Coupling quantum Monte Carlo and independent-particle calculations: Self-consistent constraint for the sign problem based on the density or the density matrix

Quantum Monte Carlo (QMC) methods are one of the most important tools for studying interacting quantum many-body systems. The vast majority of QMC calculations in interacting fermion systems require a constraint to control the sign problem. The constraint involves an input trial wave function which restricts the random walks. We introduce a systematically improvable constraint which relies on the fundamental role of the density or one-body density matrix. An independent-particle calculation is coupled to an auxiliary-field QMC calculation. The independent-particle solution is used as the constraint in QMC, which then produces the input density or density matrix for the next iteration. The constraint is optimized by the self-consistency between the many-body and the independent-particle calculations. The approach is demonstrated in the two-dimensional Hubbard model by accurately determining the ground state when collective modes separated by tiny energy scales are present in the magnetic and charge correlations. Our approach also provides an ab initio way to predict effective interaction parameters for independent-particle calculations.

Figure 1

Systematic improvement of the CPMC accuracy from the self-consistent procedure. The top panel plots staggered spin density along the $y$ direction vs site label $i_y(i_x=1)$. The bottom panel plots the corresponding hole density. The left and right columns show two self-consistent procedures starting from two different initial $|{\mathrm{\Psi }}_T\rangle$'s, the free-electron wave function and the UHF solution at $U=8t$, respectively. In the legend, the $U_{\mathrm{eff}}$ value of the IP calculation is listed for each iteration step. In (a) and (b), the differences at different stages of the iteration with respect to the reference DMRG results are shown in the insets. The system is $4\times 16$ with $U=8t,h=1/8$ doping with a pinning field applied to both edges in the $y$ direction.

Figure 2

Convergence of CPMC results in the self-consistent procedure. Three initial trial wave-functions are tested: free-electron and UHF solutions at $U=4t$ and $8t$. The system is the same as in Fig. 1. In each panel, the pink band represents the converged result and error bar. In (a), the relative error in the computed ground-state energy relative to the DMRG is shown vs the self-consistency iteration. The horizontal dotted line at zero is to aid the eye. Plotted in (b) are the mean-square error [following the definition in Eq. (3)] in the local density computed by the CPMC from the reference DMRG results.

Figure 3

Finding the optimal $U_{\mathrm{eff}}$ for the IP calculations during the iterations. The staggered spin density from a UHF calculation with $U_{\mathrm{eff}}$ is shown during each iteration of the self-consistent procedure with the CPMC. The self-consistency procedure begins with the free-electron $|{\mathrm{\Psi }}_T\rangle$ in panel (a) and with the $U=8t$ UHF solution in (b). In the lower panel, the left, middle, and right depict the spin density of the free-electron wave function, the correct ground state from the converged CPMC result, and the UHF solution with $U=8t$, respectively. The system is the same as in Fig. 1.

Figure 4

Converged CPMC results after a self-consistent procedure for a large system of $16\times 32$. In the upper panel, the staggered spin and hole densities are plotted. The red and blue horizontal lines represent zero spin density and the average hole density, respectively. In the lower pane, the spin density for the cell is shown with a color map. As in the earlier systems, $U=8t,h=1/8$, and a pinning field is applied to both edges along $L_y$.