Spectrum of the tight-binding model on Cayley Trees and comparison with Bethe Lattices

Abstract

There are few exactly solvable lattice models and even fewer solvable quantum lattice models. Here we address the problem of finding the spectrum of the tight-binding model (equivalently, the spectrum of the adjacency matrix) on Cayley trees. Recent approaches to the problem have relied on the similarity between Cayley tree and the Bethe lattice. Here, we avoid to make any ansatz related to the Bethe lattice due to fundamental differences between the two lattices that persist even when taking the thermodynamic limit. Instead, we show that one can use a recursive procedure that starts from the boundary and then use the canonical basis to derive the complete spectrum of the tight-binding model on Cayley Trees. Our resulting algorithm is extremely efficient, as witnessed with remarkable large trees having hundred of shells. We also shows that, in the thermodynamic limit, the density of states is dramatically different from that of the Bethe lattice.