Worm-improved estimators in continuous-time quantum Monte Carlo

We derive the improved estimators for general interactions and employ these for the continuous-time quantum Monte Carlo method. Using a worm algorithm we show how measuring higher-ordered correlators leads to an improved high-frequency behavior in irreducible quantities such as the one-particle self-energy or the irreducible two-particle vertex for non-density-density interactions. A good knowledge of the asymptotics of the two-particle vertex is essential for calculating nonlocal electronic correlations using diagrammatic extensions to the dynamical mean field theory as well as for calculating susceptibilities. We test our algorithm against analytic results for the multiorbital atomic limit and the Falicov-Kimball model.

Figure 1

Diagrammatic representation of the two-particle Green's function $G^{\left(2\right)}$ in terms of the disconnected contribution $G^{\mathrm{disc}}$ and the connected contribution $G^{\mathrm{conn}}$. The alternative partitioning splits the two-particle Green's function into a generalized susceptibility $\chi$ with a bubble term ${\chi }^{\left(0\right)}$.

Figure 2

Top: diagrammatic representation of the one-particle improved estimator ${\left(\mathrm{\Sigma }G\right)}_{cb}^{\left(1\right)}$ [Eq. (15)]. Bottom: diagrammatic representation of the two-particle improved estimator $H_{abcd}^{\left(2\right)}$ [Eq. (21), the last part of Eq. (19)]. The local interaction is represented explicitly by a wiggly line.

Figure 3

Schematic representation of the extended configuration space including the partition function space ${\mathcal{C}}_Z$ (red), the Green's function spaces ${\mathcal{C}}_{G^{\left(1\right)}}$ and ${\mathcal{C}}_{G^{\left(2\right)}}$ (green), and the improved estimator spaces ${\mathcal{C}}_{{\left(G\mathrm{\Sigma }\right)}^{\left(1\right)}}$ and ${\mathcal{C}}_{H^{\left(2\right)}}$ (blue). Worm spaces are only linked to one another over the partition function space with the balancing parameters ${\eta }_i$.

Figure 4

Imaginary part of the self-energy $\mathrm{\Sigma }$ vs Matsubara frequencies $i\nu$ in the atomic limit with two half-filled orbitals, $U=1.0,U^{\prime }=0.5$, and $J=0.25$. The self-energy obtained from the Dyson equation (red) shows much larger fluctuations in the high-frequency region compared to the self-energy obtained from the improved estimators (green). Error bars are calculated from 40 bootstrap samples. Inset: Comparing the second moment of the self-energy from the improved estimator (green) and the Dyson equation (red) by multiplication with the Matsubara frequency with the measurement via the one- and two-particle density matrix (blue) [23].

Figure 5

Error of the self-energy $\mathrm{\Sigma }$ vs Matsubara frequencies $i\nu$ in the atomic limit for the data given in Fig. 4. The uncertainties resulting from the Dyson equation (red) follow a quadratic scaling, while the uncertainties obtained from the improved estimator (green) follow a weak linear scaling.

Figure 6

Fermionic cut through the two-particle vertex function in the density (left) and magnetic (right) channel for the same parameters as in Fig. 4. The vertex obtained from a straight-forward calculation of the two-particle Green's function (red) displays larger fluctuations in the high-frequency region than the result obtained from the two-particle improved estimator (green). Error bars are calculated from four independent inversions. The analytic result (blue) is shown for comparison.

Figure 7

Imaginary part of the self-energy $\mathrm{\Sigma }$ vs Matsubara frequencies $i\nu$ for the FK model at $U=1.0,\beta =20,\mu =0.2$, and ${\epsilon }_f=-0.038114 \left(n_f=0.25\right)$. The self-energy obtained from the improved estimators (green) is in good agreement with the exact self-energy obtained from the RLM (blue). Error bars are calculated from 40 bootstrap samples. Inset: Comparing the second moment of the self-energy from the improved estimator (green) by multiplication with the Matsubara frequency with the analytical calculation (blue).

Figure 8

Real and imaginary part of the connected part of the Green function, $G_{\left(fc\right)}^{\mathrm{conn}}$, for different bosonic frequencies $n_{\omega }=0,2,4,6$ and the same parameters as in Fig. 7. All Matsubara frequencies are given in terms of their (integer) index.