Electronic properties of the Penrose lattice. I. Energy spectrum and wave functions

The electronic structure of the two-dimensional Penrose lattice is studied by numerical diagonalization of a tight-binding Hamiltonian for finite systems with up to 3571 sites. We have analyzed the smoothness of the energy spectrum and localization behavior of the wave functions by level statistics and a generalized participation ratio. The results show that the energy spectrum contains a singular part, and most of the wave functions are critical, i.e., neither extended nor localized. These behaviors of the electronic properties are discussed based on their quasiperiodic lattice structure.