Recursive-Green-function treatment

In this section, we construct the GF for a finite chain with an applied held (Davison et al 1997), by using the CF elements of the recursion method (Haydock 1980) and thereby build the GF atom-by-atom, in a similar way as the causal-surface GF approach (Pendry et al 1991). 

The system is defined as a linear chain of m lattice sites (labelled n = 1, .m), which are initially taken as isolated from each other (see Fig. 7.2(a)), so that the GF Gm.m for this state has simple diagonal elements 

Returning to the initial GF (7.20), we now modify the system, by adding a single bond of energy fim 1 m between sites m — l and m. We can obtain the GF i m from Gm m by means of the Dyson equation (3.3) 

The above recursive process, for the 2-atom chain, can now be repeated for the 3-atom case. The Dyson equation analogous to (7.23) is 

The recursive procedure can be applied repeatedly, with one more atom being bonded to the growing chain at each step, producing GF s in CF form similar to those of (7.29) and (7.34). After (m — 1) iterations, all m (initially isolated) atoms are joined, and the GF at the n = 1 site has CF form 

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