## 5 Typical Medium Theories of Anderson Localization: Model Studies
### Disorder in Interacting Systems
#### 5.5.2 Stat DMFT Approach  
The SOPT method described above is applicable only in the weakly interacting regime. Unfortunately for the strong coupling regime, there are very few cluster solvers available for disordered interacting electron systems. The two most extensively used solvers capable of treating a wide range of energy and length scales, and are numerically exact, are quantum Monte Carlo methods [182; 183] and exact diagonalization [19; 142; 143; 19]. Quantum Monte Carlo methods have been extended to clusters [46; 158]. However, since the typical averaging must be performed on the real-frequency spectral function, the ill-posed step of analytic continuation is required for every disorder configuration and in every TMDCA iteration, rendering them unusable. Alternatively, exact diagonalization may be used, but as is well-known,  
Figure 25: The diagrams for the first and second-order self-energy of the Hubbatur model labeled in real space. The indices \(I,J\) indicate sites in the real-space cluster, while the lines are Hartree-corrected propagators \(\mathcal{G}\). the cluster sizes that can be treated are very modest, and the associated computational expense is quite substantial. At present, the only fully non-local cluster solver available that is computationally feasible and yields a real-frequency self-energy is a straight perturbation theory. Thus, one must resort to approximate cluster solvers, especially for investigating the strong coupling regime. Such a solver may be constructed by combining a non-perturbative real-frequency single-site solver and statistical DMFT [179]. The former must be capable of treating the moment formation and Kondo physics characteristic of the strong coupling regime. It must also properly incorporate the eventual many-body screening of the local moment leading to a singlet ground state. The resulting formalism is then able to capture these local dynamical correlations due to \(U\), while treating the corresponding non-local correlations at a static level. On the other hand, the correlations due to the disorder are captured exactly up to a length scale given by the linear cluster size. There are several excellent real-frequency solvers available to treat the strong coupling regime of the single-impurity Anderson model. Among them are the NRG, non-crossing approximation and the LMA [180]. Since we have used the LMA for our investigations, we provide a brief introduction to this method here. It is a diagrammatic perturbation theory-based impurity solver, starting with the unrestricted Hartree-Fock static mean-field solution. The symmetry, broken at the mean-field level, is restored through the inclusion of transverse spin flip dynamics. This symmetry restoration step, equivalent to restoring adiabatic continuity to the non-interacting limit, leads to the emergence of a low-energy Kondo scale, \(T_{K}\). The latter is an exponentially small scale in strong coupling, proportional to \(\exp{(-\alpha U/\Gamma)}\), where \(\alpha\) is a number \(\sim\)\(\mathcal{O}(1)\), \(U\) is the local Hubbard repulsion, and \(\Gamma\) is the hybridization of the impurity with the local reservoir at the chemical potential. Since, within stat-DMFT, the hybridization is site-dependent. Rather than a single Kondo scale for the entire system, a distribution of Kondo scales, \(P(T_{K})\) is obtained. The form of such a distribution and its consequences on the properties of the disordered system have been extensively investigated using slave-boson methods and phenomenological arguments [184; 185; 186]. It has been seen in the above-mentioned studies that typical medium theory-based calculations yield a Kondo scale distribution \(P(T_{K})\) exhibiting a long tail at higher Kondo scales, while diverging at a specific, lower bound scale. This is determined by the solution of the impurity problem in the particle-hole symmetric limit [187].