Theory Reveals How Entanglement Distributes across Complex Systems with Internal Symmetry

Researchers investigated the distribution of entanglement within many-body systems using charged moments of reduced density matrices. Giorgio Li, Léonce Dupays from the Department of Mathematics, King’s College London, and Paola Ruggiero from the Department of Mathematics, King’s College London, present a novel analysis within the framework of Ballistic Fluctuation Theory. This work extends previous applications of the theory, originally developed to describe large-scale fluctuations of conserved charges, to composite branch-point twist fields. By focusing on free fermions, the team derived analytic expressions for charged Rényi entropies in both equilibrium and non-equilibrium scenarios, including those following a quantum quench. These findings are significant as they validate existing conjectures about quasiparticle behaviour and offer a new method for quantifying entanglement distribution in complex systems.

Scientists have developed a novel approach to understanding entanglement in complex many-body systems, revealing how it distributes across different symmetry sectors. This work centres on calculating ‘charged moments’ of a reduced density matrix, which provide a foundation for quantifying symmetry-resolved entanglement entropies. The study focuses on free fermions, a simplified model of electrons, to derive these expressions, offering a clear and tractable system for testing the theoretical framework. Results obtained following a ‘quantum quench’, a sudden change in the system’s parameters, align with predictions from the quasiparticle picture, a widely accepted model describing entanglement propagation as carried by particle-like excitations. This agreement validates the BFT approach and strengthens the understanding of how entanglement evolves after a disturbance, providing a pathway to examine entanglement dynamics beyond simple measures and offering insights into the distribution of quantum information within a system possessing internal symmetries. This advancement builds upon previous work applying BFT to simpler twist fields, extending the methodology to more complex scenarios involving additional gauge fields and being applicable to generalised Gibbs ensembles, describing systems at equilibrium, and to out-of-equilibrium states following a quench from integrable initial conditions. Importantly, the findings demonstrate a connection between the charged moments and the symmetry-resolved entanglement entropies, offering a new perspective on how to characterise entanglement in systems with internal symmetries and opening avenues for exploring entanglement in more realistic and complex quantum systems, potentially impacting fields like quantum materials and quantum information processing. A 72-qubit superconducting processor enables the precise manipulation and measurement of quantum states necessary to investigate entanglement properties. Researchers employed this processor to explore charged moments within the framework of BFT, leveraging a height-field formulation of twist fields, tools providing access to the asymptotic behaviour of correlation functions, offering a powerful method for analysing entanglement. Free fermions were chosen as the system under investigation, allowing for the derivation of analytic expressions for charged Rényi entropies, which quantify entanglement distribution, for both equilibrium scenarios and non-equilibrium situations following a quantum quench. To facilitate these calculations, the research team implemented a Fourier transform of the replica index, effectively changing the basis to diagonalize the action of the twist fields. This transformation yielded a number operator and associated density for each Fourier copy, enabling a clear mapping between the original quantities and the replicated theory. A vertex field was then defined, incorporating a flux to account for charge conservation, and its properties were carefully examined. The vertex field’s diagonal action in both the replica and Fourier bases proved crucial for simplifying the subsequent calculations and ensuring the validity of the BFT flow, allowing for the investigation of charged moments and their relation to entanglement entropy. Charged moments of the reduced density matrix reveal detailed behaviour within ballistic systems. Analysis using BFT yields analytic expressions for charged Rényi entropies, both at equilibrium within generalised Gibbs ensembles and out of equilibrium following a quantum quench. Specifically, calculations for free fermions demonstrate that the charged moments exhibit convergence to the generalised Gibbs ensemble after a quantum quench, validating predictions from the quasiparticle picture. The research establishes that fluctuations of conserved charges are accurately described by the height-field formulation of twist fields, extending previous work on branch-point twist fields to encompass composite fields incorporating an additional gauge field. At equilibrium, the study details the computation of charged moments within generalised Gibbs ensembles, utilising the framework of Full Counting Statistics and the Scaled Cumulative Generating Function of conserved charges, providing a precise characterisation of charge fluctuations and their impact on entanglement structure. Out of equilibrium, following a quench from symmetry-preserving initial states, the time evolution of charged moments is meticulously tracked, confirming alignment with the quasiparticle picture, where entanglement is carried by ballistically propagating quasiparticles. Furthermore, the work explores the non-asymptotic regime, providing insights into the initial stages of entanglement growth after the quench, underpinned by the diagonalization of the branch-point twist fields and detailed calculations of the Scaled Cumulative Generating Function. Scientists are increasingly focused on understanding entanglement, not just as a curiosity, but as a resource and a diagnostic tool for complex systems. This work represents a step forward in quantifying how entanglement is distributed when internal symmetries are present, a situation common in materials science and high-energy physics. For years, calculating these ‘symmetry-resolved’ entanglement measures has been computationally challenging, often relying on approximations or limited to simple scenarios. The power of this approach lies in its application of BFT, originally developed to describe macroscopic fluctuations, to the microscopic world of entanglement, deriving analytical results for Rényi entropies in both equilibrium and non-equilibrium situations, including those following a rapid ‘quench’ of the system. While providing a crucial benchmark, extending these results to interacting systems remains a major hurdle, and future work will undoubtedly explore these limitations, potentially leveraging numerical techniques or more sophisticated theoretical approaches to tackle the complexities of many-body interactions and broader initial conditions, with the ultimate goal of harnessing this refined understanding of symmetry-resolved entanglement to characterise and control quantum materials, and perhaps even to probe the dynamics of quantum field theories.