Statistical error estimates in dynamical mean-field theory and extensions thereof

We employ the jackknife algorithm to analyze the propagation of the statistical quantum Monte Carlo error through the Bethe-Salpeter equation. This allows us to estimate the error of dynamical mean-field theory calculations of the susceptibility and of dynamical vertex approximation calculations of the self-energy. We find that the different frequency components of the susceptibility are uncorrelated, whereas those of the self-energy are correlated. For improving the quality of the correlation matrix taking sufficiently many jackknife bins is key, while for reducing the standard error of the mean sufficiently many Monte Carlo measurements are necessary. We furthermore show that even in the case of the self-energy, the finite covariance does not have a sizable influence on the analytic continuation.

Figure 1

Estimated correlation matrix $\widehat{corr}[G_{\text{cut1}}^{\left(2\right)}\left({\nu }_1\right),G_{\text{cut1}}^{\left(2\right)}\left({\nu }_2\right)]$ for the two-particle Green's function at the cut $\nu ={\nu }^{\prime }={\nu }_i, \omega =0$ comparing different temperatures and numbers of bins. The $40\times 40$ matrices in the top row and the $80\times 80$ matrices in the bottom row correspond to 40 and 80 fermionic frequencies, respectively.

Figure 2

Same as Fig. 1 (top) but now for the estimated correlation matrix $\widehat{corr}[G_{\text{cut2}}^{\left(2\right)}\left({\omega }_1\right),G_{\text{cut1}}^{\left(2\right)}\left({\nu }_1\right)]$, i.e., the correlation between cut2 with ${\omega }_1, {\nu }^{\prime }=\nu =0$ and cut1 with $\omega =0, {\nu }^{\prime }=\nu ={\nu }_1$. The $41\times 40$ correlation matrices correspond to 41 bosonic and 40 fermionic frequencies. As in Fig. 1, for sufficiently many bins (right) the off-diagonal components of the correlation matrix vanish. (Note that in this figure there are no diagonal components, since there are different frequencies on the axes.)

Figure 3

Same as Fig. 2 but now at $\beta =4$. The $17\times 70$ matrices correspond to 70 fermionic and 17 bosonic frequencies (which we reduced to save computational time as there was no measurable change in the self-energy).

Figure 4

Imaginary part and jackknife SEM of the $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ self-energy at different temperatures. For better visibility, the errors bars are enlarged by a factor of 500 on the left and by 50 on the right.

Figure 5

Imaginary part and jackknife SEM of the difference between the DMFT and $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ self-energy, ${\mathrm{\Sigma }}_{\mathrm{D}\mathrm{\Gamma }\text{A},\mathbf{k}\nu }-{\mathrm{\Sigma }}_{\text{DMFT},\nu }$, at $\beta =4$. Different numbers of bins and total measurements are compared. For better visibility the error bars are enlarged by a factor of 10.

Figure 6

Real part of the estimated correlation matrix $\widehat{corr}[{\mathrm{\Sigma }}_{\mathrm{D}\mathrm{\Gamma }\text{A},\mathbf{k}{\nu }_1},{\mathrm{\Sigma }}_{\mathrm{D}\mathrm{\Gamma }\text{A},\mathbf{k}{\nu }_2}]$. As in Fig. 5 different numbers of bins and total measurements are compared. The $80\times 80$ matrices correspond to 80 fermionic frequencies.

Figure 7

Real part of the estimated correlation matrix $\widehat{corr}[{\mathrm{\Sigma }}_{\mathrm{D}\mathrm{\Gamma }\text{A},\mathbf{k}{\nu }_1},{\mathrm{\Sigma }}_{\mathrm{D}\mathrm{\Gamma }\text{A},\mathbf{k}{\nu }_2}]$ at different temperatures and $\mathbf{k}$ points. The $40\times 40$ (top) and $80\times 80$ matrices (bottom) correspond to 40 and 80 fermionic frequencies, respectively.

Figure 8

Density and magnetic DMFT susceptibility, ${\chi }_d$ and ${\chi }_m$, at different temperatures and momenta. For better visibility the error bars are enlarged by a factor of 10.

Figure 9

DMFT density susceptibility ${\chi }_d$ at $\beta =4$ for two different momenta $\mathbf{q}$, comparing different numbers of bins and total measurements.

Figure 10

Same as Fig. 9 but for the magnetic susceptibility ${\chi }_m$. For better visibility the error bars are enlarged by a factor of 10.

Figure 11

Estimated correlation matrix of the DMFT susceptibilities $\widehat{corr}[{\chi }_r\left({\omega }_1\right),{\chi }_r\left({\omega }_2\right)], r\in \{m,d\}$ at different temperatures and momenta. A total number of $256\times n_0$ measurements are binned into 256 jackknife samples. The $9\times 9$ matrices correspond to nine bosonic frequencies.

Figure 12

Same as Fig. 11 but now at $\mathbf{q}=(\pi ,0)$ and comparing different numbers of jackknife bins $n_b$ for a total number of $16\times n_0$ measurements.

Figure 13

Imaginary part of the self-energy on the real-frequency axis at different momenta $\mathbf{k}$, comparing the analytic continuation without covariance matrix, with the proper covariance matrix, and with a constant error.