Localization properties of a tight-binding electronic model on the Apollonian network

Abstract

An investigation on the properties of electronic states of a tight-binding Hamiltonian on the Apollonian network is presented. This structure, which is defined based on the Apollonian packing problem, has been explored both as a complex network and as a substrate on the top of which physical models can be defined. The Schr\"odinger equation of the model, which includes only nearest-neighbor interactions, is written in a matrix formulation. In the uniform case, the resulting Hamiltonian is proportional to the adjacency matrix of the Apollonian network. The characterization of the electronic eigenstates is based on the properties of the spectrum, which is characterized by a very large degeneracy. The $2\ensuremath{\pi}/3$ rotation symmetry of the network and large number of equivalent sites are reflected in all eigenstates, which are classified according to their parity. Extended and localized states are identified by evaluating the participation rate. Results for other two nonuniform models on the Apollonian network are also presented. In one case, interaction is considered to be dependent of the node degree, while in the other one random on-site energies are considered.