Many, perhaps most, materials of current interest for potential technological applications are far from being perfect and homogeneous. In fact, disorder is common and often unavoidable feature of real materials. It appears, that in some materials, controlled disorder induced by chemical doping, irradiation or strain from ion implantation may expose the role of disorder in stabilizing or destroying phases. For example, disorder can lead to electron localization, affect the quantum critical regime, lead to nano-scale phase separation, stabilize the integer quantum Hall effect. Some doped semiconductors rely on the intentional introduction of impurities to modulate their electrical properties.

One of the most fundamental disorder-related phenomenon is the disorder-induced spatial localization of electron, known as Anderson localization. As shown by Anderson in 1958 , increasing disorder in a metal will eventually turn it to an insulator in which the extended states of a metal become localized. Studies on Anderson localization has recently been peaked by the discovery that many of the interesting properties of the technologically important doped semiconductors (dilute magnetic semiconductor GaMnAs ) are actually due to the interplay of localization, spin-orbit coupling and magnetic double exchange.

There are few numerical methods which allow for the reliable study of Anderson localization in real materials or in interacting disordered systems. The major bottleneck is the need of very large systems for such numerical studies. Hence, various embedding schemes are necessary. Together with Mark Jarrell group at LSU, we developed the typical medium dynamical cluster approximation (TMDCA) to study localization in disordered electronic systems with both diagonal and off-diagonal disorder. The TMDCA is a multiscale method which systematically incorporates non-local correlations and converges surprisingly quickly with cluster size to the exact result, reducing the numerical complexity of treating localization by orders of magnitude when compared to the present state of the art methods. This enables study of Anderson localization in interacting disordered systems and for models of realistic materials. In future, I intend to continue the application of the developed method to treat disorder in real materials.