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 ${\mathbf{ Ω}}_n({\mathbf{ r}})$. 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).

Figures (3)

• Figure 1: Sublattice orbital magnetizations $M_A$ (red) and $M_B$ (blue); $M_A+M_B$ (black). In gray, we highlight the gap region. We observe in the gap that the slope of the black curve is quantized and satisfies exactly $\partial_\mu(M_A+M_B) = \frac{e}{h}C$, even locally in $\bm r$-space. Rows: (top) HM$_{\text{topo}}$, (middle) HM$_{\text{triv}}$,(bottom) mHM. Columns: (Left) results for Bloch electron using (\ref{['eq:MorbExtendedLocal2']},\ref{['eq:MorbExtendedLocal3']},\ref{['eq:MorbExtendedLocal4']},\ref{['eq:MorbExtendedLocal5']}), (Right) results obtained using (\ref{['eq:LocalBerry']},\ref{['eq:LocalBerry1']}), computed locally on a single site in the bulk, for a finite sample of $N=1088$ sites and open boundaries.
• Figure 2: Magnetization for a single site (chosen arbitrarily in the bulk - all bulk sites are found to give equivalent results) in each sublattice in a sample with $N=1088$ sites and open boundaries: (Left) computed using expressions (\ref{['eq:Localmag']},\ref{['eq:LocalChern']}), (right) computed using Bianco-Resta expressions. (top) HM$_{\text{topo}}$, (middle) HM$_{\text{triv}}$, (bottom) mHM.
• Figure 3: Orbital magnetization in the topological phase of the Haldane model for a finite size sample with open-boundary. Blue curves: pure band states contribution. Orange curves: pure gap terms and mixed band-gap terms contributions. Dashed-line: orbital magnetic moment contribution, dot-dashed: Berry curvature contribution. (left) single unit cell, (midle) 80% of the unit cells (excluding the edge sites contribution), (right) whole sample including edge sites. This figure shows that topology (quantized slope in the bulk) is obtained from the band states contribution locally in the bulk, and from the edge states (gap states) when averaged over the entire sample.