Complete solution of the tight binding model on a Cayley tree: strongly localised versus extended states

The complete set of Eigenstates and Eigenvalues of the nearest neighbour tight binding model on a Cayley tree with branching number $b=2$ and $M$ branching generations with open boundary conditions is derived. We find that of the $N= 1 +3 (2^M-1)$ total states only $3 M +1$ states are extended throughout the Cayley tree. The remaining $N-(3 M+1)$ states are found to be strongly localised states with finite amplitudes on only a subset of sites. In particular, there are, for $M>1$, $3 \times 2^{M-2}$ surface states which are each antisymmetric combinations of only two sites on the surface of the Cayley tree and have energy eactly at $E=0$, the middle of the band. The ground state and the first two excited states of the Cayley tree are found to be extended states with amplitudes on all sites of the Cayley tree, for all $M$. We use the results on the complete set of Eigenstates and Eigenvalues to derive the total density of states and a local density of states.