Theory of local orbital magnetization: local Berry curvature

We present a microscopic theory for the local (single site) orbital magnetization in tight-binding systems. Each occupied state of energy ${\epsilon }_n$ 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 ${\mathbf{\Omega }}_{n\mathbf{k}}^{\text{topo}}(\mathbf{r})$ and a geometric contribution ${\mathbf{\Omega }}_{n\mathbf{k}}^{\text{geom}}(\mathbf{r})$ 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 (Moiré, amorphous, quasicrystals, defects, molecules).