A universal feature of charged entanglement entropy

Rényi entropies, \(S_n\), admit a natural generalization in the presence of global symmetries. These "charged Rényi entropies" are functions of the chemical potential \(\mu\) conjugate to the charge contained in the entangling region and reduce to the usual notions as \(\mu\rightarrow 0\). For \(n=1\), this provides a notion of charged entanglement entropy. In this letter we prove that for a general \(d (\geq 3)\)-dimensional CFT, 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 \(\langle JJ\rangle \) and \(\langle TJJ \rangle\), 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 arXiv:1310.4180 and basic thermodynamic relations.