Efficiency of quantum Monte Carlo impurity solvers for the dynamical mean-field theory

Since the inception of the dynamical mean-field theory, numerous numerical studies have relied on the Hirsch-Fye quantum Monte Carlo (HF-QMC) method for solving the associated impurity problem. Recently developed continuous-time algorithms (CT-QMC) avoid the Trotter discretization error and allow for faster configuration updates, which makes them candidates for replacing HF-QMC. We demonstrate, however, that a state-of-the-art implementation of HF-QMC (with extrapolation of discretization $\Delta \tau \to 0$) is competitive with CT-QMC. A quantitative analysis of Trotter errors in HF-QMC estimates and of appropriate $\Delta \tau$ values is included.

Figure 1

(Color online) (a) Energy versus squared temperature: extrapolated HF-QMC results (circles) and estimates (Ref. 10) from CT-QMC (triangles, squares). (b) Data with leading $T$ corrections subtracted, plus reference (filled circles).

Figure 2

(Color online) Comparisons of errors in QMC energy estimates: absolute deviations of the data shown in Fig. 1 from the reference curve on a logarithmic scale.

Figure 3

(Color online) (a) HF-QMC estimate of energy at $T=1∕45$ (circles) versus discretization $\Delta \tau$ with extrapolation $\Delta \tau \to 0$ (dashed line). (b) Same data after subtraction of approximate leading $\Delta \tau$ error versus $\Delta {\tau }^2$ in comparison with reference HF-QMC data (filled circles and solid line).

Figure 4

(Color online) Imaginary part of self-energy on the imaginary axis for $T=1∕45$ as estimated by CT-QMC (Ref. 10) using the hybridization expansion (triangles) or the weak-coupling expansion (circles) and by HF-QMC (dashed lines for $\Delta \tau =0.4$, $\Delta \tau =0.2$; solid line for $\Delta \tau \to 0$).

Figure 5

(Color online) Kinetic energy versus squared temperature (minus leading $T$ corrections): estimates from HF-QMC at finite discretization (squares for $\Delta \tau =0.15$, diamonds for $\Delta \tau =0.2$) and after extrapolation $\Delta \tau \to 0$ (circles); dashed and solid lines denote corresponding reference results. Also shown are HF-QMC results extracted from Fig. 3 in Ref. 10 (crosses); thin lines are guides to the eye only.

Figure 6

(Color online) Kinetic energy versus squared temperature: estimates at fixed CPU time from CT-QMC (Ref. 10) (triangles, squares) and HF-QMC (empty circles) after subtracting leading $T$ corrections. Also shown are higher-precision results from HF-QMC (filled circles and solid line) and hybridization CT-QMC (double-dashed line).

Figure 7

(Color online) Coefficients of Trotter errors in HF-QMC estimates of the energy (see text).

Figure 8

(Color online) Coefficients of Trotter errors in HF-QMC estimates of kinetic energy. Inset: relative coefficients.

Figure 9

(Color online) Coefficients of Trotter errors in HF-QMC estimates of double occupancy.