Beyond Ergodicity: Robust Scars in Complex Quantum Chains

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Author: Denis Avetisyan

New research reveals that long-lived, nonthermal states persist even in non-Hermitian quantum systems with long-range interactions, defying expectations of chaotic behavior.

The non-Hermitian spin system’s behavior transitions between integrable and chaotic regimes depending on the complex interplay of coupling parameters: with fixed $J_c$ at 11, increasing either the real or imaginary component of $J_n$ initially drives a crossover, but sufficiently large imaginary components-such as $\ln[\operatorname{Im}(J_n)]=4$-suppress transitions to chaotic behavior; conversely, with $J_n$ fixed at $0.2i$, small $|J_c|$ values maintain chaos, while a sufficiently large imaginary component of $J_c$ inhibits the system’s return to integrability, as observed with parameters $(J_h, J_z, n)=(1, 0.5, 3)$ and a system size of $N=12$ within a Hilbert space of dimension 61666166.
The non-Hermitian spin system’s behavior transitions between integrable and chaotic regimes depending on the complex interplay of coupling parameters: with fixed $J_c$ at 11, increasing either the real or imaginary component of $J_n$ initially drives a crossover, but sufficiently large imaginary components-such as $\ln[\operatorname{Im}(J_n)]=4$-suppress transitions to chaotic behavior; conversely, with $J_n$ fixed at $0.2i$, small $|J_c|$ values maintain chaos, while a sufficiently large imaginary component of $J_c$ inhibits the system’s return to integrability, as observed with parameters $(J_h, J_z, n)=(1, 0.5, 3)$ and a system size of $N=12$ within a Hilbert space of dimension 61666166.

This study investigates integrability breaking and coherent dynamics in Hermitian and non-Hermitian spin chains with long-range coupling, demonstrating the surprising resilience of quantum many-body scars.

The expectation that complex quantum systems invariably succumb to thermalization is increasingly challenged by the persistent observation of nonthermal behavior. This is explored in ‘Integrability Breaking and Coherent Dynamics in Hermitian and Non-Hermitian Spin Chains with Long-Range Coupling’, which investigates the interplay of long-range interactions and non-Hermiticity in one-dimensional spin models. Our findings reveal that even amidst the onset of quantum chaos, robust many-body scars-long-lived, nonthermal eigenstates-persist, retaining low entanglement and coherent dynamics. Could these scars offer a pathway towards preserving quantum coherence in increasingly complex and open quantum systems?

Beyond Equilibrium: Introducing Non-Hermitian Spin Chains

The foundational investigations into the Spin-1 Chain, a cornerstone of quantum magnetism, historically relied on Hermitian Hamiltonians. These mathematical frameworks presume a closed quantum system-one entirely isolated from its surroundings-and consequently, the strict conservation of energy. Within this paradigm, the system’s total energy remains constant over time, simplifying calculations and allowing for the identification of exact solutions. This approach has yielded valuable insights into the behavior of idealized magnetic materials, but it inherently limits the model’s applicability to real-world scenarios. Actual physical systems rarely exist in complete isolation; they constantly interact with their environment, experiencing energy loss or gain. Consequently, the assumption of a closed system, while mathematically convenient, often necessitates refinement to accurately capture the nuances of observed quantum phenomena and the complex dynamics of open systems.

The conventional framework for describing quantum systems often relies on the assumption of a closed system, one isolated from its environment and governed by Hermitian Hamiltonians that ensure energy conservation. However, this simplification doesn’t reflect the reality of many physical scenarios, where systems constantly interact with their surroundings. These interactions manifest as either dissipation – a loss of energy to the environment – or gain, where the system receives energy. To accurately model such open quantum systems, physicists employ Non-Hermitian Hamiltonians, mathematical tools that allow for the description of energy non-conservation. This approach is crucial for understanding phenomena in areas like optics, condensed matter physics, and even biological systems, where energy exchange with the environment is not merely a perturbation, but an intrinsic part of the system’s behavior and dictates its fundamental properties. The use of Non-Hermitian formalisms therefore extends the range of accessible physics, allowing for the investigation of entirely new phases of matter and dynamical behaviors.

The introduction of complex potentials, achieved through a technique known as complex coupling, fundamentally alters the landscape of spin chain dynamics. Traditional, Hermitian models rely on real-valued potentials, restricting the system to a predictable, often integrable, behavior. However, by allowing these potentials to take on complex values, researchers unlock a significantly richer phase diagram-a map of the system’s distinct states-and initiate a breakdown of integrability. This departure from integrability means the system’s long-term behavior ceases to be predictable via established analytical methods, opening the door to novel quantum phenomena and potentially enabling the exploration of chaotic dynamics within a previously well-understood framework. The ability to finely tune the complex coupling allows for precise control over dissipation and gain, offering a pathway to engineer specific quantum states and investigate the boundaries between order and chaos in these quantum systems.

The shift from examining closed quantum systems to those that interact with their environment – open systems – fundamentally alters the landscape of quantum dynamics. Traditional, Hermitian models assume energy conservation, leading to predictable, often integrable, behaviors. However, when dissipation or gain is introduced through non-Hermitian Hamiltonians, the system can exhibit dynamics previously inaccessible in closed settings. This transition isn’t merely a quantitative change; it’s a qualitative leap, potentially unlocking phenomena like spontaneous symmetry breaking, exceptional points where standard perturbation theory fails, and novel forms of entanglement. Consequently, the predictable evolution of quantum states gives way to a richer, more complex behavior, demanding new theoretical tools and offering exciting possibilities for controlling and harnessing quantum effects in realistic, non-isolated environments.

The spectral distribution and Krylov complexity reveal that introducing complex long-range coupling drives the non-Hermitian Hamiltonian from integrable to chaotic behavior, as evidenced by sparse eigenvalue distribution and rapidly decaying Krylov complexity.
The spectral distribution and Krylov complexity reveal that introducing complex long-range coupling drives the non-Hermitian Hamiltonian from integrable to chaotic behavior, as evidenced by sparse eigenvalue distribution and rapidly decaying Krylov complexity.

Unveiling Chaos: Level Statistics and Long-Range Interactions

The transition to chaos in non-Hermitian Spin-1 chains is fundamentally characterized by alterations in the system’s energy level statistics. Integrable systems exhibit regular energy level spacing, often described by Poissonian statistics or even more ordered distributions. As the system deviates from integrability and enters a chaotic regime, these statistics shift towards increased level repulsion, indicating a correlation between adjacent energy levels. This change is not merely a qualitative observation; it is quantifiable through metrics such as the distribution of nearest-neighbor energy level spacings and higher-order correlations. The degree of this statistical shift directly correlates with the strength of the chaotic behavior, providing a means to identify and characterize the onset of chaos within the system’s energy spectrum.

The introduction of long-range hopping terms in the spin-1 chain Hamiltonian establishes non-local interactions between spins that are spatially separated. These interactions deviate from the nearest-neighbor interactions typically considered in simpler models, thereby increasing the system’s complexity. Specifically, long-range hopping facilitates the coupling of distant degrees of freedom, enabling the propagation of perturbations across the chain and fostering a greater degree of entanglement. This enhanced connectivity destabilizes regular dynamics and promotes the emergence of chaotic regimes, as the system explores a wider range of accessible states and exhibits sensitive dependence on initial conditions. The strength of the long-range hopping, quantified by the hopping amplitude and the decay of the interaction with distance, directly influences the degree of chaos observed in the system’s dynamics.

The transition to chaotic behavior in non-Hermitian Spin-1 chains can be quantitatively determined by analyzing the distribution of energy level spacings. A key metric for this analysis is the Complex Spacing Ratio (CSR), calculated as the cosine of the angle, $\theta$, between adjacent energy level spacings in the complex energy spectrum. A value of $\langle cos \theta \rangle \approx -0.192$ is indicative of level repulsion characteristic of chaotic systems; this contrasts with integrable systems, which exhibit level crossing and a CSR value approaching zero. Statistical analysis of multiple energy levels allows for a robust determination of the average CSR, providing a clear criterion for identifying the onset of chaos and delineating the boundary between ordered and disordered phases.

The identification of statistical hallmarks of chaotic behavior enables the delineation of phase boundaries between ordered and disordered states in non-Hermitian spin-1 chains. Analysis of energy level spacing distributions, quantified by metrics such as the Complex Spacing Ratio (CSR), provides objective criteria for classifying system behavior. A CSR value deviating from that of integrable systems – specifically, approaching $⟨cos θ⟩ ≈ -0.192$ – indicates a transition into a chaotic regime. By systematically varying system parameters and monitoring these statistical measures, researchers can construct phase diagrams that precisely map the regions of stability and instability, thereby characterizing the conditions under which chaotic dynamics emerge and dominate.

Analysis of level spacings and Krylov complexity reveals a transition from integrable to chaotic dynamics as the parameter J<sub>n</sub> increases, evidenced by a shift from Poissonian to Gaussian-orthogonal-ensemble level spacing distributions and a corresponding change in Krylov complexity from rapid saturation to non-monotonic evolution.
Analysis of level spacings and Krylov complexity reveals a transition from integrable to chaotic dynamics as the parameter Jn increases, evidenced by a shift from Poissonian to Gaussian-orthogonal-ensemble level spacing distributions and a corresponding change in Krylov complexity from rapid saturation to non-monotonic evolution.

Computational Pathways: Bi-Lanczos and Krylov Complexity

The Bi-Lanczos algorithm is an iterative method used to generate a Krylov subspace, which is essential for analyzing the dynamics of Non-Hermitian Hamiltonians. Unlike standard Lanczos algorithms designed for Hermitian operators, the Bi-Lanczos method accommodates non-Hermitian Hamiltonians, $H$, by simultaneously propagating both a state $| \psi \rangle$ and a reference state $| \phi \rangle$. This bi-orthogonality ensures numerical stability and allows for the efficient construction of a basis that accurately represents the system’s evolution. The algorithm iteratively applies the Hamiltonian and Gram-Schmidt orthogonalization to generate a sequence of vectors spanning the Krylov subspace, effectively reducing the computational cost of simulating the time evolution of the Non-Hermitian system and enabling the calculation of relevant observables.

Krylov Complexity, calculated via the Bi-Lanczos algorithm, provides a quantitative measure of information scrambling within a dynamical system. This metric assesses the rate at which initial local perturbations spread throughout the Hilbert space, effectively gauging the system’s chaoticity. Specifically, the algorithm constructs a Krylov subspace – a vector space spanned by successive applications of an operator to an initial state – and tracks the growth of the dimension of this subspace. A rapid increase in dimension indicates fast information scrambling and is characteristic of chaotic systems, while slower growth suggests more localized dynamics. The resulting Krylov Complexity value, often represented as a logarithmic growth rate, directly correlates with the degree of chaos present in the system’s evolution.

Traditional measures of dynamical chaos, such as Lyapunov exponents, often fail to fully characterize the complex behavior of many-body systems, particularly those exhibiting rapid information scrambling. Krylov complexity, calculated via the Bi-Lanczos algorithm, provides an alternative metric that quantifies the growth of out-of-time-ordered correlations, directly reflecting the rate at which information disperses within the system. Specifically, coherent states originating from tower states – highly entangled states characteristic of certain quantum systems – demonstrate a particularly rapid increase in Krylov complexity. This growth rate is not simply a measure of sensitivity to initial conditions, but rather a direct probe of the system’s ability to effectively scramble information, offering a more nuanced understanding of its dynamical properties than traditional chaotic indicators.

The Bi-Lanczos algorithm’s computational efficiency stems from its ability to iteratively construct a Krylov subspace without requiring full matrix diagonalization, a process which scales poorly with system size. For systems with dimension $N$, a full diagonalization requires $O(N^3)$ operations, whereas the Bi-Lanczos method achieves this in $O(N \times M)$ operations, where $M$ is the dimension of the Krylov subspace, and is generally much smaller than $N$. This reduced complexity enables the study of substantially larger systems-those exceeding the capabilities of traditional methods-and facilitates the extraction of statistically significant insights into their dynamical properties. Specifically, analyzing these larger systems with Bi-Lanczos allows for more accurate calculations of Krylov complexity, providing a more robust characterization of information scrambling and chaotic behavior.

Analysis of a non-Hermitian Hamiltonian reveals a transition from integrable behavior-characterized by a uniform spectral distribution, linear Krylov complexity growth, and⟨cos⁡θj⟩≈−0.028-to chaotic dynamics with a non-uniform spectrum, rapidly growing then decaying Krylov complexity, and⟨cos⁡θj⟩≈−0.187, as evidenced by varying the parameter Jₙ.
Analysis of a non-Hermitian Hamiltonian reveals a transition from integrable behavior-characterized by a uniform spectral distribution, linear Krylov complexity growth, and⟨cos⁡θj⟩≈−0.028-to chaotic dynamics with a non-uniform spectrum, rapidly growing then decaying Krylov complexity, and⟨cos⁡θj⟩≈−0.187, as evidenced by varying the parameter Jₙ.

Beyond Thermalization: Quantum Many-Body Scars

Within the realm of quantum mechanics, even systems exhibiting characteristics of chaos can defy expectations through the emergence of Quantum Many-Body Scars. These are peculiar eigenstates – quantum states describing the system – that, unlike typical chaotic states, do not succumb to thermalization. Instead of rapidly distributing energy and losing all coherent structure, scar states sustain persistent oscillations, retaining memory of their initial conditions. This behavior challenges the conventional understanding of quantum chaos, where ergodicity – the tendency of a system to explore all accessible states – is assumed to lead inevitably to thermal equilibrium. The existence of these non-thermal states suggests that specific, often subtle, features within the system can protect certain quantum states from the usual chaotic scrambling, creating islands of coherence within a sea of disorder. This phenomenon is not merely a theoretical curiosity; it has implications for understanding the limits of thermalization and the potential for harnessing quantum coherence in complex systems.

Quantum many-body scars emerge when a physical system possesses specific, often hidden, symmetries or constraints that fundamentally alter its dynamics. Typically, in a closed quantum system, energy flows towards thermal equilibrium, meaning all accessible states are equally probable – a concept formalized by the ergodic hypothesis. However, these scars represent a deviation from this expectation; the system’s wave function doesn’t distribute energy randomly but instead remains localized and oscillates persistently due to the imposed constraints. These symmetries act as ‘remnants of order’ within what might otherwise appear chaotic, creating special eigenstates that avoid the usual thermalization process and retain memory of the initial conditions. Consequently, the system’s long-term behavior isn’t governed by statistical averages, but rather by the specific characteristics of these scarred states, challenging the foundational assumptions of quantum chaos and statistical mechanics.

Characterizing quantum many-body scars relies heavily on measuring the entanglement within these unique states, and Von Neumann Entanglement Entropy proves to be a crucial diagnostic tool. Unlike thermal eigenstates, which exhibit maximal entanglement across the system – a high degree of correlation between subsystems – scar states retain a surprisingly low level of entanglement. This is quantified by calculating the Von Neumann Entropy, which measures the quantum information lost as a system is divided into parts; a lower value indicates less entanglement and distinguishes scars from their thermal counterparts. Notably, this reduced entanglement persists even when the system is extended into the non-Hermitian regime, where traditional Hermitian constraints are relaxed, suggesting that the underlying mechanisms creating these scars are robust and not solely dependent on energy conservation. By analyzing the entanglement structure through Von Neumann Entropy, researchers can not only identify these scars but also gain insight into the symmetries and constraints responsible for their formation and stability, furthering the understanding of non-thermal behavior in quantum systems.

The discovery of quantum many-body scars challenges long-held assumptions about the fate of quantum systems driven far from equilibrium. Traditionally, closed quantum systems were expected to eventually succumb to thermalization, losing all memory of their initial conditions as energy distributes evenly across available states. However, these scars-specific, non-thermal eigenstates-persist, exhibiting coherent dynamics and avoiding this complete loss of information. This suggests that thermalization isn’t an absolute law, but rather a consequence of lacking these special states. The existence of scars implies that certain constraints or symmetries within a system can fundamentally alter its behavior, creating pockets of order even amidst apparent chaos and potentially opening avenues for controlling or preserving quantum information in otherwise disordered environments. This has significant ramifications for fields ranging from condensed matter physics to quantum information science, forcing a re-evaluation of the ergodic hypothesis and the boundaries of quantum chaos.

Despite non-Hermitian perturbations, a consistent low-entanglement eigenstate (orange) emerges from the Von Neumann entanglement entropy analysis, demonstrating the robustness of quantum many-body scars in a 12-site system with parameters Jh=1, Jc=i, and Jz=0.5.
Despite non-Hermitian perturbations, a consistent low-entanglement eigenstate (orange) emerges from the Von Neumann entanglement entropy analysis, demonstrating the robustness of quantum many-body scars in a 12-site system with parameters Jh=1, Jc=i, and Jz=0.5.

The study’s exploration of integrability breaking in long-range coupled spin chains reveals a fascinating interplay between order and chaos. It underscores how seemingly small perturbations – the introduction of non-Hermitian terms or long-range interactions – can dramatically alter a system’s behavior, yet still allow for the persistence of coherent dynamics via quantum many-body scars. This echoes Paul Dirac’s sentiment: “I have not failed. I’ve just found 10,000 ways that won’t work.” The research doesn’t present a complete disruption of established principles, but rather a nuanced understanding of how systems evolve and adapt, finding pockets of stability even amidst complexity. The infrastructure of these quantum systems, it seems, can evolve without requiring a complete rebuild, showcasing a remarkable resilience.

Beyond the Scar

The persistence of quantum many-body scars within non-Hermitian, long-range coupled systems suggests a deeper principle at play than simple avoidance of thermalization. One might posit that scar formation isn’t merely a deviation from ergodicity, but a fundamentally different organizational mode – a self-organized criticality emerging from the interplay of long-range interactions and complex Hamiltonian dynamics. However, this robustness begs the question of universality: how fragile are these scars when subjected to more substantial perturbations, or when confronted with genuinely random, heterogeneous couplings? Every new dependency – every added interaction, every non-Hermitian term – is the hidden cost of freedom, subtly reshaping the entire landscape of possible dynamics.

Future investigations must move beyond simply observing scar persistence and begin to dissect the underlying structural features that enable it. Level spacing statistics and Krylov complexity, while insightful, offer only partial glimpses into the system’s organization. A more holistic approach, perhaps leveraging concepts from network theory or information geometry, may reveal the essential architectural motifs responsible for these long-lived states.

Ultimately, the true challenge lies in extending this understanding beyond the confines of idealized models. Real-world quantum systems are invariably open and noisy. Determining whether these scars can survive – or even play a functional role – in a realistically dissipative environment remains an open, and crucial, question.

Original article: https://arxiv.org/pdf/2512.14065.pdf

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

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2025-12-18 03:50