Negative sign problem in continuous-time quantum Monte Carlo: Optimal choice of single-particle basis for impurity problems

The negative sign problem in quantum Monte Carlo (QMC) simulations of cluster impurity problems is the major bottleneck in cluster dynamical mean-field calculations. In this paper, we systematically investigate the dependence of the sign problem on the single-particle basis. We explore both the hybridization-expansion and interaction-expansion variants of continuous-time QMC simulations for three-site and four-site impurity models with baths that are diagonal in the orbital degrees of freedom. We find that the sign problem in these models can be substantially reduced by using a nontrivial single-particle basis. Such bases can be generated by diagonalizing a subset of the intracluster hoppings.

Figure 1

(a) Trimer impurity model with an orbital-diagonal bath. Each site is connected to a bath with a semicircular density of states of width 4. (b) Schematic representation of the hoppings and the Coulomb interactions in the dimer+monomer basis ($t=t^{\prime }$). (c) Inequivalent realizations of links for the trimer impurity problem ($t\ne t^{\prime }$). $N_L$ is the number of links in a graph.

Figure 2

Average sign computed for the trimer model with $t^{\prime }/t=1$ and $U=5$ ($\beta =15,25,50$). (a) $t$ dependence of the average sign. We show data for the site basis, the diagonal basis, the optimal basis found in the brute-force search, and the dimer+monomer basis. (b) $\beta$ dependence of the average sign computed for the site basis, the diagonal basis, and the dimer+monomer basis at $t=0.6$. The dotted lines denote a fit by the exponential form (16).

Figure 3

On-site imaginary-time Green's function $G\left(\tau \right)$ computed for the trimer model with $t=t^{\prime }=0.5$ and $U=5$ ($\beta =50$). We symmetrize the data using the threefold rotational symmetry.

Figure 4

Distribution function of the expansion order ($n$) and expansion-order-resolved average sign computed at $t=t^{\prime }=0.6$ and $\beta =50$.

Figure 5

Average sign computed for the trimer model: (a) $t^{\prime }/t=2$ and (b) 0.5 at $\beta =50$. We show the maximum value of the average sign among single-particle bases generated by graphs with the same $N_L$. We also plot the highest value of the average sign found in a brute-force search. We show the average sign computed for each basis in Fig. 11 of Appendix pp1.

Figure 6

Tetramer impurity model (inset) and symmetrically inequivalent graphs for $t=t^{\prime }$ (a) and $t\ne t^{\prime }$ (b).

Figure 7

Average sign computed for the tetramer model: (a),(b) $t^{\prime }/t=1$ and (c),(d) $t^{\prime }/t=0.5$. In (a) and (c), we show the maximum value of the average sign among single-particle bases generated by graphs with the same $N_L$ computed at $\beta =50$. In (b) and (d), we show the $\beta$ dependence of the average sign computed at $t=0.4$. The dotted lines are fits by the exponential (16).

Figure 8

Comparison of the average sign computed for the trimer model using the site basis and the dimer+monomer basis by CT-INT. (a) $U$ dependence of the average sign at $\beta =50$. (b) $\beta$ dependence of the average sign at $U=8$.

Figure 9

Comparison of the average sign computed for the tetramer using the site basis and the $N_L=2$ (No. 2) basis by CT-INT. (a) $U$ dependence of the average sign at $\beta =50$. (b) $\beta$ dependence of the average sign at $U=8$.

Figure 10

Comparison of the self-energy $\mathrm{\Sigma }\left(i{\omega }_n\right)$ computed by the site basis and the $N_L=2$ (No. 2) basis using CT-INT at $U=8$ and $\beta =50$. We plot the site-diagonal element ${\mathrm{\Sigma }}_{11}$ and the site-off-diagonal element ${\mathrm{\Sigma }}_{12}$ in the site basis.

Figure 11

Average sign computed for the trimer model: (a) $t^{\prime }/t=2$ and (b) 0.5 at $\beta =50$. Right panel, $t^{\prime }/t=2$; left panel, $t^{\prime }/t=0.5$. We show the average sign computed for each basis.

Figure 12

Average sign computed for the tetramer model at $\beta =50$: Right panel, $t^{\prime }/t=1$; left panel, $t^{\prime }/t=0.5$. We show the average sign computed for each basis.

Figure 13

Acceptance rate for the insertion of a double vertex as a function of the time difference between two vertices ($\mathrm{\Delta }\tau$) computed for the tetramer model at $U=8, \beta =50$, and $t=t^{\prime }=1$. The data are symmetrized about $\beta /2$ and are averaged over spin-flip interactions and pair-hopping interactions.