Experiments conducted on two-dimensional twisted materials have revealed a plethora of moiré patterns with different forms and shapes. The formation of these patterns is usually attributed to the presence of small strains in the samples, which typically arise during their fabrication. In this paper we find that the superlattice structure of such systems actually depends crucially on the interplay between twist and strain. For systems composed of honeycomb lattices, we show that this can lead to the formation of practically any moiré geometry, even if each lattice is only slightly distorted. As a result, we show that under strain the moiré Brillouin zone is not a stretched irregular hexagon, but rather a primitive cell that changes according to the geometry of the strained moiré vectors. We identify the conditions for the formation of hexagonal moiré patterns arising solely due to shear or biaxial strain, thus opening the possibility of engineering moiré patterns solely by strain. Moreover, we study the electronic properties in such moiré patterns and find that the strain tends to suppress the formation of the flat moiré bands, even in the strain-induced hexagonal patterns analogous to those obtained by the twist only. Our paper explains the plethora of moiré patterns observed in experiments, and provides a solid theoretical foundation from which one can design moiré patterns by strain. Published by the American Physical Society 2024

We develop a diagrammatic perturbation theory to account for the emergence of moiré bands in the continuum model of twisted bilayer graphene. Our framework is build on treating the moiré potential as a perturbation that transfers electrons from one layer to another through the exchange of the three wave vectors that define the moiré Brillouin zone. By working in the two-band basis of each monolayer, we analyze the one-particle Green's function and introduce a diagrammatic representation for the scattering processes. We then identify the moiré-induced self-energy, relate it to the quasiparticle weight and velocity of the moiré bands, and show how it can be obtained by summing irreducible diagrams. We also connect the emergence of flat bands to the behavior of the static self-energy at the magic angle. In particular, we show that a vanishing Dirac velocity is a direct consequence of the relative orientation of the momentum transfer vectors, suggesting that the origin of magic angles in twisted bilayer graphene is intrinsically connected to its geometrical properties. Our approach provides a diagrammatic framework that highlights the physical properties of the moiré bands.