Entanglement and alpha entropies for a massive scalar field in two dimensions

We find the analytic expression of tr ρ n ( L ) for a free massive boson field in 1+1 dimensions, where ρ( L ) is the reduced density matrix corresponding to an interval of length L . This is given exactly (except for a non-universal factor) in terms of a finite sum of solutions of non-linear differential equations of the Painlevé V type. Our method is a generalization of one introduced by Myers and is based on the explicit calculation of quantities related to the Green function on a plane, where boundary conditions are imposed on a finite cut. It is shown that the associated partition function is related to correlators of exponential operators in the sine–Gordon model in agreement with a result by Delfino et al . We also compute the short and long distance leading terms of the entanglement entropy. We find that the bosonic entropic c -function interpolates between the asymptotic limits of the Dirac and Majorana fermion ones given in a previous paper. Finally, we study some universal terms for the entanglement entropy in arbitrary dimensions which, in the case of free fields, can be expressed in terms of the two-dimensional entropy functions.