2018/01/03 by Vijay Balasubramanian, Matthew DeCross, Jackson R. Fliss +4 · 3 citations
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Combinatorics #Entropy (arrow of time) #Geometry #Mathematical analysis #Mathematical physics #Mathematics #Noncommutative and Quantum Gravity Theories #Path integral formulation #Physics #Quantum #Quantum entanglement #Quantum many-body systems #Quantum mechanics #Separable space #Torus #hep-th
paper · pdf · doi:10.1007/jhep05(2018)038
published as JHEP 1805 (2018) 038 · 34+12 pages, 15 figures
arxiv created 2018/01/03 · openalex publication_date 2018/05/01 · arxiv updated 2018/05/25 · openalex created_date 2020/11/23 · openalex updated_date 2026/08/05
A bstract 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 S 3 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) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> <mml:mi>L</mml:mi> <mml:mfenced> <mml:mn>2</mml:mn> <mml:mi>ℂ</mml:mi> </mml:mfenced> </mml:math> 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.