Connecting the tensor-categorical formulation of anyon condensation with operator algebras and entropic order parameters

Abstract

Anyon condensation that describes the transition between topological quantum field theories can be formulated in the language of tensor categories or that of operator algebras. We deploy the formalism of Doplicher-Haag-Roberts bimodules over quasi-local $\mathrm{C}^{*}$-algebras recently developed in [1] to investigate anyon condensation, which is associated with an extension of a certain operator algebra. The connection between notions in the two formulations is thereby made manifest in an intuitive, diagrammatic manner. An entropic order parameter, as the quantum information-theoretic measure characterising a condensation, is naturally defined, and we give a very simple proof of a bound on it.