## 7 Applications of the Typical Medium DCA to Systems with Disorder
### Results for Interacting Models  
#### 7.4.1 Results from SOPT  
As discussed in the introduction, the interplay between disorder and interactions can be quite subtle and counterintuitive. Using the TMDCA, we explored the effect of weak interactions in a strongly disordered Anderson-Hubbard model through second order perturbation theory, described in Section 5.5.1. A thorough benchmarking study reveals excellent agreement of the perturbation theory results until \(U\lesssim 1.0\) (in units of \(4t=1\)) with results from the DCA-CTQMC results [11]. Beyond \(U\sim 1.0\), deviations begin to appear, and the SOPT does not remain reliable. One of the main results of this study was the absence of a sharp mobility edge separating the localized from the delocalized spectrum if the chemical potential is at or beyond the mobility edge of the corresponding non-interacting system. We show the result for both the particle-hole symmetric and asymmetric cases in Figure 40. The TDOS on a logarithmic scale vs. \(\omega\) on a linear scale, for a fixed cluster size of \(N_{c}=38\), various \(U\) values and a fixed disorder ratio \(W/W_{c}(U)=0.86\) is displayed. The non-interacting case shows a sharp drop of the TDOS at the band edges, thus exhibiting a sharp mobility edge. However, for \(U>0\), the TDOS is seen to have exponential tails at the band edges. We also found that the width of the mobility edge depends on the location of the chemical potential [10] and goes continuously to zero as the energy approaches the chemical potential. Here, the decay of the states via interactions is suppressed by the lack of phase space for which energy is conserved and the Pauli principle satisfied. This is similar to the situation in a Fermi liquid. However, here, the Pauli principle, together with energy and momentum conservation means that the scattering  
Figure 40: (Top) The typical DOS as a function of frequency, for the non-interacting case (\(N_{c}=38\), \(U=0.0\), units \(4t=1\)) and two weakly interacting cases (\(U=0.1,0.2\)) are shown for a disorder value \(W\) that is close to the critical disorder, i.e., \(W/W_{c}(U)=0.86\)[11] of the 3D Anderson-Hubbard model. The \(U=0\) TDOS shows a sharp band edge, while for \(U>0\), exponential tails are seen, indicating the broadening of the mobility edge. (Bottom) The typical DOS as a function of frequency, for the interacting case (\(N_{c}=38,U=0.2\), units \(4t=1\)) at various chemical potentials (\(\mu\)). As the \(\mu\) approaches the non-interacting mobility edge, the exponential tail seen in the top panel is replaced by a sharp edge. rate vanishes quadratically with the energy measured relative to the Fermi energy. As a result, the Fermi liquid has a resistivity which is quadratic in temperature, a linear in temperature electronic specific heat, etc. In our case, the momentum conservation is lost since the impurities break translational invariance. Therefore, we might expect a different power law; perhaps, a lower power reflecting the fact that the phase space will open more quickly than in a Fermi liquid, due to the reduced number of constraints. The absence of a sharp mobility edge may also be understood through a perturbation theory argument (which should be valid in weak coupling), where the starting point is the non-interacting disordered system having a clear mobility edge. A perturbation theory in \(U\) involves convolutions which mix the localized states below and extended states above the mobility edge, thus leading to a smearing of the TDOS band edge, and hence to a complete absence of a sharp division between the extended and localized states. Since only these states very close to the Fermi surface are probed by most experiments, this phenomena may be difficult to distinguish from the non-interacting case. The difficulty is that since the width goes to zero as the chemical potential approaches the remnant of the mobility edge. Therefore, that experiments (most of them) that probe only the states near the Fermi energy will see a sharp mobility edge.