## 4 Coarse-Grained Methods
### Map from the Lattice to the Cluster  
To derive the DCA, we first apply the DCA coarse-graining procedure to the diagrams in the generating functional \(\Phi(G,U)\). In the DCA, we obtain an approximate \(\Phi^{c}\) by applying the DCA Laue function to the internal vertices of the lattice \(\Phi^{l}\). This is illustrated for the second order term in Figure 21.  
Figure 20: Path-integral interpretation of the screening of a propagating particle. The single-particle lattice Green’s function, \(G^{l}\), describes the quantum phase and amplitude the particle accumulates along its path as it propagates from space-time location \(0\) to \(x\). It is poorly approximated by the cluster Green’s function from a small cluster calculation, \(G^{l}\approx G^{c}\), especially when \(x,r\leq L_{c}\), the linear cluster size. Its self-energy, which describes generally short-ranged \(r\) screening processes, is well approximated \(\Sigma^{l}\approx\Sigma^{c}\), by a small cluster calculation, especially when the cluster size \(L_{c}\) is greater than the screening length. As discussed in Section 2 this screening length \(f_{TF}\approx r\) which may be less than an Angstrom for a good metal. Therefore, rather than directly approximating the lattice Green’s function by the cluster Green’s function, the cluster self-energy is used to approximate the lattice self-energy in a Dyson equation for the lattice Green’s function \(G^{l}=G^{l}+G^{l0}+G^{l0}\Sigma^{l}G^{l}\), where \(G^{l0}\) is the bare lattice Green’s function. It is easy to see that the corresponding term in the self-energy \(\Sigma^{(2)}\) is obtained from a functional derivative of \(\Phi^{(2)}\), \(\Sigma^{(2)}_{\sigma}=\delta\Phi^{(2)}/\delta G_{\sigma}\), and the irreducible vertices \(\Gamma^{(2)}_{\sigma\sigma^{\prime}}=\delta\Sigma^{(2)}_{\sigma^{\prime}}/ \delta G_{\sigma^{\prime}}\). This is illustrated for the second order self-energy in Figure 22.  
Above, we justified these approximations in wavenumber space; however, one may also make a real-space argument. In high spatial dimensions \(D\), one may show [13; 14] that \(G(r,\tau)\) falls of exponentially quickly with increasing \(r\)\(G(r,\tau)\sim t^{r}\propto D^{-r/2}\) (here \(t\) is the hopping probability amplitude) while the interaction remains local. Thus, when \(D=\infty\) all non-local graphs vanish. In finite \(D\), due to causality, we may expect the Green's functions to fall exponentially for large time displacements; whereas, the decay of the quasiparticle ensures that it also fall exponentially with large spacial displacements. Therefore, one may safely assume that longer range graphs are \"smaller\" in magnitude. Now, consider a non-local correction to the local approximation where only graphs constructed from \(G(r=0,\tau)\) enter. The first such graph would be when all vertices are at \(r=0\) apart from one which is on a near neighbor to \(r=0\), which we will label as \(r=1\). We allow \(G(r=1)/G(r=0)\) to be the \"small\" parameter. It is easy to see that the first non-local correction to \(\Phi\) is fourth-order in \(G(r=1)/G(r=0)\). Likewise, the first such corrections to the self-energy are third order while those for the Green's function itself are first order in \(G(r=1)/G(r=0)\). Thus, the approximation where lattice quantities are approximated by cluster quantities, is much better for the self-energy than for the Green's function. Thus, the most accurate approximation is to replace the lattice generating functional with the cluster  
Figure 21: A second-order term in the generating functional of the Hubbard model. Here the undulating line represents the interaction \(U\), and on the LHS (RHS) the solid line the lattice (coarse-grained) single-particle Green’s functions. When the DCA Laue function is used to describe momentum conservation at the internal vertices, the momenta collapse onto the cluster momenta and each lattice Green’s function and interaction is replaced by the corresponding coarse-grained result.