Renyi entropies for free field theories

Renyi 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 Renyi 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 x H-d, where 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 x H-d, respectively. We calculate the Renyi 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 x H-2. Analogous calculations for massive free fields provide monotonic interpolating functions between the Renyi entropies at the Gaussian and the trivial fixed points. Finally, we discuss similar Renyi entropy calculations in d > 2.