Entanglement Entropy in (3+1)-d Free $U(1)$ Gauge Theory

TL;DR

This paper calculates the entanglement entropy for a free U(1) gauge theory in 3+1 dimensions using a gauge-invariant replica trick, revealing universal logarithmic terms linked to the conformal anomaly and differences in extractable entanglement.

Contribution

It introduces a gauge-invariant path integral approach for entanglement entropy in 3+1D U(1) theories and compares the universal and extractable parts, highlighting boundary scalar effects.

Findings

Universal logarithmic term equals the a anomaly coefficient.

Extractable entanglement's logarithmic coefficient differs from the total.

Boundary scalar accounts for the difference in entanglement measures.

Abstract

We consider the entanglement entropy for a free $U(1)$ theory in $3 + 1$ dimensions in the extended Hilbert space definition. By taking the continuum limit carefully we obtain a replica trick path integral which calculates this entanglement entropy. The path integral is gauge invariant, with a gauge fixing delta function accompanied by a Faddeev-Popov determinant. For a spherical region it follows that the result for the logarithmic term in the entanglement, which is universal, is given by the $a$ anomaly coefficient. We also consider the extractable part of the entanglement, which corresponds to the number of Bell pairs which can be obtained from entanglement distillation or dilution. For a spherical region we show that the coefficient of the logarithmic term for the extractable part is different from the extended Hilbert space result. We argue that the two results will differ in general, and this difference is accounted for by a massless scalar living on the boundary of the region of interest.