Entanglement entropy in free quantum field theory

TL;DR

This review synthesizes two robust frameworks—the Euclidean replica method and the real-time correlator approach—to compute entanglement entropy in free quantum field theories across dimensions. It presents exact 1+1 and 2+1 dimensional results for scalar and Dirac fields, including Painlevé-type structures, bosonization to sine-Gordon theories, and lattice verifications, highlighting universal terms such as the logarithmic coefficients tied to conformal anomalies. The work also develops higher-dimensional reductions to extract universal contributions and elucidates the distinct behavior of mutual information for bosons and fermions, including polygonal boundary effects and surface terms. The Outlook identifies gauge-field entanglement, continuum real-time formulations, and perturbative treatments of reduced density matrices as key open directions. Overall, the paper provides a comprehensive, methodical account of how entanglement entropy in free QFTs reveals universal, regulator-independent features linked to geometry and topology of regions.

Abstract

In this review we first introduce the general methods to calculate the entanglement entropy for free fields, within the Euclidean and the real time formalisms. Then we describe the particular examples which have been worked out explicitly in two, three and more dimensions.

Figures (15)

• Figure 1: The figure on the left shows the term $S_0$ in the entropy of circles in two dimensions as a function of the radius $R$, with a square lattice regularization. $S_0$ is obtained by fitting the data as $S(R)=c_1 R+ S_0$, and subtracting the term $c_1 R$. It is apparent that $S_0$ does not converge to any definite value. The figure on the right shows the mutual information $I(A,B)$ between two circles $A$ and $B$ of radius $R$ on the lattice, separated by a distance to each other which we have chosen to be also $R$. The calculation shows the convergence of the mutual information in the continuum limit $R\rightarrow\infty$ (with an error of $\sim 2\%$ already for $R\sim 30$). In both figures we have considered a massless scalar field and $R$ is measured in lattice units.
• Figure 2: The path integral on the lower half Euclidean space gives the vacuum wave functional. The reduced density matrix $\rho_V$ is obtained by gluing two copies of this half space along the set $-V$ complementary to $V$.
• Figure 3: tr$\rho^n_V$ is given by the path integral on a $n$-sheeted space formed by sewing the replicated Euclidean spaces with a cut along $V$. This is equivalent for free fields to $n$ decoupled multivalued fields living in a single space. These fields get multiplies by particular phase factors when crossing the cut.
• Figure 4: The integration path used in eqs. (\ref{['chuta']}) and (\ref{['chuta2']}) to define the analytic continuation of $\textrm{tr}\rho^n$ for non-integer $n$.
• Figure 5: The $c_n$ functions for a scalar field, with, from top to bottom, $n=2$, $3$, and $n\rightarrow \infty$. Here $t=m L$.