Entanglement and topological interfaces

TL;DR

This work develops a unified framework for entanglement entropy in 2D CFTs with topological interfaces, showing that the leading term $S \sim \frac{c}{3}\log L$ remains unchanged while a universal subleading constant arises as the negative KL divergence with respect to the identity interface. By mapping boundaries to chiral topological interfaces via the folding trick, the authors derive analogous left/right entanglement entropies and apply the formalism to rational CFTs (notably $su(2)_k$ WZW models) and toroidal bosons, obtaining explicit expressions such as $S = \frac{c}{3}\log L - \log|\Gamma_1/\Gamma_{12}^{\Lambda}|$ for defects and $S_{\text{LREE}} = \frac{c}{6}\log L - \log\operatorname{vol}(\pi(\Gamma^O))$ for tori. The KL interpretation clarifies when interfaces truly induce information loss (and when symmetry or duality defects leave distributions invariant), and the results connect to geometric data like $g$-factors, orbifold groups, and Narain lattices. The paper also discusses left/right entanglement for boundaries, and outlines potential extensions to supersymmetric theories and information-geometric notions of distance between CFTs.

Abstract

In this paper we consider entanglement entropies in two-dimensional conformal field theories in the presence of topological interfaces. Tracing over one side of the interface, the leading term of the entropy remains unchanged. The interface however adds a subleading contribution, which can be interpreted as a relative (Kullback-Leibler) entropy with respect to the situation with no defect inserted. Reinterpreting boundaries as topological interfaces of a chiral half of the full theory, we rederive the left/right entanglement entropy in analogy with the interface case. We discuss WZW models and toroidal bosonic theories as examples.

Figures (2)

• Figure 1: Sketch of the $K$-sheeted Riemann surface we use in the replica trick. After imposing an UV cutoff $\epsilon$, a IR cutoff $L$, the surface corresponds to the cylinder on the upper right. To derive $Z(K)$ we impose periodicity also in the direction of the real part of $z$ to obtain a torus.
• Figure 2: Plots of $-s({\cal D}_a)$ for large values of $k$, together with the asymptotic values \ref{['eq:sDainf']}. The peaks in the plots are captured by the asymptotic expression \ref{['sDkrational']}.