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 $\varepsilon_n$ contributes with a local orbital magnetic moment term ${\mathbf{ m}}n({\mathbf{ r}})$ and a local Berry-curvature term ${\mathbf{ Ω}}n({\mathbf{ r}})$. For Bloch electrons (${\mathbf{ k}}$-space), we go beyond the modern theory by revealing the sublattice texture. We identify a topological contribution ${\mathbf{ Ω}}^{\text{topo}}{n\mathbf{ k}}({\mathbf{ r}})$ and a geometric contribution ${\mathbfΩ}^{\text{geom}}{n\mathbf{ k}}({\mathbf{ r}})$ to the sublattice Berry curvature. For systems with open boundaries (${\mathbf{ r}}$-space), we derive an explicit expression of an effective onsite Berry curvature ${\mathbf{ Ω}}_n({\mathbf{ r}})$. 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).

💡 Research Summary

This paper presents a novel microscopic theory for the orbital magnetization at the scale of a single atomic site (local or onsite orbital magnetization) in tight-binding systems. Moving beyond the established modern theory of orbital magnetization which describes the bulk-averaged property, this work provides a framework to resolve how magnetization is distributed within a unit cell or across a finite sample.

The foundation of the theory is a thermodynamic derivation starting from the local grand canonical potential for a finite open-boundary system. This leads to a general real-space (r-space) expression for the onsite orbital magnetization M_orb(r) (Eq. 8). The formula reveals that each occupied energy eigenstate |n> contributes through two local quantities: a local orbital magnetic moment m_n(r), which is simply the state’s total moment weighted by the onsite probability, and an effective local Berry curvature Ω_n(r) (Eqs. 9, 10). The latter is a newly defined quantity that sums to zero over all sites but encodes local topological and geometric information.

Applying this formalism to periodic Bloch electrons (k-space), the authors derive expressions for the sublattice-resolved magnetization M_orb(r_α) (Eq. 3). A key insight is the decomposition of the local Berry curvature Ω_nk(r_α) into a topological contribution (Ω_topo) and a geometric contribution (Ω_geom) (Eqs. 5-7). Ω_topo represents how the conventional Berry curvature is projected onto sublattices, while Ω_geom consists of purely interband terms that vanish upon summing over all sublattices but are crucial for creating local magnetic textures.

The theory is numerically validated using two-band models (Haldane model in topological/trivial phases and a modified Haldane model). Calculations using the k-space and r-space formulae show perfect agreement, confirming the consistency of the approach. The results unveil distinct orbital magnetic orders: nearly orbital ferromagnetism in the topological insulator phase, and orbital antiferromagnetism or ferrimagnetism in trivial insulator phases. Notably, even in trivial gaps, a finite slope in the sublattice magnetization versus chemical potential is observed, stemming solely from the geometric Berry curvature contribution.

Finally, the paper contrasts its local magnetization and local Chern marker (Eqs. 14, 15) with the earlier Bianco-Resta (BR) approach. While the two methods are fundamentally different—scaling as N^2 vs. N^3 in computation and yielding qualitatively different sublattice profiles—they are shown to be compatible, as they yield identical results when summed over a bulk unit cell. This new local theory provides a powerful microscopic tool for investigating emergent orbital magnetic textures and their topological signatures in a wide range of contemporary systems, including moiré materials, amorphous solids, quasicrystals, and molecular structures.