## 5 Typical Medium Theories of Anderson Localization: Model Studies  
In this section, via a series of subsections, we develop a formalism which incorporates the typical medium analysis into the DCA. The resulting formalism enables the study of electron localization in models derived from first-principles DFT calculations. As summarized in Table 1, a progression of quantum cluster theories is proposed, each incorporating more chemical details of the model, including both diagonal and off-diagonal disorder, multiple bands, and electronic interactions. This culminates in a formalism able to deal with the full chemical details provided by modern electronic structure calculations.  
These developments are hampered by the lack of a limit where these mean-field theories are exact. Typically, we develop mean-field theories which are exact in some physically meaningful limit, such as the limit of infinite dimensions. The resulting theory then inherits some features due to this exactness even when applied in finite dimensions, such as thermodynamic consistency, translational invariance, etc.  
However, to be most useful, the mean-field theory must yield results that are reasonably consistent with the real solution in finite dimensions. Magnetism is a good example. Here, the Weiss mean-field theory becomes exact in infinite dimensions. With a proper scaling of the model parameters with the dimensionality \(D\), the phase diagram of the \(3D\) model can be qualitatively reproduced by the mean-field formalism. However, the details of the transition, such as the universality class, may change with \(D\), even becoming mean-field like above the upper critical dimension. Despite this, since the transition persists, the mean-field theory may be used to study it.  
For localization, the problem is complicated by the fact that the phenomena do not persist into infinite dimensions. As we have seen, the CPA/DMFT becomes exact in the infinite dimensional limit. However, as discussed in Section 2.1, they fail to capture localization due to the self-averaging nature of the average DOS used to define their effective medium. As a cluster extension of these formalisms, the DCA also fails to capture Anderson localization phenomena [8] and so fails to provide an adequate mean-field theory for localization.  
A significant step towards this goal was developed by Dobrosavljevic et al. [9]. They demonstrated that the TDOS vanishes as the disorder strength increases, and hence can serve as a proper order parameter for Anderson Localization. The authors constructed the typical medium theory (TMT), where they incorporated the geometric averaging over disorder in the CPA self-consistency loop.  
The TMT is the first successful mean-field theory for Anderson Localization. Nevertheless, because of its local single-site nature, it suffers several drawbacks. It underestimates the critical disorder strength by about twenty percent and does not capture the re-entrance features in the mobility edge (see Section 7), The lack of a non-trivial limit where it becomes exact, can make the results difficult to interpret. For example, the TMT predicts a transition in any dimension, but it is not clear a priori whether this is more likely true in high or low finite dimensions. The CPA, which is exact in high dimensions, inherits several features from this exact limit. For example, without a priori knowledge of the upper critical dimension, we might be more inclined to believe its predictions for a 3D model over those for a 1D model. This lack of an exact limit makes the imposition of any other a priori known constraints significant.