## 5 Typical Medium Theories of Anderson Localization: Model Studies
### Building Quantum Cluster Theories for the Study of Localization  
In this section, we address these difficulties associated with the construction of a mean-field theory with no known non-trivial exact limiting solution. Our approach will be to construct a theory which inherits the desirable properties of the DMFT/CPA and DCA in the weak disorder limit, while also incorporating the TDOS order parameter into the mean-field host ensuring that the method is also able to capture localization phenomena. The natural way to improve upon the local TMT is to construct a cluster extension which satisfies the constraints mentioned in Section 2.1 which when rephrased in terms of clusters are:We approximate the coupling of the clusters to their lattice environment at the single-particle level (akin to the Fermi golden rule) neglecting two-particle and higher processes. This coupling is proportional to the square of a matrix element between the cluster and its host, times an appropriate DOS which describes the states available on the surrounding clusters. 2. Since on average each cluster is equivalent to all the others, this DOS will also be proportional to some appropriate cluster density of states. In addition, since the distribution of the DOS is highly skewed, the typical DOS is quite different than the average DOS. The typical cluster DOS, which is clearly more representative of the local environment, will be used to define the effective medium. In addition, there are several additional desirable properties of a cluster theory, some of which appear in [77] which should also be satisfied if possible:  
Maintain the translational invariance of the impurity averaged cluster, i.e., there should be no distinction between, e.g., sites in the center and those at the boundary of the cluster. 4. The clusters should maintain the point group symmetries of the lattice. 5. The method should be fully causal, with positive definite spectra \(A(\mathbf{K},\omega)=-1/\pi\Im G(\mathbf{K},\omega)>0\)
6. It should recover the DCA when the disorder is weak. 7. it should recover the TMT when \(N_{c}=1\)
8. In lieu of interactions, the scatterings at different energies are completely independent of each other. 9. For large \(N_{c}\to\infty\) it should become exact while avoiding self-averaging effects. 10. It should be extensible to multiple bands, and realistic models with longer ranged diagonal and off-diagonal disorder  
Based on these criteria, we have constructed a set of TMDCA algorithms, listed in Table 1. By construction, all the algorithms listed in the table satisfy the first two criteria. Furthermore, since they each map the periodic lattice problem onto a self-consistently embedded periodic cluster, they all maintain translational invariance. The point group symmetry of the cluster is a matter of choice. By allowing the cluster to have a lower symmetry than the lattice, there are far more clusters that can be used, e.g., in cluster size scaling calculations. The quality and the selection criteria for the clusters have been addressed by D.D. Betts [168; 169; 170]. All proposed algorithms are fully causal. The first two algorithms discussed below may be shown to be causal with a proof involving two conformal maps [8; 24]. This proof is not applicable to the multiband methods; however, we have not observed any causality violations in the iteration of the resulting equations. All the algorithms recover the DCA in the weak disorder limit, whereas they do not all recover the TMT when \(N_{c}=1\). There appears to be a trade-off between this and maintaining the independence of the scatterings at different energies. The algorithms which use a Hilbert transform to calculate the imaginary part of the cluster Green's function, including the original TMT, violate this rule. The ones that calculate the cluster typical Green's function directly (and not the typical DOS), both imaginary and real parts, satisfy the rule.