On the Entanglement Entropy of Maxwell Theory: A Condensed Matter Perspective

TL;DR

This paper reviews the calculation of entanglement entropy in Maxwell gauge theories, clarifying discrepancies between high energy and condensed matter approaches, and highlights the significance of charged states and topological contributions.

Contribution

It reconciles different methods of calculating entanglement entropy in Maxwell theory and emphasizes the role of charged states and topological effects in defining entanglement.

Findings

The conformal trace anomaly determines the universal logarithmic coefficient.

Coupling to dynamical charges introduces a topological contribution to entanglement entropy.

Charged states are essential for a complete understanding of gauge theory entanglement.

Abstract

Despite the seeming simplicity of the theory, calculating (and even defining) entanglement entropy for the Maxwell theory of a $U(1)$ gauge field in (3+1) dimensions has been the subject of controversy. It is generally accepted that the ground state entanglement entropy for a region of linear size $L$ behaves as an area law with a subleading logarithm, $S = \alpha L^2 -\gamma \log L$. While the logarithmic coefficient $\gamma$ is believed to be universal, there has been disagreement about its precise value. After carefully accounting for subtle boundary corrections, multiple analyses in the high energy literature have converged on an answer related to the conformal trace anomaly, which is only sensitive to the local curvature of the partition. In contrast, a condensed matter treatment of the problem yielded a topological contribution which is not captured by the conformal field theory calculation. In this perspective piece, we review aspects of the various calculations and discuss the resolution of the discrepancy, emphasizing the important role played by charged states (the "extended Hilbert space") in defining entanglement for a gauge theory. While the trace anomaly result is sufficient for a strictly pure gauge field, coupling the gauge field to dynamical charges of mass $m$ gives a topological contribution to $\gamma$ which survives even in the $m\rightarrow\infty$ limit. For many situations, the topological contribution from dynamical charges is physically meaningful and should be taken into account. We also comment on other common issues of entanglement in gauge theories, such as entanglement distillation, algebraic definitions of entanglement, and gauge-fixing procedures.