Figure 1

For cases (a)–(d) the lower panel describes the sense of layer polarization for each spin-valley combination, while the upper panel schematically indicates the corresponding Hall conductivity contributions. (a) A valley Hall (QVH) insulator with a net layer polarization and a charge-density wave mass $\lambda {\sigma }_z$; (b) an anomalous Hall (QAH) insulator with a valley-dependent mass $\lambda {\tau }_z{\sigma }_z$; (c) a layer-antiferromagnetic (LAF) insulator with a spin-dependent mass $\lambda s_z{\sigma }_z$; (d) a quantum spin Hall (QSH) insulator (or a 2D TI) with a valley- and spin-dependent mass term $\lambda {\tau }_z{s}_z{\sigma }_z$.Reuse & Permissions

Figure 2

The magnitude of orbital magnetic moments carried by individual states versus in-plane momentum, for each spin and valley flavor in $ABC$ graphene $N$ layers. Here the moments are in units of ${\mu }_B$ and $|\lambda |=10\mathrm{meV}$.Reuse & Permissions

Figure 3

Intravalley and intervalley edge states in chirally stacked graphene systems. (a),(d) For a single layer, (b),(e) for a bilayer, and (c),(f) for a trilayer. To visualize the edge states, the intralayer and interlayer nearest neighbor hoppings are chosen as ${\gamma }_0=1$ and ${\gamma }_1=0.3$, respectively; $\lambda =0.25$ for (a)–(c) and $\lambda =0.3\sqrt{3}{\tau }_z$ for (d)–(f).Reuse & Permissions