3.5.4 Real Space Ansatz

Up to this point, the various approximations to the irreducible vertex function within the MSMB technique were implemented using the momentum-space Ansatz. We now return to the previously introduced implementation of the real-space Ansatz (see Eq. 3.11). The significant difference is that the separation of length-scales in the real-space approach is straightforward, and therefore over-counting 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 self-consistence is concerned (see Sec. 3.5.1).

Figure 3.14: Imaginary part of the self-energy at lowest Matsubara frequency as obtained by the MSMB method using the second order approximated $\Gamma$ in conjunction with the real-space and momentum-space based Ansatz at $\beta =31$, $U=W=1.0$, and $n=0.75$. Also shown is the large single-cluster QMC result. Multi-scale results are for cluster sizes $N_c^{(1)}=8$ and $N_c^{(2)}=32$.

Fig. 3.14 compares the multi-scale self-energy of the two Ansatz implementations to the large single cluster QMC result. It is quite apparent that the real-space implementation of the MSMB method is inferior. The momentum-space Ansatz provides a self-energy more closely resembling the exact result throughout the Brillouin zone.

The real-space Ansatz provides an intuitive and simple way to combine the different length-scales of the problem in contrast to a more complicated implementation in momentum space. Although inferior, the real-space 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 self-energy contributions in Eq. 3.11, which were found to be negligible in this approach. The resulting interpolated small cluster QMC self-energy (not shown) closely resembles that of the the real-space Ansatz based MSMB method.