Application to a Model Hamiltonian

One can, of course, generalize G, Ar(z) to the case of G,at(z) and in this way investigate the localization properties of a particular solution (Pff z) at a certain energy z = lim,..o ( + f ) at an arbitrary site i, by inserting Gj,jv(z) instead of Gij (z) in the ratio (4.104). The expression for Gi,N z) can be found elsewhere. In most practical cases, however, when the chains are not very long, equation (4.104) suffices for the analysis of the localization properties of a wave function. 

As an example we consider a model Hamiltonian matrix in the case of orthogonal basis functions (S = I). The Hamiltonian for these systems is assumed to possess zero matrix elements Hy when / — y 4. It is trivial to modify the computer programs to consider a greater or lesser number of diagonals. The matrix half-bandwidth of 5 was chosen arbitrarily to demonstrate the technique for something other than the usual trivial case of the tridiagonal Hamiltonian of a one-dimensional system. 

The Hamiltonian of the ordered system has matrix elements given 

FIGURE 4.31. Comparison oflogGi (A)/( vjv(A) for the ordered and disordered systems defined by equations (4.109) and (4.110). Strong localization at the band edges of the disordered system and complete delocalized behavior in the ordered system is predicted. 

---