions of reality. Transparency regarding these inherent limitations is, therefore, minimal morality, not an optional addendum.

Beyond Equilibrium: Charting a Course Forward

The persistence of oscillatory behavior in this work, particularly within a flat-band system, invites careful consideration. It is tempting to speak of ‘robustness’ or ‘protection’ against thermalization, but such language risks obscuring the fact that these systems are not simply stable; they are dynamically distinct. The absence of relaxation, while mathematically elegant, demands deeper investigation into the conditions under which such non-equilibrium dynamics become the norm, rather than the exception. This is especially pertinent as the field increasingly turns towards engineering quantum materials – a system that doesn’t ‘settle’ may not be a failure, but a feature.

Future research should address the limits of this behavior. How sensitive are these oscillations to imperfections in the model, or to the introduction of even weak interactions not captured here? Extending these investigations to genuinely many-body localized systems, where the absence of thermalization stems from a fundamentally different mechanism, could reveal critical distinctions. It is crucial to remember that technology without care for people is techno-centrism; ensuring fairness is part of the engineering discipline.

Ultimately, a nuanced understanding of these non-equilibrium dynamics is not merely an academic exercise. It represents a shift in perspective, from seeking states of minimal energy to understanding the complex, evolving landscapes that define genuine quantum matter. The study of flat bands, and systems like them, presents a chance to redefine what it means for a system to be ‘stable’-a question with implications far beyond the confines of condensed matter physics.

Original article: https://arxiv.org/pdf/2512.03341.pdf

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

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2025-12-04 23:24