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

In ref. [1], we analyzed the reflected entropy (SR) in random tensor networks motivated by its proposed duality to the entanglement wedge cross section (EW) in holographic theories, S-R = 2EW/4G. 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 Temp erley-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 is an element of Z(> 0). Each sector contributes to the reflected entropy an amount 2EW/4G weighted by its probability. We provide a gravitational interpretation in terms of fixed-area, higher-genus multiboundary wormholes with genus 2k - 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 Temp erley-Lieb algebra, will emerge from a modular flowed version of reflected entropy.

Automated summary

In this paper, we analyze a network of two random tensors to study the reflected entanglement spectrum and its connection to multiboundary wormholes. The spectrum is controlled by the representation theory of the Temp erley-Lieb algebra and can be expressed as a sum over different irreducible representations labeled by a topological index. We provide a gravitational interpretation and confirm the holographic duality of the reflected entropy away from phase transitions. Moreover, we discover non-perturbative contributions from new geometries near phase transitions and speculate the emergence of a non-trivial von Neumann algebra associated with the Temp erley-Lieb algebra in a modular flowed version of reflected entropy.