Theory of optical axion electrodynamics and application to the Kerr effect in topological antiferromagnets

Abstract Electromagnetic fields in a magneto-electric medium behave in close analogy to photons coupled to the hypothetical elementary particle, the axion. This emergent axion electrodynamics is expected to provide novel ways to detect and control material properties with electromagnetic fields. Despite having been studied intensively for over a decade, its theoretical understanding remains mostly confined to the static limit. Formulating axion electrodynamics at general optical frequencies requires resolving the difficulty of calculating optical magneto-electric coupling in periodic systems and demands a proper generalization of the axion field. Here, we introduce a theory of optical axion electrodynamics. We define the proper optical axion magneto-electric coupling through its relation to optical surface Hall conductivity and provide ways to calculate it in quasi-two-dimensional and three-dimensional lattice systems. By employing our new formulae, we show that optical axion electrodynamics can lead to a significant Kerr effect but vanishing Faraday effect in fully compensated antiferromagnets, refuting the conventional wisdom that the Kerr effect is a measure of the net magnetic moment. Our work sets the foundation for a concrete understanding of the axion magneto-optic effects and opens up a promising route to studying antiferromagnetism with light. Our theory is particularly relevant to materials like MnBi$_2$Te$_4$, a topological antiferromagnet whose mageto electric response is shown here to be dominated by the axion contribution even at optical frequencies.