## Department of Physics Papers

#### Document Type

Journal Article

#### Date of this Version

6-1-1985

#### Publication Source

Physical Review B

#### Volume

31

#### Issue

11

#### Start Page

7393

#### Last Page

7407

#### DOI

10.1103/PhysRevB.31.7393

#### Abstract

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}^{(0)} for *σ=∞* is shifted to complex energy E_{1} (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 E_{c}=ReE_{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.

#### Recommended Citation

Kim, Y.,
&
Harris, A.
(1985).
Density of States of the Random-Hopping Model on a Cayley Tree.
*Physical Review B,*
*31*
(11),
7393-7407.
http://dx.doi.org/10.1103/PhysRevB.31.7393

**Date Posted:** 12 August 2015

This document has been peer reviewed.