Symmetry-directed electronic and optical properties in a two-dimensional square-lattice ZnPc metal-organic framework

The electronic structure of materials is fundamentally governed by their crystal symmetry. While most research on two-dimensional materials has focused on hexagonal lattices, such as graphene, hexagonal boron nitride, and transition metal dichalcogenides. This work explores a square-lattice system: the experimentally realized phthalocyanine-based metal-organic framework (ZnPc-MOF). Using group representation theory, we classify the electronic bands of ZnPc-MOF monolayer, AA- and AB-stacked bilayers, and twisted bilayers in terms of the irreducible representations (irreps) of their little groups. We find that bands in the AB-stacked bilayer remain two-fold degenerate along the $Y$ and $Y^{\prime}$ high-symmetry lines, as a consequence of the sole presence of two-dimensional irreps along these directions. We further derive optical transition selection rules to interpret the optical conductivity, revealing pronounced polarization-dependent optical responses. Additionally, we investigate the quasicrystalline electronic states in the 45$^{\circ}$ twisted bilayer (ZnPc-MOF quasicrystal) using the resonant coupling Hamiltonian. Compared to graphene quasicrystals, ZnPc-MOF quasicrystal exhibits weaker resonant coupling strengths, yet its quasicrystalline states lie closer to the Fermi energy, suggesting a greater contribution to low-energy electronic phenomena.