Interface contributions to topological entanglement in abelian Chern-Simons theory

• Published in: The journal of high energy physics 2017-09, Vol.2017 (9), p.1-34, Article 56
• Main Authors: Fliss, Jackson R., Wen, Xueda, Parrikar, Onkar, Hsieh, Chang-Tse, Han, Bo, Hughes, Taylor L., Leigh, Robert G.
• Format: Article
• Language: English
• Subjects: Boundary conditions; Chern-Simons Theories; Classical and Quantum Gravitation; Elementary Particles; Entropy; Gauge Symmetry; High energy physics; Hilbert space; Mathematical analysis; Phases; Physics; Physics and Astronomy; PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Quantum entanglement; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Quantum theory; Regular Article - Theoretical Physics; Relativity Theory; String Theory; Topological Field Theories; Topological States of Matter
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP09(2017)056

A bstract We study the entanglement entropy between (possibly distinct) topological phases across an interface using an Abelian Chern-Simons description with topological boundary conditions (TBCs) at the interface. From a microscopic point of view, these TBCs correspond to turning on particular gapping interactions between the edge modes across the interface. However, in studying entanglement in the continuum Chern-Simons description, we must confront the problem of non-factorization of the Hilbert space, which is a standard property of gauge theories. We carefully define the entanglement entropy by using an extended Hilbert space construction directly in the continuum theory. We show how a given TBC isolates a corresponding gauge invariant state in the extended Hilbert space, and hence compute the resulting entanglement entropy. We find that the sub-leading correction to the area law remains universal, but depends on the choice of topological boundary conditions. This agrees with the microscopic calculation of [1]. Additionally, we provide a replica path integral calculation for the entropy. In the case when the topological phases across the interface are taken to be identical, our construction gives a novel explanation of the equivalence between the left-right entanglement of (1+1)d Ishibashi states and the spatial entanglement of (2+1)d topological phases.