On the Boundary Conformal Field Theory Approach to Symmetry-Resolved Entanglement

TL;DR

This paper develops a boundary conformal field theory approach to decompose and analyze symmetry-resolved entanglement entropies in 2D CFTs, especially with extended symmetries, by relating bipartitions to annulus geometries and boundary conditions.

Contribution

It introduces a method to obtain symmetry-resolved entanglement spectra directly from BCFT partition functions without charged moment calculations, applicable to theories with various symmetries.

Findings

Decomposition of BCFT partition functions yields symmetry resolution of entanglement spectrum.

Symmetry-resolved Rènyi entropies derived to all orders in UV cutoff expansion.

Applied to free boson theories with $U(1)$, $ ext{R}$, and $ ext{Z}_2$ symmetries.

Abstract

We study the symmetry resolution of the entanglement entropy of an interval in two-dimensional conformal field theories (CFTs), by relating the bipartition to the geometry of an annulus with conformal boundary conditions. In the presence of extended symmetries such as Kac-Moody type current algebrae, symmetry resolution is possible only if the boundary conditions on the annulus preserve part of the symmetry group, i.e. if the factorization map associated with the spatial bipartition is compatible with the symmetry in question. The partition function of the boundary CFT (BCFT) is then decomposed in terms of the characters of the irreducible representations of the symmetry group preserved by the boundary conditions. We demonstrate that this decomposition already provides the symmetry resolution of the entanglement spectrum of the corresponding bipartition. Considering the various terms of the partition function associated with the same representation, or charge sector, the symmetry-resolved R\'enyi entropies can be derived to all orders in the UV cutoff expansion without the need to compute the charged moments. We apply this idea to the theory of a free massless boson with $U(1)$, $\mathbb{R}$ and $\mathbb{Z}_2$ symmetry.