Comparison of two-quantum-cluster approximations

We provide microscopic diagrammatic derivations of the molecular coherent potential approximation (MCA) and dynamical cluster approximation (DCA) and show that both are $\Phi$ derivable. The MCA (DCA) maps the lattice onto a self-consistently embedded cluster with open (periodic) boundary conditions, and therefore violates (preserves) the translational symmetry of the original lattice. As a consequence of the boundary conditions, the MCA (DCA) converges slowly (quickly) with corrections $\mathcal{O}{(1/L}_c)$ $\left[\mathcal{O}{(1/L}_c^2\right)],$ where $L_c$ is the linear size of the cluster. These analytical results are demonstrated numerically for the one-dimensional symmetric Falicov-Kimball model.