## 4 Coarse-Grained Methods
### Map from the Lattice to the Cluster
Figure 22: A second-order term in the self-energy of the Hubbard model obtained from the first functional derivative of the corresponding term in the generating functional \(\Phi\) (Figure 21). 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. result, \(\Phi^{l}\approx\Phi^{c}\) and the lattice self-energy as the cluster result \(\Sigma^{l}(\mathbf{k})\approx\Sigma^{c}(\mathbf{K})\) and use it in the lattice Dyson's equation to form the lattice single-particle Green's function. Summarizing, the map from the lattice to the cluster is accomplished by replacing \(G(\mathbf{k})\) by \(\tilde{G}(\mathbf{K})\) and the interaction \(V(\mathbf{k})\) by \(\tilde{V}(\mathbf{K})\) in the diagrams for the generating functional. These are precisely the generating functional, self-energy and vertex diagrams of a finite-size cluster with a bare Hamiltonian defined by \(\mathcal{G}\), and an interaction determined by the bare coarse-grained \(\tilde{V}(\mathbf{K})\). In this mapping from the lattice to the cluster, the complexity of the problem has been greatly reduced since this cluster problem may often be solved exactly and with multiple methods including quantum Monte Carlo [158]. _b. Map from the Cluster Back to the Lattice_  
We may accomplish the mapping from the cluster back to the lattice problem by minimizing the lattice estimate for the self-energy. The corresponding DCA estimate for the free energy is  
\[F_{DCA}=-k_{B}T\left(\Phi^{c}-\mathrm{Tr}\left[\Sigma^{l}_{\sigma}G_{\sigma} \right]+\mathrm{Tr}\ln\left[-G_{\sigma}\right]\right) \tag{34}\]  
where \(\Phi^{c}\) is the cluster generating functional (we use superscripts \(c\) and \(l\) to denote the cluster and the lattice quantities, respectively). The trace indicates summation over frequency, momentum, and spin. We may prove that the corresponding optimal estimates of the lattice self-energy and irreducible lattice vertices are the corresponding cluster quantities. \(F_{DCA}\) is stationary with respect to \(\mathbf{G}_{\sigma}\),  
\[\frac{-1}{k_{B}T}\frac{\delta F_{DCA}}{\delta G_{\sigma}(\mathbf{k})}=\Sigma^{ c}_{\sigma}(M(\mathbf{k}))-\Sigma^{l}_{\sigma}(\mathbf{k})=0, \tag{35}\]  
which means that \(\Sigma^{l}(\mathbf{k})=\Sigma^{c}(M(\mathbf{k}))\) is the proper approximation for the lattice self-energy corresponding to \(\Phi^{c}\). The corresponding lattice single-particle propagator is then given by  
\[G^{l}(\mathbf{k},z)=\frac{1}{z-\varepsilon_{\mathbf{k}}-\Sigma^{c}(M(\mathbf{ k}),z)}\,, \tag{36}\]  
here is the lattice dispersion, \(z\) is the imaginary frequency. A similar procedure is used to construct the two-particle quantities needed to determine the phase diagram or the nature of the dominant fluctuations that can eventually destroy the quasiparticle. This procedure is a generalization of the method of calculating response functions in the DMFT [17; 159]. In the DCA, the introduction of the momentum dependence in the self-energy will allow one to detect some precursor to transitions which are absent in the DMFT; but for the actual determination of the nature of the instability, one needs to compute the response functions. These susceptibilities are thermodynamically defined as second derivatives of the free energy with respect to external fields. \(\Phi^{c}(G)\) and \(\Sigma^{c}_{\sigma}\), and hence \(F_{DCA}\) depend on these fields only through \(G_{\sigma}\) and \(G_{\sigma}^{0}\). Following Baym [160; 161] it is easy to verify that, the approximation  
\[\Gamma_{\sigma,\sigma^{\prime}}\approx\Gamma^{c}_{\sigma,\sigma^{\prime}} \equiv\delta\Sigma^{c}_{\sigma}/\delta G_{\sigma^{\prime}} \tag{37}\]  
yields the same estimate that would be obtained from the second derivative of \(F_{DCA}\) with respect to the applied field.