The mapping to a cluster is accomplished by coarse-graining the lattice problem in reciprocal space. The DMFA and other local approximations like the CPA are equivalent to neglecting momentum conservation at all internal vertices of the self energy, so that the problem is coarse-grained over the entire Brillouin zone. The DCA systematically restores momentum conservation by introducing a coarse-graining scale . As a result, non-local dynamical correlations of range are treated accurately, since the lattice problem is mapped onto a periodic cluster of roughly this size. The cluster problem may be solved using QMC.
We have used the DCA to study pairing, the phase diagram, the pseudogap, and the metal-insulator transition in the Hubbard model. For simulations that occur in the parameter regime away from the critical point of the half filled 2D model, we find very rapid convergence with cluster size.
Results obtained with the DCA and other similar Quantum Cluster Methods, together with the associated formalisms are discussed in our recent Review of Modern Physics article.