Electrical Polarization and Orbital Magnetization: The Position Operator Tamed

Abstract

Macroscopic polarization P and magnetization M are the most fundamental concepts in textbook treatments of condensed media. They are intensive vector quantities that intuitively carry the meaning of dipole per unit volume. But for many years, both P and the orbital term in M evaded even a precise microscopic definition and severely challenged quantum mechanical calculations. Contrary to a widespread incorrect belief, P has nothing to do with the periodic charge distribution in the bulk of a polarized crystal; analogously, the orbital term in M has nothing to do with the bulk current distribution. When a bounded sample is addressed, P and M can indeed be expressed in terms of charge and current distributions, but the boundary contributions are essential. The field has undergone a genuine revolution since the early 1990s. The modern theory of polarization, based on a Berry phase, is a mature topic since the late 1990s; it is now implemented in most first-principle electronic structure codes. Many calculations have addressed various phenomena (ferroelectricity, piezoelectricity, lattice dynamics, infrared spectra of liquid, and amorphous systems) in several materials and are in spectacular agreement with experiments; they have provided thorough understanding of the behavior of ferroelectric and piezoelectric materials. The modern theory of orbital magnetization started in 2005, but some fundamental issues are still in development at the time of writing (2017). Only a few first-principle calculations have appeared so far.