Entanglement entropy of a Maxwell field on the sphere

TL;DR

The authors address the universal logarithmic term in the entanglement entropy for a Maxwell field on a four-dimensional sphere, showing that the coefficient is given by $c^M_{ m log}=-16/45$ and does not match the trace anomaly's Euler-density coefficient. They derive a reduction to two copies of a massless scalar with the $l=0$ mode removed, establishing the exact relation $c^M_{ m log}=2(c^S_{ m log}-c^{S_{l=0}}_{ m log})$ with $c^S_{ m log}=-1/90$ and $c^{S_{l=0}}_{ m log}=1/6$, and confirm it through lattice EE and MI calculations. The paper emphasizes that mutual information, unlike EE with centers, is free of boundary ambiguities and yields a universal result, while the discrepancy with the anomaly persists, inviting exploration of charged theories and boundary contributions. Overall, the work clarifies the entanglement structure of gauge fields in curved geometries and highlights the nuanced relationship between universal entropic terms and conformal anomalies. $c^M_{ m log}=-16/45$ is the robust, MI-supported universal value reported here, with substantial implications for how gauge theories encode information at boundaries.

Abstract

We compute the logarithmic coefficient of the entanglement entropy on a sphere for a Maxwell field in $d=4$ dimensions. In spherical coordinates the problem decomposes into one dimensional ones along the radial coordinate for each angular momentum. We show the entanglement entropy of a Maxwell field is equivalent to the one of two identical massless scalars from which the mode of $l=0$ has been removed. This shows the relation $c^M_{\log}=2 (c^S_{\log}-c^{S_{l=0}}_{\log})$ between the logarithmic coefficient in the entropy for a Maxwell field $c^M_{\log}$, the one for a $d=4$ massless scalar $c_{\log}^S$, and the logarithmic coefficient $c^{S_{l=0}}_{\log}$ for a $d=2$ scalar with Dirichlet boundary condition at the origin. Using the accepted values for these coefficients $c_{\log}^S=-1/90$ and $c^{S_{l=0}}_{\log}=1/6$ we get $c^M_{\log}=-16/45$, which coincides with Dowker's calculation, but does not match the coefficient $-\frac{31}{45}$ in the trace anomaly for a Maxwell field. We have numerically evaluated these three numbers $c^M_{\log}$, $c^S_{\log}$ and $c^{S_{l=0}}_{\log}$, verifying the relation, as well as checked they coincide with the corresponding logarithmic term in mutual information of two concentric spheres.

Figures (5)

• Figure 1: Two parallel planes separated by a distance $L$ in the $x^1$ direction. These define the entangling surfaces for regions $A$ and $B$.
• Figure 2: The sphere entanglement entropy for a Maxwell field where we have subtracted the area and constant terms. The fitting curve is $0.17763 \log(R)$
• Figure 3: Two sets, $A$ and $B$: $A$ is a sphere of radius $R_1=R-\epsilon/2$ and $B$, the complimentary region of a sphere of radius $R_2=R+\epsilon/2$. The averaged radius is $R=(R_1+R_2)/2$ and the annulus section is $\epsilon=R_2-R_1$.
• Figure 4: Mutual information between $A$ and $B$ (Fig.(\ref{['set']})) for the scalar $l=0$ mode. The solid interpolating curve is $f(\eta)$.
• Figure 5: Mutual information between $A$ and $B$ (Fig.(\ref{['set']})) for a Maxwell field for a single set of modes $(B^m, E^r)$. In the plot, the area term has been subtracted from the data. The fit shown is $-0.9899 (f(\eta)+\frac{2}{90}\log(\eta))$ with $f$ the interpolating function of the MI of the $l=0$ massless scalar (Fig.(\ref{['zeromode']}))