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 $E^2$>$E_c^2$, where $E_c$ is a critical energy described below. The singularity in the Green’s function which occurs at energy $E_1^{\mathrm{}\left(0\right)\mathrm{}}$ for σ=∞ is shifted to complex energy $E_1$ (on the unphysical sheet of energy E) for small ${\sigma }^{\mathrm{-}1}$. This calculation shows that the density of states is a smooth function of energy E around the critical energy $E_c$=Re$E_1$, 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 $E_1$ 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.