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

TL;DR

This paper investigates entanglement entropy in 3D Chern-Simons theory with various gauge groups, revealing bounds on entropy growth and periodic behaviors in linked torus states, with implications for topological quantum invariants.

Contribution

It provides asymptotic bounds on Rénnyi entropy for linked boundaries in Chern-Simons theory with generic gauge groups, and analyzes the periodic structure in torus link entanglement.

Findings

Rénnyi entropy cannot grow faster than ln(k) or ln(r) in large coupling or rank limits.

Entanglement entropy exhibits a periodic structure in the linking number n, vanishing at specific moduli.

Refined Chern-Simons invariants can eliminate the periodic structure in the entanglement pattern.

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 R\'enyi 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 R\'enyi 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 R\'enyi entropy for these links shows a periodic structure in $n$ and vanishes whenever $n = 0 \text{ (mod } \textsf{p})$, where the integer $\textsf{p}$ 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 $n$.