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

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/correlation measures such as the mutual information and the entanglement negativity. In particular, it is found that the entanglement negativity of two adjacent noncontractible regions on a torus provides a simple way to distinguish Abelian and non-Abelian topological orders.

Figure 1

Various setups discussed in Sec. 3a. (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 antiquasiparticle $\overline{a}$. 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. 2, the bipartition is along the other noncontractible 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.

Figure 6

Top row: A $S^2$ with four quasiparticles, with the subsystem $A$ containing two quasiparticles $a$ and $\overline{a}$. Each red solid line represents a Wilson line connecting quasiparticles $\overline{a}$ and $a$. The two configurations represent two states $|{\mathrm{\Psi }}_1\rangle$ and $|{\mathrm{\Psi }}_2\rangle$, respectively. Bottom row: A $S^2$ with four quasiparticles, with two quasiparticles $\overline{a}$ and $\overline{a}$ in the subsystem $A$, and the other two quasiparticles $a$ and $a$ in the subsystem $B$. Each red solid line represent a Wilson line that connects $\overline{a}$ and $a$. The two configurations represent two states $|{\mathrm{\Psi }}_1^{\prime }\rangle$ and $|{\mathrm{\Psi }}_2^{\prime }\rangle$, respectively.

Figure 7

Top row: A $S^2$ with four quasiparticles, with two quasiparticles $\overline{a}$ and $\overline{b}$ in subsystem $A$, and the other two quasiparticles $a$ and $b$ in subsystem $B$. The red solid lines are Wilson lines which connect $\overline{a}\left(\overline{b}\right)$ and $a\left(b\right)$. The two configurations represent two states $|{\mathrm{\Psi }}_1\rangle$ and $|{\mathrm{\Psi }}_2\rangle$, respectively. Bottom row: A $S^2$ with four quasiparticles, with two quasiparticles $\overline{a}$ and $\overline{a}$ in subsystem $A$, and the other two quasiparticles $a$ and $a$ in subsystem $B$. The two configurations correspond to the two states $|{\mathrm{\Psi }}_1^{\prime }\rangle$ and $|{\mathrm{\Psi }}_2^{\prime }\rangle$, respectively.

Figure 8

Four setups in calculating the mutual information and the entanglement negativity. Two adjacent noncontractible regions $A_1$ and $A_2$ on a torus with noncontractible [(a) and (b)] and contractible (c) $B$. (d) Two disjoint noncontractible regions $A_1$ and $A_2$ on a torus with noncontractible region $B$. The red solid line represents a Wilson loop threading through the interior of the torus.