Symmetry-Resolved Entanglement Entropy from Heat Kernels

TL;DR

This work develops an improved heat-kernel framework to compute symmetry-resolved entanglement entropies in charged quantum systems, addressing limitations of the traditional Sommerfeld approach when gauge fields and chemical potentials are present. By introducing a sign-dependent phase regulator, the authors obtain a globally convergent expansion that reconciles discrete residue sums with continuous spectral decompositions, enabling charged Rényi entropies in arbitrary dimensions and recovering neutral results in the $ obreak mu o 0$ limit. The framework is validated through exact matches with $(1+1)$D CFT twist-operator results and holographic entropy calculations on $S^1 imes H^{d-1}$, and is extended to Gaussian cMERA, establishing a charged entanglement flow equation that connects RG scale to symmetry-resolved entanglement. The work unifies charged and neutral entanglement, extends to curved backgrounds, and lays groundwork for a direct cMERA formulation of SREE, with potential implications for holography and quantum information in conformal field theories.

Abstract

We develop a systematic framework for computing symmetry-resolved entanglement entropies (SREE) in charged quantum systems based on an improved heat kernel approach. Although the conventional Sommerfeld formula proves effective for neutral systems, it encounters limitations when gauge fields or chemical potentials are introduced due to incomplete residue prescriptions and violations of asymptotic boundary conditions. By reconstructing the analytic structure of the heat kernel using a sign-dependent phase factor, we derive a globally convergent expansion that reconciles discrete residue summations with continuous spectral decompositions. We further apply this framework to Gaussian continuous multi-scale entanglement renormalization ansatz (cMERA) states and show that the entanglement entropy (EE) can be expressed in terms of the cMERA flow functions. In particular, we obtain a symmetry-resolved entanglement entropy flow equation in the presence of a chemical potential. This formulation extends naturally to arbitrary spacetime dimensions and recovers established results for neutral systems in the mu -> 0 limit. We validate our framework through two settings: (1) exact agreement with (1+1)-dimensional conformal field theory (CFT) predictions using twist-operator techniques, and (2) consistency with holographic entropy calculations on S1 x H^(d-1) geometries. Our results both unify the treatment of charged and neutral entanglement entropy and extend this treatment to real-space renormalization frameworks, providing a robust tool for probing symmetry-resolved entanglement in conformal field theories, their holographic duals, and cMERA representations.

Figures (1)

• Figure 1: Integration contours $\Gamma_0$, $\Gamma_1$, and $\Gamma_2$ in the complex $w$-plane. All contours are topologically equivalent, enclosing the same set of poles except for the one at the origin.