## 4 Coarse-Grained Methods
### The DCA
#### 4.3.2 DCA: A Diagrammatic Derivation
This is illustrated for a fourth-order term in the self-energy shown in Figure 19. Each internal leg \(G(\mathbf{k})\) in a diagram is replaced by the coarse-grained Green's function \(\tilde{G}(\mathbf{M}(\mathbf{k}))\), defined by  
\[\tilde{G}(\mathbf{K})\equiv\frac{N_{c}}{N}\sum_{\tilde{\mathbf{k}}}G(\mathbf{ K}+\tilde{\mathbf{k}})\quad, \tag{32}\]  
and each interaction in the diagram is replaced by the coarse-grained interaction  
\[\tilde{V}(\mathbf{K})\equiv\frac{N_{c}}{N}\sum_{\tilde{\mathbf{k}}}V(\mathbf{ K}+\tilde{\mathbf{k}})\quad, \tag{33}\]  
where \(N\) is the number of points of the lattice, \(N_{c}\) is the number of cluster \(\mathbf{K}\) points, and the \(\tilde{\mathbf{k}}\) summation runs over the momenta of the cell about the cluster momentum \(\mathbf{K}\) (see Figure 17). For the Anderson model, where the scattering potential is local, the interaction is unchanged by coarse-graining. The diagrammatic sequences for the self-energy and its functional derivatives are unchanged; however, the complexity of the problem is greatly reduced since \(N_{c}\ll N\). Provided that the propagators are sufficiently weakly momentum dependent, this is a good approximation. If \(N_{c}\) is chosen to be small, the cluster problem can be solved using conventional techniques such as QMC. This averaging process also establishes a relationship between the systems of size \(N\) and \(N_{c}\). When \(N_{c}=N\) a finite-size simulation is recovered. Therefore, there are no mean-field embedding effects, etc. _b. Map from the Cluster Back to the Lattice_  
Once the cluster problem is solved, we use the solution of the cluster problem to approximate the lattice problem. This may be done in several ways, and it is not _a priori_ clear which way is optimal. At the single-particle particle level, we could, e.g., calculate the cluster single-particle Green's function and use it to approximate the lattice result, \(G^{l}(\mathbf{k},\omega)\approx G^{c}(M(\mathbf{k}),\omega)\). Or, at the other extreme, we  
Figure 19: Use of the DCA Laue function \(\Lambda_{DCA}\) leads to the replacement of the lattice propagators \(G(\mathbf{k}_{1})\), \(G(\mathbf{k}_{2})\),... by coarse-grained propagators \(\tilde{G}(\mathbf{K})\), \(\tilde{G}(\mathbf{K}^{\prime})\),... The impurity scattering dashed lines and unchanged by coarse-graining since the scatterings are local. could calculate the self-energy on the cluster, and use it to first approximate the lattice result \(\Sigma^{l}(\mathbf{k},\omega)\approx\Sigma^{c}(M(\mathbf{k}),\omega)\), and then use the Dyson equation \(G^{l}(\mathbf{k},\omega)=\left(1-\Sigma^{c}(M(\mathbf{k}),\omega)G^{l,0}( \mathbf{k},\omega)\right)^{-1}\) to calculate the lattice Green's function (\(G^{l,0}(\mathbf{k},\omega)\) is the bare lattice Green's function). The second way is far better. We will motivate this mapping with more rigor in the next part, where we calculate and minimize the free energy, but here we offer a physically intuitive motivation. Physically, this is justified by the fact that irreducible terms such as the self-energy are short-ranged, while reducible quantities the \(G\) must be able to reflect the long length and time scale of physics. This is motivated in Figure 20. As the particle propagates from the origin to space-time location \(x\), the quantum phase and amplitude it accumulates is described by the single-particle Green's function \(G(x)\). Consequently if \(x\) is larger than the size of the DCA cluster, then \(G(x)\) is poorly approximated by the cluster Green's function. However, the self-energy \(\Sigma\) describes the many-body processes that produce the screening cloud surrounding the particle. As we saw in Section 2.3 these distances are typically very short, on the order of an Angstrom or less, so the lattice self-energy is often well approximated by the cluster quantity.