Microscopic polarization and magnetization fields: Including spin effects

We extend a microscopic theory of polarization and magnetization to include the spin degree of freedom of the electrons. The microscopic polarization and magnetization fields are introduced throughout an extended system using a set of orthogonal orbitals associated with each site. As well free charge and current density fields are introduced associated with charge movement from site to site. The sites act as natural expansion points when evaluating the perturbative response of the multipole moments to applied electromagnetic fields that can be at arbitrary frequency and vary over the unit cell. Associated with the dipole moments are the respective macroscopic polarization and magnetization fields, from which we can extract the various material response tensors. For example, in the limit of uniform fields, we recover the magnetoelectric polarizability tensor, which contains the accepted expression for the orbital magnetoelectric polarizability as well as a straightforward spin dependent contribution. This general framework can also be extended to treat the spin current response in materials.