Untangling Quantum Chaos: How States Spread Reveals Localization

by

Author: Denis Avetisyan

New research explores the structure of quantum states to differentiate between systems that exhibit chaotic behavior and those trapped in a localized, non-thermal state.

Analysis of Krylov-space anatomy and infinite-time spread complexity provides a robust indicator of many-body localization in disordered quantum spin chains.

Distinguishing between genuinely disordered quantum systems exhibiting ergodic behavior and those undergoing many-body localization remains a central challenge in condensed matter physics. This is addressed in ‘Krylov-space anatomy and spread complexity of a disordered quantum spin chain’, where the authors investigate the structure of quantum states within Krylov space to characterize these distinct phases. They demonstrate that the infinite-time spread complexity-a measure of state delocalization optimized to the Krylov basis-serves as a robust indicator of localization, scaling linearly with system size in the ergodic phase but sublinearly in the many-body localized phase. Does this Krylov-space perspective offer a new pathway to understanding the fundamental limits of quantum thermalization and the emergence of complex behavior in disordered systems?

Disorder’s Embrace: Unraveling Quantum Complexity

The enduring puzzle of how quantum systems evolve over extended periods is fundamentally complicated by the introduction of disorder – imperfections or variations within the system itself. Unlike pristine, predictable environments, disordered quantum systems exhibit behaviors that defy simple descriptions, as interactions between particles become entangled with the irregularities of the landscape they inhabit. This creates a complex interplay where energy transport can be suppressed, leading to phenomena like many-body localization, or conversely, sustained, albeit unusual, dynamics. Determining the long-time fate of these systems isn’t merely an academic exercise; it impacts fields ranging from the design of novel materials with tailored electronic properties to understanding the fundamental limits of quantum computation, as maintaining quantum coherence over time is crucial for practical applications. The challenge lies in developing theoretical frameworks and experimental techniques capable of accurately capturing these subtle, often counterintuitive, effects.

The investigation of quantum systems plagued by disorder presents a significant hurdle for conventional analytical techniques. These methods, frequently successful with pristine, ordered systems, often fail to adequately account for the intricate correlations that emerge when randomness is introduced. This limitation is particularly acute in the study of many-body localization (MBL), a phase of matter where disorder prevents thermalization and leads to an unusual lack of energy transport. The failure of traditional approaches stems from their reliance on perturbation theories or mean-field approximations which struggle to capture the collective behavior and emergent phenomena arising from the complex interplay between disorder and quantum entanglement. Consequently, a deeper understanding requires novel theoretical frameworks and computational tools capable of accurately describing the system’s intricate dynamics and accessing the full scope of quantum complexity.

Krylov Space: A Quantum Microscope

The Krylov space, constructed by applying a Hermitian operator H iteratively to an initial state |ψ_0⟩ – generating the basis vectors |ψ_0⟩, H|ψ_0⟩, H^2|ψ_0⟩, and so forth – provides a subspace onto which any quantum state can be efficiently projected. This projection reduces the dimensionality of the problem, enabling complexity analysis with significantly fewer computational resources than methods operating in the full Hilbert space. Traditional approaches often require representing the entire state vector, scaling exponentially with system size, whereas Krylov subspace methods focus on representing the state within a limited, dynamically constructed basis, achieving a comparable level of accuracy with reduced computational cost. This efficiency is particularly advantageous when dealing with large quantum systems or when tracking the time evolution of quantum states, as it allows for the approximation of complex dynamics with tractable computational demands.

Krylov Spread Complexity (KSC) measures the rate at which a quantum state, initially localized, disperses within the Krylov subspace. This dispersion is quantified by tracking the entanglement entropy of a reduced density matrix obtained by restricting the state to a growing subspace spanned by the Krylov basis. A larger KSC indicates a faster rate of information scrambling and a more complex quantum dynamics, while a slower rate suggests a more localized and potentially integrable system. Specifically, KSC is calculated as the growth rate of the entanglement entropy S = -\text{Tr}(\rho \log \rho), where ρ is the reduced density matrix, and provides a quantifiable metric for characterizing the complexity of quantum states and their evolution over time.

Krylov space methods are especially effective in characterizing the behavior of disordered quantum systems due to their ability to capture the influence of randomness on quantum state evolution. In these systems, the Hamiltonian itself contains random elements, leading to a complex energy landscape and localization effects. The Krylov subspace, constructed by repeatedly applying the Hamiltonian to an initial state, effectively probes this landscape and reveals how randomness affects the spreading and entanglement of the quantum state. Specifically, the dimensionality of the Krylov subspace, and metrics derived from it like Krylov Complexity, provide quantifiable measures of the degree of localization and the rate of information propagation within the disordered system, offering insights unattainable through traditional approaches that average over the disorder.

Decoding Order from Chaos: A Complexity Metric

Krylov Spread Complexity functions as a diagnostic tool for differentiating between ergodic and localized quantum phases of matter. In ergodic systems, quantum states exhibit exploration of the entire accessible phase space, which is reflected in a linear scaling of Krylov complexity with system size \propto N , where N represents the system size. Conversely, localized phases are characterized by restricted state exploration, resulting in a sublinear dependence of Krylov complexity on system size \propto N^{\alpha} , with \alpha < 1 . This difference in scaling behavior directly relates to the rate at which quantum information spreads within the system, providing a quantifiable metric for identifying the transition between these distinct phases.

Analysis of disordered spin chains using Krylov Spread Complexity demonstrates a direct relationship between disorder and the propagation of quantum information. Specifically, in the ergodic phase, the infinite-time Krylov spread complexity scales linearly with system size, expressed as \propto N^{H} , where N represents the system size and H is a scaling exponent. This linear scaling indicates that quantum information effectively explores the Hilbert space, with the rate of exploration increasing proportionally to the system size. This behavior contrasts with localized phases and provides a quantitative metric for distinguishing ergodic from non-ergodic behavior in disordered quantum systems.

Krylov Spread Complexity, used to quantify eigenstate complexity, reveals the impact of rare resonances on system dynamics. In the many-body localized (MBL) phase, this complexity exhibits a sublinear scaling with system size ∝ N^{\alpha}H, where α < 1 and H represents the local Hilbert space dimension. This sublinear behavior indicates that quantum information spreads only within a limited region of the system, constrained by the presence of strong disorder and the resulting localization of eigenstates. The exponent α directly reflects the efficiency of information propagation and is significantly reduced in the MBL phase compared to ergodic systems, demonstrating a fundamental difference in how quantum information is encoded and processed.

Statistical Whispers: Unveiling Hidden Quantum Rules

Large-deviation analysis offers a robust methodology for investigating the likelihood of infrequent occurrences within the intricate dynamics of quantum systems. Unlike traditional approaches focused on average behavior, this technique centers on the tails of probability distributions, allowing researchers to quantify the chance of substantial deviations from typical outcomes. By examining these rare events-such as unusually high energy fluctuations or atypical entanglement patterns-scientists gain insight into the system’s fundamental limits and resilience. This is achieved through the calculation of a ‘large deviation function’, which governs the exponential decay of the probability of observing events far from the mean, and provides a pathway to understanding the underlying mechanisms driving complex quantum phenomena – even when those phenomena are statistically improbable. The power of this method lies in its ability to reveal hidden structural properties and critical transitions within the system, offering a more complete picture than traditional statistical analyses.

Investigating the statistical distribution of Eigenstate Complexity provides a crucial window into the dynamics of many-body quantum systems. Rather than focusing on individual quantum states, this technique examines the collective behavior of a vast ensemble, revealing patterns and underlying mechanisms otherwise obscured. By characterizing how complexity is distributed across these states, researchers can discern the factors influencing information storage and processing within the system. This statistical approach highlights that complexity isn’t a uniform property, but instead emerges from specific arrangements and interactions, demonstrating that even in seemingly disordered systems, there are predictable statistical features governing the distribution of complexity and ultimately, the system’s behavior. The analysis demonstrates that the breadth and shape of this distribution provides insights into the fundamental limits of information storage and the emergence of novel quantum phases.

Analysis of many-body localized (MBL) systems reveals a surprising characteristic in the decay of quantum state amplitudes along the Krylov chain: a stretched exponential profile, rather than the typical exponential decay expected in ergodic systems. This indicates a fundamentally different dynamic where information persists for extraordinarily long times, yet is distributed across an immense number of quantum states. Despite an exponentially growing number of contributing eigenstates, the analysis demonstrates that only a vanishingly small fraction actually dominate the long-time complexity of the system. This counterintuitive result, stemming from the combined application of large-deviation analysis and Krylov chain techniques, highlights the unique statistical properties of MBL phases and provides a powerful method for characterizing their complex behavior – revealing how localization drastically alters the usual relationship between the number of states and the persistence of quantum information.

The exploration of Krylov space anatomy, as detailed in this work, mirrors a systematic dismantling of established quantum paradigms. The researchers don’t simply observe the behavior of disordered quantum spin chains; they actively probe the limits of predictability, essentially reverse-engineering the system’s response to disorder. This methodical deconstruction aligns with the assertion by Jean-Paul Sartre: “Existence precedes essence.” The system’s ‘essence’ – its localization properties – isn’t predetermined but revealed through the process of observation and analysis within the Krylov space, much like how an individual defines themselves through action and experience. The infinite-time spread complexity, as a metric, becomes the tangible manifestation of that ‘existence’.

Where Do We Go From Here?

The mapping of many-body localization onto the geometry of Krylov space offers a compelling, if unsettling, proposition. One begins to wonder if the very notion of “localization” isn’t simply a convenient label for a particular, predictable pattern of state-space exploration. Perhaps what appears as arrested diffusion is merely a highly constrained, fractal-like trajectory. The infinite-time spread complexity, so neatly correlated with localization in this work, might then be less a diagnostic and more a descriptor of the system’s intrinsic dimensionality – a measure of how much space it has to not explore.

The assumption of a sharp transition between ergodic and localized phases feels increasingly brittle. The study of finite-size scaling, while revealing, skirts the question of whether such transitions truly exist in the thermodynamic limit, or if instead there’s a continuous spectrum of behaviors, each with its own unique Krylov anatomy. What happens when disorder itself becomes dynamical, or when interactions beyond the simple, nearest-neighbor model are introduced? Does the fractal scaling of entanglement entropy remain a faithful guide, or does it break down, revealing a hidden order within the apparent chaos?

The true test, of course, lies in extending these ideas beyond the relatively clean confines of disordered spin chains. Can this Krylov-space perspective illuminate the behavior of genuinely complex, interacting systems-those found not in carefully crafted models, but in the messy reality of materials science or even biological systems? One suspects the answer is not a simple ‘yes’ or ‘no’, but a challenge to redefine what “localization” and “ergodicity” even mean when the rules themselves are subject to constant renegotiation.

Original article: https://arxiv.org/pdf/2603.25724.pdf

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

See also:

2026-03-29 22:20