A 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 work we show 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 $h(A:B) = S_R(A:B) - I(A:B)$ between two parties, we argue that for $A,B$ being adjacent subregions in the bulk, $h=\frac{c_+}{3}\log 2$, 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 $h=\frac{|C|}{3}\log 2$ for a Chern-$C$ insulator. Our work  establishes a unique bulk entanglement criteria for the presence of a conformal field theory on the boundary.