Entanglement entropy and nonabelian gauge symmetry

TL;DR

This paper reviews a method to define and compute entanglement entropy in gauge theories, including nonabelian cases, by embedding the physical Hilbert space into a product space with boundary degrees of freedom, and tests it in 2D Yang-Mills theory.

Contribution

It introduces a gauge-invariant definition of entanglement entropy for nonabelian gauge theories and demonstrates its consistency with known results in 2D Yang-Mills theory.

Findings

Surface degrees of freedom contribute to entanglement entropy.

The definition matches thermal entropy in de Sitter space.

Results agree with Euclidean replica trick calculations.

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.