Reflected entropy in random tensor networks. Part II. A topological index from canonical purification

A bstract In ref. 1, we analyzed the reflected entropy (S R) in random tensor networks motivated by its proposed duality to the entanglement wedge cross section (EW) in holographic theories, SR=2EW4G S R = 2 EW 4 G. In this paper, we discover further details of this duality by analyzing a simple network consisting of a chain of two random tensors. This setup models a multiboundary wormhole. We show that the reflected entanglement spectrum is controlled by representation theory of the Temperley-Lieb algebra. In the semiclassical limit motivated by holography, the spectrum takes the form of a sum over superselection sectors associated to different irreducible representations of the Temperley-Lieb algebra and labelled by a topological index k ∈ ℤ >0. Each sector contributes to the reflected entropy an amount 2kEW4G 2 k EW 4 G weighted by its probability. We provide a gravitational interpretation in terms of fixed-area, higher-genus multiboundary wormholes with genus 2 k – 1 initial value slices. These wormholes appear in the gravitational description of the canonical purification. We confirm the reflected entropy holographic duality away from phase transitions. We also find important non-perturbative contributions from the novel geometries with k ≥ 2 near phase transitions, resolving the discontinuous transition in S R. Along with analytic arguments, we provide numerical evidence for our results. We finally speculate that signatures of a non-trivial von Neumann algebra, connected to the Temperley-Lieb algebra, will emerge from a modular flowed version of reflected entropy.