Symmetry in auxiliary-field quantum Monte Carlo calculations

We show how symmetry properties can be used to greatly increase the accuracy and efficiency in auxiliary-field quantum Monte Carlo (AFQMC) calculations of electronic systems. With the Hubbard model as an example, we study symmetry preservation in two aspects of ground-state AFQMC calculations, the Hubbard-Stratonovich transformation and the form of the trial wave function. It is shown that significant improvement over state-of-the-art calculations can be achieved. In unconstrained calculations, the implementation of symmetry often leads to shorter convergence time and much smaller statistical errors and thereby a substantial reduction of the sign problem. Moreover, certain excited states become possible to calculate which are otherwise beyond reach. In calculations with constraints, the use of symmetry can reduce the systematic error from the constraint. It also allows release-constraint calculations, leading to essentially exact results in many cases. Detailed comparisons are made with exact diagonalization results. Accurate ground-state energies are then presented for larger system sizes in the two-dimensional repulsive Hubbard model.

Figure 1

Statistical error bar (log scale) vs projection time for different HS transformations. FP calculations are shown. The error bars increase exponentially with projection time, but the optimal choice of the background $\overline{n}$ in Eq. (23) greatly reduces the fluctuation and improves efficiency. The system shown is a $4\times 4$ lattice with $N_{\uparrow }=3$, $N_{\downarrow }=3$, and $U=4$.

Figure 2

Illustration of the effect of preserving symmetry in the HS transformation: Hirsch spin [Eq. (18)] vs. Gaussian charge [Eqs. (22) and (23)]. (a) The energy from FP vs projection time, with the inset showing a magnified view of $\beta \in (3,3.5)$. The system is a $4\times 4$ lattice, with $N_{\uparrow }=N_{\downarrow }=8$ and $U=4$. A multideterminant trial wave function is chosen (Sec. 3b). The number of walkers was ${10}^4$, with a total of 10 separate runs for statistics. (b) The statistical error bar as a function of projection time in a semilog plot. The system is the $8\times 8$ Hubbard model, with $N_{\uparrow }=N_{\downarrow }=32$ and $U=8$. A Hartree-Fock trial wave function is chosen. The number of walkers was ${10}^5$, with a total of 20 separate runs to obtain the final averages and estimate the error bars.

Figure 3

Comparison of discrete spin and Gaussian charge decomposition in the presence of a sign/phase problem. The logarithm of the statistical error bars from FP is plotted vs projection time for a $4\times 4$ lattice with $N_{\uparrow }=N_{\downarrow }=7$ and $U=8$. A symmetry multideterminant trial wave function is used. The number of walkers is $5\times {10}^5$, with a total of 100 separate runs to estimate the error bars.

Figure 4

Total energy vs projection time in FP calculations using discrete spin and continuous charge decompositions. The system is an $8\times 8$ lattice under periodic boundary conditions, with $N_{\uparrow }=N_{\downarrow }=13$ and $U=4$. A FE trial wave function is used. The number of walkers is $5\times {10}^5$, with a total of 60 runs to obtain the average and estimate the statistical error bars. The red horizontal line gives the final CPMC result. The inset shows a magnified view of the last part of the projection. The spin decomposition calculation provides a reliable upper bound to the ground-state energy.

Figure 5

Effect of symmetry decomposition in release-constraint calculations. The main figure shows the convergence of CPMC energy with projection time compared with the exact ground-state energy. The CPMC calculation uses the discrete spin decomposition. The inset shows RC calculations, starting from a converged CP state, using two different forms of the HS decomposition. The system is the same as in Fig. 3: $4\times 4$, $N_{\uparrow }=N_{\downarrow }=7$, and $U=8$, using a symmetry multideterminant trial wave function. The number of walkers is $1\times {10}^5$, with a total of 100 runs in the RC portion to collect statistics.

Figure 6

The effect of symmetry trial wave functions in FP calculations compared to exact results in $4\times 4$ lattices with $U=4$: (a) $N_{\uparrow }=N_{\downarrow }=8$, no sign problem; (b) $N_{\uparrow }=N_{\downarrow }=7$, severe phase problem. The UHF trial wave function is generated by a UHF calculation with $U=0.5$, $0s0kx0ky1D$ is a single-determinant noninteracting trial wave function with $S^2=0$ and $k_x=k_y=0$, and $0s0kx0kyMD$ is a multideterminant trial wave function which has rotational symmetry in momentum space in (a) and $B_1$ symmetry in (b), in addition to $S^2=0,k_x=0,k_y=0$.

Figure 7

FP calculations for the ground state and three excited states. The energy is plotted vs projection time for a $4\times 4$ lattice with $N_{\uparrow }=N_{\downarrow }=3$ and $U=4$. The exact results for the ground state and the three excited state energies are shown for comparison. The symmetry of each energy level is labeled. The trial wave functions are chosen with the correct symmetry using a multideterminant. The error bars are shown but are smaller than symbol size at most points.

Figure 8

Accuracy of CPMC in the Hubbard model as a function of interaction strength. Results are shown for the $4\times 4$ lattice with $N_{\uparrow }=N_{\downarrow }=7$ as a function of $U$ and are compared with exact diagonalization. When the trial wave function preserves symmetry, the systematic bias in the calculated energy from the CP approximation is reduced.

Figure 9

The structure factor $S\left(K\right)$ of the spin-spin correlation function for three interaction strengths. The system is $4\times 4$ with $N_{\uparrow }=N_{\downarrow }=7$, and the horizontal axis labels of $K$ are in units of $\pi /2$. The symmetry trial wave function has $S^2=0$, $K_x=K_y=0$, and $B_1$ symmetry. CPMC has 10 000 walkers, with back propagation $\beta =1$.

Figure 10

RC calculations with and without symmetry trial wave functions. The system is $3\times 3$ with $N_{\uparrow }=N_{\downarrow }=2$ and $U=4$. The symmetry trial wave function has $S^2=2,K_x=0,K_y=0$, while the UHF wave function breaks these symmetries. CP/UHF is very accurate, but RC/UHF has nonmonotonic behavior and slow convergence, as shown by the explicit propagation. RC/SYM converges rapidly and monotonically. The explicit propagation (EP) result of RC/UHF is shown to large projection time in the inset.

Figure 11

RC and CP results for ground and the first excited state energies vs crystal momentum. RC greatly improves the calculation of excited states and band structures. The system is $4\times 4$ with $N_{\uparrow }=N_{\downarrow }=5$ and $U=4$. QMC statistical error bars are smaller than symbol size. The horizontal axis gives $\left|k\right|$ along a line cut $k_y=2k_x$. Exact diagonalization (ED) results are shown for comparison. The line is to aid the eye.