## 5 Typical Medium Theories of Anderson Localization: Model Studies
### Typical Medium Dynamical Cluster Approximation (TMDCA)  
In this section, we develop a cluster extension of the TMT, the typical medium DCA formalism (TMDCA) for the single-band Anderson model in 3D with diagonal disorder (the Hamiltonian was given in Section 4.1). Due to the lack of a limit where the formalism becomes exact, the defining Ansatz for this formalism is not uniquely defined. In consideration of this, we will be guided by the desirable properties listed above. We found two Ansatzes which satisfy most of these desirable properties. * Ansatz 1 \[\rho_{typ}^{c}(\mathbf{K},\omega)=exp\frac{1}{N_{c}}\sum_{I}^{N_{c}}\langle \ln\rho_{I}^{c}(\omega,V)\rangle\left\langle\frac{\rho^{c}(K,\omega,V)}{\frac{ 1}{N_{c}}\sum_{I}\rho_{I}^{c}(\omega,V)}\right\rangle.\] (45) When the cluster size \(N_{c}=1\), this Ansatz [10] recovers the local TMT with \(\rho_{typ}(\omega)=e^{\langle\ln\rho(w,V)\rangle}\). For weak disorder, the TMDCA recovers the average DCA results, with \(\rho_{typ}(\mathbf{K},\omega)\approx\langle\rho(\mathbf{K},w,V)\rangle\). In addition, in the limit of \(N_{c}\rightarrow\infty\), the TMDCA becomes exact. Hence, between these limits, this Ansatz 1 of the TMDCA systematically incorporates non-local correlations into the local TMT. Since, this Ansatz uses the TDOS, to get typical cluster Green's function \(G_{typ}^{c}(\mathbf{K},\omega)\), we use a Hilbert transformation, with \[G_{typ}^{c}(\mathbf{K},\omega)=\int d\omega^{\prime}\frac{\rho_{typ}^{c}( \mathbf{K},\omega^{\prime})}{\omega-\omega^{\prime}}\,.\] (46)
* Ansatz 2 While Ansatz 1 works rather well for simple single-band models with local and non-local disorder, we find that it can suffer from numerical instabilities when applied to complex first-principle effective Hamiltonians with many orbitals and non-local disorder potentials. Such numerical instabilities arise due to the Hilbert transformation which is used to calculate the Green's function from the TDOS \(\rho_{typ}^{c}(\mathbf{K},\omega)\). To avoid such numerical instabilities, we constructed the following Ansatz 2 [172] where we calculate \(G_{typ}^{c}(\mathbf{K},\omega)\) directly as \[G_{typ}^{c}(\mathbf{K},\omega)=exp\frac{1}{N_{c}}\sum_{I}^{N_{c}}\langle\ln \rho_{I}^{c}(\omega,V)\rangle\left\langle\frac{G^{c}(\mathbf{K},\omega,V)}{ \frac{1}{N_{c}}\sum_{I}\rho_{I}^{c}(\omega,V)}\right\rangle.\] (47) This Ansatz 2 again incorporates the typical value of the local DOS, the resulting formalism again becomes exact in the limit of \(N_{c}\rightarrow\infty\), promotes numerical stability of the algorithm, and converges quickly with cluster size. As noted in Table 1 it does not reproduce the TMT when \(N_{c}=1\). This is due to the lack of a limit where the formalism is exact so that the Ansatz may be uniquely defined. These two Ansatzes will be used below as paradigms for the development of Ansatzes for more realistic systems and will be referred to as Ansatz 1 and 2, respectively. The main modification of the DCA self-consistency loop for the TMDCA involves the calculation of the cluster typical Green's function \(G_{typ}^{c}(\mathbf{K},\omega)\) using Equations (45) and (46) or Equation (47). The typical Green's function is then used to complete the self-consistency loop. A schematic diagram of the TMDCA self-consistency loop is shown in Figure 23. The TMDCA iterative procedure is described as follows:  
1. We start with a guess for the cluster self-energy \(\Sigma(\mathbf{K},\omega)\), usually set to zero. 2. Then we calculate the coarse-grained cluster Green's function \(\tilde{G}(\mathbf{K},\omega)\) as \[\tilde{G}(\mathbf{K},\omega)=\frac{N_{c}}{N}\sum_{k}\frac{1}{\omega+\mu-\varepsilon (\tilde{k}+\mathbf{K})-\Sigma(\mathbf{K},\omega)}\,.\] (48)
3. The cluster problem is now set up by calculating the cluster-excluded Green's function \(\mathcal{G}(\mathbf{K},\omega)\) as \[\mathcal{G}(\mathbf{K},\omega)=\frac{1}{\frac{1}{\mathcal{G}(\mathbf{K},\omega )}+\Sigma(\mathbf{K},\omega)}\,.\] (49)
4.