Reflected entropy in random tensor networks II: a topological index from the canonical purification

TL;DR

This paper explores the duality between reflected entropy and entanglement wedge cross section in holography, revealing a topological index from the Temperley-Lieb algebra and its gravitational interpretation involving multiboundary wormholes.

Contribution

It introduces a topological index in reflected entropy via TL algebra representation theory and connects it to gravitational multiboundary wormholes, extending the holographic duality analysis.

Findings

Reflected entanglement spectrum is governed by TL algebra representations.

The spectrum includes superselection sectors labeled by a topological index k.

Non-perturbative geometries resolve phase transition discontinuities.

Abstract

In arXiv:2112.09122, 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, $S_R=2 \frac{EW}{4G}$. 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 Temperley-Lieb (TL) 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 TL algebra and labelled by a topological index $k\in \mathbb{Z}_{\geq 0}$. Each sector contributes to the reflected entropy an amount $2k \frac{EW}{4G}$ 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 $k\geq 2$ near phase transitions, resolving the discontinuous transition in $S_R$. Along with analytic arguments, we provide numerical evidence for our results. We comment on the connection between TL algebras, Type II$_1$ von Neumann algebras and gravity.