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We study the electronic structure of quasicrystals composed of incommensuratestacks of atomic layers. We consider two systems: a pair of square latticeswith a relative twist angle of $\theta=45^\circ$ and a pair of hexagonallattices with a relative twist angle of $\theta=30^\circ$, with variousinterlayer interaction strengths. This constitutes every two-dimensionalbilayer quasicrystal system. We investigate the resonant coupling governing thequasicrystalline order in each quasicrystal symmetry, and calculate thequasi-band dispersion. The resonant interaction emerges in bilayerquasicrystals if all the dominant interlayer interactions occur between theatomic orbitals that have the same magnetic quantum number. Thus, not only thequasicrystal composed of the widely studied graphene, but also those composedof transition metal dichalcogenides will exhibit the quasicrystalline states.We find that some quasicrystalline states, which are usually obscured bydecoupled monolayer states, are more prominent, i.e., "exposed", in the systemswith strong interlayer interaction. We also show that we can switch the statesbetween quasicrystalline configuration and its layer components, by turning onand off the interlayer symmetry.

Quasicrystalline electronic states in twisted bilayers and the effects of interlayer and sublattice symmetries