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When two identical two-dimensional (2D) periodic lattices are stacked inparallel after rotating one layer by a certain angle relative to the otherlayer, the resulting bilayer system can lose lattice periodicity completely andbecome a 2D quasicrystal. Twisted bilayer graphene with 30-degree rotation is arepresentative example. We show that such quasicrystalline bilayer systemsgenerally develop macroscopically degenerate localized zero-energy states(ZESs) in strong coupling limit where the interlayer couplings areoverwhelmingly larger than the intralayer couplings. The emergent chiralsymmetry in strong coupling limit and aperiodicity of bilayer quasicrystalsguarantee the existence of the ZESs. The macroscopically degenerate ZESs areanalogous to the flat bands of periodic systems, in that both are composed oflocalized eigenstates, which give divergent density of states. For monolayers,we consider the triangular, square, and honeycomb lattices, comprised ofhomogenous tiling of three possible planar regular polygons: the equilateraltriangle, square, and regular hexagon. We construct a compact theoreticalframework, which we call the quasiband model, that describes the low energyproperties of bilayer quasicrystals and counts the number of ZESs using asubset of Bloch states of monolayers. We also propose a simple geometric schemein real space which can show the spatial localization of ZESs and count theirnumber. Our work clearly demonstrates that bilayer quasicrystals in strongcoupling limit are an ideal playground to study the intriguing interplay offlat band physics and the aperiodicity of quasicrystals.

Macroscopically degenerate localized zero-energy states of quasicrystalline bilayer systems in strong coupling limit