A MSMB approach based on a first order approximated vertex is similar to the limited FLEXhybrid approach previously considered by J. Hague et. al. (3). In their work the intermediate lengthregime was addressed using the FLEX which was incorporated within a multiscale approach using the fully selfconsistent momentum space Ansatz (see Eqs. 3.12 and 3.13). However, in the regime of stronger couplings and/or lower temperatures, where a significant contribution of longer ranged correlations is to be expected, the method fails. In this section we propose changes to the implementation of the multiscales method which significantly improves its range of applicability by 1) restriction of the Ansatz to the large cluster, 2) modification of the cluster conversion, and 3) removal of the selfconsist implementation of the Ansatz on the large cluster.

In the original inception(3), the Ansatz is evaluated fully selfconsistently within both the QMC and FLEX as discussed in section 3.2.4. The selfenergy contributions of these two clusters solvers are linked by the momentum space Ansatz. However, long rangedcorrelations, as are described by the large cluster FLEX, are assumed weak. Hence, their presence in the effective medium of the small cluster only minutely effects the QMC selfenergy. We thus neglect the Ansatz on the small cluster (Eq. 3.12). With this modification, the small cluster problem can be solved independently of the large cluster problem. This results in a significant reduction in numerical complexity.
With the removal of the small cluster Ansatz condition, only a unidirectionally cluster conversion of small cluster selfenergies to the large cluster remains. In this work, a periodic cubic spline interpolation ^{3.1} is employed. This provides a good approximation for the multiscale selfenergy in the high temperature/small limit. Fig. 3.7 shows the imaginary part of the exact, QMC calculated and FLEX selfenergies on both clusters calculated at high . In this regime, correlations are short ranged in nature and thus well described by the small cluster itself, while remaining longranged features, which are provided by the large cluster FLEX, are insignificant. The interpolated small cluster selfenergy accurately replicates that of the explicit large single cluster QMC result. This results in a vanishing FLEX contribution within the Ansatz as the large and interpolated small cluster FLEX results are identical and thus cancel each other (see superimposed, lower set of curves in Fig. 3.7).

In this high temperature regime (e.g. Fig. 3.7) the fully selfconsistent Ansatz as well as our modified one remain numerically stable. However, one inherent limitation of the approach is already apparent. The observed magnitude of the negative imaginary part of the QMC selfenergy is significantly smaller than the FLEX result. This overestimation of the selfenergy by the FLEX deems a first order approximation to the vertex to be insufficient. A selfconsistent implementation of the Ansatz aggregates the problem as the Ansatz selfenergy used to initialize each FLEX iteration is similar in magnitude to the QMC selfenergy. This results in an insufficient damping of the FLEX potentials and the subsequent overestimation of the FLEX selfenergy renders the large cluster selfenergy calculation numerically unstable. This is an inherent problem in the selfconsistent approach. In an attempt to enhance the numeric stability of the MSMB method we remove the selfconsistent implementation of the Ansatz on the large cluster. This leaves two independent DCA calculations for the small and large cluster which, after the individual problems are converged, are combined by the Ansatz. The resulting selfconsistency scheme is depicted in Fig. 3.8. However, the effective media embedding the two cluster problems now are unaware of correlations as determined by the other cluster solver technique. Therefore, the effective medium of the large cluster now lacks the explicit short ranged correlations as provided by the QMC. Similarly, the effective medium of the small cluster only contains longranged correlations as provided by the dynamical meanfield approximation.
The third modification enters in the determination of the small cluster FLEX selfenergy . Rather than calculating it explicitly, we obtain it by coarse graining the large cluster selfenergy onto the small cluster. The combination of all changes introduced up to this point significantly increase the range of low temperatures which can be addressed by a perturbative MSMB approach.
Throughout the remainder of the paper we will restrict ourselves to the nonselfconsistent momentum space Ansatz as outlined in this section. It indeed succeeds in addressing some of the problems encountered in the original implementation and yields a numerically stable MSMB method.