## 4 Coarse-Grained Methods
### A Few Fundamentals  
In this section, we will introduce two central paradigms in the physics of many-body systems: the Anderson and Hubbard models of disordered and interacting electrons on a lattice, respectively. We will then use perturbation theory to prove and demonstrate some fundamental ideas. Consider an Anderson model with diagonal disorder, described by the Hamiltonian  
\[H=-\sum_{\langle ij\rangle,\sigma}t\left(c_{i,\sigma}^{\dagger}c_{j,\sigma}+c_{ j,\sigma}^{\dagger}c_{i,\sigma}\right)+\sum_{i\sigma}(V_{i}-\mu)n_{i,\sigma} \tag{17}\]  
where \(c_{i,\sigma}^{\dagger}\) creates a quasiparticle on site \(i\) with spin \(\sigma\), and \(n_{i,\sigma}=c_{i,\sigma}^{\dagger}c_{i,\sigma}\). The disorder occurs in the local orbital energies \(V_{i}\), which we assume are independent quenched random variables distributed according to some specified probability distribution \(P(V)\). The effect of the disorder potential \(\sum_{i\sigma}V_{i}n_{i,\sigma}\) can be described using standard diagrammatic perturbation theory (although we will eventually sum to _all_ orders). It may be rewritten in reciprocal space as  
\[H_{\text{dis}}=\frac{1}{N}\sum_{i,\mathbf{k},\mathbf{k}^{\prime},\sigma}V_{i}c _{\mathbf{k},\sigma}^{\dagger}c_{\mathbf{k}^{\prime},\sigma}e^{i\mathbf{r}_{i }(\mathbf{k}-\mathbf{k}^{\prime})}, \tag{18}\]  
here \(N\) is the total number of lattice sites. The corresponding irreducible (skeleton) contributions to the self-energy may be represented diagrammatically [77] and the first few are displayed in Figure 12. Here each \(\circ\) represents the scattering of an electronic Bloch state from a local disorder potential at some site \(X\). The dashed lines connect scattering events that involve the same local potential. In each graph, the sums over the sites are restricted so that the different \(X\)'s represent scattering from _different_ sites. No graphs representing a single scattering event are included since these may simply be absorbed as a renormalization of the chemical potential \(\mu\) (for single-band models). Translational invariance and momentum conservation are restored by averaging over all possible values of the disorder potentials \(V_{i}\). For example [8], consider the second diagram in Figure 12, given by  
\[\frac{1}{N^{3}}\sum_{i,\mathbf{k}_{3},\mathbf{k}_{4}}\langle V_{i}^{3}\rangle G (\mathbf{k}_{3})G(\mathbf{k}_{4})e^{i\mathbf{r}_{i}\cdot(\mathbf{k}_{1}- \mathbf{k}_{3}+\mathbf{k}_{3}-\mathbf{k}_{4}+\mathbf{k}_{4}-\mathbf{k}_{2})}\,, \tag{19}\]  
where \(G(\mathbf{k})\) is the disorder-averaged single-particle Green's function for state \(\mathbf{k}\). The average over the distribution of scattering potentials \(\langle V_{i}^{3}\rangle=\langle V^{3}\rangle\) is independent of the position \(i\) in the lattice. After summation over the remaining labels, this becomes  
\[\langle V^{3}\rangle G(\mathbf{r}=0)^{2}\delta_{\mathbf{k}_{1},\mathbf{k}_{2 }}\,, \tag{20}\]  
where \(G(\mathbf{r}=0)\) is the local Green's function. Thus, the second diagram's contribution to the self-energy involves only local correlations. Since the internal momentum labels always cancel in the exponential, the same is true for all non-crossing diagrams shown in the top half of Figure 12.  
Only the diagrams with crossing dashed lines have non-local contributions. Consider the fourth-order diagrams such as those shown on the bottom left and upper right of Figure 12. During the disorder averaging, we generate potential terms \(\langle V^{4}\rangle\) when the scattering occurs from the same local potential (i.e., the third diagram) or \(\langle V^{2}\rangle^{2}\) when the scattering occurs from different sites, as in the fourth diagram. When the latter diagram is evaluated, to avoid overcounting, we need to subtract a term proportional to \(\langle V^{2}\rangle^{2}\) but corresponding to scattering from the same site. This term is needed to account for the fact that the fourth diagram should only be evaluated for sites \(i\
eq j\).