3.5.1 First Order in $U$

We begin the evaluation of the MSMB method by considering the first order approximation to the irreducible vertex. The previous section introduced a numerically stable approach to the perturbative MSMB method. This allows for the FLEX based MSMB treatment of stronger coupling/lower temperature regimes.

The major limitation of this method however remains in the still significant difference in magnitude of the QMC and FLEX self-energies leading to an overestimation of long length-scale features introduced by the FLEX. In Fig. 3.9 it is quite apparent that at lower temperatures the FLEX MSMB implementation overestimates the size of the long length-scale features i.e. amplitude of the oscillations in the imaginary part of the self-energy. This is yet further indication that a bare approximation to the vertex is inadequate to address the intermediate length regime.

Figure 3.9: Imaginary part of the self-energy at lowest Matsubara frequency as obtained by the MSMB method using first (FLEX) and second order approximated irreducible vertices $\Gamma$ in comparison to the large single-cluster QMC results at $\beta =31$, $U=W=1.0$, and $n=0.75$. Multi-scale results are for cluster sizes $N_c^{(1)}=8$ and $N_c^{(2)}=32$.