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

Abstract

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, $$ {S}_R=2\frac{EW}{4G} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>R</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mfrac> <mml:mi>EW</mml:mi> <mml:mrow> <mml:mn>4</mml:mn> <mml:mi>G</mml:mi> </mml:mrow> </mml:mfrac> </mml:math> . 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 &gt;0 . Each sector contributes to the reflected entropy an amount $$ 2k\frac{EW}{4G} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>2</mml:mn> <mml:mi>k</mml:mi> <mml:mfrac> <mml:mi>EW</mml:mi> <mml:mrow> <mml:mn>4</mml:mn> <mml:mi>G</mml:mi> </mml:mrow> </mml:mfrac> </mml:math> 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.