Ishibashi States, Topological Orders with Boundaries and Topological Entanglement Entropy

TL;DR

This work analyzes how gapped boundaries and interfaces in 2+1D bosonic topological orders affect topological entanglement entropy (TEE). By leveraging Ishibashi states and anyon condensation, the authors build ground-state bases and conformal boundary conditions for Abelian and non-Abelian (D(G)) theories, linking TEEs to the pattern of condensation and to intermediate phases that extend or enhance chiral symmetry. They demonstrate explicit entanglement results for cylinders with and without interfaces, recovering known lattice results and deriving new interface-sensitive contributions governed by condensation data and branching structures. The framework unifies boundary/interface physics with entanglement in topological phases, offering a CFT/anyon-condensation perspective with potential applications to more complex interfaces and endpoint entanglement problems.

Abstract

In this paper, we study gapped edges/interfaces in a 2+1 dimensional bosonic topological order and investigate how the topological entanglement entropy is sensitive to them. We present a detailed analysis of the Ishibashi states describing these edges/interfaces making use of the physics of anyon condensation in the context of Abelian Chern-Simons theory, which is then generalized to more non-Abelian theories whose edge RCFTs are known. Then we apply these results to computing the entanglement entropy of different topological orders. We consider cases where the system resides on a cylinder with gapped boundaries and that the entanglement cut is parallel to the boundary. We also consider cases where the entanglement cut coincides with the interface on a cylinder. In either cases, we find that the topological entanglement entropy is determined by the anyon condensation pattern that characterizes the interface/boundary. We note that conditions are imposed on some non-universal parameters in the edge theory to ensure existence of the conformal interface, analogous to requiring rational ratios of radii of compact bosons.

Figures (4)

• Figure 1: The $D(\mathbb{Z}_N)-D(\mathbb{Z}_M)$ interface on a cylinder. The doubled red and blue lines at the interface denotes the pair of left and right moving modes in the upper and lower entanglement cut.
• Figure 2: For a cylinder topology with non-trivial GSD, the ground states can be specified either by anyon line connecting the upper and lower physical boundary, or by anyon loop winding around non-contractible cycle.
• Figure 3: A cylindrical region $R$ is embedded in a cylinder. The entanglement cut (white dashed line) separates region $R$ and $\bar{R}$. An anyon line (yellow) connects the upper and lower boundary.
• Figure 4: There is an interface between region $R$ and $\bar{R}$, which coincides with the entanglement cut (white dashed line). Different anyons are matched across the interface.