Self-similar ground-state wave function for electrons on a two-dimensional Penrose lattice.

Abstract

I define a hopping Hamiltonian for independent electrons on a two-dimensional, infinite, quasiperiodic Penrose lattice with a particular on-site potential, depending upon a parameter r. I then find the exact ground-state wave function for this Hamiltonian. The wave function is shown numerically to have a power-law decay from the origin, and the exponent is determined numerically. The wave function may or may not be normalizable over the infinite lattice, depending upon the parameter r. The wave function is then demonstrated to be self-similar, in that the wave function for two identical regions of the lattice is the same, except for a scale factor. The scaling of the wave function is discussed, and a bound for the decay of the wave function established. Finally, I determine exactly the scaling distribution for the wave function, and thus calculate exactly the previously introduced decay exponent. The wave function is shown to have a critical value of the parameter r, above which the wave function is normalizable and thus localized, while below the wave function has a power-law decay but is not normalizable.