  The dynamical cluster approximation (DCA) is a systematic extension beyond
the single site approximation in dynamical mean field theory (DMFT), to include
spatially non-local correlations in quantum many-body simulations of strongly
correlated systems. We extend the DCA with a continuous lattice self-energy in
oder to achieve better convergence with cluster size. The new method, which we
call DCA$^+$, cures the cluster shape dependence problems of the DCA, without
suffering from causality violations of previous attempts to interpolate the
cluster self-energy. A practical approach based on standard inference
techniques is given to deduce the continuous lattice self-energy from an
interpolated cluster self-energy. We study the pseudogap region of a hole-doped
two-dimensional Hubbard model and find that in the DCA$^+$ algorithm, the
self-energy and pseudo-gap temperature $T^*$ converge monotonously with cluster
size. Introduction of a continuous lattice self-energy eliminates artificial
long-rage correlations and thus significantly reduces the sign problem of the
quantum Monte Carlo cluster solver in the DCA$^+$ algorithm compared to the
normal DCA. Simulations with much larger cluster sizes thus become feasible,
which, along with the improved convergence in cluster size, raises hope that
precise extrapolations to the exact infinite cluster size limit can be reached
for other physical quantities as well.

Strongly Correlated Electrons