## 4 Coarse-Grained Methods
### A Few Fundamentals
For example, the fourth diagram yields  
\[\frac{1}{N^{4}}\sum_{i\
eq j\mathbf{k}_{3}\mathbf{k}_{4}\mathbf{k}_{5}}V_{i}^{ 2}V_{j}^{2}e^{i\mathbf{r}_{i}\cdot(\mathbf{k}_{1}+\mathbf{k}_{4}-\mathbf{k}_{5 }-\mathbf{k}_{3})}e^{i\mathbf{r}_{j}\cdot(\mathbf{k}_{5}+\mathbf{k}_{3}- \mathbf{k}_{4}-\mathbf{k}_{2})}G(\mathbf{k}_{5})G(\mathbf{k}_{4})G(\mathbf{k}_ {3}) \tag{21}\]  
Evaluating the disorder average \(\langle...\rangle\), we get the following two terms:  
\[\frac{1}{N^{4}}\sum_{ij\mathbf{k}_{3}\mathbf{k}_{4}\mathbf{k}_{5 }}\langle V^{2}\rangle^{2}e^{i\mathbf{r}_{i}\cdot(\mathbf{k}_{1}+\mathbf{k}_{4 }-\mathbf{k}_{5}-\mathbf{k}_{3})}e^{i\mathbf{r}_{j}\cdot(\mathbf{k}_{5}+ \mathbf{k}_{3}-\mathbf{k}_{4}-\mathbf{k}_{2})}G(\mathbf{k}_{5})G(\mathbf{k}_ {4})G(\mathbf{k}_{3})\] \[-\frac{1}{N^{4}}\sum_{i\mathbf{k}_{3}\mathbf{k}_{4}\mathbf{k}_{5 }}\langle V^{2}\rangle^{2}e^{i\mathbf{r}_{i}\cdot(\mathbf{k}_{1}-\mathbf{k}_{ 2})}G(\mathbf{k}_{5})G(\mathbf{k}_{4})G(\mathbf{k}_{3}) \tag{22}\]  
Momentum conservation is restored by the sum over \(i\) and \(j\); i.e., over all possible locations of the two scatterers. It is reflected by the Laue functions, \(\Lambda=N\delta_{\mathbf{k}+...}\), within the sums  
\[H_{\text{dis}}=\frac{1}{N}\sum_{i,\mathbf{k},\mathbf{k}^{\prime},\sigma}V_{i}c _{\mathbf{k},\sigma}^{\dagger}c_{\mathbf{k}^{\prime},\sigma}e^{i\mathbf{r}_{i }(\mathbf{k}-\mathbf{k}^{\prime})}, \tag{18}\]0  
Since the first term in Equation (23) involves convolutions of \(G(\mathbf{k})\) it reflects non-local correlations. Local contributions such as the second term in Equation (23) can be combined together with the contributions from the corresponding local diagrams such as the third diagram in Figure 12 by replacing \(\langle V^{4}\rangle\) in the latter by the cumulant \(\langle V^{4}\rangle-\langle V^{2}\rangle^{2}\). Given the fact that different \(X\)'s must correspond to different sites, it is easy to see that all crossing diagrams must involve non-local correlations. The developed formalism also works for interacting systems. Again, we will use perturbation theory to illustrate some of these ideas. Consider the Hubbard model [153] which is the simplest model of a correlated electronic lattice system. Both it and the \(t\)-\(J\) model are thought to describe at least qualitatively some of the properties of transition-metal oxides, and high temperature superconductors [154]. The Hubbard model Hamiltonian is given as  
\[H_{\text{dis}}=\frac{1}{N}\sum_{i,\mathbf{k},\mathbf{k}^{\prime},\sigma}V_{i}c _{\mathbf{k},\sigma}^{\dagger}c_{\mathbf{k}^{\prime},\sigma}e^{i\mathbf{r}_{i }(\mathbf{k}-\mathbf{k}^{\prime})}, \tag{18}\]1  
\[H_{\text{dis}}=\frac{1}{N}\sum_{i,\mathbf{k},\mathbf{k}^{\prime},\sigma}V_{i}c _{\mathbf{k},\sigma}^{\dagger}c_{\mathbf{k}^{\prime},\sigma}e^{i\mathbf{r}_{i }(\mathbf{k}-\mathbf{k}^{\prime})}, \tag{18}\]2 The first few graphs in the irreducible self-energy of a diagonally disordered system. Each \(\circ\) represents the scattering of a state \(\mathbf{k}\) from sites (marked \(X\)) with a local disorder potential distributed according to some specified probability distribution \(P(V)\). The numbers label the \(\mathbf{k}\) states of the fully dressed Green’s functions, represented by solid lines with arrows. where \(c_{j\sigma}^{\dagger}\) (\(c_{j\sigma}\)) creates (destroys) an electron at site \(j\) with spin \(\sigma\), \(n_{i\sigma}=c_{i\sigma}^{\dagger}c_{i\sigma}\) stands for the particle number at a given site \(i\). The first term describes the hopping of electrons between nearest-neighboring sites \(i\) and \(j\), and the \(U\) term describes the interaction between two electrons once they meet at a given site \(i\). As for the disordered case described above, the effect of the local Hubbard \(U\) potential can be described using standard diagrammatic perturbation theory.