Universal tripartite entanglement signature of ungappable edge states

Gapped two-dimensional topological phases can feature ungappable edge states which are robust even in the absence of protecting symmetries. In this Letter we argue that a multipartite entanglement measure recently proposed in the context of holography, the Markov gap, provides a universal diagnostic of ungappable edge states. Defined as a difference of the reflected entropy and mutual information $h\left(A\text{:}B\right)=S_R\left(A\text{:}B\right)-I\left(A\text{:}B\right)$ between two parties, we argue that for $A,B$ being adjacent subregions in the bulk $h=\frac{c_{+}}{3}ln2$, where $c_{+}$ is the minimal total central charge of the boundary theory. As evidence, we prove that $h=0$ for string-net models and numerically verify that $h=\frac{\left|C\right|}{3}ln2$ for a Chern-$C$ insulator. Our Letter establishes a unique bulk entanglement criteria for the presence of a conformal field theory on the boundary.

Figure 1

(a) Tripartition of a 1D system on the circle. (b) A 2D system on a square lattice.

Figure 2

(a) The tripartition of a two-dimensional system with punctures illustrated by shaded regions. There are left- and right-moving modes on the boundary of the shaded regions. Starting from the geometry in Fig. 1, we act with unitary disentanglers on the larger circular regions $N$ and $S$ (including the shaded interiors). Intuitively, in systems with a gapped bulk, such a unitary can be chosen to turn the degrees of freedom in the shaded interior into a product state. (b) An open cylinder with a tripartition where the three parties are strips that connect partitions of the two circles. Viewing the product state regions as punctures, the two geometries are topologically equivalent and can be deformed into each other using finite-depth local unitaries.

Figure 3

(a) Toric code on a mesh covering the surface of a sphere. Degrees of freedom (0 or 1, denoted by circles) live on links, and the ground state is a uniform superposition over configurations of 0's and 1's satisfying the constraint that at each vertex an even number of 1's must meet. (b) The configuration in (a) may be reduced to this one using local unitaries acting on $A,B$, and $C$.

Figure 4

$h$ for the Hofstadter model with Chern number $C=1$. The dashed line denotes the theoretical value $\frac{1}{3}ln2$.