Duality between atomic configurations and Bloch states in twistronic materials

The relative orientation (twist) of successive layers of stacked two-dimensional (2D) materials creates variations in the interlayer atomic registry. The variations often form a super lattice, called a moir\'e pattern, which can alter electronic properties. In this work we introduce a classification of the single-particle electronic structures that can occur in twisted stacks of 2D layers by characterizing them as "moir\'e molecules" or "moir\'e crystals". The molecules generate localized electronic states and moir\'e flat bands, while the crystals are sometimes unconventional and produce electronic banding in the configuration basis. The underpinning of this classification is the duality between interlayer configuration and monolayer Bloch momentum in moir\'e Hamiltonians. We apply this understanding to diagrams of local electron density in untwisted geometries to produce intuitive and quantitative predictions of twistronic properties. We provide a conceptual introduction to this framework through a one-dimensional model, and then apply it to 2D twisted bilayers of the semi-metal graphene, and of MoS$_2$, a representative material of the transition metal dichalcogenide (TMDC) family of semiconductors. This level of thorough understanding of twistronic phenomena is vital in the search for new material platforms for localized moir\'e electrons.