## 2 Background: Electron Localization in Disordered Medium
### Anderson Localization  
Strong disorder may have dramatic effects upon the metallic state [38]: the extended states that are spread over the entire system become exponentially localized, centered at one position in the material. In the most extreme limit, this is obviously true. Consider for example a single orbital that is shifted in energy so that it falls below (or above) the continuum in the density of states (DOS). Clearly, such a state cannot hybridize with other states since there are none at the same energy. Thus, any electron on this orbital is localized, via this (deep) trapped states mechanism, and the electronic DOS at this energy will be a delta function. Of course, this is an extreme limit. Even in the weak disorder limit, the resistivity of ideal metallic conductors decreases with lowering temperature. In reality, at very low temperatures, the resistivity saturates to a residual value. This is due to the imperfections in the formation of the crystal. If the disorder is not too strong, the perfect crystal remains a good approximation. The imperfections can be considered as the scattering centers for the current-carrying electrons. Hence, the scattering processes between the electrons and defects lead to the reduction in the conduction of electrons. For low dimensional systems, the scattering can induce substantial change even for weak disorder. Within the weak localization theory, based on the Langer-Neal maximally crossed graphs, the correction to the conductivity can be rather large [79; 80; 81]. It can drive a metal into an insulator for dimension \(D\leq 2\) (D is a dimensionality of the system) if the impurity does not break time reversal symmetry. Historically, it was first shown by Anderson that finite disorder strength can lead to the localization of electronic states in his seminal 1958 paper [25]. The technique involved can be considered as a locator expansion for the effective hopping element of Anderson model Hamiltonian around the limit of the localized state. He found a region of disorder strength in which the expansion is convergent and thus the localized state endures. Please note that the probability distribution of the effective hopping element, instead of its average value, was discussed in the original paper by Anderson. The importance of the distribution in disordered system is a critical insight in the development of the typical medium theory [82]. Subsequently, Mott argued that the extended states would be separated from the localized states by a sharp mobility (localization) edge in energy [83; 84; 85]. His argument is that scattering from disorder is elastic, so that the incoming wave and the scattered wave have the same energy. On the other hand, nearly all scattering potentials will scatter electrons from one wavevector to all others, since the strongest scattering potentials are local or nearly so. If two states, corresponding to the same energy  
Figure 3: The TMDCA may be used to study electron localization in both simple model Hamiltonians as well as those extracted from first-principles calculations. and different wavenumbers exist, then the scattering potential will cause them to mix, causing both to become extended. An important development of the localization theory was the introduction of the concept of scaling. In 1972, Edwards and Thouless performed a numerical analysis on the dependence between the degree of localization and the boundary condition of the eigenstate of the Anderson model. They argued that the ratio of the energy shift from the change in the boundary conditions (\(\Delta E\)) to the energy spacing (\(\eta\)) can be used as a measure for the degree of localization [86]. The ratio \(\Delta E/\eta\) now known as the Thouless energy is identified as a dimensionless conductance, \(g(L)\), where \(L\) is the linear dimension of a system [87]. For a localized state, the Thouless energy decreases as the system size increases and tends to zero in the limit of a large system.