Strictly localized eigenstates on a three-dimensional Penrose lattice

Abstract

The existence of the strictly localized states in a three-dimensional (3D) Penrose lattice is demonstrated for a simple tight-binding model. In the center model there exist degenerate states at an energy E=2; the corresponding wave functions are strictly localized and have the form of tenfold rings. In the vertex model the degenerate states at E=0 were found; the corresponding wave functions have the form of rhombitriaconta icosidodecahedra. The degeneracy of these states is proportional to the system size and, therefore, is infinite in the infinite system.