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

TL;DR

This work develops a BCFT framework to compute symmetry-resolved entanglement entropies directly from the spectrum of the entanglement BCFT, avoiding charged moments. By mapping the entangling cut to an annulus and carefully choosing boundary conditions that preserve a target symmetry, the BCFT partition function is decomposed into irreducible representations, yielding all-order expressions for symmetry-resolved entropies in free boson theories with U(1), ℝ, and Z2. The authors demonstrate equipartition of entanglement for U(1)/ℝ sectors and identify a controlled breaking of equipartition for Z2, with explicit results for both compact and non-compact cases and multiple boundary configurations (NN, DD, DN). The approach provides a transparent, efficient alternative to charged moments and offers a platform for generalizing to non-abelian symmetries and holographic contexts, with potential applications to lattice realizations and quantum holography. Overall, the paper advances the understanding of how symmetries shape entanglement spectra in BCFTs and delivers exact, all-orders results in key test models.

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 Rényi 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.

Figures (1)

• Figure 1: The BCFT setup of the entanglement entropy. Small disks of raidus $\epsilon$ are excised around the entangling point (left panel). The resulting manifold is mapped into an annulus (right panel) by a conformal transformation in such a way that the small disks encircling the entangling points become the boundaries of the annulus (blue circles).