## 4 Coarse-Grained Methods
### The DCA
#### 4.3.2 DCA: A Diagrammatic Derivation  
This coarse-graining procedure and the relationship of the DCA to the local approximations (DMFT/CPA) is illustrated by a microscopic diagrammatic derivation [8] of the DCA. We chose disorder case for the demonstration. Quantum cluster theories are defined by two mappings: one from the lattice to the cluster and the other from the cluster back to the lattice. _a. Map from the Lattice to the Cluster_  
To define the first mapping, we start from the diagrams in the irreducible self-energy \(\Sigma(V,G)\) of the Anderson model illustrated in Figure 12. We saw above, that when we completely neglect momentum conservation by first coarse-graining the interactions and Green's functions over the entire first Brillioun zone, the diagrams corresponding to non-local corrections vanish, leaving the reduced set of local diagrams which constitute the CPA illustrated in Figure 15. The resulting approximation shares the limitations of a local approximation, described above, including the neglect of non-local correlations. The DCA systematically incorporates such neglected non-local correlations by systematically restoring the momentum conservation at the internal vertices of the self-energy \(\Sigma\). To this end, the Brillouin zone is divided into \(N_{c}=L_{V}^{D}\) cells of size \(\Delta k=2\pi/L_{c}\) (c.f. Figure 17 for \(N_{c}=8\)). Each cell is represented by a cluster momentum \(\mathbf{K}\) in the center of the cell. We require that momentum conservation is (partially) observed for momentum transfers between cells, i.e., for momentum transfers larger than \(\Delta k\), but neglected for momentum transfers within a cell, i.e., less than \(\Delta k\). This requirement can be established by using the Laue function [24]  
\[\Lambda_{DCA}(\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{k}_{3},\mathbf{k}_{4})=N_ {c}\delta_{\mathbf{M}(\mathbf{k}_{1})+\mathbf{M}(\mathbf{k}_{2}),\mathbf{M}( \mathbf{k}_{3})+\mathbf{M}(\mathbf{k}_{4})}, \tag{31}\]  
where \(\mathbf{M}(\mathbf{k})\) is a function which maps \(\mathbf{k}\) onto the momentum label \(\mathbf{K}\) of the cell containing \(\mathbf{k}\) (see Figure 17). This choice for the Laue function systematically interpolates between the exact result, Equation (25), which it recovers when \(N_{c}\to N\) and the DMFT result, Equation (26), which it  
Figure 18: The DCA many-to-few mapping of an arbitrary point in the first Brillioun zone to one of \(N_{c}=8\) cluster momenta \(\mathbf{K}\). recovers when \(N_{c}=1\). With this choice of the Laue function the momenta of each internal leg may be freely summed over the cell. This procedure accurately reproduces the physics on short length scales and provides a cutoff of longer length scales where the physics is approximated with the mean field. For short distances \(r\lesssim L_{c}/2\), where \(L_{c}\) is now the linear size of the cluster, the Fourier transform of the Green's function \(\tilde{G}(r)\approx G(r)+\mathcal{O}((r\Delta k)^{2})\), so that short-ranged correlations are reflected in the irreducible quantities constructed from \(\tilde{G}\); whereas, longer ranged correlations \(r>L_{c}/2\) are cut off by the finite size of the cluster [24]. Longer ranged interactions are also cut off when the transformation is applied to the interaction. To see this, consider an extended Hubbard model on a (hyper)cubic lattice with the addition of a near-neighbor interaction \(V\sum_{\langle ij\rangle}n_{i}n_{j}\) where \(\langle ij\rangle\) denotes near-neighbor pairs. When the point group of the cluster is the same as the lattice the coarse-grained interaction takes the form \(V\sin(\Delta k/2)/(\Delta k/2)\sum_{\langle ij\rangle}n_{i}n_{j}\). It vanishes when \(N_{c}=1\) so that \(\Delta k=2\pi\). If \(N_{c}\) is larger than one, then non-local corrections of length \(\approx\pi/\Delta k\) to the DMFT/CPA are introduced. When applied to the DCA, the cluster self-energy will be constructed from the _coarse-grained average_ of the single-particle Green's function within the cell centered on the cluster momenta.