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 - 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

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

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 -