Reflected entropy in random tensor networks. Part II. A topological index from canonical purification
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13130_2023_Article_19972.pdf
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Author(s)
Akers, Chris
Faulkner, Thomas
Lin, Simon
Rath, Pratik
Date Issued
January 13, 2023
Publisher
Springer Berlin Heidelberg
Citation
Journal of High Energy Physics. 2023 Jan 13;2023(1):67
Version
Final published version
Abstract
Abstract
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
=
2
EW
4
G
$$ {S}_R=2\frac{EW}{4G} $$
. 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 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
2
k
EW
4
G
$$ 2k\frac{EW}{4G} $$
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 ≥ 2 near phase transitions, resolving the discontinuous transition in SR. 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.
MIT Department
Massachusetts Institute of Technology. Center for Theoretical Physics
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DOI of Published Version
https://doi.org/10.1007/JHEP01(2023)067
