Electric polarization and magnetization in metals

TL;DR

This paper compares different theoretical approaches to defining electric polarization and magnetization in metals, highlighting disagreements and clarifying their physical implications, especially regarding conductivity and Hall effects.

Contribution

It introduces a new approach to defining microscopic polarization and magnetization in metals that aligns with classical electrodynamics and clarifies differences with the modern theory.

Findings

The new approach reproduces the classical conductivity tensor in the long-wavelength limit.

Disagreements with the extended modern theory of magnetization for metals are elucidated.

In the trivial insulator limit, the expressions recover expected polarization and magnetization values.

Abstract

A feature of the "modern theory" is that electric polarization is not well-defined in a metallic ground state. A different approach invokes the general existence of a complete set of exponentially localized Wannier functions, with respect to which general definitions of microscopic electronic polarization and magnetization fields, and free charge and current densities are always admitted. These definitions assume no particular initial electronic state of the crystal, and the set of microscopic fields satisfy the usual relations of classical electrodynamics. Notably, when applied to a trivial insulator initially occupying its $T=0$ ground state, the expressions for the unperturbed polarization and orbital magnetization, and for the orbital magnetoelectric polarizability tensor obtained from these different approaches can agree. However, the "modern theory of magnetization" has been extended via thermodynamic arguments to include metals and Chern insulators. We here compare with that generalization and find disagreement; the manner in which the expressions differ elucidates the distinct philosophies of these approaches. Our approach leads to the usual electrical conductivity tensor in the long-wavelength limit; in the absence of any scattering mechanisms, the dc divergence of that tensor is due to the free current density and the finite-frequency generalization of the anomalous Hall contribution arises from a combination of bound and free current densities. As well, in the limit that the electronic ground state is that of a trivial insulator, our expressions reduce to those expected for the unperturbed polarization and magnetization, and the electric susceptibility.