Density of states of the Cayley tree

Abstract

The density of states (DOS) for nearest-neighbor hopping on Cayley trees is studied with a simple tight-binding model. A Cayley tree characterized by the coordination number Z and size S has two types of energy states: linear chain states and confinement states. The former, which constitute the overwhelming majority in number, are degenerate by a power law of Z − 1 to the power of the number of shells constituting a tree, whereas the latter, the wavefunctions of which spread out all over the tree, are merely S + 1 non-degenerate states for site-centered trees or 2S non-degenerate states for bond-centered trees, independently of Z. The polynomial equations that express the DOS of the confinement states are obtained, and the relation between the confinement states and the DOS of the Bethe lattice is discussed.