## 8 Conclusions  
Over the past couple of decades, DMFT and its generalization, the DCA have become a major paradigm in the field of computational strongly correlated systems. They provide a new framework for the study of strong interaction. Interesting phenomena such as the metal-Mott insulator transition can be studied in a controllable fashion. A glaring shortcoming of the CPA (a DMFT analog for disordered systems) is its limitation for treating strong disorder. The Anderson insulator due to disorder is completely absent not only because of the local nature of the method but also because the average DOS used in the CPA does not serve as an order parameter for Anderson localized states. There have been cluster extensions of the CPA, including the DCA and MCPA. The DCA is the momentum-space quantum cluster theory, which is based on a mapping from the lattice models onto the quantum cluster embedded in self-consistently determined effective medium. Such mapping involves the concept of coarse-graining, and has been used in the CPA, DMFT and their cluster extensions. A very important feature of the DCA is that it is a controllable approximation with a small parameter of \(1/L_{c}\) (\(L_{c}\) is the linear cluster size), and its ability to provide systematic non-local corrections to the CPA and DMFT. This is significant, since while the CPA and DMFT are exact in the infinite dimensional \(D\) limit, a physically meaningful systematic expansion in \(1/D\) has yet to be formulated. Thus, when viewed as an extension of the DMFT/CPA, the DCA is significant in that it adds a control or small parameter to these quantum cluster approaches. When applied to disordered systems, the DCA incorporates the non-local correlations missed in the CPA, and as a result it provides a better qualitative description of the average spectra, it still cannot capture the large disorder effects, including Anderson localization. This limitation from the fact that the average DOS used in the DCA is not critical at the transition, and hence cannot serve as an order parameter. The proposal to identify the TDOS (with the geometrical not algebraic averaging over disorder) as the order parameter of Anderson localization has inspired the development of the TMT which incorporates the TDOS within the CPA formalism. The TMT is an important development in generalizing the CPA for capturing the Anderson MIT. However, a single-site approximation cannot  
Figure 46: DOS (blue) and typical DOS (red) of Ga\({}_{1-x}\)Mn\({}_{x}\)N for various Mn concentrations: \(x\) = 0.02, 0.03, 0.05, 0.1, with \(N_{c}\) = 32, showing that the impurity band is completely localized for \(x\leq\) 0.03. The chemical potential is set to be zero and denoted as the dash line. Inset: Zoom in of the DOS and TDOS around the chemical potential. Reprint from [172]. provide a quantitatively accurate calculation at finite dimensions. In addition, thus, a cluster extension along the lines of the DCA which can handle both strong interactions and disorder is desired. The TMDCA, which is a main focus of this review, is such cluster extensions, for disordered and interacting systems. Inheriting some properties from the DCA, the TMDCA is a controlled approximation with a small parameter of \(1/L_{c}\), and it systematically includes the non-local corrections to the TMT results. We discuss various benchmarks of the accuracy of the TMDCA against other conventional methods for the Anderson model, including KPM and TMM methods. The versatility of the TMDCA makes it a superior choice when dealing with more complicated models and systems. We survey a series of extensions of the TMDCA to include more chemical details of the model, including off-diagonal disorder, multiple orbitals, long ranged disorder potential and electronic interactions. These extensions make it possible to incorporate the TMDCA with first-principles calculations to study the localization in a material-specific way.