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

TL;DR

This paper studies electronic states in a tight-binding model on the Apollonian network, revealing how network symmetry and site properties influence localization and spectrum characteristics.

Contribution

It introduces a detailed analysis of electronic eigenstates on the Apollonian network, including uniform, degree-dependent, and disordered models, highlighting spectral degeneracies and state classifications.

Findings

Eigenstates reflect network symmetry and site equivalence.

Participation rates distinguish extended and localized states.

Spectrum exhibits high degeneracy influenced by network structure.

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 defined. The Schrodinger 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 $2\pi /3$ 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 non-uniform 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.