Renyi Entropies for Free Field Theories

TL;DR

This paper computes Renyi entropies for free massless and massive fields in various dimensions using thermal partition functions and path integrals, revealing their behavior across fixed points and extending to higher dimensions.

Contribution

It provides explicit calculations of Renyi entropies for free fields in multiple dimensions using zeta-function regularization and explores their properties beyond two dimensions.

Findings

Agreement between branched covering and hyperbolic space calculations.

Massive fields interpolate between fixed points.

Extension of methods to higher dimensions.

Abstract

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 S^{d-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.