SOME PHYSICAL INFERENCES FROM THE QUASI-CRYSTALLINE TOPOLOGY OF PENROSE LATTICES

The vertex problem of a Penrose lattice is analyzed, using a nearest neighbor tight-binding Hamiltonian. It is discovered that there are connectivity rules, which suffice to predict some important features of the eigenvalue spectrum, such as the bandwidth, the existence and size of a central gap, and the wave functions of the highly degenerate state at the central peak (E = 0). The theoretical predictions are tested numerically, by using a novel renormalization method to diagonalize the Hamiltonian in Penrose lattice of large size. In particular, the existence of connected paths between sites of certain coordination determines the nature of the states around the center of spectrum.