Orbital magnetization of correlated electrons with arbitrary band topology

Spin-orbit coupling introduces chirality into the electronic structure. This can have profound effects on the magnetization induced by the orbital motion of electrons. Here we derive a formula for the orbital magnetization of interacting electrons in terms of the full Green's function and vertex functions. The formula is applied within dynamical mean-field theory to the Kane-Mele-Hubbard model that allows both topological and trivial insulating phases. We study the insulating and metallic phases in the presence of an exchange magnetic field. In the presence of interactions, the orbital magnetization of the quantum spin Hall insulating phase with inversion symmetry is renormalized by the bulk quasiparticle weight. The orbital magnetization vanishes for the in-plane antiferromagnetic phase with trivial topology. In the metallic phase, the enhanced effective spin-orbit coupling due to the interaction sometimes leads to an enhancement of the orbital magnetization. However, at low doping, magnetization is suppressed at large interaction strengths.

Figure 1

Orbital magnetization for the noninteracting KM model. Panels (a) and (b) show the partial (band) orbital magnetization for the trivial band insulating phase ${\lambda }_{\mathrm{SO}}=0.1t,\lambda =0.8t$ and the QSH insulating phase ${\lambda }_{\mathrm{SO}}=0.1t,\lambda =0$, respectively. The partial orbital magnetization in the presence of a time-reversal symmetry-breaking exchange field $h=-0.04t$ acting on spins only is shown by solid lines. Dashed lines show the partial orbital magnetization in the absence of an exchange field. The shaded area shows the bulk spectrum gap. Symbols for valence and conduction bands are identified in panel (b). In panels (c) and (d), $h=-0.04t$. The total orbital magnetization as a function of $\mu$ is in (c). In (d) orbital magnetization with ${\lambda }_{\mathrm{SO}}=0.1t$ as a function of $\lambda /t$ for electron densities $n=1.0,1.1$, and 1.25. The semimetal phase at the boundary between the QSH insulator and the trivial band insulator is broadened slightly by the applied exchange field. All data are in units of $(e{a}^{2}t/2\hslash )$, where $a$ is the lattice constant.

Figure 2

$M_{\mathrm{orb}}$ of the interacting KMH model as a function of $U/t$. Panel (a) at half-filling. The shaded area shows the correlated QSH phase. In panel (b), $M_{\mathrm{orb}}$ with ${\lambda }_{\mathrm{SO}}=0.1t,\lambda =0$ as a function of $U/t$ for electron densities $n=1.1$ (top) and $n=1.25$ (bottom). A small exchange field $h=-0.04t$ is applied. There is an out-plane AFM phase for $n=1.1$ at $U/t\simeq 5.4$.

Figure 3

Diagrammatic expansion of the change in total energy due to the presence of a magnetic field, evaluated in the zero-field limit. Lines show the fully dressed Green's function ${\mathit{\lambda }}^E\equiv [{\mathbf{H}}_0+(i{\omega }_m-\mu )\mathbf{1}]$ is the energy vertex, and ${\mathbf{k}}_{-}\equiv \mathbf{k}-\mathbf{q}/2,{\mathbf{k}}_{+}\equiv \mathbf{k}+\mathbf{q}/2$. The second diagram on the first line is independent of $\mathbf{q}$, and its derivative with respect to $\mathbf{q}$ vanishes. Evaluating the diagrams in the limits $\mathbf{q}\to \mathbf{0}$ and ${\mathbf{A}}_{\mathbf{q}}\to \mathbf{0}$ the derivative with respect to ${\mathbf{A}}_{\mathbf{q}}$ is replaced by $-(e/\hslash ){\partial }_{\mathbf{k}}$, whereas the derivative with respect to $\mathbf{q}$ is replaced by $(\pm 1/2){\partial }_{\mathbf{k}}$ depending on the momentum of the propagator line. The first two diagrams on the second line are equal in this limit and give the first line of Eq. (2).

Figure 4

Panel (a): The honeycomb lattice with lattice constant $a$ consists of two sublattices $A$ and $B$ and is spanned by the basis vectors ${\mathbf{a}}_1=a/2(\sqrt{3},3),{\mathbf{a}}_2=a/2(\sqrt{3},-3)$. Nearest-neighbor lattice sites are connected by the vectors ${\mathit{\delta }}_1=a(0,1),{\mathit{\delta }}_2=a/2(\sqrt{3},-1)$, and ${\mathit{\delta }}_3=a/2(-\sqrt{3},-1)$. Panel (b): The hexagonal first Brillouin zone contains the two nonequivalent Dirac points $\mathbf{K}=(4\pi /3\sqrt{3}a)(1,0)$ and ${\mathbf{K}}^{\prime }=-(4\pi /3\sqrt{3}a)(1,0)$.