Numerical determination of the entanglement entropy for free fields in the cylinder

TL;DR

This work computes the universal logarithmic contribution to the entanglement entropy for a cylindrical region in $(3+1)$ dimensions for free scalar and Dirac fields. By dimensionally reducing the cylinder to a tower of massive disks and evaluating disk entropies with lattice correlator methods, the authors extract the coefficients of the large-mass expansion and relate the cylinder's logarithmic coefficient to the disk data. They report $s_s=\frac{L}{240R}$ for scalars and $s_f=\frac{6L}{240R}$ for Dirac fields, with complementary disk-mass corrections $\Delta S_m$ that agree with analytical expectations; the results are cross-checked through two independent numerical paths. The findings reinforce the link between entanglement entropy and the conformal anomaly in four dimensions and validate the dimensional-reduction approach as a robust numerical tool for free-field theories.

Abstract

We calculate numerically the logarithmic contribution to the entanglement entropy of a cylindrical region in three spatial dimensions for both, free scalar and Dirac fields. The coefficient is universal and proportional to the type $c$ conformal anomaly in agreement with recent analytical predictions. We also calculate the mass corrections to the entanglement entropy for scalar and Dirac fields in a disk. These apparently unrelated problems make contact through the dimensional reduction procedure valid for free fields whereby the entanglement entropy for the cylinder can be calculated as an integral over masses of the disk entanglement entropies. Coming from the same numerical evaluation in the lattice, each coefficient is cross checked by the other, testing in this way the two results simultaneously.

Figures (4)

• Figure 1: Scalar field: The points correspond to the coefficient of the linear term in $r$ in the disk entanglement entropy for different masses. The linear coefficient in $m$ in the fit, drawn with a solid line, is $-0.52359\sim -\frac{2\pi}{12}$. This is the value of $c_1$ in (\ref{['expansion']}).
• Figure 2: Scalar field: $\hat{c}_{-1}(m)=-c_{-1}(m)$. The points correspond to the coefficient of the term $\frac{1}{r}$ in the disk entanglement entropy for different masses. The coefficient of the term proportional to $\frac{1}{m}$ in the fit drawn with a solid line is $0.01342 \sim \frac{\pi}{240}$. This is the value of $-c_{-1}$ in (\ref{['expansion']}).
• Figure 3: Dirac field: The points correspond to the coefficient of the linear term in $r$ in the disk entanglement entropy for different masses. The linear coefficient in $m$ in the fit, drawn with a solid line, is $-1.04658 \sim-2\times\frac{2 \pi}{12}$. This is $c_1$ in (\ref{['expansion']}). The extra factor of two is due to the fermion doubling in the lattice.
• Figure 4: Dirac field: $\hat{c}_{-1}(m)=-c_{-1}(m)$. The points correspond to the coefficient of the term $\frac{1}{r}$ in the disk entanglement entropy for different masses. The coefficient of the term proportional to $\frac{1}{m}$ in the fit, drawn with a solid line, is $0.07754 \sim \frac{6\pi}{240}$. This is the value of $-c_{-1}$ in (\ref{['expansion']}). The factors $2$ and $1/2$ due to the fermion doubling and the spin multiplicity in the dimensional reduction formula (\ref{['sm']}) respectively, compensate each other.