Revisiting the symmetry-resolved entanglement for non-invertible symmetries in $1{+}1$d conformal field theories

TL;DR

This paper investigates how boundary conditions affect the calculation of symmetry-resolved entanglement entropy in 1+1D conformal field theories with non-invertible symmetries, revealing modifications to the theoretical framework and providing numerical validation.

Contribution

It identifies the impact of boundary conditions on the construction of projectors for non-invertible symmetries and revises the existing theoretical approach with numerical support.

Findings

Boundaries alter the fusion algebra of non-invertible symmetries.

The modified formalism differs from previous proposals.

Numerical simulations support the revised predictions.

Abstract

Recently, a framework for computing the symmetry-resolved entanglement entropy for non-invertible symmetries in $1{+}1$d conformal field theories has been proposed by Saura-Bastida, Das, Sierra and Molina-Vilaplana [Phys. Rev. D109, 105026]. We revisit their theoretical setup, paying particular attention to possible contributions from the conformal boundary conditions imposed at the entangling surface -- a potential subtlety that was not addressed in the original proposal. We find that the presence of boundaries modifies the construction of projectors onto irreducible sectors, compared to what can be expected from a pure bulk approach. This is a direct consequence of the fusion algebra of non-invertible symmetries being different in the presence or absence of boundaries on which defects can end. We apply our formalism to the case of the Fibonacci category symmetry in the three-state Potts and tricritical Ising model and the Rep($S_3$) fusion category symmetry in the $SU(2)_4$ Wess-Zumino-Witten conformal field theory. We numerically corroborate our findings by simulating critical anyonic chains with these symmetries as a finite lattice substitute for the expected entanglement Hamiltonian. Our predictions for the symmetry-resolved entanglement for non-invertible symmetries seem to disagree with the recent work by Saura-Bastida et al.