Relaxation-driven topological domains in moir\'e materials

Spatial control of topology is highly desirable for realizing tunable quantum functionalities in materials. Moir\'e superlattices formed by twisting van der Waals heterostructures provide a natural platform for spatially modulated electronic phases, yet the emergence of tunable topological domains in these systems remains largely unexplored. Here we show that structural relaxation in twisted bilayer BiSb drives the formation of a distinct moir\'e topological phase, characterized by coexisting topologically nontrivial (Z$_2$ = 1) and trivial (Z$_2$ = 0) domains within a single moir\'e unit cell. As the twist angle is reduced, relaxation-induced modulation of the interlayer separation stabilizes an expanding network of topological regions embedded within trivial backgrounds of the moir\'e unit cell. The resulting internal domain boundaries host topologically-protected gapless 1D edge states that are directly visible in our simulated scanning-tunnelling microscopy maps.Furthermore, we demonstrate that the real-space topological domain structure and associated gapless edge states can be reversibly tuned by an out-of-plane electric field. Together, these results establish twisted BiSb as a promising platform for programmable topological domain patterning, where intrinsic networks of edge channels can be continuously tuned and electrically reconfigured in moir\'e materials.