To account for the omitted set of short length-scale self-energy diagrams in the -approximation, we substitute appropriate diagrams from the small cluster QMC result. This diagrammatic substitution between the different length-scales in the MSMB method is done by means of an analytic Ansatz.
One possible implementation of the MSMB method which yields a self-energy containing correlations on both long and short length-scales is given by the real-space Ansatz:
In this formalism, the MSMB self-energy is constructed by taking all contributions of lengths up to the linear sizes of the small cluster ( ) from the exact QMC result and the remaining longer ranged contributions are complemented by the -approximated large cluster self-energy. Note, the multi-scale self-energy obtained in Eq. 3.10 has lost its spatial continuity. Fourier-transforming the individual length-scale contributions in Eq. 3.10 yields
J. Hague et. al. in ref. (3) considered the following, alternative momentum-space Ansatz to combine the different length-scales:
While the real space implementation of the Ansatz was a straight forward combination of different length-scale elements, the momentum implementation interpretation is more involved. The -method provides an estimate for the set of self-energy diagrams which convey the long-ranged correlations of the system. However, since in reciprocal space there isn't an explicit separation of length-scales, the remaining short length-scale diagrams which are to be supplied by the QMC calculation have to be identified. This can be accomplished by removing the sub-set of -approximated self-energy diagrams on the small cluster from the complete set of the QMC calculation hence avoiding a double counting of the corresponding self-energy contributions.
Within the traditional form of the DCA, all self-energies are inherently causal for each individual cluster. One consequence of causality is that . In this Ansatz based implementation of the MSMB method however, causality is not inherently guaranteed. The combination of self-energy contributions of various length-scales in the Ansatz is only assured to yield a causal multi-scale self-energy in the limit of the cluster sizes approaching one another. Therefore, causality in these schemes cannot be guaranteed and hence has to be monitored closely throughout. For a more detailed discussion of this and the entire momentum-space Ansatz see Ref. (3).
The remaining self-consistent implementation of the multi scale method (i.e. two cluster DCA) is similar to that of the traditional, single cluster DCA. Fig. 3.5 depicts a flow chart of the implementation of both Ansätze in the scheme of the overall self-consistency loop. The bottom loop shows the already discussed DCA self-consistency loop using the QMC as a small cluster solver. For the large cluster solver (-method) the self-consistency is similar. In the overall scheme of the Ansatz, the two cluster problems are combined in a fully self-consistent approach: After each iteration of the QMC/DCA loop, the corresponding /DCA contribution on the large cluster is evaluated. The Ansatz is used after each step to calculate new estimates for the self-energies on both the small and large cluster, thus yielding a fully self-consistent solution. The corresponding paths are shown in Fig. 3.5. The converged Ansatz self-energy will be dominated by the small cluster QMC self-energy which contributes the strongest, short-ranged correlations. The remaining weaker long-ranged correlations are incorporated in the difference of -approximated self-energies between the two clusters. In moving away from a fully self-consistent approach, the self-consistency restrictions may be lessened to various degrees. Some possible implementations will be discussed in further detail in section 3.4.