Emergent geometry from the entanglement of fractional charges of zigzag graphene nanoribbon

The topological order in zigzag graphene nanoribbons arises from the interplay between edge termination, electron–electron interactions, and disorder [1]. The topological order manifests in several ways, including a universal value of the topological entanglement entropy and the presence of e^-/2 fractional charges localized at the zigzag edges [2]. In this work, we investigate the long-range entanglement between these fractional charges. We embed the ribbon into an AdS-like bulk geometry and employ the mutual information to quantify the entanglement. We demonstrate the geometric emergence in the entanglement where the edge-to-edge mutual information plays a role as geodesic distance in hyperbolic space. Our results offer a holographic picture of fractionalized degrees of freedom in quasi-one-dimensional systems [3].