Properties of one-dimensional quasilattices

We study the properties of one-dimensional quasilattices numerically and analytically. The geometrical properties of general one-dimensional quasilattices are discussed. The Ising model on these lattices is studied by a decimation transformation: The critical temperature and critical exponents do not differ from those for a regular periodic chain. The vibrational spectrum in the harmonic approximation is analyzed numerically. The system exhibits characteristics of both a regular periodic system and a disordered system. In the low-frequency region, the system behaves as a regular periodic system; wave functions appear extended. In the high-frequency region, the spectrum is self-similar and there is no unique behavior for the wave functions. The spectrum shows many gaps and Van Hove singularities. The gaps in the spectrum are also obtained analytically by examining the convergence of a continued-fraction expansion plus decimation transformation. The energy spectrum of a tight-binding electron Hamiltonian on the Fibonacci chain is also analyzed; it shows similar characteristics to those of the lattice vibration spectrum