Entanglement entropy and the colored Jones polynomial
Abstract
We study the multiparty entanglement structure of states in ChernSimons theory created by performing the path integral on 3manifolds 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) ChernSimons theory where these are stabilizer states, a fact we use to rederive an explicit formula for the entanglement entropy across a general link bipartition. We then present the following results for SU(2) ChernSimons theory: (i) The entanglement entropy for a bipartition of a link gives a lower bound on the genus of surfaces in the ambient S ^{3} separating the two sublinks. (ii) All torus links (namely, links which can be drawn on the surface of a torus) have a GHZlike 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 Wlike entanglement — i.e., partial traces leave a nonseparable state. (iii) Finally, we consider hyperbolic links in the complexified SL(2,C) ChernSimons 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 NeumannZagier potential on the moduli space of hyperbolic structures on the link complement.
 Publication:

Journal of High Energy Physics
 Pub Date:
 May 2018
 DOI:
 10.1007/JHEP05(2018)038
 arXiv:
 arXiv:1801.01131
 Bibcode:
 2018JHEP...05..038B
 Keywords:

 ChernSimons Theories;
 Topological Field Theories;
 Wilson;
 't Hooft and Polyakov loops;
 Conformal Field Theory;
 High Energy Physics  Theory
 EPrint:
 34+12 pages, 15 figures