Rényi Entropies for Free Field 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 field theories, the Rényi entropies across S may be extracted from the thermal partition functions of these theories on either (d+1)-dimensional de Sitter space or R×H, where H 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 × H, 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 and on S × H. 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. November 2011 ∗On leave from Joseph Henry Laboratories and Center for Theoretical Science, Princeton University. ar X iv :1 11 1. 62 90 v1 [ he pth ] 27 N ov 2 01 1