## 5 Typical Medium Theories of Anderson Localization: Model Studies
### Disorder in Interacting Systems  
In this section, we review the modifications of the TMDCA that are required for the study of interacting disordered systems. To model the interplay between disorder and electron-electron interactions, we consider the Anderson-Hubbard model given by the Hamiltonian,  
\[H=-\sum_{<ij>,\sigma}t_{ij}\left(c_{i\sigma}^{\dagger}c_{j\sigma}+h.c.\right) +\sum_{i\sigma}\left(V_{i}-\mu\right)n_{i\sigma}+U\sum_{i}n_{i\uparrow}n_{i \downarrow}, \tag{66}\]  
here as before, \(V_{i}\) describes the random disorder potential, and \(U\) is the strength of the electron-electron interaction between electrons at site \(i\). Electron-electron interactions are unavoidable in any realistic situation and might have a dramatic effect on the MIT [175; 176; 177; 123]. The important question is, to what extent do they change the nature of the localization transition. In fact, as we have seen, near the transition, the hybridization between the cluster and its host vanishes, so that \(U/\Delta\) becomes large suggesting that interaction effects become more important near the transition. Great care must be taken while calculating disorder-averaged quantities in the presence of interactions. This is especially true when there is a need to mix linear and non-linear operations. Examples include the calculation of typical (as opposed to arithmetically averaged) spectra, or when performing measurements in a QMC simulation when there is a minus sign problem. This problem arises since disorder averaging is inherently different than the thermodynamic averaging used in the calculation of the partition function \(Z\). The latter is always linear but only applied to the arguments of \(Z\). The situation is somewhat less clear when we must also perform averaging over disorder. However, we may be guided by our desire to formulate a theory which properly describes experiments. Nearly all experimental measurements are described by response functions, which may be expressed as derivatives of the free energy. Furthermore, to obtain a large signal, most experiments, such as light scattering, are done on relatively large samples. If the sample is disordered, then this means that the response function, \(A(\mathbf{k},\omega)\) in our example, is averaged over the sample which has many local disorder configurations. The same is true for most experiments, including bolometry, nearly any scattering experiment including ARPES, neutrons, etc. Therefore, to describe these experiments, we disorder average not the partition function, but the logarithm [167; 178] of the partition function and its functional derivatives which include all the observable response functions. This rule may easily be applied to quantum cluster calculations. We start by generating disorder configurations of the cluster potential \(V\) stochastically. For a given interaction strength \(U\) and randomly chosen disorder configuration \(V\), we solve the interacting cluster problem, obtaining a set of response functions, e.g., \(G^{c}(\mathbf{K},\omega,V)\). When we have the final response functions for each disorder configuration \(V\), we then take the average over the disorder. One of the prominent advantages of the TMDCA is that electron-electron interactions can be included in a very straightforward way while respecting these rules for disorder averaging. Within the TMDCA, the only modification to the algorithm for the inclusion of interactions is through the calculation of the cluster Green's function for each disorder configuration  
\[G^{c}(V,U)=\left(\mathcal{G}^{-1}-V-\Sigma^{\text{Int}}(U)+U/2\right)^{-1}\,, \tag{67}\]  
where \(\Sigma^{\text{Int}}(U)\) is a thermodynamically averaged self-energy matrix that may be derived through a real-space, real-frequency cluster solution of the electron-electron interaction term \(U\) in the Hamiltonian of Equation (66).