Rényi entropies for free field theories

Abstract

Rényi entropies S _q are useful measures of quantum entanglement; they can be calculated from traces of the reduced density matrix raised to power q, with q ≥ 0. For (d + 1)-dimensional conformal field theories, the Rényi entropies across S ^d−1 may be extracted from the thermal partition functions of these theories on either (d + 1)-dimensional de Sitter space or $\mathbb{R} \times {\mathbb{H}^d}$, where ${\mathbb{H}^d}$ is the d-dimensional hyperbolic space. These thermal partition functions can in turn be expressed as path integrals on branched coverings of the (d + 1)-dimensional sphere and S ¹×${\mathbb{H}^d}$, respectively. We calculate the Rényi entropies of free massless scalars and fermions in d = 2, and show how using zeta-function regularization one finds agreement between the calculations on the branched coverings of S ³ and on S ¹ × ${\mathbb{H}^2}$. Analogous calculations for massive free fields provide monotonic interpolating functions between the Rényi entropies at the Gaussian and the trivial fixed points. Finally, we discuss similar Rényi entropy calculations in d > 2.