The emergence of topologically nontrivial flat bands in moiré materials provides an opportunity to explore the interplay between topological physics and correlation effects, leading to the recent experimental realization of interacting topological phases, e.g., fractional Chern insulators. In this work, we propose a general mechanism of band inversion induced by the moiré Brillouin zone folding of atomic bands for engineering topological minibands in moiré materials. We illustrate this mechanism via two classes of model Hamiltonians, namely the Rashba model and the Bernevig-Hughes-Zhang (BHZ) model, under the moiré superlattice potentials. We find moiré minibands with the nontrivial band topology, including the ${\mathbb{Z}}_2$ number, mirror Chern number, and fragile topology, and the topological phase diagram is constructed for these moiré models. A general theory based on band representations in the morié Brillouin zone is also developed for a generalization of this mechanism to all 2D plane groups. Possible experimental realizations of our model Hamiltonian are discussed.

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• Flat bands
• Topological materials
• Topological phases of matter
• Twisted heterostructures