Renyi Entropies for Free Field Theories

TL;DR

This work develops a framework to compute Rényi entropies $S_q$ for free field theories in $d=2$ by mapping to thermal partition functions on curved backgrounds, notably $S^1 imes ext{H}^2$ and $q$-fold branched coverings of $S^3$, and verifies their equivalence via zeta-function regularization. It provides explicit, high-precision results for free conformal scalars and Dirac fermions, both massless and massive, detailing integral and series representations for ${ m F}_q$ and exact $S_q$ values for several $q$. The paper also introduces massive interpolating functions $f_q(m)$ that capture RG flow between Gaussian and trivial fixed points, and it discusses generalizations to $d>2$, highlighting the role of the hyperbolic density of states in these calculations. Overall, it clarifies how Rényi entropies in free theories depend on geometry, regularization, and mass, offering precise benchmarks for analytic and numerical checks in higher dimensions and interacting theories.

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.

Figures (4)

• Figure 1: The Rényi entropies $S_q$ for the complex conformal scalar field. Note that the function $S_q$ is a monotonic function of $q$. The black dashed line is the asymptotic value $S_\infty$.
• Figure 2: The Rényi entropies $\tilde{S}_q$ for the massless Dirac field. Note that the function $\tilde{S}_q$ is a monotonic function of $q$. The black dashed line is the asymptotic value $\tilde{S}_\infty$.
• Figure 3: The Rényi entropy interpolating function $f_q(m)$ for a massive complex scalar field, plotted as a function of $m$ for $q = 1,2,3, 4, 5$ (plots are darker for larger $q$).
• Figure 4: The Rényi entropy interpolating function $\tilde{f}_q(m)$ for a massive Dirac field, plotted as a function of $m$ for $q = 1,2,3,4,5$ (plots are darker for larger $q$).