TY - JOUR

T1 - Reflected entropy in random tensor networks. Part II. A topological index from canonical purification

AU - Akers, Chris

AU - Faulkner, Thomas

AU - Lin, Simon

AU - Rath, Pratik

N1 - Publisher Copyright:
© 2023, The Author(s).

PY - 2023/1

Y1 - 2023/1

N2 - 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, SR=2EW4G. 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 2kEW4G 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.

AB - 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, SR=2EW4G. 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 2kEW4G 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.

KW - AdS-CFT Correspondence

KW - Gauge-Gravity Correspondence

UR - http://www.scopus.com/inward/record.url?scp=85146261944&partnerID=8YFLogxK

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U2 - 10.1007/JHEP01(2023)067

DO - 10.1007/JHEP01(2023)067

M3 - Article

AN - SCOPUS:85146261944

SN - 1126-6708

VL - 2023

JO - Journal of High Energy Physics

JF - Journal of High Energy Physics

IS - 1

M1 - 67

ER -