Entanglement entropy and nonabelian gauge symmetry

TL;DR

This work addresses the challenge of defining entanglement entropy in gauge theories where region-based Hilbert space factorization fails. It proposes a canonical embedding of the physical Hilbert space into regionally split spaces that include boundary degrees of freedom acting as surface charges, yielding a gauge-invariant entanglement entropy with explicit boundary and bulk contributions. In two-dimensional gauge theories, the authors demonstrate that the entropy matches known Euclidean and thermal results: in de Sitter space it reproduces the thermal entropy, and via the replica trick it aligns with established calculations in YM2. The lattice extension shows a three-term decomposition into boundary, boundary-dimension, and bulk entropies, highlighting edge modes as a universal feature of gauge entanglement with potential implications for gravity and holography.

Abstract

Entanglement entropy has proven to be an extremely useful concept in quantum field theory. Gauge theories are of particular interest, but for these systems the entanglement entropy is not clearly defined because the physical Hilbert space does not factor as a tensor product according to regions of space. Here we review a definition of entanglement entropy that applies to abelian and nonabelian lattice gauge theories. This entanglement entropy is obtained by embedding the physical Hilbert space into a product of Hilbert spaces associated to regions with boundary. The latter Hilbert spaces include degrees of freedom on the entangling surface that transform like surface charges under the gauge symmetry. These degrees of freedom are shown to contribute to the entanglement entropy, and the form of this contribution is determined by the gauge symmetry. We test our definition using the example of two-dimensional Yang-Mills theory, and find that it agrees with the thermal entropy in de Sitter space, and with the results of the Euclidean replica trick. We discuss the possible implications of this result for more complicated gauge theories, including quantum gravity.