Electric polarization and magnetization in metals and topologically non-trivial matter*

A feature of the modern theory of polarization is that metallic systems do not admit a well-defined electric polarization, P. This is predicated on the assumption that P is well-defined if and only if the electronic ground state is “localized”. If instead one takes the view that P is more fundamentally related to the general existence of a complete set of exponentially localized Wannier functions, which follows from topological considerations, a definition is always admitted. This is the perspective we have adopted in the unified theory of microscopic polarization and magnetization fields that we have previously developed. Interestingly, when the modern theory admits a well-defined P, in particular for “trivial” insulators, these philosophically different approaches agree. Comparison with the modern theory of magnetization is somewhat different; we find agreement for the orbital magnetization in “trivial” insulators and as well the predicted magnetoelectric effect, but disagree with later thermodynamic extensions to include metals and Chern insulators in that description. In addition to considering such quantities in metals and Chern insulators, we also provide a novel perspective on the distinct contributions to the electrical conductivity tensor in the long-wavelength limit.