Edge theory approach to topological entanglement entropy, mutual information and entanglement negativity in Chern-Simons theories

TL;DR

This work develops an edge-theory framework to compute topological entanglement entropy, mutual information, and entanglement negativity for (2+1)D Chern-Simons theories on general manifolds. By regularizing boundary Ishibashi states per topological sector and exploiting interference between superposed Wilson-line configurations, the approach reveals universal topological data such as quantum dimensions, R-symbols, and monodromy in entanglement signals. It reproduces known topological entanglement entropy results across sphere, torus, and genus-g manifolds, and provides new results for entanglement negativity, including a practical criterion to distinguish Abelian from non-Abelian orders via ground-state dependence. The method offers a more direct, boundary-based route than surgery, extends to higher genus, and yields actionable insights for diagnosing topological phases through entanglement measurements.

Abstract

We develop an approach based on edge theories to calculate the entanglement entropy and related quantities in (2+1)-dimensional topologically ordered phases. Our approach is complementary to, e.g., the existing methods using replica trick and Witten's method of surgery, and applies to a generic spatial manifold of genus $g$, which can be bipartitioned in an arbitrary way. The effects of fusion and braiding of Wilson lines can be also straightforwardly studied within our framework. By considering a generic superposition of states with different Wilson line configurations, through an interference effect, we can detect, by the entanglement entropy, the topological data of Chern-Simons theories, e.g., the $R$-symbols, monodromy and topological spins of quasiparticles. Furthermore, by using our method, we calculate other entanglement measures such as the mutual information and the entanglement negativity. In particular, it is found that the entanglement negativity of two adjacent non-contractible regions on a torus provides a simple way to distinguish Abelian and non-Abelian topological orders.

Figures (8)

• Figure 1: Various setups discussed in Sec. \ref{['SphereP']}. (a) A $S^2$ is bipartited into two subsystems $A$ and $B$, with the interface labeled by $b$. (b) A $S^2$ with a quasiparticle $a$ and an anti-quasiparticle $\bar{a}$. A Wilson line connecting the two quasiparticles threads through the interface $b$. The two quasiparticles correspond to two punctures, and therefore the geometry in (b) is equivalent to a cylinder in topology. (c) A $S^2$ with two pairs of quasiparticles. (d) A $S^2$ with $N$ pairs of quasiparticles.
• Figure 2: A $T^2$ with a two-component $AB$ interface. The region $B$ is connected in (a) and disconnected in (b). $b_1$ and $b_2$ denote the interface that separates $A$ from $B$. The red solid line represents a Wilson loop which may fluctuate among different topological sectors.
• Figure 3: A $T^2$ with a two-component $AB$ interface labeled by $b_1$ and $b_2$. Compared to Fig. \ref{['torus']}, the bipartition is along the other non-contractible cycle on $T^2$. The red (magenta) solid line represents the Wilson loop threading through the interior(exterior) of the torus along the longitudinal(meridional) circle.
• Figure 4: A manifold of genus $g=2$. We have three components of $AB$ interfaces labeled by $b_1$, $b_2$ and $b_3$, respectively. The red solid lines $a$ and $b$ represent two independent Wilson loops threading through the interior of the double torus along the longitudinal circles.
• Figure 5: A manifold of genus $g$ with $g=N$. We have a $(N+1)$-component interface labeled by $b_1, b_2, \cdots, b_{N+1}$, respectively. We consider $N$ independent Wilson loops that thread through the interior of the manifold along the longitudinal circles. Each Wilson loop (red solid lines) can fluctuate among different topological sectors independently.