Dynamical Cluster Approximation

The Dynamical Cluster Approximation (DCA) is a fully causal approach which systematically incorporates non-local corrections to the DMFA by mapping the lattice problem onto an embedded periodic cluster of size $N_c$. For $N_c=1$ the DCA is equivalent to the DMFA and by increasing $N_c$ the dynamic correlation length can be gradually increased while the calculation remains in the thermodynamic limit. In contrast to previous approaches, the DCA is fully causal, preserves the full translational and point group symmetry of the lattice, and is easy to implement.

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 $\Delta k$. As a result, non-local dynamical correlations of range $\sim \pi/\Delta k$ 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 $T=0$ 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.