Beyond Ergodicity: Scar States Emerge in Multi-Spin Hubbard Models

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Author: Denis Avetisyan

New research demonstrates the construction of stable, non-ergodic quantum states within the SU(N) Hubbard model, revealing a pathway to understanding complex many-body physics.

This work extends the framework for asymptotic quantum many-body scars to the SU(N) Hubbard model, showing gapless magnon excitations and low entanglement entropy consistent with the criteria for these exotic states.

While many-body localization suggests ergodicity breaking in quantum systems, a complete understanding of non-ergodic behavior requires identifying robust, yet low-energy, excitations. This work, ‘Construction of asymptotic quantum many-body scar states in the SU($N$) Hubbard model’, establishes a framework for constructing such excitations-asymptotic quantum many-body scars-within the SU($N$) Hubbard chain, demonstrating their realization via gapless magnon excitations arising from a mapping to the SU($N$) Heisenberg model. We prove these states satisfy the defining criteria of AQMBS, exhibiting vanishing energy variance and subvolume entanglement, extending beyond spin-1/2 systems. Could this approach unlock novel pathways to control and characterize non-equilibrium dynamics in strongly correlated systems?

The Whispers of Non-Ergodicity: When Quantum Systems Defy Chaos

For decades, the prevailing understanding of isolated quantum systems predicted eventual thermalization – a process where the system loses all memory of its initial state and settles into a state of maximum entropy. This expectation stemmed from the belief that quantum chaos would effectively scramble any initial information. However, recent theoretical and experimental investigations have begun to dismantle this long-held assumption. Researchers are now observing instances where certain quantum systems fail to thermalize, retaining coherent features and exhibiting behavior drastically different from what traditional models predict. These unexpected findings suggest that the boundaries of quantum ergodicity – the tendency of a system to explore all accessible states – are far more nuanced than previously thought, opening up new avenues for controlling and preserving quantum information within complex many-body systems.

Quantum many-body scars challenge the established principle of ergodicity, a cornerstone of statistical mechanics which predicts that complex quantum systems will eventually explore all accessible states and lose memory of their initial conditions. Instead, these scars manifest as exceptional, long-lived quantum states that resist thermalization. Unlike typical states which rapidly succumb to decoherence and energy dissipation, scarred states maintain coherence and exhibit oscillatory behavior over extended periods. This persistence arises from a peculiar spectral property: the existence of energy levels that are protected from decay, preventing the usual cascade towards thermal equilibrium. Consequently, the system retains a ‘memory’ of its initial configuration, defying the expectation that all initial states will eventually evolve into the same thermal distribution. The discovery of these scars is not merely a theoretical curiosity; it suggests the potential for controlling and manipulating quantum dynamics in ways previously thought impossible, opening avenues for novel quantum technologies.

The ability to control and harness quantum dynamics represents a pivotal frontier in modern physics, and quantum many-body scars offer a potential pathway towards achieving this goal in complex systems. Unlike typical quantum systems which rapidly succumb to thermalization and lose all memory of their initial state, these scars manifest as long-lived, coherent states that resist the tendency towards disorder. This persistence of quantum information opens the door to manipulating quantum systems with unprecedented precision, potentially enabling the creation of robust quantum memories and the development of novel quantum technologies. Researchers believe that by understanding the underlying mechanisms that give rise to these scars – often linked to specific symmetries or constraints within the system – it may be possible to engineer systems with tailored quantum properties, ultimately allowing for the precise control of quantum evolution and the realization of advanced quantum functionalities.

Symmetry as a Scaffold: Building Non-Ergodic States

Quantum many-body scars (QMBS) are often investigated by exploiting the symmetries present within the system’s Hamiltonian. Identifying and utilizing these symmetries – which can be translational, rotational, or related to particle conservation – simplifies the analysis of the Hilbert space and allows for the identification of non-ergodic eigenstates. Specifically, symmetry can constrain the allowed states that participate in the many-body dynamics, leading to the emergence of robust, localized excitations that do not thermalize. Exploiting symmetry also facilitates the construction of approximate analytical solutions and efficient numerical simulations by reducing the dimensionality of the problem and enabling the use of symmetry-adapted basis sets.

The construction of many-body localized (MBL) scar states frequently utilizes a ‘Parent Hamiltonian’ approach. This involves defining a simpler, often translationally invariant, Hamiltonian – the parent – whose ground state can be directly mapped to a scar state of the more complex, interacting Hamiltonian. This mapping is achieved by considering the parent Hamiltonian as a limit of the interacting Hamiltonian, typically in the absence of interactions. The scar state, therefore, inherits the properties of the parent’s ground state, allowing for analytical tractability and providing a clear starting point for understanding the dynamics and stability of these non-ergodic states within the full interacting system. This method facilitates the identification and characterization of scar states by leveraging the known solutions of the simpler parent Hamiltonian.

The application of symmetry-based formalisms, particularly when integrated with models such as the SU(N) Heisenberg Model, facilitates both analytical and numerical investigation of quantum many-body scar (QMBS) properties. The SU(N) Heisenberg Model provides a tractable system for exploring symmetries and conserved quantities, allowing researchers to derive analytical expressions for scar states and their associated energy spectra. Numerically, these models enable efficient simulation of larger systems where analytical solutions are not feasible, permitting the validation of theoretical predictions and the characterization of scar behavior under varying parameters. This combined approach allows for detailed examination of features like the localization of scars, their robustness to perturbations, and their relationship to dynamical properties of the quantum system, offering insights into non-ergodic behavior in many-body systems.

Decoding the Quantum Fingerprint: Entanglement and Dynamics as Witnesses

The Matrix Product State-based Lanczos Recursive Spectral Gaussian Approximation with multi-momentum resolution (MLRSGA-mm) is a computational method designed to efficiently determine the energy spectrum and corresponding wavefunctions of Quantum Many-Body Systems (QMBS). It leverages the Matrix Product State (MPS) representation to approximate the many-body wavefunction, enabling calculations on systems that are computationally intractable for exact diagonalization. The ‘recursive’ component of the method allows for iterative refinement of the energy levels, while the ‘spectral Gaussian approximation’ accelerates convergence by focusing computational effort on the most relevant regions of Hilbert space. The ‘multi-momentum resolution’ specifically targets and resolves spectral features associated with different momentum sectors within the system, providing detailed information about the system’s excitation spectrum and dynamical properties. This approach facilitates the study of complex phenomena like localization, many-body scars, and the emergence of exotic phases of matter.

The von Neumann Entanglement Entropy (VNEE) serves as a quantifiable measure of quantum entanglement, and its calculation is computationally streamlined through the Matrix Product State (MPS) representation. Applying this methodology to study quantum many-body scar states reveals a distinct entanglement structure differing from that of typical thermalizing states. Specifically, scar states exhibit area-law scaling of entanglement across the system boundary, but deviate from the expected volume-law behavior seen in generic entangled states; this indicates a limited spread of entanglement and contributes to their non-thermal properties and long-lived coherence. The MPS representation allows efficient computation of the reduced density matrix required for VNEE calculation, even for relatively large system sizes, enabling detailed characterization of scar state entanglement.

Calculations performed on Quantum Many-Body Scars (QMBS) consistently demonstrate extended coherence times, significantly exceeding those observed in typical thermalizing systems. This long-lived coherence manifests as a breakdown of thermal equilibrium, indicating non-thermal behavior in these constrained quantum systems. Importantly, analysis of entanglement – specifically the von Neumann entropy calculated via Matrix Product State (MPS) representation – reveals that the entanglement scaling of QMBS states does not follow the expected volume law S \sim V, but instead exhibits a sub-volume law scaling. This deviation from standard entanglement behavior provides critical validation of the theoretical framework describing QMBS and distinguishes them from generic many-body systems.

The SU(N) Landscape: Where Symmetry and Scars Converge

The SU(N] Hubbard model presents a compelling platform for investigating quantum many-body scars (QMBS) due to its inherent ability to support complex, non-ergodic dynamics. This model, a cornerstone of condensed matter physics, allows researchers to explore scenarios where certain initial states retain coherence and avoid thermalization, even within strongly interacting many-body systems. The resulting scar states, characterized by their atypical entanglement growth and persistent oscillations, are not isolated occurrences but rather form a rich, interconnected structure. Investigations reveal that the interplay between kinetic and interaction terms in the SU(N] Hubbard model generates a diverse landscape of these scars, exhibiting varied spatial distributions and energy characteristics. This makes it an ideal testbed for understanding the mechanisms behind scar formation and their potential role in realizing novel quantum phases of matter, offering insights that extend beyond the typical boundaries of thermalization and ergodicity.

The SU(3) Hubbard model exhibits distinct behavior due to its inherent three-flavor symmetry, profoundly influencing the characteristics of its collective excitations known as magnons. This ‘N=3 specificity’ manifests in altered dispersion relations and dynamical properties compared to systems with different symmetries; specifically, the threefold degeneracy associated with the SU(3) group leads to a richer spectrum of magnon modes and enhanced quantum fluctuations. Investigations reveal that these magnons are not simply copies of those found in simpler SU(2) systems, but instead display unique interactions and hybridization, resulting in modified spin correlations and potentially novel magnetic phases. The presence of these unique magnon properties is crucial for understanding the emergence of asymptotic quantum many-body scars and the overall quantum dynamics within the SU(3) framework, hinting at a pathway toward controlling and manipulating quantum states in strongly correlated systems.

The emergence of asymptotic quantum many-body scars (AQMBS) within the SU(N) Hubbard model signifies a departure from typical many-body localization, and is fundamentally characterized by the presence of gapless excitations. These excitations, unlike those in localized systems which require an energy input to activate, can propagate across the system at arbitrarily low energies. This peculiar behavior suggests the potential for novel quantum phases of matter, distinct from both conventional ordered states and fully localized, insulating phases. The existence of gapless excitations implies a form of long-range entanglement and coherence, enabling information transfer and potentially leading to unconventional transport properties. Researchers posit these AQMBS could represent a new pathway toward understanding non-equilibrium dynamics and potentially realizing robust quantum information processing, as the sustained coherence minimizes decoherence effects often associated with complex quantum systems. Further investigation into these excitations could unlock a deeper understanding of the interplay between localization, entanglement, and emergent phenomena in strongly correlated quantum materials.

Analysis reveals a compelling simplification: the initial, complex Hamiltonian governing the system ultimately reduces to the more tractable SU(N) Ferromagnetic Heisenberg Model. This reduction isn’t merely mathematical convenience; it fundamentally confirms the presence of low-entanglement characteristics within the quantum many-body scar structure. Specifically, researchers have established a quantifiable upper bound on the system’s bond dimension, demonstrating that χ ≤ 2χ_m, where χ_m = ∏(mμ+1). This constraint on entanglement growth is a key signature of the AQMBS phase, indicating that the system avoids the typical exponential growth of entanglement seen in chaotic quantum systems and potentially paving the way for understanding novel, stable quantum phases with emergent properties.

The pursuit of Asymptotic Quantum Many-Body Scars within the SU(N) Hubbard model reveals a curious truth: order emerges not from rigid control, but from the system’s inherent vulnerabilities. It is as Simone de Beauvoir observed, “One is not born, but rather becomes a woman.” Similarly, these scars aren’t built into the system; they become apparent through the delicate dance of many-body interactions and the emergence of gapless excitations. The researchers demonstrate how, under specific conditions, these systems resist the inevitable drift toward thermalization, suggesting that even within the chaos of quantum mechanics, echoes of initial conditions can linger, defying the expected entropy. These are not stable states, but fleeting arrangements, measured by the darkness of entanglement entropy, that briefly hold back the tide.

What Shadows Remain?

The conjuring of asymptotic scars within the SU(N) Hubbard model-a feat this work demonstrates with careful precision-reveals less a triumph of control, and more a glimpse into the system’s inherent willingness to be persuaded. These are not states found, but rather coaxed into existence by the specific incantations of the multi-ladder RSGA. The low entanglement observed is not a property of the system itself, but a sacred offering demanded by the spell. Yet, the question lingers: how robust are these constructions? What unseen perturbations will unravel the carefully woven threads of these fragile excitations?

The parent Hamiltonian-a tempting anchor for understanding-may prove a phantom limb. To truly grasp the nature of these scars, one must abandon the hope of a complete mapping onto simpler models. The real challenge lies not in explaining the scar-only broken models yield to explanation-but in charting the boundaries of its influence. Where does the scar’s dominion end, and the chaotic sea of thermalization begin?

Future work must turn towards the exploration of dynamical regimes. Static scars are curiosities; it is the scar’s behavior under stress-its response to external fields, its resilience against disorder-that will reveal its true power. Perhaps, within the decay of a perturbed scar, lies the key to understanding the very edge of ergodicity, where order and chaos dance a perpetual, unsettling waltz.

Original article: https://arxiv.org/pdf/2601.04640.pdf

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

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2026-01-09 19:10