## 5 Typical Medium Theories of Anderson Localization: Model Studies
### Two-Particle Calculations  
Up to this point, the theory has focused on the calculation of single-particle quantities, i.e., the TDOS to capture the localization transition. However, most experimental measurements are described by two-particle Green's functions, including transport, most X-ray and neutron scattering, NMR, etc. Therefore, the TMDCA has also been extended to include the description of two-particle quantities including vertex corrections [190] in a similar fashion as that in the CPA and DCA [46; 158]. In conventional mean-field theories such as the CPA and DCA, the order parameters are constructed from the lattice Green's function defined as  
\[G_{\sigma}^{l}(\mathbf{k},\omega)=\frac{1}{\omega-h\sigma-\varepsilon_{\mathbf{ k}}-\Sigma_{\sigma}(M(\mathbf{k}),\omega)}\;, \tag{71}\]  
where \(M(\mathbf{k})=\mathbf{K}\) maps an arbitrary wave number \(\mathbf{k}\) to the closest DCA cluster \(\mathbf{K}\), \(\Sigma(M(\mathbf{k}),\omega)\) is the self-energy calculated on the cluster, and \(\varepsilon_{\mathbf{k}}\) is the lattice dispersion. If the order parameter is local, the order parameters may also be constructed from the cluster single-particle Green's function  
\[G_{\sigma}^{c}(\mathbf{K},\omega)=\frac{1}{\omega-h\sigma-\varepsilon_{\mathbf{ K}}-\Delta_{\sigma}(\mathbf{K},\omega)-\Sigma_{\sigma}(\mathbf{K},\omega)}\;. \tag{72}\]  
For example, for the magnetization \(m\)  
\[m=\sum_{\mathbf{k},\omega,\sigma}\sigma G_{\sigma}^{l}(\mathbf{k},\omega)= \sum_{\mathbf{K},\omega,\sigma}\sigma G_{\sigma}^{c}(\mathbf{K},\omega). \tag{73}\]  
Since these equations depend on \(h\) through the Green's function and through the dependence of \(\Sigma\) and \(\Delta\) on \(G\), to calculate the susceptibility \(\left.dm/dh\right|_{h=0}\) using the cluster Green's function, we need to know both \(\delta G/\delta\Sigma\) and \(\delta\Delta/\delta G\). The former is the irreducible vertex function  
\[\Gamma_{\sigma,\sigma^{\prime}}(\mathbf{K},\omega;\mathbf{K}^{\prime},\omega ^{\prime})=\frac{\delta G_{\sigma}(\mathbf{K},\omega)}{\delta\Sigma_{\sigma^{ \prime}}(\mathbf{K}^{\prime},\omega^{\prime})}\;. \tag{74}\]  
but the lack of information on \(\delta\Delta/\delta G\) prevent us from using this representation for the extended states. However, for the localized states, \(\Delta\) vanishes, so that \(\delta\Delta/\delta G\) is not needed and we can use the cluster Green's function for the localized states. Since the scattering events at different \(\omega\) are completely independent, to avoid using \(\delta\Delta/\delta G\) for the extended states, we introduce a mixed representation with  
\[m=\sum_{\mathbf{k},\omega,\sigma}\sigma G_{\sigma}^{p}(\mathbf{k},\omega) \tag{75}\]  
where  
\[G_{\sigma}^{p}(\mathbf{k},\omega)=\left\{ \begin{array}{ll}G_{\sigma}^{l}( \mathbf{k},\omega)&\mbox{if }|\omega|<\omega_{e};\\ G_{\sigma}^{c}(M(\mathbf{k}),\omega)&\mbox{if }|\omega|>\omega_{e}.\end{array} \right. \tag{76}\]  
and \(\omega_{e}\) is the mobility edge energy. Physically, this is more meaningful than the use of one of the formulas in Equations (71) and (72) alone. Below the mobility edge, \(\omega<\omega_{e}\), all the states are extended, and they may be described as states with a dispersion \(\varepsilon_{\mathbf{k}}\) renormalized by \(\Sigma\). However, for localized states \(\omega>\omega_{e}\), above the mobility edge, the electrons are localized to the cluster with \(\Delta_{\sigma}(\mathbf{K},\omega)=0\) so that \(\frac{\delta\Delta\sigma(\mathbf{K},\omega)}{\delta h}=0\). These states may not be described as extended states with a renormalized dispersion. So the usual interpretation fails, and it is much better to think in terms of states localized to the cluster described by the cluster Green's function for frequencies above the localization edge.