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Title: Entanglement entropy and the colored Jones polynomial

Abstract

We study the multi-party entanglement structure of states in Chern-Simons theory created by performing the path integral on 3-manifolds with linked torus boundaries, called link complements. For gauge group SU(2), the wavefunctions of these states (in a particular basis) are the colored Jones polynomials of the corresponding links. We first review the case of U(1) Chern-Simons theory where these are stabilizer states, a fact we use to re-derive an explicit formula for the entanglement entropy across a general link bipartition. We then present the following results for SU(2) Chern-Simons theory: (i) The entanglement entropy for a bipartition of a link gives a lower bound on the genus of surfaces in the ambient S3 separating the two sublinks. (ii) All torus links (namely, links which can be drawn on the surface of a torus) have a GHZ-like entanglement structure — i.e., partial traces leave a separable state. By contrast, through explicit computation, we test in many examples that hyperbolic links (namely, links whose complements admit hyperbolic structures) have W-like entanglement — i.e., partial traces leave a non-separable state. (iii) Finally, we consider hyperbolic links in the complexified SL(2,C) Chern-Simons theory, which is closely related to 3d Einstein gravity with a negativemore » cosmological constant. In the limit of small Newton constant, we discuss how the entanglement structure is controlled by the Neumann-Zagier potential on the moduli space of hyperbolic structures on the link complement.« less

Authors:
 [1];  [2];  [3]; ORCiD logo [2];  [3];  [2]
  1. Univ. of Pennsylvania, Philadelphia, PA (United States). David Rittenhouse Lab.; Vrije Univ. Brussel (VUB) and International Solvay Institutes, Brussels (Belgium). Theoretische Natuurkunde
  2. Univ. of Pennsylvania, Philadelphia, PA (United States). David Rittenhouse Lab.
  3. Univ. of Illinois, Urbana, IL (United States). Dept. of Physics
Publication Date:
Research Org.:
Duke Univ., Durham, NC (United States); Univ. of Illinois at Urbana-Champaign, IL (United States)
Sponsoring Org.:
USDOE Office of Science (SC)
OSTI Identifier:
1507583
Grant/Contract Number:  
FG02-05ER41367; SC0015655
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
Journal of High Energy Physics (Online)
Additional Journal Information:
Journal Volume: 2018; Journal Issue: 5; Journal ID: ISSN 1029-8479
Publisher:
Springer Berlin
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Chern-Simons Theories; Topological Field Theories; Wilson; 't Hooft and Polyakov loops; Conformal Field Theory

Citation Formats

Balasubramanian, Vijay, DeCross, Matthew, Fliss, Jackson, Kar, Arjun, Leigh, Robert G., and Parrikar, Onkar. Entanglement entropy and the colored Jones polynomial. United States: N. p., 2018. Web. doi:10.1007/jhep05(2018)038.
Balasubramanian, Vijay, DeCross, Matthew, Fliss, Jackson, Kar, Arjun, Leigh, Robert G., & Parrikar, Onkar. Entanglement entropy and the colored Jones polynomial. United States. doi:10.1007/jhep05(2018)038.
Balasubramanian, Vijay, DeCross, Matthew, Fliss, Jackson, Kar, Arjun, Leigh, Robert G., and Parrikar, Onkar. Mon . "Entanglement entropy and the colored Jones polynomial". United States. doi:10.1007/jhep05(2018)038. https://www.osti.gov/servlets/purl/1507583.
@article{osti_1507583,
title = {Entanglement entropy and the colored Jones polynomial},
author = {Balasubramanian, Vijay and DeCross, Matthew and Fliss, Jackson and Kar, Arjun and Leigh, Robert G. and Parrikar, Onkar},
abstractNote = {We study the multi-party entanglement structure of states in Chern-Simons theory created by performing the path integral on 3-manifolds with linked torus boundaries, called link complements. For gauge group SU(2), the wavefunctions of these states (in a particular basis) are the colored Jones polynomials of the corresponding links. We first review the case of U(1) Chern-Simons theory where these are stabilizer states, a fact we use to re-derive an explicit formula for the entanglement entropy across a general link bipartition. We then present the following results for SU(2) Chern-Simons theory: (i) The entanglement entropy for a bipartition of a link gives a lower bound on the genus of surfaces in the ambient S3 separating the two sublinks. (ii) All torus links (namely, links which can be drawn on the surface of a torus) have a GHZ-like entanglement structure — i.e., partial traces leave a separable state. By contrast, through explicit computation, we test in many examples that hyperbolic links (namely, links whose complements admit hyperbolic structures) have W-like entanglement — i.e., partial traces leave a non-separable state. (iii) Finally, we consider hyperbolic links in the complexified SL(2,C) Chern-Simons theory, which is closely related to 3d Einstein gravity with a negative cosmological constant. In the limit of small Newton constant, we discuss how the entanglement structure is controlled by the Neumann-Zagier potential on the moduli space of hyperbolic structures on the link complement.},
doi = {10.1007/jhep05(2018)038},
journal = {Journal of High Energy Physics (Online)},
issn = {1029-8479},
number = 5,
volume = 2018,
place = {United States},
year = {2018},
month = {5}
}

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