Electronic and vibrational modes on a Penrose lattice: Localized states and band structure

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Abstract
We define a one-parameter family of hopping Hamiltonians with an on-site potential, for independent electrons on a two-dimensional quasiperiodic Penrose lattice. The resulting models include the vibrational modes of the lattice. This problem is then investigated numerically—exploiting the symmetries of the model including scale invariance, up to systems of 3126 sites—and for various boundary conditions. We find the following results for the density of states. (1) There is a peak of zero width at a known energy. (2) We calculate the fraction of states in this central peak, assuming point (5) below. (3) These states are strictly localized; we calculate the wave functions explicitly. (4) The remainder of the states lie in various bands separated by band gaps; this band structure is discussed in the limit of a large on-site potential. (5) There is strong numerical evidence that the energy eigenvalues do not cross, as the on-site potential strength is varied, in the thermodynamic limit.