Theory of local orbital magnetization: local Berry curvature

Abstract

We present a microscopic theory for the local (single site) orbital magnetization in tight-binding systems. Each occupied state of energy $\varepsilon_n$ contributes with a local orbital magnetic moment term ${\mathbf{ m}}_n({\mathbf{ r}})$ and a local Berry-curvature term ${\mathbf{ \Omega}}_n({\mathbf{ r}})$. For Bloch electrons (${\mathbf{ k}}$-space), we go beyond the modern theory by revealing the sublattice texture. We identify a topological contribution ${\mathbf{ \Omega}}^{\text{topo}}_{n\mathbf{ k}}({\mathbf{ r}})$ and a geometric contribution ${\mathbf{\Omega}}^{\text{geom}}_{n\mathbf{ k}}({\mathbf{ r}})$ to the sublattice Berry curvature. For systems with open boundaries (${\mathbf{ r}}$-space), we derive an explicit expression of an effective onsite Berry curvature ${\mathbf{ \Omega}}_n({\mathbf{ r}})$. Considering two band models, the $\mathbf{ k}$-space and $\mathbf{ r}$-space onsite magnetizations coincide numerically but differ from the Bianco-Resta approach. They reveal orbital ferromagnetism in topological insulators, and orbital antiferro- and ferrimagnetism in trivial insulators. This theory can be used to investigate orbital magnetic textures and their topological properties in many systems of current interest (Moir\'e, amorphous, quasicrystals, defects, molecules).