## 8 Conclusions
We also discuss the calculation of two-particle response functions, such as the conductivity, which can be directly measured in experiments. A prominent advantage of the TMDCA is that it can include electronic interactions and treat the disorder and interaction on equal footing. Since in the TMDCA a geometric average of the local DOS is used for the self-consistency, it requires a real-frequency cluster solver to provide reliable spectra for each disorder configuration. A general real-frequency cluster solver that can cover the whole range of electronic interaction will greatly improve the TMDCA results to study the interplay between disorder and correlation effect. We presented two calculations for the Anderson-Hubbard model using two perturbation-based cluster solvers each of which is suitable for weak or strong interaction, respectively. Most significantly, we show that in the limits of strong disorder and weak interactions treated perturbatively, that the phenomena of 3D localization, including a mobility edge, remains intact. However, the metal-insulator transition is pushed to larger disorder values by the local interactions. We also study the limits of strong disorder and strong interactions capable of producing moment formation and screening, with a non-perturbative local approximation. Here, we find that the Anderson localization quantum phase transition is accompanied by a quantum-critical fan in the energy-disorder phase diagram. The TMDCA has been successfully combined with the Density-Functional Framework to study functional materials including the iron-based superconductors and DMS. This opens a broad venue of various applications of the developed method to realistic systems with disorder. In the future it can be applied to systems where disorder plays an important role, such as intermediate band semiconductors, topological Anderson insulators [225; 226]. Combinations of this method with other first-principle methods, including multiple-scattering theory for disordered systems is underway. Although some systems can be successfully described, as we show in our contribution, in combination with the DFT, additional methodological developments such as the full self-consistency are needed to be able to better describe the experimental observations. Conceptualization, M.J., H.T., Y.Z., N.S.V., T.B. ; methodology, H.T., Y.Z., M.J., N.S.V., T.B. ; software, H.T., Y.Z., N.S.V., T.B. ; validation, H.T., Y.Z., K.T. ; formal analysis, H.T., Y.Z., N.S.V., K.T., T.B., L.C., M.J., recourses, M.J., T.B. ; supervision, M.J.; writing-review and editing, H.T., M.J., Y.Z., K.T., T.B., N.S.V., L.C. ; funding acquisition, M.J. This research (H.T., Y.Z., and M.J.) was funded by the National Science Foundation under the NSF EPSCoR Cooperative Agreement No. EPS-1003897 with additional support from the Louisiana Board of Regents. M.J. gratefully acknowledges support from NSF Materials Theory grant DMR1728457 for the work involving quantum criticality in interacting systems, and the US DOE DE-SC0017861 for the work involving the first-principles study of strongly disordered energy materials. We thank Vladimir Dobrosavljevic, Dieter Vollhardt, Juana Moreno, Wei Ku, Vaclav Janis, Richard Scaltzer for useful conversations. A portion of the work (T.B), was conducted at the Center for Nanophase Materials Sciences, which is a DOE Office of Science User Facility. L.C. acknowledges the financial support by the Deutsche Forschungsgesellschaft (DFG) through the Transregional Collaborative Research center TRR80/F6. We thank D. Vollhardt and J. Jarrell for careful read of the manuscript. The authors declare no conflict of interest.