## 7 Applications of the Typical Medium DCA to Systems with Disorder
### Results for Two-Particle Calculations  
The typical analysis has been applied to the single-band Anderson model to calculate the DC conductivity [190]. As shown in Figure 38, the DC conductivity vanishes in the region where the TDOS is zero. This is expected since when the TDOS is zero, meaning all states are localized on the cluster, the hybridization function also becomes zero and all clusters are isolated.  
Figure 36: Evolution of the mobility edge of the a-b two-orbital model as \(t^{ab}\) increases, while \(V^{aa}\) and \(V^{bb}\) are fixed. The results are calculated for \(N_{c}=64\). A dome-like shape shows up around the band center, signaling the closing of the TDOS gap. Reprint from [12].  
Figure 37: Comparison of the ADOS and TDOS of the a-b two-orbital model calculated with the DCA, TMDCA and KPM with fixed disorder strength \(V^{aa}=V^{bb}=0.8\) with impurity concentration \(x=0.5\) and various values of the inter-band hopping \(t^{ab}\). The KPM uses 2048 moments on a cubic lattice of size \(48^{3}\) and 200 independent realizations generated with 32 sites randomly sampled from each realization. Reprint from [12].  
The convergence of the critical disorder strength \(W_{c}\) with the cluster size \(N_{c}\) is also studied. Figure 39 shows the DC conductivity at zero chemical potential as a function of disorder strength \(W\) for several \(N_{c}\). \(W_{c}\) is defined by the vanishing of the DC conductivity. The results show that as cluster size \(N_{c}\) increases for \(N_{c}\geq 12\), the \(W_{c}\) systematically increases until it converges to \(W_{c}\approx 2.1\). This is consistent with the values reported in the literature [209]. From this cluster onward, \(W_{c}\) converges to \(\approx\)2.1. The TMDCA results are also compared with the KPM [42; 212; 213; 214] which leads to excellent agreement for most values of the disorder strength. The results get noisy near the transition (Figure 39), but the deviation from the KPM calculations is in the correct direction given that the KPM is a finite-sized approximation and the conductivity vanishes near the critical disorder strength.  
Figure 39: DC conductivity of the 3D Anderson model at \(T=0\) and \(\mu=0\) (band center) vs. disorder \(W\) for different cluster size \(N_{c}=1,10,12,64,92\). The DC conductivity vanishes at \(W_{c}\) where all states become localized. For \(N_{c}=1\) (TMT), the critical disorder strength \(W_{c}^{\lambda_{c}=1}\approx 1.65\) (units \(4t=1\)). As the cluster size increases, \(W_{c}\) systematically increases with \(W_{c}^{\lambda_{c}\gg 12}\approx 2.10\pm 0.10\) (in units of \(4t=1\)), showing a quick convergence with cluster size to the KPM result. Reprint from [190].  
Figure 38: The evolution of the ADOS, TDOS and DC conductivity of the single-band 3D Anderson model at various disorder strengths \(W\) for the single-site TMT and the TMDCA with cluster size \(N_{c}=64\). Here, for the DC conductivity, \(\omega\) corresponds to the chemical potential used in the calculation. Arrows indicate the position of the mobility edge, which separates the extended electronic states from the localized ones. Reprint from [190].