Topological insulator in twisted transition metal dichalcogenide heterotrilayers

The quantum spin Hall effect has been predicted in twisted homobilayer transition metal dichalcogenides (TMDs) owing to the layer-pseudospin magnetic field. Recently, experimental observations have also confirmed such topological states of matter. However, the topological electronic properties in multilayer moir\'e superlattices remain to be further explored. In twisted TMDs heterotrilayers, the realization of a moir\'e potential with various symmetries becomes feasible. Here, we demonstrate that twisted trilayer TMDs can enter a topological insulator phase under the influence of a moir\'e potential with ${C}_{6}$ symmetry. Specifically, we built two types of trilayer heterostructures, where the low-energy valence band electrons are contributed by the middle layer. In the AA-stacked moir\'e ${\mathrm{WS}}_{2}/{\mathrm{WSe}}_{2}/{\mathrm{MoS}}_{2}$ heterotrilayers where only the middle layer is twisted, the maxima of the moir\'e potential exhibit an approximate ${C}_{6}$ symmetry. The ${C}_{6}$ symmetry effectively compensates for the spatial inversion symmetry breaking in the ${\mathrm{WSe}}_{2}$ layer, leading to a twist-angle-dependent topological phase transition. Leveraging a Green's function approach, we calculate the local state density of edge states at topological minigaps, confirming their nature as moir\'e edge states. In the helical twisted AA-stacked moir\'e ${\mathrm{MoS}}_{2}/{\mathrm{WSe}}_{2}/{\mathrm{MoS}}_{2}$ heterotrilayers, we observed a mosaic pattern of topological and trivial insulators. The emergence of a topological mosaic is attributed to the maxima of the local moir\'e potential possessing ${C}_{6}$ symmetry. The results provide a way for the experimental realization of topological phases in TMDs heterojunctions.