## 5 Typical Medium Theories of Anderson Localization: Model Studies
### Off-Diagonal Disorder
#### 5.3.1 DCA with Off-Diagonal Disorder  
This poses problems when formulating a Green's function formalism. Even after averaging over the disorder, the Green's functions depend on the type, \(A\) or \(B\), of the sites involved. Blackman, Esterling and Berk [173] (BEB) extended the CPA to systems with off-diagonal disorder. They developed an elegant formalism to address the problem in multicomponent alloys. BEB showed  
Figure 24: For off-diagonal disorder the hopping amplitude depends on the occupancy of the neighboring sites. the scalar CPA equation becomes a \(2\times 2\) matrix equation. For example, for our binary alloy model, the BEB single-particle Green's function is a \(2\times 2\) matrix  
\[ \begin{array}{ll}\underline{G(\mathbf{k},\omega)}&=\left(\begin{array}{cc}G ^{AA}(\mathbf{k},\omega)&G^{AB}(\mathbf{k},\omega)\\ \\ G^{BA}(\mathbf{k},\omega)&G^{BB}(\mathbf{k},\omega)\end{array}\right)\.\end{array}  \tag{53}\]  
Since physically the Green's function describes the amplitude and phase the particle accumulates as it propagates, we can expect, i.e., \(\int d\omega\frac{-1}{\pi}\Im G^{AA}(\mathbf{k},\omega)=c_{A}\), \(\int d\omega\frac{-1}{\pi}\Im G^{BB}(\mathbf{k},\omega)=c_{B}\), etc. In momentum space, if there is only nearest-neighbor hopping between all ions as in our simple example, the bare dispersion can be written as (the under-bar denotes matrices)  
\[ \begin{array}{ll}\underline{\varepsilon}_{\mathbf{k}}&=\left(\begin{array}[] {cc}t^{AA}&t^{AB}\\ \\ t^{BA}&\\ \end{array}\right)\varepsilon_{\mathbf{k}}\end{array}  \tag{54}\]  
where in three dimensions for our simple model \(\varepsilon_{k}=-2t(\cos(k_{x})+\cos(k_{y})+\cos(k_{z}))\) with \(4t=1\) which sets our unit of energy, and \(t^{AA}\), \(t^{BB}\), \(t^{AB}\), and \(t^{BA}\) are unitless prefactors. Using this, we may define a bare lattice propagator, and a corresponding diagrammatic perturbation theory for the lattice single-particle propagator \(G(\mathbf{k},\omega)\). As done in the previous sections, the CPA or BEB formalism may be derived by replacing the Laue function by one at each internal vertex of the irreducible quantities, including the generating functional, and its functional derivatives the self-energy and the vertex functions. However, being single-site approximations, the CPA and the BEB theories neglect all disorder-induced non-local correlations. The DCA systematically incorporates such missing non-local corrections by mapping the lattice problem onto a self-consistently embedded cluster problem. The mapping is accomplished by replacing the Laue function in the internal vertices of the irreducible quantities by the DCA Laue function. This causes all the Green's functions and vertices to be replaced by their coarse-grained counterparts. The remaining details of the DCA formalism for off-diagonal disorder may then be defined by following the same procedures discussed in Section 4.3. To define the mean-field coupling between the cluster and its host, we introduce a DCA hybridization matrix \(\Delta\). \[ \begin{array}{ll}\underline{\Delta(\mathbf{K},\omega)}=\left(\begin{array}[] {cc}\Delta^{AA}(\mathbf{K},\omega)&\Delta^{AB}(\mathbf{K},\omega)\\ \Delta^{BA}(\mathbf{K},\omega)&\Delta^{BB}(\mathbf{K},\omega)\end{array}\right) \end{array}  \tag{55}\]  
which is related to the cluster Green's function, through the \(2\times 2\) matrix equation  
\[ \begin{array}{ll}\underline{G}^{c}(\mathbf{K},\omega)=\left(\omega- \underline{\varepsilon}_{\mathbf{k}}-\underline{\Delta(\mathbf{K},\omega)}- \underline{\Sigma(\mathbf{K},\omega)}\right)^{-1}\end{array}  \tag{56}\]  
With this result, the mapping between the lattice and the cluster is established, and the cluster problem may be solved with a variety of methods. We choose to average over the disorder configurations stochastically. It is possible to enumerate all configurations of the cluster. For a binary alloy, there are \(2^{N_{c}}\) such configurations, and an algorithm which enumerates all of them would scale exponentially in \(N_{c}\).