The previous two sections considered initially a first order approximated irreducible vertex function (i.e. FLEX) and then proceeded to show the significant improvement one can achieve by including second order corrections to within the MSMB method. Both of these perturbative approaches however resulted in an overestimation of the long length-scale features of the self-energy at low temperatures. We now return to the non-perturbative -approximation using the irreducible QMC vertex. This approximation is expected to significantly improve the results, especially at lower temperatures where higher order corrections to the bare vertex are important and measurably contribute to the self-energy.
Fig. 3.13 compares the imaginary part of the multi-scale self-energies obtained by considering both the second order vertex and the full QMC vertex, to the numerically exact large single-cluster QMC self-energy. While the resulting multi-scale self-energy utilizing the full vertex function gives a better approximation for some momenta, the second order vertex solution remains preferable for others. This behavior may be somewhat unexpected since the inclusion of higher order corrections in the vertex was expected to further improve the overall quality of the multi-scale solution. However, we believe that this discrepancy arises from difficulties in extracting the exact vertex related to problems with high frequency conditioning in the QMC and is not an intrinsic problem of the method. As a consequence, the consideration of second order corrections in the -approximation remains the most useful at this time. It successfully allows for the exploration of lower temperatures - a regime where the reducible vertex function develops a richness in features.