Large-party limit of topological entanglement entropy in Chern-Simons theory

TL;DR

This work analyzes the large-party limit of topological entanglement entropy in (2+1)-D Chern-Simons theory with compact gauge group $G$ and level $k$, focusing on $d$-party states associated with torus-link complements $S^3\setminus T_{dm,dn}$. By expressing the $d$-party state coefficients through modular $\mathcal{S}$ and $\mathcal{T}$ data and performing a basis change, the authors show that in the limit $d\to\infty$ only Abelian anyons (tied to the center $Z_G$) contribute to entanglement, yielding an upper bound $EE_{\mathrm{LP}}\le\ln|Z_G|$. They provide an explicit SU(2) example with $M_R=| (\mathcal{S}^*\mathcal{T}^n\mathcal{S})_{R0} |^2$, where the large-$d$ entropy reduces to a two-eigenvalue problem and exhibits parity-dependent behavior in $n$ and $k$; they further compute the large-$k$ semiclassical limit and obtain closed-form expressions involving coefficients $C(r,n)$ and $D(r,n)$. These results illuminate the Abelian-dominated structure of topological entanglement in large-party limits and offer a framework for extending the analysis to more general link complements in topological quantum field theories.

Abstract

We investigate the topological entanglement entropy of quantum states arising in the context of three-dimensional Chern-Simons theory with compact gauge group $G$ and Chern-Simons level $k$. We focus on the quantum states associated with the $T_{dm,dn}$ torus link complements, which is a $d$-party pure quantum state, and analyze its large-party limit, i.e., $d\to \infty$ limit. We show that the entanglement measures in this limit will receive contributions only from the Abelian anyons, and non-Abelian sectors are suppressed in the large-party limit. Consequently, the large-party limiting value of the entanglement entropy has an upper bound of $\ln |Z_G|$, where $|Z_G|$ is the order of the center of the group $G$. As an explicit example, we perform quantitative analysis for the simplest case of the SU(2) group and $T_{d,dn}$ torus link to obtain the large-party limit of the entanglement entropy. We further investigate the semiclassical ($k \to \infty$) limit of the entropies after taking the large-party limit for this particular example.

Figures (1)

• Figure 1: Variation of $\text{EE}_{\text{LP}}$ vs $k$ for quantum states $\ket{T_{d,\,dn}}$ and SU(2) group. The plots are shown for $n=2,4,6,8$. The dotted red line indicates the maximum possible value, which is $\ln 2$.