  We present the algorithmic details of the dynamical cluster approximation
(DCA), with a quantum Monte Carlo (QMC) method used to solve the effective
cluster problem. The DCA is a fully-causal approach which systematically
restores non-local correlations to the dynamical mean field approximation
(DMFA) while preserving the lattice symmetries. The DCA becomes exact for an
infinite cluster size, while reducing to the DMFA for a cluster size of unity.
We present a generalization of the Hirsch-Fye QMC algorithm for the solution of
the embedded cluster problem. We use the two-dimensional Hubbard model to
illustrate the performance of the DCA technique. At half-filling, we show that
the DCA drives the spurious finite-temperature antiferromagnetic transition
found in the DMFA slowly towards zero temperature as the cluster size
increases, in conformity with the Mermin-Wagner theorem. Moreover, we find that
there is a finite temperature metal to insulator transition which persists into
the weak-coupling regime. This suggests that the magnetism of the model is
Heisenberg like for all non-zero interactions. Away from half-filling, we find
that the sign problem that arises in QMC simulations is significantly less
severe in the context of DCA. Hence, we were able to obtain good statistics for
small clusters. For these clusters, the DCA results show evidence of non-Fermi
liquid behavior and superconductivity near half-filling.
