We have introduced a numerically feasible MSMB extension to the DCA. In this method the lattice problem is mapped onto that of two embedded clusters, dividing the problem into three length-scales. Correlations on each of the length-scales are approximated commensurate with the strength of the correlation on the respective scale. The intermediate length regime, which bridges the explicit treatment of short ranged correlations by means of the QMC to the long ranged dynamical mean-field one, is addressed in a diagrammatic long-wave length approximation based on the two particle irreducible vertex of the small cluster. The first order approximation to the vertex results in the FLEX but including higher order corrections result in a substantial better multi-scale result when compared to explicit large cluster QMC calculations. This can be attributed to the significance of higher order diagrams at lower temperatures. We proceeded to show that our MSMB results indicate spin and charge separation and the obtained velocities compare favorably to significantly larger finite size QMC calculation. The inclusion of the explicit QMC calculated vertex is currently still limited but further work in this direction looks promising. However, in any of the introduced implementations, the MSMB approach provides a means to adequately address large cluster problems on all length-scales at significant lower computational expense.
During the completion of this paper we learned of two related studies Ref. (24) and (25) where long-ranged correlations are addressed in a non-self-consistent approach.