## 2 Background: Electron Localization in Disordered Medium
### Order Parameter of Anderson Localization  
As we discussed in the previous section, effective medium theories have been used to study Anderson localization; however, progress has been hampered partly due to ambiguity in identifying an appropriate order parameter for Anderson localization, allowing for a clear distinction between localized and extended states [9]. An order parameter function had been suggested about three decades ago, in the study of Anderson localization on the Bethe lattice [103; 104]. It has been shown that the parameter is closely related to the distribution of on-site Green's functions, in particular the local density of states [105]. Recently, following the work of Dobrosavljevic et al. [9], there has been tremendous progress along these ideas, with the local typical DOS identified as the order parameter. To demonstrate how the local DOS and its typical (most probable value) can be used as an order parameter for Anderson localization, we consider a thought experiment. We imagine dividing the system up into blocks, as illustrated in Figure 4. Later, when we construct our quantum cluster theory of localization, each of the blocks should be thought of as a cluster, and we construct the system by periodically stacking the blocks. We make two controllable approximations. 1. We approximate the effect of coupling the block to the reminder of the lattice via Fermi's golden rule--coupling \(\Delta\) which is proportional to the density of accessible states. 2. Since on average each cluster is equivalent to all the others, this density will also be proportional to some appropriate block DOS. Furthermore, imagine that the average level spacing of the states in a block is \(\delta E\). If \(\Delta\gg\delta E\), then we have a metal since the states at this energy have a significant probability of escaping from this block, and the next one, etc. Alternatively, if \(\Delta\ll\delta E\) the escape probability of the electrons is low, so that an insulator forms. So what does this mean in terms of the local electronic density of states (LDOS) that is measured, i.e., via STM at one site in the system, and the average DOS (ADOS) measured, i.e., via tunneling (or just by averaging the LDOS)? In Figure 5 we calculate the ADOS and typical density of states (TDOS) for a simple (Anderson) single-band model on a cubic lattice with near-neighbor hopping \(t\) (bare bandwidth \(12t=3\) to establish an energy unit) and with a random site \(i\) local potential \(V_{i}\) drawn from a \"box\" distribution of width \(2W\), with \(P(V_{i})=\frac{1}{2W}\Theta(W-|V_{i}|)\). As can be seen from the Figure 5, as we increase the disorder strength \(W\), the global average DOS (dashed lines) always favors the metallic state (with a finite DOS at the Fermi level \(\omega=0\)) and it is a smooth (not critical) function even above the transition. In contrast to the global average DOS, the local density of states (solid lines), which measures the amplitude of the electron wave function at a given site, undergoes significant qualitative changes as the disorder strength \(W\) increases, and eventually becomes a set of the discrete delta-like functions as the transition is approached. Figure 4: To help understand localization, we divide the system into blocks. The average spacing of the energy levels of a block is \(\delta E\) and the Fermi golden rule width of the levels is \(\Delta\). If \(\Delta\gg\delta E\) then we have a metal and if \(\Delta\ll\delta E\), an insulator. This must mean that the probability distributions of the local DOS for a metal and for an insulator is also very different. This is illustrated in Figure 6. In particular, the most probable (typical) value of the local DOS in a metal is very different than the typical value in an insulator. Consider again the local DOS in the metal and insulator. In the metal, the probability distribution function is Gaussian-like form. The local DOS at any one energy the DOS at each site is a continuum.