## 5 Typical Medium Theories of Anderson Localization: Model Studies
### TMDCA for Multi-Orbital Systems  
Since realistic materials also have multiple orbitals, the TMDCA formalism has been generalized to multi-orbital system at the simple model level [12] as well as for realistic materials [172]. For the standard DCA, where the Green's function is averaged over disorder algebraically, the multi-orbital generalization is as simple as replacing all the quantities in the single orbital system with their matrix form. This is because all the linear operations performed in the single orbital system are also valid in the matrix system. However, in the TMDCA, the order parameter is constructed from the typical values of the LDOS i.e., the TDOS, approximated as the geometric average of the LDOS. Therefore, we need to construct a multi-orbital generalization of the typical Green's function with an imaginary part that can properly reflect the TDOS so that it captures the localization of electrons. Since the off-diagonal elements of the LDOS are not positive definite, an extension of single-band TMDCA to multi-orbital systems is not straightforward. Despite the difficulty described above, it has been shown that [12] the critical behavior of the TDOS is independent of the local basis and the vanishing of the TDOS is equivalent to the vanishing of the typical value of the LDOS for all the orbitals, leaving some freedom to construct the appropriate typical Green's function. For the simple multi-orbital Anderson model with local diagonal disorder and guided by the selection criteria discussed in Section 5.1, we construct the following Ansatz for the typical DOS for the multi-orbital case [12]:  
\[\rho^{c,nn^{\prime}}_{typ}(\mathbf{K},\omega)=\left\{ \begin{array}{ll}e^{ \frac{1}{Nc}\sum_{l}\left\langle ln\rho^{m^{\prime}}_{ll}(\omega)\right\rangle }\left\langle\frac{\rho^{m^{\prime}}(\mathbf{K},\omega)}{\frac{1}{Nc}\sum_{l} \rho^{m^{\prime}}_{ll}(\omega)}\right\rangle,&if\ n=n^{\prime}\\ \\ e^{\frac{1}{Nc}\sum_{l}\left\langle ln\left|\rho^{m^{\prime}}_{ll}(\omega) \right|\right\rangle}\left\langle\frac{\rho^{m^{\prime}}(\mathbf{K},\omega)}{ \frac{1}{Nc}\sum_{l}\left|\rho^{m^{\prime}}_{ll}(\omega)\right|}\right\rangle,&if\ n\
eq n^{\prime}\end{array} \right. \tag{61}\]  
with  
\[\rho^{m^{\prime}}_{ll}(\omega)=-\frac{1}{\pi}\text{Im}[G^{c,nn^{\prime}}_{ll} (\omega)]\,. \tag{62}\]  
Here, \(n\) and \(n^{\prime}\) are orbital indices. As one can see, the orbital diagonal part (\(n=n^{\prime}\)) takes the same form as the single-band TMDCA Ansatz 1, while the orbital off-diagonal part (with \(n\
eq n^{\prime}\)) is of a similar form but involves the absolute value of the off-diagonal 'local' DOS. The typical cluster Green's function is then constructed through a Hilbert transformation  
\[G^{c,nn^{\prime}}_{typ}(\mathbf{K},\omega)=\int d\omega^{\prime}\frac{\rho^{c, nn^{\prime}}_{typ}(\mathbf{K},\omega^{\prime})}{\omega-\omega^{\prime}} \tag{63}\]This Ansatz has been tested in the two-band Anderson model and it was shown that it successfully captures the localization of electrons with relatively fast convergence with the cluster size (more details are described in Section 7.2.2). However, for more complicated materials such as (Ga,Mn)N, where the disorder potential contains both diagonal and off-diagonal parts, if a direct generalization of the Blackman off-diagonal disorder Ansatz above is applied, severe numerical instabilities arise when solving the self-consistent TMDCA equations. The main source of the instability comes from the Hilbert transformation used to calculate the full typical Green's function from the TDOS \(\rho_{typ}^{c,nn^{\prime}}\) of Equation (63). Since the Hilbert transformation connects the typical Green's function at all the frequencies and makes the real component of the typical Green's function a functional of its imaginary part, this means a small error at certain frequency can spread to its neighbor frequencies, which makes the calculation numerically unstable, especially for systems with multiple bands and complicated disorder potentials.