## 5 Typical Medium Theories of Anderson Localization: Model Studies
### Typical Medium Dynamical Cluster Approximation (TMDCA)
Since the cluster problem is solved in real space, we then Fourier transform \(\mathcal{G}(\mathbf{K},\omega)\) to real space: \(\mathcal{G}_{I,J}=\sum_{\mathbf{K}}\mathcal{G}(\mathbf{K})\exp(i\mathbf{K} \cdot(\mathbf{R}_{I}-\mathbf{R}_{J}))\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**System/Ansatz** & **Characteristics** & **ODP** & **VDP** \\ \hline Single Band & Recovers TMT at \(N_{c}=1\). & & \\ Local (diagonal) Disorder & Recovers DCA for \(W<<W_{c}\) & 8 & 7 \\ Ansatz Equation (45) & Calculate \(p_{typ}\) & & \\  & Hilbert trans. for \(G_{typ}^{c}\) & & \\ \hline Single Band & Not TMT when \(Nc=1\). & & \\ Local (diagonal) Disorder & Recovers DCA for \(W<<W_{c}\) & 7 & 8 \\ Ansatz Equation (47) & Calculate \(G_{typ}^{c}\) directly & & \\ \hline Single Band & \(2\times 2\) matrix & & \\ Off-Diagonal Disorder & Calculate \(\rho_{typ}\) matrix & 8 & 7 \\ Ansatz Equation (59) & HT to get \(G_{typ}^{c}\) matrix & & \\ \hline Multiband Systems & Matrix in orbital space & & \\ Local Disorder & Calculate \(\rho_{typ}\) matrix & 8 & 7 \\ Ansatz Equation (61) & HT to get \(G_{typ}^{c}\) matrix & & \\  & Recovers DCA for \(W<<W_{c}\) & & \\ \hline Realistic Material Systems & Matrix in orbital space \(G_{typ}^{c}\) & & \\ Complex Disorder Potentials & Recovers DCA for \(W<<W_{c}\) & 7 & 8 \\ with full DFT detail & & & \\ Ansatz Equation (47) & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: A progression of TMDCA algorithms, with each one able to incorporate greater chemical detail as we go down the list. The first column lists systems that may be studied together with the label of the defining Ansatz. The second column lists some additional characteristics including a brief discussion of the desirable properties. The columns labeled VDP and ODP identify the desirable properties, discussed above, which are notably violated and observed. 5. We solve the cluster problem using, e.g., a random sampling simulation. Here, we stochastically generate random configurations of the disorder potential \(V\). For each disordered configuration, we construct the new fully dressed cluster Green's function as \[G^{c}(V)=(\mathcal{G}^{-1}-V)^{-1}.\] (50) We then calculate the disorder-averaged, typical cluster Green's function \(G^{c}_{typ}(\mathbf{K},\omega)\) via the Hilbert transform using Equation (46) for Ansatz 1, or we can directly calculate the \(G^{c}_{typ}(\mathbf{K},\omega)\) from Equation (47) if we use Ansatz 2. 6. With the cluster problem solved, we use the obtained typical cluster Green's function \(G^{c}_{typ}(\mathbf{K},\omega)\) to obtain a new estimate for the cluster self-energy \[\Sigma(\mathbf{K},\omega)=\mathcal{G}^{-1}(\mathbf{K},\omega)-(G^{c}_{typ}( \mathbf{K},\omega))^{-1}\] (51)
7. We repeat this procedure starting from 2, until \(\Sigma(\mathbf{K},\omega)\) converges to the desired accuracy. We note that instead of using the self-energy in the self-consistency, one can also use the hybridization function \(\Delta(\mathbf{K},\omega)\). Both procedures are observed to converge to the same solution. The current implementation of the TMDCA is limited to zero temperature analysis, where the real-frequency formalism with the positive local DOS allows to perform the geometrical averaging over the disorder with \(exp<\ln\rho(w,V)>\). Such constraint is a limiting factor of the finite temperature extensions of the TMDCA with the use of the Matsubara frequency formalism, which requires to perform the analytical continuation to obtain the real-frequency spectrum so that the geometric averaging can be employed. Since the geometric averaging is performed in every iteration of the calculation, the errors brought by the analytical continuation will propagate through the iterations and make the calculations numerically unstable.