Tight-binding models on branched structures

Abstract

In this paper we analyze the properties of electrons in noncrystalline structures, mathematically described by graphs. We consider a tight-binding model for noninteracting quantum particles and its perturbative expansion in the hopping parameter, which can be mapped into a random-walk problem on the same graph. The model is solved on a wide class of structures, called bundled graphs, which are used as models for the geometrical structure of polymers and are obtained joining to each point of a ``base'' graph a copy of a ``fiber'' graph. The analytical calculation of the Green's functions is obtained through an exact resummation of the perturbative series using graph combinatorial techniques. In particular, our result shows that when the base graph is a d-dimensional crystalline lattice, the fibers generate a self-energy of pure geometrical origin in the base Green's functions.