Topological Entanglement Entropy in Chern-Simons Theories and Quantum Hall Fluids

TL;DR

This work derives the universal topological entanglement entropy for 2+1D Chern-Simons theories by a surgery/replica approach and connects gamma to the modular data of the accompanying CFT, namely the S-matrix, quantum dimensions, and fusion rules. It provides explicit results for Abelian and non-Abelian theories, including U(1)_m Laughlin states, SU(2)_k and coset constructions, and Moore-Read/Read-Rezayi parafermion states, showing how gamma emerges from the topology of the observed region and the fusion channels of punctures. A key finding is that, apart from degenerate-vacuum cases, the entanglement entropy depends primarily on quantum dimensions and fusion rules rather than the full S-matrix, and that in degenerate situations the state choice can influence the entropy. The results offer a principled way to extract topological data from entanglement measurements and have implications for interferometry and topological quantum computation in fractional quantum Hall systems.

Abstract

We compute directly the entanglement entropy of spatial regions in Chern-Simons gauge theories in 2+1 dimensions using surgery. We use these results to determine the universal topological piece of the entanglement entropy for Abelian and non-Abelian quantum Hall fluids.

Figures (17)

• Figure 1: Conceptual picture of $\rho_A$. The trace over B corresponds to gluing $\tau=0$ to $\tau=\beta$ in the B region, leaving a cut open in the A region.
• Figure 2: $\textrm{tr}\rho_A^3$ is obtained by gluing three copies of the diagram in Fig. \ref{['fig:Cut_Tube_No_Fields']} back to back along the cut in the A region.
• Figure 3: Shading implies a solid 3-ball. With a one-component interface, the A and B regions are disks. It is useful in the following constructions to view the 3-ball as a disk rotated about an axis passing through the origin, as shown at right.
• Figure 4: For spatial topology $S^2$ with one interface component, we explicitly show the construction of $\textrm{tr}\rho_A^2$. The overall manifold is generated by four pieces of disks glued together one after another and rotated along the same axis as in Fig. \ref{['fig:S2single']}.
• Figure 5: The space $X_{n=2}$, obtained by gluing $n=2$ copies of the space shown in Fig. \ref{['fig:T2single']}. The result is an $S^3$ joined to two copies of $S^2\times S^1$ along $S^2$'s. For general $n$, the glued geometry $X_n$ consists of an $S^3$ joined in this way to $n$$S^2\times S^1$'s.