## 5 Typical Medium Theories of Anderson Localization: Model Studies
### Disorder in Interacting Systems
#### 5.5.2 Stat DMFT Approach
Extensions to statistical DMFT combined with the slave-boson solver yields a \(P(T_{K})\) that also has a long tail at larger \(T_{K}\) but is not divergent at lower scales [125]. Instead, it is highly skewed, has a maximum at a specific scale, and has either a vanishing or a finite intercept depending on whether the disorder is below or above a critical disorder value. Such a distribution with a finite intercept has been shown to be a sufficient condition for the system to exhibit non-Fermi liquid (nFL) behavior in transport and thermodynamics. Thus, these theories have provided a route to explain the crossover from conventional metallic behavior at low disorder to singular, non-Fermi liquid behavior at strong disorder [177; 188]. Nevertheless, since slave-boson methods are just a renormalized version of the non-interacting limit, and hence fail to capture dynamics at all energy scales, the above theories do not provide an insight into the role of dynamics in the Fermi liquid to non-Fermi liquid crossover. Additionally, since the stat-DMFT does not incorporate an embedding of the disordered cluster into a translationally invariant medium, it does not allow access to Anderson localization unless the cluster is prohibitively large. The TMDCA combined with a cluster solver based on stat-DMFT and the LMA does not suffer from the two shortcomings of the previous work. A rapid convergence with increasing cluster size, ensured by the embedding of the cluster in a medium, ensures the feasibility of the solver, thus allowing the replacement of the slave-boson solver by a non-perturbative, albeit more expensive method such as the LMA. Additionally, the TMDCA captures Anderson localization almost exactly in the non-interacting case, as discussed in the previous sections. The stat-DMFT-based TMDCA algorithm is illustrated in Figure 26[187]. The input to the cluster solver is the real-space hybridization matrix, derived through the real-space host Green's function. It, in turn, can be obtained through a Fourier transform of the **K**-space host Green's function, \(\mathcal{G}(\textbf{K},\omega)\)The cluster solver begins with a solution of \(N_{c}\) impurity problems, for which the two required inputs are the local orbital energy, \(\epsilon_{I}=-U/2+V_{I}\), and the local hybridization function, \(\Delta_{II}^{(\mathrm{o})}(\omega)\). The output of this step is a diagonal self-energy matrix, \(\underline{\Sigma}(\omega)\). The second step uses the modified Dyson's equation, namely  
\[\underline{\Omega}^{c}(V,\omega)=\left[\underline{\mathcal{G}}^{-1}- \underline{\Sigma}-\underline{\epsilon}\right]^{-1} \tag{70}\]  
which incorporates the effects of interactions and disorder on an equal footing and yields the real-space cluster Green's function. This can now be inverted as shown in Step-3 of Figure 26, to get a new local hybridization function, \(\Delta_{II}^{(n)}(\omega)\). The final step involves a stat-DMFT self-consistency check of the local hybridization function, as shown in Step-4. If \(\Delta_{II}(\omega)\) is found to have been converged (within a numerical tolerance), the cluster solver is exited, with the output being the cluster Green's function found at Step-2, else the new local hybridization function is plugged back into the Step-1 of the cluster solver, and these steps are repeated until the convergence is reached [189]. The last box in Figure 26 shows that the output of the cluster solver is the converged (within stat-DMFT) cluster Green's function for a single disorder configuration (as obtained in Step-2). Subsequently this is then Fourier transformed to cluster momentum space, and the disorder average is carried out, as in the standard TMDCA algorithm (see Section 5). In practice, since the number of disorder realizations is very large (\(\sim\)3000) and the largest cluster size used was \(N_{c}=38\), a very large set of impurities (\(\sim\)10\({}^{5}\)) need to be solved.