Theory of Quasicrystals in Twisted Bilayer Graphene on Hexagonal Boron Nitride*

We discuss the theoretical aspects of lattice relaxation in twisted bilayer graphene on hexagonal boron nitride (hBN) that are inspired by experimental investigations conducted by X. Lai et al. Despite the incommensurate nature of the graphene-hBN moiré interference pattern with respect to the twisted bilayer graphene (TBG) lattice, except for a few isolated commensurate twists, we discover a broad spectrum of relative alignments where lattice relaxation leads to the reestablishment of commensuration. Beyond a specific range of relative twists, where there is no separation between the moiré pattern and the emergent moiré-of-moiré scale, relaxation can no longer restore commensurability. In this regime, we observe the formation of a moiré quasicrystal characterized by the superposition of incommensurate periodicities. We also explore the implications for transport experiments involving these devices. Adjusting the alignment of TBG and hBN presents an intriguing means to manipulate and explore interactive physics within moiré patterns with new emergent periodicities.