Microscopic Theory of the Magnetic Susceptibility of Insulators

We present a general theory of the magnetic susceptibility of insulators that can be extended to treat spatially varying and finite frequency fields. We find that for the zero frequency response our method agrees with existing approaches in the literature. We clarify some of the differences that have been noted in the different approaches and prove that in the insulating regime all methods agree with with the identification of various sum rules. We clarify details about the gauge-dependence of the magnetic susceptibility tensor, and prove it is invariant to U(N) gauge transformations. We find that while the theory can be written with diagonal elements of the Berry connection these terms can be repackaged and written as an explicitly gauge-invariant ``geometric" contribution to the magnetic susceptibility that depends on the Abelian Berry curvature. Our formalism identifies a Hermitian spontaneous magnetization matrix element (distinct from existing approaches) which features prominently in the expressions for the susceptibility tensor, and identifies a natural partitioning of the response in terms of atomic and itinerant features in going from the atomic limit to the general insulating crystal. Additionally, we consider an h-BN model to explore some of the features of the magnetic susceptibility tensor and practical considerations for computing.