## 7 Applications of the Typical Medium DCA to Systems with Disorder
### Results for the Anderson Model  
#### 7.1.1 Typical DOS as an Order Parameter for Anderson Localization  
We start our discussion of the results by presenting the application of the TMDCA to a single-site Anderson model in 3D. First we demonstrate that the typical and not the average DOS can serve as a proper order parameter for defining the Anderson localization transition. In Figure 30, we compare the algebraically averaged DOS (ADOS) calculated using the conventional DCA scheme (dashed lines) and the TDOS (solid lines) obtained from both a single-site TMT (left panel, \(N_{c}=1\)) and finite clusters obtained from the TMDCA (right panel, \(N_{c}=38\)). The TMDCA employed Ansatz 1 for various disorder strengths \(W\) for the box disorder distribution with \(P(V)=\frac{2}{10}\theta(W-|V|)\).  
As seen from Figure 30, as the disorder strength increases, the ADOS broadens but remains finite while the TDOS obtained from both the TMT (\(N_{c}=1\)) and the TMDCA (\(N_{c}=38\)) continuously decreases. It eventually vanishes even at the band center at the critical disorder strength with \(W_{c}(N_{c}=1)\)\(\approx\) 1.65 and \(W_{c}(N_{c}=38)\)\(\approx\) 2.25 (in units \(4t=1\)). Below the transition, for \(W<W_{c}\), the part of the spectrum with vanishing TDOS corresponds to localized states, while the part of spectrum with a finite TDOS corresponds to the extended states. As one can see the band tail localize first. Also, notice that at small disorder with \(W<<W_{c}\), e.g., \(W=0.4\) the ADOS and the TDOS are almost the same. This indicates that at small disorder the TMDCA reduces to the standard DCA scheme, which is consistent with the analysis used to construct Ansatz 1 in Section 5.2.  
Comparing the local TMT (\(N_{c}=1\)) and the non-local TMDCA (\(N_{c}>1\)) results, one observes a crucial difference between them. For the local TMT, the mobility edge (indicated by arrows) delineating the region with extended states where the TDOS is finite, always becomes narrower with increasing disorder strength \(W\). For a finite cluster TMDCA, the mobility edge first expands and then decreases, hence giving rise to the re-entrance behavior, missing in the single-site TMT.  
The resulting \(W-\omega\) (disorder-energy) phase diagram is shown in Figure 31. Here, we show the mobility edge trajectories, (obtained by the frequencies \(\omega\) where the TDOS vanishes at a given disorder strength \(W\)), and the band edge trajectories, (where the ADOS calculated within the DCA scheme vanishes). To benchmark our results, we also present the mobility edge trajectories obtained from the TMM. The finite cluster TMDCA trajectories gradually approach the TMM results with the re-entrance behavior, (missing in \(N_{c}=1\) case) recovered with increasing cluster size. For a large clusters \(N\geq 92\) our TMDCA results converge to TMM trajectories within the errors of both approaches.  
Figure 30: (Left) TMT (\(N_{c}=1\)); (Right) TMDCA (\(N_{c}=38\)). The ADOS (dash line) and the TDOS (solid lines) of the 3D Anderson model for different disorder strengths \(W\) in units where \(4t=1\). At low disorder, the ADOS and TDOS coincide. This also indicates that the TMDCA and the DCA solutions are very similar. As disorder strength \(W\) increases the TDOS becomes suppressed and vanishes above the transition. In the local TMT (\(N_{c}=1\)) scheme, the mobility edge (indicated by arrows) moves towards the band center \(\omega=0\) monotonically, while in the TMDCA the mobility edge first moves to higher energy, and roughly around \(W>1.75\) it starts moving towards the band center, indicating that TMDCA can successfully capture the re-entrance behavior missing in the TMT scheme. Reprint from [174].