Symmetry-resolved Entanglement Entropy from Heat Kernels Accurately Computes Charged System Entropies Via a Globally Convergent Expansion

Understanding entanglement, a key feature of quantum mechanics, requires increasingly sophisticated methods when applied to complex systems, particularly those with internal symmetries and external forces. Yuan-Chun Jing, Chao Niu, and Zhuo-Yu Xian, from Jinan University and Freie Universität Berlin, have developed a new framework for calculating symmetry-resolved entanglement entropy, a measure of quantum connectedness that accounts for these symmetries. Their approach improves upon existing methods by accurately handling charged systems where traditional calculations often fail, and importantly, connects this calculation to real-space renormalization techniques. The team achieves this by reconstructing the mathematical structure of the heat kernel, a central tool in quantum field theory, allowing for a globally accurate and convergent calculation of entropy. This advancement not only unifies the treatment of charged and neutral systems, but also provides a powerful new tool for investigating quantum entanglement in diverse physical settings, including conformal field theories and their holographic counterparts.

Systems based on an improved heat kernel approach are investigated. While the conventional Sommerfeld formula effectively calculates entanglement entropy for neutral systems, it encounters limitations when dealing with systems possessing gauge fields or chemical potentials, due to incomplete mathematical prescriptions and violations of boundary conditions. Scientists overcame these challenges by reconstructing the analytic structure of the heat kernel using a sign-dependent phase factor, deriving a globally convergent expansion that accurately combines discrete summations with continuous spectral decompositions. This framework successfully calculates entanglement entropy for Gaussian continuous multi-scale entanglement renormalization ansatz, or cMERA, states.

Entanglement, Thermalization and Holographic Tensor Networks

This collection of research focuses on quantum information theory, condensed matter physics, quantum gravity, quantum field theory, and numerical methods. Key themes include entanglement, entanglement entropy, symmetry-resolved entanglement, quantum thermalization, and the AdS/CFT correspondence, a theoretical framework linking gravity to quantum field theory. Early foundational work by White and Steven established the Density Matrix Renormalization Group, a cornerstone numerical method for studying strongly correlated systems, while Susskind’s Holographic Principle proposed that a volume of space can be described by information on its boundary. Recent research emphasizes symmetry-resolved entanglement, investigating how entanglement is distributed among different symmetry sectors of a quantum system.

Experiments by Azses and colleagues detected symmetry-protected topological states and symmetry-resolved entanglement on quantum computers, while Neven and others developed methods for detecting this type of entanglement using partial transpose moments. Numerical methods, such as Matrix Product States and the continuous Multiscale Entanglement Renormalization Ansatz, provide powerful tools for representing and studying quantum states. Investigations into quantum quenches and dynamics, led by Parez and colleagues, explore how symmetry-resolved entanglement evolves after a sudden change in the system’s parameters. This body of work represents a comprehensive exploration of entanglement and its role in understanding complex quantum systems.

Entanglement Entropy via Heat Kernel Reconstruction

Scientists have developed a novel framework for computing symmetry-resolved entropies in systems with electric charge, building upon an improved heat kernel approach. Traditional methods for calculating entanglement entropy, like the Sommerfeld formula, prove limited when applied to charged systems due to inaccuracies in mathematical prescriptions and violations of boundary conditions. This work overcomes these limitations by reconstructing the analytic structure of the heat kernel using a sign-dependent phase factor, resulting in a globally convergent expansion that accurately combines discrete summations with continuous spectral decompositions. The team demonstrated that entanglement entropy can be expressed in terms of the flow functions of Gaussian continuous multi-scale renormalization ansatz, or cMERA, states.

Crucially, they derived a symmetry-resolved entropy flow equation applicable even when a chemical potential is present, extending the framework’s utility to arbitrary spacetime dimensions and recovering established results for neutral systems as the chemical potential approaches zero. Validation of the framework involved achieving exact agreement with predictions from (1+1)-dimensional conformal field theory using twist-operator techniques and confirming consistency with holographic entropy calculations on S1 × H^(d-1) geometries. This method unifies the treatment of both charged and neutral entanglement entropy, extending its application to real-space renormalization frameworks. The results provide a robust tool for probing symmetry-resolved entanglement in conformal field theories, their holographic duals, and cMERA representations, delivering a significant advancement in understanding quantum correlations and their relationship to symmetry.

Symmetry-Resolved Entropy in Charged Quantum Systems

Scientists developed a framework for calculating symmetry-resolved entropies in systems with electric charge, building upon an improved heat kernel approach. This work addresses limitations encountered when applying standard formulas to charged environments, ensuring accurate calculations even with the presence of gauge fields or chemical potentials. The team successfully reconciled discrete and continuous mathematical techniques, creating a globally convergent expansion for these calculations. The framework was applied to complex quantum states known as Gaussian continuous multi-scale renormalization ansatz states, revealing a novel equation describing how symmetry-resolved entropy changes as the system evolves. Validating this approach, the researchers demonstrated agreement with predictions from conformal field theory and holographic calculations, confirming the framework’s accuracy and broad applicability. This work unifies the treatment of charged and neutral systems and extends it to real-space renormalization methods, offering a robust tool for investigating symmetry-resolved entanglement in diverse physical contexts.