Microscopic polarization and magnetization fields in extended systems

We introduce microscopic polarization and magnetization fields at each site of an extended system, as well as free charge and current density fields associated with charge movement from site to site, by employing an approach based on a set of orthogonal orbitals at each site. These microscopic fields are defined using a single-particle electron Green function. For an infinite crystal we choose the orbitals to be maximally-localized Wannier functions, and in the longwavelength limit we recover the expected linear response of an insulator, including the zero frequency transverse conductivity. For a topologically trivial insulator we recover the expected expressions for the macroscopic polarization and magnetization in the ground state, and find that the linear response to excitation at arbitrary frequency is described solely by the microscopic polarization and magnetization fields. For very general optical response calculations the microscopic fields automatically satisfy charge conservation, even under basis truncation, and do not suffer from the false divergences at zero frequency that can plague response calculations using other approaches.