Symmetry-projected wave functions in quantum Monte Carlo calculations

We consider symmetry-projected Hartree-Fock trial wave functions in constrained-path Monte Carlo (CPMC) calculations. Previous CPMC calculations have mostly employed Hartree-Fock (HF) trial wave functions, restricted or unrestricted. The symmetry-projected HF approach results in a hierarchy of wave functions with increasing quality: the more symmetries that are broken and restored in a self-consistent manner, the higher the quality of the trial wave function. This hierarchy is approximately maintained in CPMC calculations: the accuracy in the energy increases and the statistical variance decreases when further symmetries are broken and restored. Significant improvement is achieved in CPMC with the best symmetry-projected trial wave functions over those from simple HF. We analyze and quantify the behavior using the two-dimensional repulsive Hubbard model as an example. In the sign-problem-free region, where CPMC can be made exact but a constraint is deliberately imposed here, spin-projected wave functions remove the constraint bias. Away from half filling, spatial symmetry restoration in addition to that of the spin leads to highly accurate results from CPMC. Since the computational cost of symmetry-projected HF trial wave functions in CPMC can be made to scale algebraically with system size, this provides a potentially general approach for accurate calculations in many-fermion systems.

Figure 1

Performance of different symmetry-projected trial wave functions in CPMC. (a) The statistical uncertainty of the computed ground-state energy in a semilog plot (arbitrary scale). (b) The percentage error, $(E_{\mathrm{CP}}-E_0)/\left|E_0\right|$, between the CPMC energy and the exact energy. The system is the $4\times 4$ Hubbard model, with $N_{\uparrow }=N_{\downarrow }=8$ and $U=4$. The number of walkers was 1000, an equilibrium phase of ${\beta }_{\mathrm{eq}}=4$ was discarded, and $\Delta \tau =0.01$ was used in the runs.

Figure 2

Size dependence of the computed energies in the half-filled Hubbard model at $U=4$. The behaviors of two different forms of trial wave functions are compared. The variational energies of the trial wave functions are shown in the upper curves. The dashed lines are polynomial fits to the data. CPMC results using the two trial wave functions as (artificial) constraints are shown in the lower curves, together with exact free-projection results. (Note the different vertical scales between the upper and lower curves.) Our best estimate of the ground-state energy in the thermodynamic limit is given by the horizontal lines. Its statistical uncertainty is indicated by the width of the line in the inset.

Figure 3

(a) Variational energies of the UHF and three UHF-based symmetry-projected wave functions, and (b) the corresponding CPMC energies using these as trial wave functions. The energies are shown in terms of relative errors in the correlation energy (see text). Note the different scales between (a) and (b). Four systems are studied, each represented by a different color/pattern of vertical bar. The CPMC results were obtained with a population size of 1000 and 10 independent blocks of size $\beta \sim 1$ each. The statistical error bars are indicated. Trotter error is negligible.

Figure 4

Illustration of GHF-based symmetry-projected trial wave functions in CPMC. The relative errors in the correlation energy from CPMC using GHF-type wave functions (SG,S-GHF) are compared with the corresponding UHF-type (SG,S-UHF). The latter are taken from the rightmost group in Fig. 3. The same computational parameters are used for the two types of calculations.

Figure 5

Momentum distribution $n\left(\mathbf{k}\right)$ vs $\left|\mathbf{k}\right|$. Results from CPMC using different trial wave functions are compared with each other and with exact diagonalization (ED). The inset shows the error from ED results. The mixed estimator is used instead of back propagation to magnify the effect of the trial wave function. The system is $4\times 4$ with $7\uparrow ,7\downarrow$ and $U=8$. The same run parameters are used as in Fig. 3.