Disk entanglement entropy for a Maxwell field

TL;DR

The paper analyzes disk entanglement entropy along the RG flow between a 3D Maxwell theory with compact U(1) and its dual free compact scalar, showing the renormalized entropy F(r) decreases monotonically with radius and interpolates between UV logarithmic growth and IR conformal-scalar value. Using the replica trick in the scalar formulation, it derives Renyi entropies with a winding-sector sum that reduces to a Riemann-Siegel theta function, and demonstrates analytic control in the UV/IR limits with a robust numerical extrapolation to obtain the von Neumann entropy at intermediate scales. It further extends the analysis to higher dimensions and clarifies how discrete gauging alters the reduced density matrix, establishing that the compact theory yields a purer state and smaller entropies via a majorization argument. The work highlights subtle factorization issues in gauge theories, connects the entanglement structure to topological sectors, and provides techniques that may be applied to other flows and higher-form dualities.

Abstract

In three dimensions, the pure Maxwell theory with compact U(1) gauge group is dual to a free compact scalar, and flows from the Maxwell theory with non-compact gauge group in the ultraviolet to a non-compact free massless scalar theory in the infrared. We compute the vacuum disk entanglement entropy all along this flow, and show that the renormalized entropy F(r) decreases monotonically with the radius r as predicted by the F-theorem, interpolating between a logarithmic growth for small r (matching the behavior of the S^3 free energy) and a constant at large r (equal to the free energy of the conformal scalar). The calculation is carried out by the replica trick, employing the scalar formulation of the theory. The Renyi entropies for n>1 are given by a sum over winding sectors, leading to a Riemann-Siegel theta function. The extrapolation to n=1, to obtain the von Neumann entropy, is done by analytic continuation in the large- and small-r limits and by a numerical extrapolation method at intermediate values. We also compute the leading contribution to the renormalized entanglement entropy of the compact free scalar in higher dimensions. Finally, we point out some interesting features of the reduced density matrix for the compact scalar, and its relation to that for the non-compact theory.

Figures (5)

• Figure 1: Absolute value of the error in the predicted value of $s_1$ from (top curve) degree $2p$ polynomial interpolating functions and (bottom curve) degree $(p,p)$ rational interpolating functions, taking as input the values of $s_n$ for $n=2,\ldots,2p+2$.
• Figure 2: $\Delta S_n(r)$ for $n=1,2,3,4$ (bottom to top). The functions for $n=2,3,4$ are given by equation \ref{['Theta']}, while $\Delta S(r)=\Delta S_1(r)$ is obtained by the rational extrapolation method described in the text.
• Figure 3: $F_n(r)$ for $n=1,2,3,4$ (top to bottom). The functions for $n=2,3,4$ are calculated from $\Delta S_n(r)$ using \ref{['Fncalc']}, while $F(r)=F_1(r)$ is obtained by the rational extrapolation method described in the text.
• Figure 4: Relative error in \ref{['match']} when $T$ is truncated to an $N\times N$ matrix, for a few sample values of $k$.
• Figure 5: $J(\beta)$ as given by \ref{['Jbeta2']} (top curve) and as calculated by numerical inversion of truncated matrices of dimension $N=2,8,32,128,512,2048$ (lower curves, bottom to top). The functions are invariant under $\beta\to1-\beta$, so only the region $0<\beta<1/2$ is shown. Inset: Difference between \ref{['Jbeta2']} and truncated matrix result (same values of $N$, top to bottom).