Magnetoelectric polarizability: A microscopic perspective

We extend a field theoretic approach for the investigation of the electronic charge-current density response of crystalline systems to arbitrary electromagnetic fields. The approach leads to the introduction of microscopic polarization and magnetization fields, as well as free charge and current densities, the dynamics of which are described by a lattice gauge theory. The spatial averages of such quantities constitute the fields of macroscopic electrodynamics. We implement this formalism to study the modifications of the orbital electronic properties of a class of insulators due to uniform dc electric and magnetic fields, at zero temperature. To first order in the electric and magnetic fields, the free charge and current densities vanish; thus the linear effect of such fields is captured by the first-order modifications of the microscopic polarization and magnetization fields. Associated with the dipole moment of the microscopic polarization (magnetization) field is a macroscopic polarization (magnetization), for which we extract various tensors relating it to the electric and magnetic fields. We focus on the orbital magnetoelectric polarizability (OMP) tensor, and find the accepted expression as derived from the modern theory of polarization and magnetization. Since our results are based on the spatial averages of microscopic polarization and magnetization fields, we can identify the distinct contributions to the OMP tensor from the perspective of this microscopic theory, and we establish the general framework in which extensions to finite frequency can be made.