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

TL;DR

This work shows that symmetry-resolved entanglement for non-invertible symmetries in 1+1d CFTs must be treated with boundary-aware projections, since the boundary fusion algebra can differ from the bulk. By developing defect-network methods and explicit boundary projectors $P_r^{[a,b]}$, the authors reveal that equipartition breaking in the SREE depends on the boundary data and yields $\Delta S_n(q,r)=\log \frac{d_r}{d_a d_b}$ (or $\log \frac{d_r}{\sqrt{\mathcal{C}}}$ in symmetric boundary setups), deviating from bulk-only predictions. The paper applies these ideas to the Fibonacci category (three-state Potts and tricritical Ising) and Rep$(S_3)$ in SU$(2)_4$, with numerical checks on open anyonic chains that corroborate the boundary-modified predictions and highlight discrepancies with prior Cat-SREE results. Collectively, the results underscore the need to incorporate boundary conditions and defect fusion data when linking entanglement measures to non-invertible symmetries, with implications for SymTFT frameworks and lattice realizations.

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.

Figures (13)

• Figure 1: Two slits of radius $\varepsilon$ are inserted at the entangling surface and boundary conditions $a$ and $b$ are imposed. By a conformal transformation, the region $A$ is mapped to an annulus of width $w$ with boundary states $\ket{a}$ and $\ket{b}$.
• Figure 2: The $F$-symbols allow manipulation of defect networks.
• Figure 3: Fusing a defect parallel with a boundary naturally results in normalisation factors of the junction between the defect and boundary. These normalisation factors are denoted with a bold circle, as shown in the second line.
• Figure 4: With the normalisation factors in \ref{['fig:junction-norm']}, defects can be shrunk on a boundary with no additional factors incurred.
• Figure 5: The boundary conditions $a$ and $b$ are separated from the reference, identity boundary $1$. The boundary condition changing operator $\psi_i$ is traded for a defect line with the same label $i$, incurring a factor of $\alpha_i^{ab}$.