Entanglement on linked boundaries in Chern-Simons theory with generic gauge groups

Abstract

We study the entanglement for a state on linked torus boundaries in $3d$ Chern-Simons theory with a generic gauge group and present the asymptotic bounds of Rényi entropy at two different limits: (i) large Chern-Simons coupling $k$, and (ii) large rank $r$ of the gauge group. These results show that the Rényi entropies cannot diverge faster than $\ln k$ and $\ln r$, respectively. We focus on torus links $T(2,2n)$ with topological linking number $n$. The Rényi entropy for these links shows a periodic structure in $n$ and vanishes whenever $n=0\text{ (mod }\text{p})$, where the integer $\text{p}$ is a function of coupling $k$ and rank $r$. We highlight that the refined Chern-Simons link invariants can remove such a periodic structure in $n$.

31 pages, 5 figures