Ishibashi states, topological orders with boundaries and topological entanglement entropy. Part I

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.

Ishibashi states, topological orders with boundaries and topological entanglement entropy. Part II. Cutting through the boundary

“…In physical terms, the Lagrangian algebra A corresponds to a set of anyons that condense at the boundary -they are not conserved across the boundary. This is reviewed in detail in section 2 of our companion paper [5]. We pick out a few important points here.…”

“…In the current situation, the added complication is the physical boundaries that the entanglement cut ends on. There is some appropriate boundary condition at each end, and they are precisely conformal boundary conditions [5]. Now for a given pair of boundary condition {x, y}, it determines the Hilbert space H xy .…”

“…Each entanglement cut then corresponds to a generalized Ishibashi state matching Φ i L and Φ i R , in a way completely analogous to the boundary Ishibashi state in [5] except that…”

“…There has been a lot of recent work exploring the effect topological boundaries have on the entanglement entropy of topological phases of matter [1,2,3,4,5,6,7]. In our last paper [5] we computed topological entanglement entropies of 2+1 dimensional topological order using Ishibashi states at the entanglement cut. This paper is a companion paper where we inspect the situation where the entanglement cut touches the topological boundaries.…”

Bulk entanglement entropy for photons and gravitons in AdS$_3$

“…We believe our results may have applications to topological phases in condensed matter theory. In particular, our work addresses in the context of a gapless edge theory issues similar to those discussed for a gapped edge theory in [34,35].…”