Unlocking Quantum Secrets with Charge and Entanglement

Author: Denis Avetisyan

New research leverages ballistic fluctuation theory to map the behavior of charged particles and quantum entanglement in complex systems, even after a disruptive event.

This study provides analytical results for charged moments and symmetry-resolved entanglement using ballistic fluctuation theory and the generalized Gibbs ensemble framework.

Understanding how entanglement is distributed among conserved charges in many-body systems remains a central challenge in modern condensed matter physics. This work, ‘Charged moments and symmetry-resolved entanglement from Ballistic Fluctuation Theory’, leverages Ballistic Fluctuation Theory to compute charged moments and, consequently, symmetry-resolved Rényi entropies in both equilibrium and out-of-equilibrium settings. We derive analytic results for free fermions following a quantum quench, demonstrating agreement with predictions from the quasiparticle picture and generalizing established techniques to encompass generalized Gibbs ensembles. These findings pave the way for a deeper understanding of non-equilibrium dynamics and the role of symmetry in complex quantum systems – but how broadly applicable are these techniques to strongly interacting models?

Unveiling Hidden Order: Beyond Simple Averages

The behavior of any system driven away from equilibrium is rarely captured by its overall, averaged properties. While global quantities like total energy or average particle density provide a broad overview, they frequently mask the critical, localized fluctuations that dictate the system’s true dynamics. These fluctuations, often arising from microscopic variations, can dramatically alter how a system evolves and relaxes – a phenomenon obscured by traditional methods focused solely on macroscopic observables. Consequently, a more detailed examination of these internal variations is essential to truly understand non-equilibrium processes, requiring analytical techniques sensitive to the system’s internal structure and the distribution of its constituent parts rather than solely relying on coarse-grained, averaged values.

Following a rapid alteration to a physical system – a ‘quantum quench’ or similar disturbance – conventional analytical approaches frequently fall short in detailing the redistribution of fundamental conserved quantities. These quantities, such as total energy, particle number, or momentum, aren’t merely shifted as a bulk property; instead, their fluctuations and local distributions become critical to understanding the system’s evolving state. Traditional methods, often reliant on averaging these quantities globally, effectively mask these vital details, presenting a blurred picture of the post-quench dynamics. This limitation arises because the system doesn’t immediately reach a uniform distribution of these conserved charges; instead, localized excesses and deficits emerge, dictating the system’s behavior and influencing the emergence of novel phases or correlations. Consequently, a more refined approach is necessary to accurately map the intricate dance of these conserved quantities as the system navigates towards a new, potentially far-from-equilibrium, steady state.

Symmetry-resolved entanglement represents a paradigm shift in understanding complex quantum systems far from equilibrium. Rather than simply measuring the total entanglement between subsystems, this approach dissects it according to the conservation of physical quantities like energy, particle number, or momentum. By quantifying how entanglement is distributed within specific sectors defined by these conserved charges, researchers gain unprecedented insight into the dynamics of quantum many-body systems. This technique reveals hidden order within seemingly chaotic states, pinpointing how information is stored and processed even when global averages fail to capture the full picture. $\text{For example, it can distinguish between states where entanglement is concentrated in low-energy sectors versus those where it's broadly distributed}$ . Ultimately, symmetry-resolved entanglement provides a powerful new framework for characterizing and predicting the behavior of systems ranging from condensed matter physics to quantum field theory, offering a more nuanced understanding of non-equilibrium dynamics.

Mapping the Flow: Ballistic Fluctuation Theory in Action

Ballistic Fluctuation Theory (BFT) addresses the behavior of conserved quantities – such as electric charge, energy, or particle number – in systems operating under non-equilibrium conditions. These systems, driven far from equilibrium by external forces or gradients, exhibit fluctuations in charge density that extend across macroscopic scales. Unlike systems where transport is diffusive and fluctuations are localized, BFT models these fluctuations as arising from ballistic propagation – essentially, unimpeded movement of information. This allows for the analysis of how these large-scale charge fluctuations emerge and evolve, providing a framework to understand transport phenomena in systems exhibiting ballistic behavior. The theory is particularly relevant in scenarios where traditional hydrodynamic approaches, relying on local equilibrium assumptions, fail to accurately describe the observed dynamics.

Ballistic transport, central to the understanding of dynamic charge distributions, describes a regime where carriers – such as electrons or holes – move across a material with minimal scattering events. This is in contrast to diffusive transport, where frequent collisions randomize the carrier’s trajectory. The absence of significant scattering allows information about charge distribution to propagate effectively without degradation, maintaining coherence over longer distances. This characteristic is typically observed in low-dimensional systems, or at high frequencies where the mean free path exceeds the relevant length scale. Consequently, ballistic transport enables a clearer and more predictable mapping of charge fluctuations, forming the basis for analyzing the system’s response to external stimuli and facilitating the application of $ln(Zm(α)) ~ x \int dk H_m(k)$ .

Ballistic Fluctuation Theory quantifies charge fluctuations through the calculation of ‘charged moments’, denoted as Z_m(α). These moments exhibit a linear relationship with system size, ‘x’, as expressed by the equation $\ln(Z_m(\alpha)) \sim x \in t dk H_m(k)$ . Here, $H_m(k)$ represents the power spectrum of the conserved charge. This scaling behavior allows for a direct measurement of fluctuation strength as a function of system size, providing a quantitative metric for understanding charge dynamics in non-equilibrium systems. The logarithmic form, $\ln(Z_m(\alpha))$ , facilitates analysis by converting multiplicative fluctuations into additive quantities proportional to system size.

Dissecting the Complex: Replica Method and Height Field Formulation

The replica method is a mathematical technique used to compute entanglement entropies in many-body systems, particularly those exhibiting strong interactions where traditional approaches fail. This method addresses the difficulty of directly calculating the entanglement entropy by introducing a mathematical trick: it considers $n$ identical copies, or “replicas,” of the system and calculates the free energy of the combined system. The entanglement entropy is then obtained by analytically continuing the result to $n = 1$ . Because entanglement entropy is directly related to the scaling of quantities like the free energy, this procedure effectively allows calculation of entanglement in systems lacking simple, closed-form solutions. Crucially, the entanglement entropy is linked to the calculation of charged moments – which describe the fluctuations of conserved quantities – making the replica method valuable for studying transport and non-equilibrium phenomena in interacting systems.

Branch-point twist fields function as mathematical tools that establish a direct correspondence between the calculation of entanglement entropy and the full counting statistics (FCS) of conserved charges within a system. Specifically, introducing these fields effectively alters the boundary conditions of the system, creating a ‘branch-point’ in the geometry. This alteration allows entanglement, normally characterized by quantities like the Rényi entropy, to be mapped onto the probability distribution of counting the total number of conserved charges – such as energy or particle number – that cross a given boundary. The FCS, denoted as $P(n)$ , provides the probability of observing n charges crossing the boundary, and its derivative, related to the cumulants, directly contributes to the calculation of the charged moments derived from entanglement.

The height field formulation facilitates the analytical calculation of charged moments by representing branch-point twist fields as a height function $h(x)$ . This representation transforms the problem from a complex quantum field theory calculation into an analysis of the statistical properties of this height field. Specifically, charged moments are expressed as integrals over the Scaled Cumulant Generating Function (SCGF), $\mathcal{G}_s(\lambda)$ , which characterizes the fluctuations of the height field and is directly related to the full counting statistics of the conserved charge. This allows for the computation of these moments through functional integration techniques, providing a tractable approach to understanding charge transport and correlations in interacting systems.

Revealing the Signatures: From Theory to Predictive Power

The statistical behavior of charge fluctuations in many-body systems isn’t always predictable by simple models; often, deviations from a Poisson distribution-the expected outcome for independent events-indicate underlying correlations and non-equilibrium dynamics. Researchers have successfully combined ballistic fluctuation theory, which describes the propagation of disturbances, with the replica method-a mathematical technique for calculating disordered system averages-to predict these distributions. This approach doesn’t merely confirm deviations from Poissonian behavior, but provides analytical expressions that quantify the subtle changes in charge fluctuations, revealing insights into the system’s response to external stimuli and its journey toward or away from equilibrium. By accurately mapping these statistical signatures, it becomes possible to characterize the complex interplay of interactions and dynamics that govern the system’s behavior, even when traditional equilibrium assumptions break down.

Researchers have successfully formulated analytical expressions to quantify symmetry-resolved entropies, providing a powerful tool to characterize both systems in established equilibrium and those undergoing dynamic shifts following a quantum quench. These entropies, which detail the entanglement structure of a system while accounting for conserved symmetries, reveal how information is distributed under various conditions. The derived formulas allow for precise calculations of these entropies, going beyond traditional measures and offering insights into the fundamental properties of quantum systems experiencing non-equilibrium dynamics. This approach allows for the characterization of entanglement not just overall, but specifically within symmetry sectors, offering a more nuanced understanding of quantum information processing and the behavior of interacting quantum systems-particularly relevant in areas like condensed matter physics and quantum computing, where symmetry plays a crucial role in determining system properties.

The study reveals that when systems experience only minor disturbances from equilibrium, a Gaussian approximation provides a surprisingly accurate and analytically tractable description of charge fluctuations. This simplification allows researchers to derive closed-form solutions for quantities like Rényi entropies – measures of entanglement and information – which would otherwise require complex numerical simulations. Importantly, the research confirms that the established scaling relationships for these entropies remain valid even when the system is slightly perturbed, bolstering confidence in their use as indicators of underlying physical behavior. This finding is crucial because it bridges the gap between idealized theoretical models and the more realistic, often messy, conditions found in experiments, offering a pathway to understand non-equilibrium dynamics with greater precision.

The pursuit of analytical results, as demonstrated in this work utilizing Ballistic Fluctuation Theory, echoes a fundamental desire for clarity and understanding. It is a refinement, a distillation of complex interactions into comprehensible forms. This mirrors the sentiment expressed by René Descartes: “It is not enough to have a good mind; the main thing is to use it well.” The paper’s success in computing charged moments and symmetry-resolved entanglement, even in out-of-equilibrium scenarios, isn’t merely a technical feat. It represents a considered application of intellect, a dedication to unraveling complexity with elegant precision-a principle that aligns with a belief that true mastery lies in the artful simplification of the intricate.

Where Do We Go From Here?

The present work, while offering a compelling analytical framework, inevitably highlights the gaps in understanding. The elegance of Ballistic Fluctuation Theory lies in its ability to extract universal features; however, applying it to systems lacking the requisite symmetries-or those with interactions stubbornly resistant to perturbation-remains a challenge. One suspects the true complexity isn’t necessarily more calculation, but rather a more insightful mapping of interacting systems onto simpler, solvable models. A persistent question involves the limits of the Generalized Gibbs Ensemble itself; does it truly capture the long-time behavior, or is it merely a convenient approximation masking a more subtle, slowly-evolving state?

Future effort will likely center on extending these techniques to spatially inhomogeneous systems-the messy reality of most materials. The connection between symmetry-resolved entanglement and transport properties deserves further scrutiny; a quantitative understanding of how entanglement dictates current flow could yield novel device designs. One anticipates a growing emphasis on out-of-equilibrium phenomena, particularly those driven by non-unitary processes, where the standard tools of equilibrium statistical mechanics falter. Consistency in these explorations-a dedication to rigorous analysis-will be essential, for future researchers will judge not what was calculated, but how it was done.

Ultimately, the pursuit of charged moments and entanglement is not merely a mathematical exercise. It is an attempt to decipher the fundamental language of quantum matter, to understand how collective behavior emerges from the interactions of countless particles. The answers, when they arrive, will likely be less about grand, unifying theories and more about recognizing the inherent limitations of any given description.

Original article: https://arxiv.org/pdf/2602.12185.pdf

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

Unveiling Hidden Order: Beyond Simple Averages

Mapping the Flow: Ballistic Fluctuation Theory in Action

Dissecting the Complex: Replica Method and Height Field Formulation

Revealing the Signatures: From Theory to Predictive Power

Where Do We Go From Here?

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