Non-Abelian effective mechanism from layer pseudospin and lattice pseudospin in twisted bilayer graphene

We identify a valley-polarized Chern insulator in the van der Waals heterostructure, Pt$_{2}$HgSe$_{3}$/CrI$_3$, for potential applications with interplay between electric, magnetic, optical, and mechanical effects. The interlayer proximity magnetic coupling nearly closes the band gap of Pt$_{2}$HgSe$_{3}$ and the strong intra-layer spin-orbit coupling further lifts the valley degeneracy by over 100 meV leading to positive and negative band gaps at opposite valleys. In the valley with negative gap, the interfacial Rashba spin-orbit coupling opens a topological band gap of 17.8 meV, which is enlarged to 30.8 meV by adding an $h$-BN layer. We find large orbital magnetization in Pt$_{2}$HgSe$_{3}$ layer that is much larger than spin, which can induce measurable optical Kerr effect. The valley polarization and Chern number are coupled to the magnetic order of the nearest neighboring CrI$_3$ layer, which is switchable by electric, magnetic, and mechanical means in experiments. The presence of $h$-BN protects the topological phase allowing the construction of superlattices with valley, spin, and layer degrees of freedoms.