  The dynamical cluster approximation (DCA) is a quantum cluster extension to
the single-site dynamical mean-field theory that incorporates spatially
nonlocal dynamic correlations systematically and nonperturbatively. The DCA$^+$
algorithm addresses the cluster shape dependence of the DCA and improves the
convergence with cluster size by introducing a lattice self-energy with
continuous momentum dependence. However, we show that the DCA$^+$ algorithm is
plagued by a fundamental problem when its self-consistency equations are
formulated using the bare Green's function of the cluster. This problem is most
severe in the strongly correlated regime at low doping, where the DCA$^+$
self-energy becomes overly metallic and local, and persists to cluster sizes
where the standard DCA has long converged. In view of the failure of the
DCA$^+$ algorithm, we propose to complement DCA simulations with a
post-interpolation procedure for single-particle and two-particle correlation
functions to preserve continuous momentum dependence and the associated
benefits in the DCA. We demonstrate the effectiveness of this practical
approach with results for the half-filled and hole-doped two-dimensional
Hubbard model.
