Topological Entanglement Entropy from the Holographic Partition Function

TL;DR

This work establishes that the universal topological entanglement entropy in chiral 2+1D topological phases can be captured by a holographic partition function built from edge RCFT data, unifying edge and bulk entropies. Quantum dimensions and the total quantum dimension are computed via the fusion algebra and modular S-matrix, with the Verlinde formula providing the link between RCFT and bulk topological data. The authors show that the edge entropy equals $-\ln \mathcal{D}$ and the bulk entropy arises from fusion-channel degeneracies, and demonstrate how these contributions are encoded in the holographic partition function $Z=\sum_a T_{0a}\chi_a(q)$, which also governs entropy changes under a point contact. They further relate entropy loss at a point contact to a flow between UV and IR fixed points and interpret it as a topological entanglement entropy effect, with concrete examples in Laughlin, Moore-Read, and Read-Rezayi states. Overall, the paper provides a powerful CFT-based framework to quantify and interpret topological entanglement in 2+1D systems via their edge theories, with implications for diagnosing topological order and understanding impurity-like perturbations.

Abstract

We study the entropy of chiral 2+1-dimensional topological phases, where there are both gapped bulk excitations and gapless edge modes. We show how the entanglement entropy of both types of excitations can be encoded in a single partition function. This partition function is holographic because it can be expressed entirely in terms of the conformal field theory describing the edge modes. We give a general expression for the holographic partition function, and discuss several examples in depth, including abelian and non-abelian fractional quantum Hall states, and p+ip superconductors. We extend these results to include a point contact allowing tunneling between two points on the edge, which causes thermodynamic entropy associated with the point contact to be lost with decreasing temperature. Such a perturbation effectively breaks the system in two, and we can identify the thermodynamic entropy loss with the loss of the edge entanglement entropy. From these results, we obtain a simple interpretation of the non-integer `ground state degeneracy' which is obtained in 1+1-dimensional quantum impurity problems: its logarithm is a 2+1-dimensional topological entanglement entropy.

Figures (2)

• Figure 1: The edge modes with spacetime a cylinder (illustrated on the left) have only one chirality. The physics of these is closely related to that of a non-chiral system with spacetime a strip (illustrated on the right), where the boundary conditions couple the left and right movers.
• Figure 2: In the UV, bulk quasiparticles can be anywhere. In the IR, they must end up on one of the two sides. For two bulk quasiparticles of type $a$, they can both end up on one side, or on separate sides.