Entanglement and alpha entropies for a massive scalar field in two dimensions

TL;DR

This work derives the entanglement and alpha entropies for a massive scalar field in 1+1 dimensions by reformulating ${\rm tr}(\rho_A^n)$ on an $n$-sheeted plane with a finite cut, and connects the partition function to correlators in the Sine-Gordon model via a Painlevé V equation. The authors obtain an exact expression for integer $\alpha$ entropies in terms of Painlevé-V solutions, together with short- and long-distance expansions; the entropic c-function for the scalar interpolates between Dirac and Majorana fermion values and exhibits a cusp at the conformal point due to a zero mode. They further show how universal terms in entanglement entropy in arbitrary dimensions can be constructed from the two-dimensional results by KK reduction, providing explicit constants $k_S$ and $k_D$ in 2+1 dimensions and validating them numerically on lattices. The results deepen the link between entanglement in quantum field theory and integrable structures (Painlevé equations and SG correlators), with potential implications for an entropic c-theorem and higher-dimensional universal entropy terms.

Abstract

We find the analytic expression of the trace of powers of the reduced density matrix on an interval of length L, for a massive boson field in 1+1 dimensions. This is given exactly (except for a non universal factor) in terms of a finite sum of solutions of non linear differential equations of the Painlevé V type. Our method is a generalization of one introduced by Myers and is based on the explicit calculation of quantities related to the Green function on a plane, where boundary conditions are imposed on a finite cut. It is shown that the associated partition function is related to correlators of exponential operators in the Sine-Gordon model in agreement with a result by Delfino et al. We also compute the short and long distance leading terms of the entanglement entropy. We find that the bosonic entropic c-function interpolates between the Dirac and Majorana fermion ones given in a previous paper. Finally, we study some universal terms for the entanglement entropy in arbitrary dimensions which, in the case of free fields, can be expressed in terms of the two dimensional entropy functions.

Figures (2)

• Figure 1: The $c_{n}$ function for a real scalar field with, from top to bottom, $n=2$, $3$ and $50$. At the origin they take the value $\frac{1+n}{6n}$ and they decay exponentially fast for large $t$.
• Figure 2: The entropic $c$-function $c(t)=L dS(L)/dL$ for a real scalar field (dotted line) and the Dirac and Majorana fermion ones (top and bottom dashed lines respectively). The bosonic $c$-function interpolates between the Dirac one at the origin, where both tend to the conformal value $c=1/3$, and the Majorana $c$-function for large $t$, where they decay exponentially fast. The cusp at the origin of the bosonic function is due to a $1/\log(t)$ term. These curves were obtained by numerical computation on a lattice of up to $600$ points.