3.2.4 Ansatz

To account for the omitted set of short length-scale self-energy diagrams in the $\lambda$-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:

$$\begin{split}&\Sigma^{(N_c^{(2)})}(x_i,x_j)=\\ &\left\{\begin... ...{N_c^{(2)}}{2} \\ 0, & otherwise \end{array}\right. \end{split}$$ (3.10)

In this formalism, the MSMB self-energy is constructed by taking all contributions of lengths up to the linear sizes of the small cluster ( $N_c^{(1)}/2$) from the exact QMC result and the remaining longer ranged contributions are complemented by the $\lambda$-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

$$\begin{split}\Sigma^{(N_c^{(2)})}&(K_2,i\omega_n) = \Sigma^{(... ...~K_2) \Sigma^{(N_c^{(2)})}_{\lambda}(x=j,i\omega_n) \end{split}$$ (3.11)

where only diagonal parts of the Fourier transformed self-energy are considered.

J. Hague et. al. in ref. (3) considered the following, alternative momentum-space Ansatz to combine the different length-scales:

$$\begin{split}\Sigma ^{(N_c^{(1)})}(K_1,i\omega _n ) &= \Sigma... ...r \Sigma _{\lambda}^{(N_c^{(2)})} (K_1,i\omega _n ) \end{split}$$ (3.12)

$$\begin{split}\Sigma ^{(N_c^{(2)})} (K_2,i\omega _n )& = \Sigm... ...ar \Sigma _{\lambda}^{(N_c^{(1)})}(K_2,i\omega _n ) \end{split}$$ (3.13)

where $\Sigma _{\lambda}^{(N_c^{(1)})}$ is the self-energy obtained in the $\lambda$-approximation when implemented on the small cluster. The self-energies $\Sigma^{(N_c^{(1)})}$ and $\Sigma^{(N_c^{(2)})}$ on the small and large cluster, respectively, exist on different grid sizes, and it becomes necessary to convert self-energies from one to the other. This conversion is denoted by a bar over the self-energy which denotes an interpolation when going from a coarser grid to a finer one and a coarse-graining step otherwise. For example, the large cluster self-energy $\Sigma^{(N_c^{(2)})}$ is constructed from the explicit $\lambda$-approximated self-energy on the large cluster, and two interpolated small cluster (denoted by the superscript $(N_c^{(1)})$) self-energies $\bar \Sigma _{QMC}^{(N_c^{(1)})}$ and $\bar \Sigma _{\lambda}^{(N_c^{(1)})}$. It is important to note that the coarse graining in going from large- to small-cluster self-energies is not an averaging over electron propagators but the term is used in this context to imply an averaging of the self-energy within a cluster-cell.

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 $\lambda$-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 $\lambda$-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 $-\frac{1}{\pi}Im\Sigma(k,\omega)>0$. 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 ($\lambda$-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 $\lambda$/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 $\lambda$-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.

Figure 3.5: Flow chart for the Ansatz self-consistent implementation of the MSMB/DCA.