Figure 1

(Color online) Illustration of the diagrams generated by the continuous-time impurity solvers. (i) Weak-coupling method: third order diagram consisting of three vertices (diamonds) $U{n}_{\uparrow }\left(\tau \right)n_{\downarrow }\left(\tau \right)$ linked by lines representing the function ${\mathcal{G}}_{0,\sigma }({\tau }_i-{\tau }_j)$. (ii) Hybridization expansion method: here, the orders of the diagrams for up and down spins can be different. Each creation operator $c_{\sigma }^{†}\left(\tilde{\tau }\right)$ (empty dot) is connected to an annihilation operator $c_{\sigma }\left(\tau \right)$ (filled dot) by a line representing the hybridization function $F_{\sigma }(\tau -\tilde{\tau })$. The black lines correspond to a particle number 1 and empty spaces to particle number 0, so the overlaps between the lines for up and down spins yield the potential energy. In both approaches, the diagrams corresponding to different connecting ${\mathcal{G}}_0$ or $F$ lines are summed up into a determinant, and these determinants are sampled by a Monte Carlo procedure.Reuse & Permissions

Figure 2

(Color online) Scaling of the matrix size with inverse temperature and interaction strength. Upper panel: temperature dependence for $U∕t=4$. In the case of Hirsch-Fye, the resolution $N=\beta U$ has been chosen as a compromise between reasonable accuracy and acceptable speed, while the average matrix size is plotted for the continuous-time solvers. Lower panel: dependence on $U∕t$ for fixed $\beta t=30$. The solutions for $U⩽4.5$ are metallic, while those for $U⩾5.0$ are insulating. The much smaller matrix size in the relevant region of strong interactions is the reason for the higher efficiency of the hybridization expansion method.Reuse & Permissions

Figure 3

(Color online) Upper panel: kinetic energy $E_{\mathit{kin}}=2t^2{\int }_0^{\beta }d\tau G\left(\tau \right)G(-\tau )$ obtained using the three QMC impurity solvers for $U∕t=4.0$ and $\beta t=10,15,\dots ,50$. The Hirsch-Fye simulations for $\Delta \tau =1∕U$ (as in Fig. 2) yield systematically higher energies. The inset shows results obtained with the continuous-time solvers for $\beta t=35$, 40, 45, and 50. Lower panel: potential energy $U⟨n_{\uparrow }{n}_{\downarrow }⟩$ for the same interaction strength.Reuse & Permissions

Figure 4

(Color online) Lowest Matsubara frequency value for $G$ for $U∕t=4.0$ using measurements in both imaginary time and frequency space in the weak-coupling case. The upper panel shows the Green’s function, and the lower panel the relative error on the measurement. Unlike in the Hirsch Fye algorithm, there are essentially no systematic errors in the continuous-time algorithms. In the case of the hybridization expansion algorithm, results for measurements in $\tau$ and $\omega$ are plotted. Both measurements yield a similar accuracy at low frequency. The hybridization expansion algorithm gives very accurate results, and the error bars show no dependence on $\beta$. This indicates that in the measured temperature range, two competing effects essentially cancel: the efficiency of the matrix updates, which decreases at lower temperatures, and the efficiency of the measurement procedure [Eq. (22)], which yields better results for larger matrix sizes.Reuse & Permissions

Figure 5

(Color online) Self-energy $\mathrm{Im}\Sigma \left(i{\omega }_0\right)∕{\omega }_0$ as a function of $\beta$ for $U∕t=4.0$. The Hirsch-Fye results exhibit large discretization errors, while the continuous-time methods agree within error bars. The hybridization expansion method is particularly suitable for measuring quantities which depend on low-frequency components, such as the quasiparticle weight.Reuse & Permissions

Figure 6

(Color online) Upper panel: low-frequency region of the self-energy $\Sigma \left(i\omega \right)$ for $U∕t=4.0$, $\beta t=45$. Noise in the higher frequencies is clearly visible for the values measured in $\tau$, while the values measured in $\omega$ in the weak-coupling algorithm converge smoothly to the high-frequency tail. Lower panel: high-frequency region of the self-energy $\Sigma \left(i\omega \right)$ for $U∕t=4.0$, $\beta t=45$. Noise in the higher frequencies is clearly visible for the values measured in $\tau$, while the values measured in $\omega$ in the weak-coupling algorithm converge smoothly to the high-frequency tail, ${\lim }_{\omega \to 0}\Sigma \left(i{\omega }_n\right)=U^2(1-n)n∕\left(i{\omega }_n\right)$.Reuse & Permissions

Figure 7

(Color online) Intermediate frequency region of the self-energy $\Sigma \left(i\omega \right)$ for $U∕t=4.0$, $\beta t=45$. Noise in the higher frequencies is clearly visible for the values measured in $\tau$, while the values measured in $\omega$ in the weak-coupling algorithm converge smoothly to the high-frequency tail.Reuse & Permissions

Figure 8

(Color online) Matrix sizes away from half filling: the matrix size decreases for the weak-coupling algorithm, while the one for the hybridization expansion algorithm increases as one dopes a Mott-insulating state.Reuse & Permissions