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 \geq 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} \times \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} \times \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).