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

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.