## 7 Applications of the Typical Medium DCA to Systems with Disorder
### Results for Models with More Realistic Parameters
#### 7.2.1 Off-Diagonal Disorder  
So far, we have presented the TMDCA results for systems with local disorder having potentials coupling only the density operators. As they are diagonal in the creation and annihilation operators, this is called diagonal disorder. However, in many materials, the disorder not only affects the strength of the local potential, but it also impacts the strength of the hopping of electrons between different sites. Since this involves the creation of an electron on one site and the annihilation on another site, the associated disorder is called non-local or off-diagonal disorder. To demonstrate that our TMDCA scheme can properly treat such generalized cases of disorder and to understand how the off-diagonal disorder affects the electron localization, we first present the results for the 3D single-band Anderson model with disorder and hopping defined by the Hamiltonian Equation (17). To illustrate the method, we return to our simple model of an AB binary alloy. In Figure 33, we present the results for the TDOS obtained from the generalized TMDCA and the ADOS obtained from the DCA schemes for several values of the diagonal disorder strength \(V_{A}=0.15,0.6,1.0\) at fixed off-diagonal disorder amplitudes \(t^{AA}=1.5\), \(t^{BB}=0.5\), \(t^{AB}=1.0\). Figure 33: ADOS and TDOS of the A-B binary alloy model with off-diagonal disorder. The left panel displays results for the TMT \(N_{c}=1\) and the right panel for the TMDCA with \(N_{c}>1\). The average DOS (dash-dotted line) and the TDOS (shaded regions) for \(N_{c}=1\) (left panel), \(N_{c}=4^{3}\) (right panel) and blue dash lines for \(N_{c}=5^{3}\) (left panel) for various values of the local potential \(V_{A}\) with off-diagonal disorder parameters: \(t^{AA}=1.5\), \(t^{BB}=0.5\), \(t^{AB}=0.5\)(\(t^{AA}=t^{BB}\)), and \(c_{A}=0.5\). We show the TDOS for several cluster sizes \(N_{c}=1\), \(4^{3}\), and \(=5^{3}\) to show its systematic convergence with increasing cluster size \(N_{c}\). The ADOS converges within our numerical precision for cluster sizes beyond \(N_{c}=4^{3}\). The TDOS is finite for the extended states and zero for localized states. Reprint from [211]. We also present data for the local \(N_{c}=1\) case, to demonstrate the effect of non-local correlations captured within the finite cluster \(N_{c}=4^{3}\) and \(5^{3}\) DCA and TMDCA algorithms. The ADOS data for \(N_{c}>1\) shows that non-local multisite effects lead to the development of finite detailed structures in the DOS and the partial filling of the gap at larger values of disorder strength. Comparing TDOS and ADOS, we observe that for small disorder \(V_{A}\), both are practically the same. This is consistent with our analytical construction of the Ansatz (Equation (59)), where for small disorder strength, the TMDCA should converge to the DCA scheme. As the disorder strength \(V_{A}\) increases, significant differences start to emerge. Increasing \(V_{A}\) leads to the gradual opening of a gap which is more pronounced in the \(N_{c}=1\), For weaker disorder, \(V_{A}=0.6\), it is partially filled for the \(N_{c}>1\) clusters. As compared to the diagonal disorder case [174], the average DOS and TDOS become asymmetric with respect to zero frequency due to the off-diagonal randomness. We again observe that the local TMT (\(N_{c}=1\)) underestimates the extended states regime by having a narrower TDOS as compared to the case when \(N_{c}>1\). We performed a similar analysis for a range of \(V_{A}\) values, and our final result for the \(V_{A}-\omega\) parameter space is shown in Figure 34. Here for comparison we present the mobility edge boundaries (extracted from boundaries where the TDOS vanishes) from the single TMT (\(N_{c}=1\)) and the non-local TMDCA (\(N_{c}>1\)) results, and benchmark with the TMM results. The mobility edges shown in Figure 34 were extracted from the TDOS, with boundaries being defined by zero TDOS.