Universal Feature of Charged Entanglement Entropy

Rényi entropies, <math display="inline"><msub><mi>S</mi><mi>n</mi></msub></math>, admit a natural generalization in the presence of global symmetries. These “charged Rényi entropies” are functions of the chemical potential <math display="inline"><mi>μ</mi></math> conjugate to the charge contained in the entangling region and reduce to the usual notions as <math display="inline"><mrow><mi>μ</mi><mo stretchy="false">→</mo><mn>0</mn></mrow></math>. For <math display="inline"><mi>n</mi><mo>=</mo><mn>1</mn></math>, this provides a notion of charged entanglement entropy. In this Letter, we prove that for a general <math display="inline"><mi>d</mi><mo stretchy="false">(</mo><mo>≥</mo><mn>3</mn><mo stretchy="false">)</mo></math>-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 <math display="inline"><mi>d</mi></math>-dependent constants) by the coefficients <math display="inline"><msub><mi>C</mi><mi>J</mi></msub></math> and <math display="inline"><msub><mi>a</mi><mn>2</mn></msub></math>. These fully characterize, for a given theory, the current correlators <math display="inline"><mrow><mo stretchy="false">⟨</mo><mrow><mi>J</mi><mi>J</mi></mrow><mo stretchy="false">⟩</mo></mrow></math> and <math display="inline"><mrow><mo stretchy="false">⟨</mo><mrow><mi>T</mi><mi>J</mi><mi>J</mi></mrow><mo stretchy="false">⟩</mo></mrow></math>, 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 (<math display="inline"><mrow><mi>d</mi><mo>-</mo><mn>2</mn></mrow></math>) form in general dimensions as well as for free fields in <math display="inline"><mi>d</mi><mo>=</mo><mn>4</mn></math>. 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.