Orbital Magnetization in Periodic Insulators

Working in the Wannier representation, we derive an expression for the orbital magnetization of a periodic insulator. The magnetization is shown to be comprised of two contributions, an obvious one associated with the internal circulation of bulklike Wannier functions in the interior, and an unexpected one arising from net currents carried by Wannier functions near the surface. Each contribution can be expressed as a bulk property in terms of Bloch functions in a gauge-invariant way. Our expression is verified by comparing numerical tight-binding calculations for finite and periodic samples.

Figure 1

(a) Four unit cells of the Haldane model. Solid (open) circles denote sites with $E_0=-2$ $(+2)$. Arrows indicate sign of the phase $\phi$ for second-neighbor hopping. (b) Net currents $-e{\mathbf{v}}_i$ associated with WFs, plotted at their centers ${\overline{\mathbf{r}}}_i$, for a $10\times 10$ sample with $\phi =\pi /4$. Currents decrease rapidly into the bulk, so that only surface currents are visible.

Figure 2

Numerical results for the magnetization (in Bohr magnetons per unit area) for the Haldane model. Symbols denote results from a finite $30\times 30$ sample, while curves represent results of a $k$-space calculation on a periodic system.

Figure 3

Horizontal slice from a sample that extends indefinitely in the vertical direction but is otherwise similar to the one in Fig. 1. Vertical dashed lines delimit bulk and surface regions in which WFs are labeled by $s$ and $s^{\prime }$, respectively.