Compatibility relationships in van der Waals quasicrystals

The incommensurate twist structure and the interlayer coupling induce van der Waals quasicrystals (vdW-QCs). Replacing conventional band theory requiring translational symmetry, the resonant coupling Hamiltonian endowing the quasiband structure in $\mathbit{k}$ space is adopted to describe electronic properties of vdW-QCs [Moon et al., Phys. Rev. B 99, 165430 (2019)]. Here we investigate the symmetries of the resonant coupling Hamiltonians in dodecagonal and octagonal vdW-QCs. Through symmetry analyses we derive compatibility relationships (CRs) between $\mathrm{\ensuremath{\Gamma}}$ point and other irreducible pathways and predict the symmetry changes and band splits. Especially, we find that from $\mathrm{\ensuremath{\Gamma}}$ point to Brillouin zone corner points of monolayers, arbitrary twofold degenerate states are split into one ${A}^{\ensuremath{'}}$ and one ${A}^{\ensuremath{''}}$ state, and from $\mathrm{\ensuremath{\Gamma}}$ point to the intersection points of two Brillouin zones of monolayers, arbitrary twofold degenerate states are split into one $A$ and one $B$ state. Instead of projection operation analyses, we discuss the CRs of different point groups between the coupled bilayers and uncoupled monolayers to construct the interlayer hybridization selection rules (IHSRs) [Yu et al., Phys. Rev. B 105, 125403 (2022)], which govern how the interlayer states interact with each other in the resonant coupling systems of dodecagonal and octagonal vdW-QCs. These derived IHSRs indicate that the first two main resonant couplings allow the nonequivalent hybridizations only between ${B}₁$ and ${B}₂$ states and the equivalent hybridizations for $A$, ${A}_i$, ${A}^{\ensuremath{'}}$, ${A}^{\ensuremath{''}}$, $E$, or ${E}_i$ states.