Universal Feature of Charged Entanglement Entropy

Renyi entropies, S-n, admit a natural generalization in the presence of global symmetries. These "charged Renyi entropies" are functions of the chemical potential mu conjugate to the charge contained in the entangling region and reduce to the usual notions as mu -> 0. For n = 1, this provides a notion of charged entanglement entropy. In this Letter, we prove that for a general d(>= 3)-dimensional conformal field theory, the leading correction to the uncharged entanglement entropy across a spherical entangling surface is quadratic in the chemical potential, positive definite, and universally controlled (up to fixed d-dependent constants) by the coefficients C-J and a(2). These fully characterize, for a given theory, the current correlators < JJ > and < TJJ >, as well as the energy flux measured at infinity produced by the insertion of the current operator. Our result is motivated by analytic holographic calculations for a special class of higher-curvature gravities coupled to a (d - 2) form in general dimensions as well as for free fields in d = 4. A proof for general theories and dimensions follows from previously known universal identities involving the magnetic response of twist operators introduced in A. Belin et aL [J. High Energy Phys. 12 (2013) 059.] and basic thermodynamic relations.