The DCA is a systematic quantum cluster theory that maps the lattice problem onto a selfconsistently embedded cluster problem. It is an extension of the dynamical meanfield theory (DMFT)(6,5) which systematically incorporates nonlocal correlations. In the limit when the cluster size is one (i.e. single site), it recovers the purely local DMFT solution, systematically incorporates nonlocal corrections as the cluster size increases, and finally becomes exact when the cluster size equals the size of the lattice.
The respective approximations for the DMFT and DCA may be derived by approximating the Laue function which describes momentum conservation at the vertices of the irreducible diagrams:
In the DMFT, the Laue function is approximated with for all combinations of , , and . In doing so, all electron propagators in the selfenergy diagrams may be averaged over the first Brillouin Zone (BZ); thus relinquishing any momentum dependence of the selfenergy. Hence, the DMFT lattice Green's function contains local correlations of the system but is unable to capture nonlocal correlations. The DCA sets out to systematically include these nonlocal contributions. This is accomplished by partially restoring momentum conservation of the irreducible vertices. We divide the BZ into identical discrete subcells as illustrated in Fig. 3.1. The center of each cell is labeled by , and the surrounding points by , so that any arbitrary . In the DCA, this partial momentum conservation is expressed by the Laue function:
(3.4) 

Therefore, all propagators may freely be summed over intracell momenta , yielding the coarse grained Green's function
In so doing, only momentum conservation of magnitude is neglected, while larger intercell transfers are preserved. The resulting selfenergy diagrams are now those of a finite cluster of size where each lattice propagator has been replaced by its coarsegrained analog, and the remaining cluster problem is defined by . We can write for the DCA lattice Green's function
(3.6) 
where is a function which maps momentum residing in a certain subcell of the BZ to its cluster momentum and the lattice selfenergy is approximated by that of the cluster problem.
The remaining embedded cluster problem must be solved with a selfconsistency requirement that the Green's function calculated on the cluster . Fig. 3.2 depicts the corresponding DCA algorithm: Starting with an initial guess for the selfenergy, we construct the coarsegrained Green's function from the corresponding lattice (see Eq. 3.5). In the next step, we utilize one of the many available cluster solvers to determine the cluster selfenergy. This is the numerically most involved step and a variety of numerical techniques may be applied. At this point, we use the new estimate for the selfenergy to reinitialize the selfconsistency loop. It is important to notice that in this procedure only the irreducible lattice quantities are approximated by their cluster equivalent  i.e. the selfenergy.
The DCA has been successfully implemented with a variety of cluster solvers of which some are exact but limited in cluster size, while others are applicable up to larger lengthscales but involve varying degrees of approximation. Some of the cluster solvers which have been used in conjunction with the DCA include the noncrossing approximation (NCA)(7), the fluctuation exchange approximation (FLEX)(3,8) and the Quantum Monte Carlo (QMC)(2) method. While NCA and FLEX involve various levels of approximations, QMC is of special interest since it provides an essentially numerically exact solution to the problem. Although the QMC constitutes a precise cluster solver, it becomes prohibitively expensive for large clusters. The range of applicability of exact calculations is thus restricted to relatively short lengthscales. However, various properties of stronglycorrelated systems are not accounted for (e.g. MerminWagner theorem(9)) due to the absence of longranged fluctuations in these solutions.