Orbital magnetization in crystalline solids: Multi-band insulators, Chern insulators, and metals

We derive a multi-band formulation of the orbital magnetization in a normal periodic insulator (i.e., one in which the Chern invariant, or in two dimensions (2D) the Chern number, vanishes). Following the approach used recently to develop the single-band formalism [Thonhauser, Ceresoli, Vanderbilt, and Resta, Phys. Rev. Lett. 95, 137205 (2005)], we work in the Wannier representation and find that the magnetization is 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. Unlike the single-band case, where each of these contributions is separately gauge invariant, in the multi-band formulation only the sum of both terms is gauge invariant. Our final expression for the orbital magnetization can be rewritten as a bulk property in terms of Bloch functions, making it simple to implement in modern code packages. The reciprocal-space expression is evaluated for 2D model systems and the results are verified by comparing to the magnetization computed for finite samples cut from the bulk. Finally, while our formal proof is limited to normal insulators, we also present a heuristic extension to Chern insulators (having nonzero Chern invariant) and to metals. The validity of this extension is again tested by comparing to the magnetization of finite samples cut from the bulk for 2D model systems. We find excellent agreement, thus providing strong empirical evidence in favor of the validity of the heuristic formula.

Figure 1

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

Figure 2

$2\times 2$ four-site square lattice used in the numerical tests. The absolute value of the hopping parameter $t$ is set to 1. ${\Phi }_{1\cdots 4}$ are the threading fluxes through the four plaquettes.

Figure 3

Band structure of the square lattice for $\varphi =\pi ∕10$. The flux pattern is $({\Phi }_1,{\Phi }_2,{\Phi }_3,{\Phi }_4)=(2\varphi ,-\varphi ,0,-\varphi )$, and the on-site energies are $(E_A,E_B,E_C,E_D)=(-3,0,-3,0)$ (see also Fig. 2). The two lower bands are treated as occupied.

Figure 4

Orbital magnetization of the square-lattice model as a function of the parameter $\varphi$. The two lower bands are treated as occupied. Open circles: extrapolation from finite-size samples. Solid line: discretized $\mathbf{k}$-space formula (31).

Figure 5

Orbital magnetization of the square-lattice model as a function of the chemical potential $\mu$ for $\varphi =\pi ∕3$. The shaded areas correspond to the two groups of bands. Open circles: extrapolation from finite-size samples. Solid line: discretized $\mathbf{k}$-space formula (48).

Figure 6

Four unit cells of the Haldane model. Filled (open) circles denote sites with $E_0=-\Delta$ $(+\Delta )$. Solid lines connecting nearest neighbors indicate a real hopping amplitude $t_1$; dashed arrows pointing to a second-neighbor site indicates a complex hopping amplitude $t_2{e}^{i\varphi }$. Arrows indicate sign of the phase $\varphi$ for second-neighbor hopping.

Figure 7

Orbital magnetization of the Haldane model as a function of the parameter $\varphi$. The lowest band is treated as occupied. Open circles: extrapolation from finite size samples. Solid line: Eq. (48). The system has nonzero Chern number in the region in between the two vertical lines.

Figure 8

Orbital magnetization of the Haldane model as a function of the chemical potential $\mu$ for $\varphi =0.7\pi$. The shaded areas correspond the position of the two bands. Open circles: extrapolation from finite-size samples. Solid line: discretized $\mathbf{k}$-space formula (48).