Up to this point, the various approximations to the irreducible vertex function within the MSMB technique were implemented using the momentumspace Ansatz. We now return to the previously introduced implementation of the realspace Ansatz (see Eq. 3.11). The significant difference is that the separation of lengthscales in the realspace approach is straightforward, and therefore overcounting of diagrams is not an issue. The second order vertex approach was shown to be widely successful and will hence be used to investigate this Ansatz implementation as well. It should be noted that both versions of the Ansatz are treated identically as far as the implementation of selfconsistence is concerned (see Sec. 3.5.1).

Fig. 3.14 compares the multiscale selfenergy of the two Ansatz implementations to the large single cluster QMC result. It is quite apparent that the realspace implementation of the MSMB method is inferior. The momentumspace Ansatz provides a selfenergy more closely resembling the exact result throughout the Brillouin zone.
The realspace Ansatz provides an intuitive and simple way to combine the different lengthscales of the problem in contrast to a more complicated implementation in momentum space. Although inferior, the realspace Ansatz remains a viable, numerically stable alternative. It furthermore provides an alternative means of interpolating the small cluster self energy by neglecting the large cluster selfenergy contributions in Eq. 3.11, which were found to be negligible in this approach. The resulting interpolated small cluster QMC selfenergy (not shown) closely resembles that of the the realspace Ansatz based MSMB method.