## 4 Coarse-Grained Methods
### The DCA
#### 4.3.1 Coarse-Graining  
Like the DMFT/CPA, in the DCA the mapping from the lattice to the cluster diagrams is accomplished via a coarse-graining transformation. In the DMFT/CPA, the propagators used to calculate \(\Sigma\) and its functional derivatives are coarse-grained over the entire Brillouin zone, leading to local (momentum independent) irreducible quantities. In the DCA, we wish to relax this condition, and systematically restore momentum conservation and non-local corrections.  
Thus, in the DCA, the reciprocal space of the lattice (Figure 17) which contains \(N\) points is divided into \(N_{c}\) cells of identical linear size \(\Delta k\). The geometry and point groups of these clusters may be determined by considering real-space finite-size clusters of size \(N_{c}\) that are able to tile the lattice of  
Figure 16: The DMFT/CPA self-consistency algorithm.  
size \(N\). The tiling momenta \(\mathbf{K}\) are conjugate to the location of the sites in the cell labeled by \(\mathbf{X}\), while the coarse-graining wavenumbers \(\mathbf{\tilde{k}}\) label the wavenumbers within each cell surrounding \(\mathbf{K}\) and are conjugate to the real-space labels of the cell centers \(\tilde{x}\).  
The coarse-graining transformation is set by averaging the function within each cell as illustrated in Figure 18. For an arbitrary function \(f(\mathbf{k})\) (with \(\mathbf{k}=\mathbf{K}+\mathbf{\tilde{k}}\)), this corresponds to  
\[\tilde{f}(\mathbf{K})=\frac{N_{c}}{N}\sum_{\mathbf{\tilde{k}}}f(\mathbf{K}+ \mathbf{\tilde{k}}) \tag{29}\]  
where \(\mathbf{\tilde{k}}\) label the wavenumbers within the coarse-graining cell adjacent to \(\mathbf{K}\). According to Nyquist's sampling theorem Nyquist (1957), to reproduce the function \(f\) at lengths \(\lesssim L/2\) (\(L\) is a linear system size) in Equation (29), we only need to sample the reciprocal space at intervals of \(\Delta k\approx 2\pi/L\). Equation (29) may be interpreted as the sum of \(N/N_{c}\) such samplings.  
Knowledge of \(f\) on a finer scale in momentum than \(\Delta k\) is unnecessary and may be discarded to reduce the complexity of the problem. For example, convolutions of periodic functions \(f\) may be approximated as  
\[g(\mathbf{q})=\frac{1}{N}\sum_{\mathbf{k}}f(\mathbf{k}+\mathbf{q})f(\mathbf{ k})\approx\frac{1}{N_{c}}\sum_{\mathbf{K}}\tilde{f}(\mathbf{K}+\mathbf{Q}) \tilde{f}(\mathbf{K})+\mathcal{O}(\Delta k^{2})\,, \tag{30}\]  
where \(\mathbf{Q}=M(\mathbf{q})\). Equation (30) is an approximation where we first average the function over a set of D dimensional cells and then perform a sum over the cells. Thus, reducing the numerical complexity from order \(N\) to order \(N_{c}\) floating point operations.  
Figure 17: Coarse-graining cells for \(N_{c}=8\) (differentiated by alternating fill patterns) that partition the first Brillouin zone (dashed line). Each cell is centered on a cluster momentum \(\mathbf{K}\) (filled circles). To construct the DCA cluster (e.g., for \(N_{c}=8\)) we map a generic \(\mathbf{k}\) to the nearest cluster point \(\mathbf{K}=\mathbf{M}(\mathbf{k})\) (c.f. Figure 18) so that \(\mathbf{\tilde{k}}=\mathbf{k}-\mathbf{K}\) remains in the cell around \(\mathbf{K}\).