Abstract: The discovery of correlated phases in twisted moiré superlattices has spurred the search for low-dimensional materials with extraordinary properties. One promising route is the use of engineered substrates that impose specific periodic strain profiles, creating large-scale patterns without relying on interlayer rotations. Yet, tailoring substrates for specific functionalities remains challenging, as the link between substrate geometry and the resulting electronic structure is still poorly understood. 

 

Using effective models of graphene under periodic deformations with various crystalline profiles, we identify strong C_(2z) symmetry breaking as the key geometric ingredient required to open energy gaps and generate quasi-flat bands. We further show that continuous strain profiles, which create connected pseudo-magnetic field landscapes, play a decisive role in triggering specific valley-band topologies. 

 

For finite systems, band structures for ribbons with zigzag terminations mirror those of nanoribbons in external periodic magnetic fields, as they exhibit gap crossing bands with states at the edges. Although these edge states lack strict topological protection (the total Chern number vanishes), they appear to be robust and ubiquitous, persisting across a range of ribbon widths, strain strengths, and even under realistic disorder realizations. 

 

We also explore the impact of an out-of-plane electric field with the periodicity of an hBN-engineered substrate. By shaping the field near sample edges, we demonstrate that edge states can be selectively created or suppressed, resulting in semiconductor-like behavior. Extending this idea, we design gate profiles that split the system into distinct topological regions, producing junctions with true topological edge states that can be positioned anywhere in the sample.

 

Finally, we demonstrate that these electronic and topological features are tunable with circularly polarized light, which provides clear experimental signatures for identifying strain-imprinted band topologies.