Asymptotic Analysis of von Neumann Entropy in Conformal Field Theory

Abstract

AbstractGiven a QFT net $${{\mathcal {A}}}$$ A of local von Neumann algebras $${{\mathcal {A}}}(O)$$ A ( O ) , we consider the von Neumann entropy $$S_{{\mathcal {A}}}(O, {\widetilde{O}})$$ S A ( O , O ~ ) of the restriction of the vacuum state to the canonical intermediate type I factor for the inclusion of von Neumann algebras $${{\mathcal {A}}}(O)\subset {{\mathcal {A}}}({\widetilde{O}})$$ A ( O ) ⊂ A ( O ~ ) (split property). This canonical entanglement entropy $$S_{{\mathcal {A}}}(O, {\widetilde{O}})$$ S A ( O , O ~ ) is finite for the chiral conformal net on the circle generated by finitely many free Fermions (here double cones are intervals). The finiteness property is derived by an explicit formula of entropy and an observation that the operators in the definition are closely related to Hankel operators. In this paper we give further analysis of this entropy using a variety of techniques that have been developed in different context, and in particular we show that there is an upper bound given by a positive constant multiply by $$|\ln \eta |$$ | ln η | , where $$\eta $$ η is the cross ratio of the underlying system, when $$\eta \rightarrow 0$$ η → 0 .