3.5.2 Second order in $U$

We move to include second order corrections in the vertex which is expected to mitigating the effects of the underestimation to the vertex in the FLEX. This lessens the difference in magnitude between the QMC and $\lambda$ self-energy within the Ansatz (as observed in Fig. 3.7) and thus increase the applicability of the MSMB method to lower temperatures.

Fig. 3.9 compares the imaginary part of the self-energy at lowest Matsubara frequency for various degrees of approximation. In contrast to the gross overestimation of the first order approximation to $\Gamma$, the inclusion of second order in $U$ corrections within the MSMB method successfully captures the long length-scale features of the large single-cluster QMC self-energy throughout most of the Brillouin zone. The largest deviations in the multi-scale self-energy are found about the corners of the Brillouin zone i.e. $k=\pm \pi$. Within the Ansatz, the difference of the small and large cluster $\lambda$ self-energy provides the long length-scale features thus partially restoring the self-energy information which was lost by coarse graining to the small QMC cluster. However, using a $\lambda$ self-energy with a negative imaginary part which is considerably larger than that of the small cluster QMC result in an overestimation of this correction, predominately in regions encompassed by steep gradients in the self-energy.

Fig. 3.10 further illustrates the pathology of the Ansatz associated with the difference in magnitude of the cluster self-energies. We show the imaginary part of the self-energy as obtained by the various $\lambda$-approximations on the large cluster in comparison to the large single-cluster QMC result. The magnitude of the negative imaginary part of the self-consistent FLEX (first order in $U$ approximation) self-energy is, as was previously indicated, very large compared to the magnitude of the QMC result. The new approach of including second order corrections in $U$ in the vertex function, succeeds in yielding a self-energy which resembles that of the QMC more closely. Similarly to the numeric instability encountered with the FLEX approximation, the self-consistent determination of the $\lambda$ self-energy using the large cluster Ansatz also yields a larger self-energy but doesn't encounter the catastrophic divergence of the FLEX. We find that a second order $\Gamma$ combined with a non-self-consistent Ansatz provides the best multi-scale solution to the problem.

Figure 3.10: Imaginary part of the $\lambda$-approximated self-energy at lowest Matsubara frequency for the large cluster ( $N_c^{(2)}=32$) using various cluster solvers in comparison to the single cluster QMC result at $\beta =31$, $U=W=1.0$, and $n=0.75$. Also shown is the overestimated Ansatz self-consistent self-energy of the MSMB method for the second order approximated $\Gamma$.

Now that we have established the viability of the MSMB technique we take a closer look at the dependence of the small cluster size on the quality of the multi-scale result. Within the MSMB method the QMC constitutes the computationally most expensive part. Hence, we want to restrict the calculations to the smallest possible QMC cluster ($N_c^{(1)}$) without any significant loss of quality in the multi-scale results. It is therefore important to study the dependence of the multi-scale results on the size of the small cluster.

Fig. 3.11 shows the multi-scale self-energy for a variety of small cluster sizes as obtained by the MSMB method using the second order approximation to the $\lambda$ vertex. It is apparent that a cluster size of $N_c^{(1)}=4$ is too small to adequately capture the main $k$ dependence (overall magnitude) of the QMC's self-energy. Considering such a small QMC cluster grossly misrepresents the range of the QMC self-energy and is not recovered in the MSMB method. This is an indication that correlations beyond the length-scales of the small cluster are still significant and not sufficiently well approximated by the $\lambda$-method. A small cluster size of $N_c^{(1)}=8$ on the other hand, appears adequate and only slightly underestimates the self-energy in the vicinity of $k=\pm \pi$.In this area we continually observe significant remnants of the coarse graining of the QMC which is inadequately restored by the MSMB method.

Figure 3.11: Imaginary part of the self-energy at lowest Matsubara frequency as obtained by the MSMB approach using the second order $\lambda$-approximation and large single-cluster QMC. Results are for various small cluster sizes $N_c^{(1)}$ at $\beta =31$, $U=W=1.0$, and $n=0.75$.

Up to this point we have focused our investigation on the momentum dependence of the self energy at small Matsubara frequency. Since the visual representation of the self-energy features at multiple, larger Matsubara frequencies would be cumbersome, we proceed to further illustrate the strengths of this MSMB technique by focusing on the presence of spin-charge separation in the system which is manifest in the full frequency dependent self-energy.

One dimensional systems have been shown to be non-Fermi liquids. Amongst other unique features they are known to exhibit spin-charge separation(19,21,20). This very intriguing property manifests itself by the complete decoupling of spin and charge degrees of freedom. The single-particle spectra of such systems exhibits two unique peaks corresponding to either spin or charge excitations which move independently of each other. Due to the involved nature of extracting spectra from the MSMB method, we are at this time unable to directly identify the presence of any such feature in our results. One characteristic of such a separation however, is the presence of two distinct velocities corresponding to charge and spin respectively. Following an approach by Zacher(22), we examine the MSMB Matsubara frequency Green's function to the possible presence of spin-charge separation. This is done by fitting Green's function obtained from the MSMB method to that of the Luttinger model solution

$$\begin{split}&G^{(LM)}_{v_1, v_2, \kappa _{\rho}}(x,\tau)=\\ ... ...tau^2) ^{-(\kappa _ \rho +1/\kappa _ \rho-2)/8}}~~, \end{split}$$ (3.26)

where $v_1$ and $v_2$ are the spin and charge velocities and $c$ is a normalization constant. We use the approximation for the correlation exponent $\kappa _{\rho}=1$ which is deemed sufficient by Zacher for purposes of identifying the presence of two different velocities. In order to accurately perform a fit with our data we need to additionally coarse grain (see Eq. 3.5) the Luttinger liquid Green's function to obtain a fitting function in-line with the idea of the DCA. This fit yields values for both the spin and charge velocities when fitted at $k=\pi /2$ which is the DCA momentum closest to the Fermi wave vector. This specific choice of $k$ value is motivated by fact that the Luttinger model solution is based on a low energy approximation of the Hubbard model where a linearized dispersion around the Fermi vector is assumed. For the parameters in Fig. 3.12 the Fermi wave vector ( $k_F \approx 1.2$) falls into the cluster cell about $k=\pi /2$.

Figure 3.12: Spin and charge velocities ($v1$ and $v2$ respectively) obtained by fitting the different results with the Luttinger Green's functions (Eq. 3.26) about cluster momentum $k=\pi /2$ for $\beta =31$, $U=W=1.0$, and $n=0.75$. Multi-scale results are for a small cluster size $N_c^{(1)}=8$ and different large cluster sizes, and the QMC velocities were obtained from a single 32 site cluster calculation.

Fig. 3.12 shows the spin and charge velocities ($v_1$ and $v_2$ respectively) obtained by using the single cluster QMC results for smaller cluster sizes and multi-scale results for larger ones. For intermediate cluster sizes, where both the QMC and the MSMB method are feasible, we see a good match between the two methods. For larger cluster sizes where the QMC approach becomes infeasible the multi-scale results fall close to the QMC extrapolated values. Furthermore, as we extrapolate the multi-scale results to the infinite cluster size limit we find two different velocities: $v_1=0.311$ and $v_2=0.674$. These values compare rather well to those obtained by Zacher's et. al.(22) grand-canonical QMC calculation for a cluster size of 64, $\beta=80$ and all other parameters equal to ours: $v_1=0.293 \pm 0.019$ and $v_2=0.513 \pm 0.023$. Although the temperature in Fig. 3.12 is slightly higher compared to that of Zacher's results, we observed only a small temperature dependence of the fitted velocities in our calculations. The similarity between the two results are quite remarkable considering that our calculations were based on a substantially smaller 8 site QMC cluster and hence less subject to the sign-problem ^3.2. These are yet further indications that the MSMB method indeed successfully captures the long length-scale physics of the model and is in good quantitative agreement with large single-cluster QMC calculations.