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

TL;DR

The paper develops a nonperturbative framework to compute entanglement and alpha-entropies for a massive Dirac field in two dimensions by mapping the replica-traced density matrix to an external gauge field and employing bosonization. For the massless case it reproduces known conformal results and exposes a simple, universal mutual information structure, while for the massive case it formulates exact integral/differential representations in terms of Painlevé V equations via sine-Gordon correlators. It provides explicit short- and long-distance expansions, demonstrates strong agreement with lattice calculations, and discusses the possibility of extending the c-theorem to alpha-entropies. These results illuminate universal features of entanglement in 2D QFTs and establish a bridge between integrable models, special function theory, and numerical lattice checks.

Abstract

We present some exact results about universal quantities derived from the local density matrix, for a free massive Dirac field in two dimensions. We first find the trace of powers of the density matrix in a novel fashion, which involves the correlators of suitable operators in the sine-Gordon model. These, in turn, can be written exactly in terms of the solutions of non-linear differential equations of the Painlevé V type. Equipped with the previous results, we find the leading terms for the entanglement entropy, both for short and long distances, and showing that in the intermediate regime it can be expanded in a series of multiple integrals. The previous results have been checked by direct numerical calculations on the lattice, finding perfect agreement. Finally, we comment on a possible generalization of the entanglement entropy c-theorem to the alpha-entropies.

Figures (3)

• Figure 1: The plane with cuts along the intervals $(u_i,v_i)$, $i=1$,..., $p$. The circuits $C_{u_i}$ and $C_{v_i}$ are used in the text to discuss the boundary conditions.
• Figure 2: Solid lines are plots of $c_n(x)$, obtained by solving differential equations (\ref{['uni']}-\ref{['doblew']}) numerically. The values of $n$ are, from top to bottom, $n=2$, $3$, $5$ and $50$. These functions take the value $(n+1)/(6n)$ at the origin, which cumulates at $1/6=0.166...$ for large $n$. For large $x$, they decay exponentially fast. Dotted lines correspond to the $c_n(x)$ that results from putting the model on a lattice, for $n=2$ and $n=3$. This points are evaluated for set sizes ranging from $200$ to $600$ lattice points and inverse mass values ranging from $200$ to $3200$ lattice units. The fact that these points, computed for several different lattice mass values, tend to lie in a single continuous curve shows the universal character of $c_n(x)$.
• Figure 3: The dotted curve is the function $c(r)$ evaluated on a lattice, with points obtained for different values of the mass and lattice distance (see caption of figure (2)). The continuity of the plot agrees with the universal character of $c(r)$, which is in fact a function of $x=mr$. The solid-line curves are the short and long distance leading terms we evaluated analytically.