Universal tripartite entanglement signature of ungappable edge states

Abstract

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 $h(A\text{:}B)={S}_{R}(A\text{:}B)\ensuremath{-}I(A\text{:}B)$ between two parties, we argue that for $A,B$ being adjacent subregions in the bulk $h=\frac{{c}_{+}}{3}ln\phantom{\rule{0.16em}{0ex}}2$, 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 $h=\frac{|C|}{3}ln\phantom{\rule{0.16em}{0ex}}2$ for a Chern-$C$ insulator. Our Letter establishes a unique bulk entanglement criteria for the presence of a conformal field theory on the boundary.