Density of States of the Random-Hopping Model on a Cayley Tree

We formulate an integral equation and recursion relations for the configurationally averaged one-particle Green’s function of the random-hopping model on a Cayley tree of coordination number σ+1. This formalism is tested by applying it successfully to the nonrandom model. Using this scheme for 1≪σ<∞, we calculate the density of states of this model with a Gaussian distribution of hopping matrix elements in the energy range E2>Ec2, where Ec is a critical energy described below. The singularity in the Green’s function which occurs at energy E1(0) for σ=∞ is shifted to complex energy E1 (on the unphysical sheet of energy E) for small σ−1. This calculation shows that the density of states is a smooth function of energy E around the critical energy Ec=ReE1, in accord with Wegner’s theorem. In this formulation the density of states has no sharp phase transition on the real axis of E because E1 has developed an imaginary part. Using the Lifschitz argument, we calculate the density of states near the band edge for the model when the hopping matrix elements are governed by a bounded probability distribution. This case is also analyzed via a mapping similar to those used for dynamical systems, whereby the formation of energy band can be understood.