TY - JOUR

T1 - Entanglement entropy and the colored Jones polynomial

AU - Balasubramanian, Vijay

AU - DeCross, Matthew

AU - Fliss, Jackson

AU - Kar, Arjun

AU - Leigh, Robert G.

AU - Parrikar, Onkar

N1 - Funding Information:
Article funded by SCOAP3.
Funding Information:
We would like to thank Pawel Caputa, Ron Donagi, Nathan Dunfield, Sergei Gukov, Taylor Hughes, Mark Mezei, Jessica Purcell, Eric Sharpe and Tadashi Takayanagi for useful conversations or email communications. We are particularly grateful to Tudor Dimofte and Alex Maloney for several useful conversations and communications, and to Alex Maloney for brief initial collaboration. Research funded by the Simons Foundation (#385592, VB) through the It From Qubit Simons Collaboration, the US Department of Energy contract #FG02-05ER-41367 and the US Department of Energy contract #DE-SC0015655 (RGL).
Publisher Copyright:
© 2018, The Author(s).

PY - 2018/5/1

Y1 - 2018/5/1

N2 - 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 S L (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.

AB - 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 S L (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.

KW - Chern-Simons Theories

KW - Conformal Field Theory

KW - Topological Field Theories

KW - Wilson

KW - ’t Hooft and Polyakov loops

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UR - http://www.scopus.com/inward/citedby.url?scp=85046662612&partnerID=8YFLogxK

U2 - 10.1007/JHEP05(2018)038

DO - 10.1007/JHEP05(2018)038

M3 - Article

AN - SCOPUS:85046662612

SN - 1126-6708

VL - 2018

JO - Journal of High Energy Physics

JF - Journal of High Energy Physics

IS - 5

M1 - 38

ER -