Hyperinflation and Self-Similarity in Quasiperiodic One-Dimensional Lattices

Abstract

According to the existence or lack of translational symmetry, the structure of condensed matter is commonly classified into two classes, regular crystals and disordered systems. Translational symmetry in regular crystals plays a pivotal role in determining their properties. For example, one-electron states in crystals described by a hamiltonian with a periodic potential energy must be Bloch states, i.e. an eigenfunction of the hamiltonian must be a product of a plane wave and a periodic function of the lattice, which is extended in the entire crystal. This makes the energy band periodic in the reciprocal lattice space and, in turn, the density of states shows the van Hove singularities.¹ On the other hand, randomness in the hamiltonian yields properties qualitatively different from those of crystals. In fact, it is known that eigenfunctions may be localized when the randomness exceeds a critical strength.² It is also known that the van Hove singularities are smeared out by randomness.³