Impurity problems self-consistency

We can now solve self-consistently for the effective medium From the electronic structure calculation we get the tls for the different atomic kinds and the S matrix. Then we solve Eq.(5.6) to get the tc for the effective medium, which we use in Eq.(5.4) to get the scattering path operator for the effective medium. This then gives the new scattering path operators for the different atomic kinds by using Eq.(3.44) for the impurity problem. These are what is needed to restart the calculations for the next iteration. 

Let us finally point out that we have focused on bulk quantum phase transitions. Impurity quantum phase transitions" require a separate discussion that is beyond the scope of this chapter (Note, however, that within the DMFT method a bulk quantum many-particle system is mapped onto a self-consistent quantum impurity model.) Some of the methods discussed here such as quantum Monte Carlo can be adapted to impurity problems. Moreover, there are powerful special methods dedicated to impurity problems, most notably Wilson s numerical renormalization group. " " ... 

Several calculations were devoted to the study of the influence of a single, isolated vacancy on the electronic structure of an otherwise perfect crystal. It is assumed that the results obtained by investigating a single vacancy are also characteristic for crystals with somewhat higher vacancy concentrations. Ivanovsky et al. (131) considered an isolated vacancy as an impurity and proposed a self-consistent solution for this problem. These authors started from the LMTO procedure and the KKR-GF equations and developed an LMTO-GF method, which they applied to the calculation of the electronic structure of TiC, TiN, TiO, VC, VN, and VO containing... 

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