## 4 Coarse-Grained Methods
### The Laue Function and the Limit of Infinite Dimension  
The local approximation for the self-energy was used by various authors in perturbative calculations as a simplification of the k-summations which render the problem intractable. It was only after the work of Metzner and Vollhardt [13; 155] and Muller-Hartmann [14; 15] who showed that this approximation becomes exact in the limit of infinite dimension that it received extensive attention. Precisely in this limit, the spatial dependence of the self-energy disappears, retaining only its variation with time. Please see the reviews by Pruschke et al. [18] and Georges et al. [19] for a more extensive treatment. In this section, we will show that the DMFT and CPA share a common interpretation as coarse-graining approximations in which the propagators used to calculate the self-energy \(\Sigma\) and its functional derivatives are coarse-grained over the entire Brillouin zone. Muller-Hartmann [14; 15] showed that it is possible to completely neglect momentum conservation so that this coarse-graining becomes exact in the limit of infinite dimensions. For simple models such as the Hubbard and Anderson models, the properties of the bare vertex are completely characterized by the Laue function \(\Lambda\) which expresses the momentum conservation at each vertex. In a conventional diagrammatic approach  
\[\Lambda(\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3},\mathbf{k}_{4})=\sum_{ \mathbf{r}}\exp\left[i\mathbf{r}\cdot(\mathbf{k}_{1}+\mathbf{k}_{2}-\mathbf{ k}_{3}-\mathbf{k}_{4})\right]=N\delta_{\mathbf{k}_{1}+\mathbf{k}_{2},\mathbf{k}_{3}+ \mathbf{k}_{4}}\,, \tag{25}\]  
where \(\mathbf{k}_{1}\) and \(\mathbf{k}_{2}\) (\(\mathbf{k}_{3}\) and \(\mathbf{k}_{4}\)) are the momenta entering (leaving) each vertex through its legs of Green's function \(G\). However, as the dimensionality \(D\to\infty\), Muller-Hartmann showed that the Laue function reduces to [14]  
\[\Lambda_{D\to\infty}(\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3},\mathbf{k}_{ 4})=1+\mathcal{O}(1/D)\quad. \tag{26}\]  
The DMFT/CPA assumes the same Laue function, \(\Lambda_{DMFT}(\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3},\mathbf{k}_{4})=1\), even in the context of finite dimensions. More generally, for an electron scattering from an interaction (boson) pictured in Figure 14, \(\Lambda_{DMFT}(\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3})=1\). Thus, the conservation of momentum at internal vertices is neglected. We may freely sum over the internal momentum labels of each Green's function leg and interaction leading to a collapse of the momentum dependent contributions leaving only local terms. Figure 13: The first few diagrams for the Hubbard model single-particle Green’s function. Here, the solid black line with an arrow represents the single-particle Green’s function and the wavy line the Hubbard \(U\) interaction. These arguments may then be applied to the self-energy \(\Sigma\), which becomes a local (momentum-independent) function. For example, in the CPA for the Anderson model, non-local correlations involving different scatterers are ignored. Thus, in the calculation of the self-energy, we ignore all the crossing diagrams shown on the bottom of Figure 12; and retain only the class of diagrams such as those shown on the top representing scattering from a single local disorder potential. These diagrams are shown in Figure 15.  
It is easy to show this reduction in the number and complexity of the graphs is fully equivalent to the neglect of momentum conservation at each internal vertex. This is accomplished by setting each Laue function within the sum (e.g., in Equation (23) to 1. We may then freely sum over the internal momenta, leaving only local propagators. All non-local self-energy contributions (crossing diagrams) must then vanish. For example, consider again the fourth graph at the bottom of Figure 12. If we replace the Laue function \(N\delta_{\mathbf{k}_{1}+\mathbf{k}_{4},\mathbf{k}_{5}+\mathbf{k}_{3}}\to 1\) in Equation (23), then the two contributions cancel and this diagram vanishes.