Rényi entropies for free éld theories

Rényi entropies Sq 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 éld theories, the Rényi entropies across Sd-1 may be extracted from the thermal partition functions of these theories on either (d+1)-dimensional de Sitter space or R×H ^d, is where Hd 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 S1×Hd, respectively. We calculate the Rényi entropies of free massless scalars and fermions in d = 2, and show how using zeta-function regularization onefinds agreement between the calculations on the branched coverings of S ³ and on S ¹ × H ². Analogous calculations for massive free élds provide monotonic nterpolating functions between the Rényi entropies at the Gaussian and the trivial fixed points. Finally, we discuss similar Rényi entropy calculations in d < 2.