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

M. Ostilli, Claudionor G. Bezerra, G. M. Viswanathan

Published 2021-06-12Version 1

There are few exactly solvable lattice models and even fewer solvable quantum lattice models. Here we address the problem of finding the exact energy 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 trees 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. We show detailed solutions for small Cayley trees and provide and algorithm which solves the general case very efficiently. Our analysis, in particular, allows us to extrapolate the density of states in the thermodynamic limit, which turns out to be dramatically different from that of the Bethe lattice.