## 2 Background: Electron Localization in Disordered Medium  
Disorder is a common feature of many materials and often plays a key role in changing and controlling their properties. As a ubiquitous feature of real systems, it can arise in varying degrees in the crystalline host for several reasons. As shown in Figure 1, disorder may range from a few impurities or defects in perfect crystals, (vacancies, dislocations, interstitial atoms, etc. ), chemical substitutions in alloys and random arrangements of electron spins or glassy systems. One of the most important effects of disorder is that it can induce spatial localization of electrons and lead to a metal-insulator transition, which is known as Anderson localization. Anderson predicted [25] that in a disordered medium, electrons scattered off randomly distributed impurities can become localized in certain regions of space due to interference between multiple-scattering paths. Besides being a fundamental solid-state physics phenomena, Anderson localization has a profound consequences on many functional properties of materials. For example, the substitution of P or B for Si may be used to dope holes or particles into Si increasing its functionality. Disorder appears to play a crucial role also in formation of inhomogeneities in commercially important colossal magnetoresistance materials [26]. At the same time, in dilute magnetic semiconductors such as GaMnAs, there is a subtle interplay between magnetism and Anderson localization [27; 28; 29; 30; 31]. Intermediate band semiconductors are another type of material where disorder may play an important role in manipulating their properties. These materials hold the promise to significantly improve  
Figure 1: Examples of various types of disorder, including substitution and interstitial impurities, and vacancies. In addition (not shown), disorder can originate from other ways of breaking the translational symmetry, including the external disorder potentials, amorphous systems, random arrangement of spins, etc. solar cell efficiency, but only if the electrons in the impurity band are extended [32; 33; 34]. Also recently, Anderson localization of phonons has been suggested as the basis of relaxor behavior [35]. These examples show that Anderson localization has profound consequences for functional materials that we need to understand and try to control for a positive outcome. In 1977 P. W. Anderson and N. Mott shared one third each of the Nobel prize [36]. Both were, at least in part, for rather different perspectives on the localization of electrons. In Mott's picture, localization is driven by interactions, albeit originally only at the level of Thomas-Fermi screening of impurities [4]. The transition is first order, with the finite temperature second order terminus. In Anderson's picture, localization is a quantum phase transition driven by disorder. Despite more than five decades of intense research [37; 38], a completely satisfactory picture of Anderson localization does not exist, especially when applied to real materials. Several standard computationally exact numerical techniques including exact diagonalization, transfer matrix method [39; 40; 41], and kernel polynomial method [42] have been developed. They are extensively applied to study the Anderson model (a tight-binding model with a random local potential). While these are very robust methods for the Anderson model, their application to real modern materials is highly non-trivial. This is due to the computational difficulty in treating simultaneously the effects of multiple orbitals and complex real disorder potentials (Figure 2) for large system sizes. In particular, it is very challenging to include the electron-electron interaction. Practical calculations are limited to rather small systems. Also, the effects from the long-range disorder potential which happens in real materials, such as semiconductors, are completely absent.