Deconvolving the components of the sign problem

Auxiliary field quantum Monte Carlo simulations of interacting fermions require sampling over a Hubbard-Stratonovich field $h$ introduced to decouple the interactions. The weight for a given configuration involves the products of the determinant of matrices $M_{\sigma }\left(h\right)$, where $\sigma$ labels the species, and hence is typically not positive definite. Indeed, the average sign $\langle \mathcal{S}\rangle$ of the determinants goes to zero exponentially with increasing spatial size and decreasing temperature for most Hamiltonians of interest. This statement, however, does not explicitly separate two possible origins for the vanishing of $\langle \mathcal{S}\rangle$. Does $\langle \mathcal{S}\rangle \to 0$ because randomly chosen field configurations have $\det \left[M\right(h\left)\right]<0$, or does the sign problem arise because the specific subset of configurations chosen by the weighting function have a greater preponderance of negative values? In the latter case, the process of weighting the configurations with $\left|\det \right[M\left(h\right)\left]\right|$ might steer the simulation to a region of configuration space of $h$ where positive and negative determinants are equally likely, even though randomly chosen $h$ would preferentially have determinants with a single dominant sign. In this paper, we address the relative importance of these two mechanisms for the vanishing of $\langle \mathcal{S}\rangle$ in quantum simulations.

Figure 1

The average sign $\langle \mathcal{S}\rangle$ vs $\langle n\rangle$ at $U=8$ as a function of $g$ on an $8\times 8$ lattice for four values of $\beta$. As $\beta$ increases, the minimum in $\langle \mathcal{S}\rangle$ becomes wider and deeper, as expected, displaying a minimum around ${\langle n\rangle }_c\sim 0.85$. While the sign remains close to 1 for $g\to 0$ at $\beta \lesssim 3$, it starts reducing noticeably as $\beta$ is increased further, eventually forming a valley-peak-valley structure, as seen in the bottom panels. This suggests that, while the bulk of the sign problem at low temperature originates from the restricted configuration space available to the system, random configurations of the Hubbard-Stratonovich (HS) fields also result in a perceptible reduction in $\langle \mathcal{S}\rangle$ at low enough temperatures. The bottom panels, where the sign is the worst, show a sharp upturn near half-filling since the total sign is constrained to be equal to 1 at $\mu =0$ due to particle-hole symmetry.

Figure 2

Scaling of the total and spin-resolved sign with temperature and system size at $\mu =-3.0$, $U=8$ on an $8\times 8$ lattice. Left column shows $\ln (\langle \mathcal{S}\rangle )$ and $\ln \left(\langle {\mathcal{S}}_{\sigma }\rangle \right)$ vs the inverse temperature $\beta$ for a range of values of $g$. We find a faster-than-exponential reduction in the sign with decreasing temperature, both for the total and spin-resolved quantities. The inset in the top panel extends this to temperatures $\beta \sim 9$ for $g=0.0$ and $0.05$, demonstrating that even the randomly sampled case shows a faster-than-exponential scaling over a large temperature range. The right column plots the same quantities as a function of the lattice size $L^2$. The scaling in this case is exponential for all $g$ values.

Figure 3

Plots of the average sign $\langle \mathcal{S}\rangle$ (first row) and observables, including the density $\langle n\rangle$ (second row), kinetic energy $\langle \mathcal{K}\rangle$ (third row), and nearest neighbor spin-spin correlator $C_{zz}$ (fourth row), vs $\mu$ as a function of $g$ for three different values of $\beta$. These physical observables clearly demonstrate signatures of a reduced effective interaction (see text for more details).

Figure 4

The average sign $\langle \mathcal{S}\rangle$ vs $\langle n\rangle$ at $U=8$ on an $8\times 8$ lattice as a function of $\beta$ for four different $g$ values. Upper panels for $g<0.4$ demonstrate finite-sized oscillations at lower temperatures.

Figure 5

The average sign $\langle \mathcal{S}\rangle$ vs $g$ at $L=8$, $U=8$ as a function of $\beta$ for four different $\mu$ values.

Figure 6

The average sign $\langle \mathcal{S}\rangle$ vs $U$ at $\mu =0.2$, for the Hubbard model on a honeycomb lattice with $L=9$, with varying $U$ as a function of $g$, for four values of $\beta$. At $g=1$, $\langle \mathcal{S}\rangle$ shows a minimum around $U_c$. Reducing $g$ lowers the sign, shifting the minimum to the right (see text for details).

Figure 7

The average sign $\langle \mathcal{S}\rangle$ vs $U$ at half-filling $\mu =0$, as a function of $g$ on an $8\times 8$ lattice for four values of $\beta$ in the ionic Hubbard model. The staggered potential $\mathrm{\Delta }=1.0$. Results are qualitatively like the honeycomb model (see text).

Figure 8

The ratio $\langle \mathcal{S}\rangle (g=1)/\langle \mathcal{S}\rangle (g=0)$ for the square lattice Hubbard model (left), the Hubbard model on the honeycomb lattice (middle), and the ionic Hubbard model (right). The horizontal dashed black lines show $\langle {\mathcal{S}}_1\rangle /\langle {\mathcal{S}}_0\rangle =1$, providing a reference for the other plots.

Figure 9

The average sign $\langle S\rangle$ vs chemical potential $\mu$ at fixed $\beta =3$ as a function of reweight factor $g>1$ at $U=8$ on different lattice sizes $L=8,10$, and 12. The sign problem (SP) gets worse with increasing lattice size. Increasing $g$ makes the sign shallower (i.e., lessens the SP) and pushes the chemical potential at which the minimum $\langle \mathcal{S}\rangle$ occurs to lower values. With fixed lattice size $L=10$, lowering temperature results in smaller $\langle \mathcal{S}\rangle$.

Figure 10

Oversampling case: electron density $\langle n\rangle$, double occupancy $\langle n_{\uparrow }{n}_{\downarrow }\rangle$, and kinetic energy $\langle \mathcal{K}\rangle$ vs $\mu$ as a function of $g$ at $U=8$ on a $10\times 10$ square lattice. Bottom right panel displays the temperature dependence of $\langle \mathcal{K}\rangle$ at $g$ = $1.2$ (density and double occupancy have negligible lattice and temperature dependence in this regime).

Figure 11

Double occupancy $\langle n_{\uparrow }{n}_{\downarrow }\rangle$ (top) and kinetic energy $\langle \mathcal{K}\rangle$ (bottom) vs onsite interaction strength $U$ using different reweight factors $g$. Here, $L=10$ and $\beta =5$. The dashed horizontal line intersects the $g=1$ curve at $U=4$. To get the same value of $\langle n_{\uparrow }{n}_{\downarrow }\rangle$ for $g=0.7$ requires $U=5.1$, and for $g=1.2$ requires $U=3$. This implies $U_{\mathrm{eff}}=4$ for $g=1.2,U=3.0$ and for $g=0.7,U=5.1$. A similar analysis of $\langle \mathcal{K}\rangle$ implies that $U_{\mathrm{eff}}=4$ for $g=1.2$, $U=2.8$ and for $g=0.7$, $U=9.2$.

Figure 12

Effective onsite interaction $U_{\mathrm{eff}}$ determined from double occupancy $\langle n_{\uparrow }{n}_{\downarrow }\rangle$ (solid lines) and kinetic energy $\mathcal{K}$ (dotted lines) vs reweight factor $g$. Inverse temperature is fixed at $\beta =5$ (left panel) and $\beta =2.5$ (right panel), and data were extracted using an $L=10$ lattice. For $g>1$, the inferred values are roughly equal, while for small $g$, there is a marked disagreement. $U_{\mathrm{eff}}$ seems to be roughly temperature independent, except for small $g$ and large $U$.

Figure 13

The average sign of the matrices for individual spins, $\langle {\mathcal{S}}_{\sigma }\rangle$ vs $\langle n\rangle$ at $U=8$ as a function of $g$ on an $8\times 8$ lattice for four values of $\beta$. As seen for the total sign in the main text, the minimum in $\langle {\mathcal{S}}_{\sigma }\rangle$ becomes wider and deeper with increasing $\beta$. The nonmonotonic behavior of the sign with reducing $g$, where ${\mathcal{S}}_{\sigma }$ becomes worse initially and then gets progressively better, is also evident here. In contrast to the total sign, however, the spin-resolved sign is not constrained to be equal to 1 at half-filling, and thus, the sharp upturn seen in Fig. 1 near $\mu =0$ is absent here for most $g$ values.

Figure 14

Density $\langle n\rangle$ vs $\mu$ at $U=8$, $\beta =3.0$ on an $8\times 8$ lattice as a function of $g$ for four different values of the lattice size $L$. The oscillatory behavior at low $g$ in the $4\times 4$ lattice is due to the reemergence of the shell effect in small lattices. As explained in detail in the main text, a completely random sampling of the Hubbard-Stratonovich (HS) fields in quantum Monte Carlo (QMC) is more involved than a standard rescaling of $T$; in this case, pushing the system to weaker coupling by reducing the effect of $U$, resulting in a reappearance of finite-sized effects originally suppressed by the interaction.

Figure 15

$\langle \mathcal{S}\rangle$ vs $\langle n\rangle$ at $\beta =3$, $U=8$ as a function of $L$ for different $g$ values. The main observation is the finite-sized oscillations for $L=4$, with the sign reaching a maximum at the magic density $\langle n\rangle \sim 0.625$ associated with a closed-shell filling for this system size.

Figure 16

Left: $\langle \mathcal{S}\rangle$ vs $\mu$ at $g=0$, $U=8$, $L=8$ for different $\beta$ values. As $\beta$ is increased >4, $\langle \mathcal{S}\rangle$ shows strong oscillatory behavior reminiscent of shell effects. Right: values of $\langle n\rangle$ at the maxima of the sign. Dotted black lines mark the magic values of the density at this lattice size (see text).

Figure 17

The $t=0$ limit. Top left panel shows the number density $\langle n\rangle$ vs chemical potential $\mu$ at different $g$ values, showing the Mott plateau near $g=1$ disappearing as $g\to 0$. Top right panel plots $\langle n\rangle$ vs $\mu$ for $U=4.0$, $g=0$ at different $\beta$. The inset fits these curves to a Fermi function with ${\beta }_{\mathrm{eff}}$. Bottom left shows the same for fixed $\beta =3.0$, $g=0$ at different coupling $U$, with inset plotting the same at $g=1$. The final plot shows the double occupancy $\langle n_{\uparrow }{n}_{\downarrow }\rangle$ for different $g$ values along with the $U=0$ plot for comparison.