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Bulletin of the American Physical Society

Symmetry-projected Hartree-Fock wave functions in quantum Monte Carlo calculations

Hao Shi, Carlos Hoyos, Rayner Rodrı́guez, Gustavo E. Scuseria, Shiwei Zhang

Published Mar 6, 2014• 0 citations
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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.

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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.
Two methods are used, Hartree-Fock and a variational wave function, to investigate spiral states in the three-band Hubbard model. Our variational wave function is a Gutzwiller-projected Hartree-Fock spiral state. We use quantum Monte Carlo methods to calculate the variational energy exactly. In order to avoid boundary defects in the spiral spin structure, we work in a frame where the spin-quantization axis is locally rotated. Both methods show that upon doping the antiferromagnetic structure present at half-band-filling is unstable against a (1,1) spiral state. For moderate (large) values of the Cu on-site Hubbard repulsion, there is a transition to a (1,0) spiral (ferromagnetic) state. The Hartree-Fock calculations show that most of the (1,1) spiral phase is unstable. On the other hand, the (1,0) spiral state is stable.
The use of an approximate reference state wave function mid R:Phi(r) in electronic many-body methods can break the spin symmetry of Born-Oppenheimer spin-independent Hamiltonians. This can result in significant errors, especially when bonds are stretched or broken. A simple spin-projection method is introduced for auxiliary-field quantum Monte Carlo (AFQMC) calculations, which yields spin-contamination-free results, even with a spin-contaminated mid R:Phi(r). The method is applied to the difficult F(2) molecule, which is unbound within unrestricted Hartree-Fock (UHF). With a UHF mid R:Phi(r), spin contamination causes large systematic errors and long equilibration times in AFQMC in the intermediate, bond-breaking region. The spin-projection method eliminates these problems and delivers an accurate potential energy curve from equilibrium to the dissociation limit using the UHF mid R:Phi(r). Realistic potential energy curves are obtained with a cc-pVQZ basis. The calculated spectroscopic constants are in excellent agreement with experiment.