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arXiv:1402.0018 · Physical Review B

Symmetry-projected wave functions in quantum Monte Carlo calculations

Hao Shi, Carlos A. Jiménez-Hoyos, R. R. Rodríguez-Guzmán, Gustavo E. Scuseria, Shiwei Zhang

Published Mar 31, 2014• 60 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.89.125129

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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.
We benchmark constrained-path Monte Carlo (CPMC) on the triangular-lattice Hubbard model for several fillings and UUU values and show that symmetry-adapted trial wave functions substantially improve quantitative accuracy. Away from half-filling, simple free-electron-based trials that preserve the ground state symmetry yield energy deviations ≲1%\lesssim 1\%≲1% from exact diagonalization and density matrix renormalization group results. At half-filling, strong frustration in the intermediate to large UUU regimes necessitates symmetry-projected trials to reach comparable accuracy, where both free-electron and symmetry-broken Hartree-Fock trials incur substantial constraint bias. Since the computational cost of CPMC with symmetry projection scales polynomially with system size, our results motivate its use as a practical route for studying competing ground states in strongly correlated, frustrated systems.
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.
I will present our recent results on the calculation of symmetry-projected wave functions [1,2] for electronic structure theory. For Hartree-Fock and spin projection, this is a 50-year old problem in quantum chemistry, going all the way back to Lowdin and his “extended HF” theory. For Hartree-Fock-Bogoliubov and number projection, our approach offers new perspectives on Antisymmetrized Geminal Power (AGP) wavefunctions that were the focus of much attention in the 1980s. In our work, all molecular symmetries (electron number, spin S 2 and Sz , point group, and complex conjugation) are deliberately broken and restored in a self-consistent variation-after-projection approach. The resulting method yields a comprehensive black-box treatment of static correlation with one-electron (mean-field) computational cost. The ensuing wave function is of high quality multireference character competitive with exact diagonalization. The method can also be applied to calculate excited states and spectral functions [3]. The curse of the thermodynamic limit and the quest for a low-cost treatment of residual correlations will also be addressed.
We introduce an efficient and numerically stable technique to make use of a BCS trial wave function in the computation of correlation functions of strongly correlated quantum fermion systems. The technique is applicable to any projection approach involving paths of independent-fermion propagators, for example, in mean-field or auxiliary-field quantum Monte Carlo (AFQMC) calculations. Within AFQMC, in the absence of the sign problem, the methodology allows the use of a BCS reference state which can greatly reduce the required imaginary time of projection and improves Monte Carlo sampling efficiency and statistical accuracy for systems where pairing correlations are important. When the sign problem is present, the approach provides a powerful generalization of the constrained-path AFQMC technique which usually uses Slater determinant trial wave functions. As a demonstration of the capability of the methodology, we present benchmark results for the attractive Hubbard model, both spin balanced (no sign problem) and with a finite spin polarization (with sign problem).
Generalized Hartree Fock (GHF) is a long-established electronic structure method that can lower the energy (compared to spin-restricted variants) by breaking physical wavefunction symmetries, namely S^2 and Sz. After an exposition of GHF theory, we assess the use of GHF trial wavefunctions in phaseless auxiliary field quantum Monte Carlo (ph-AFQMC-G) calculations of strongly correlated molecular systems including symmetrically stretched hydrogen rings, carbon dioxide, and dioxygen. We find comparable or better accuracy than CCSD(T) with unrestricted HF and GHF references, and consistently smooth dissociation curves -- a remarkable result given the relative scalability of ph-AFQMC-G to larger system sizes. The present exploration of model strongly correlated systems marks a promising starting point for future studies of more chemically-relevant molecules, and demonstrates that ph-AFQMC-G provides a highly accurate (and, in contrast to active-space-based trials, relatively black box and always size-consistent) description of challenging systems exhibiting, e.g., antiferromagnetic coupling and/or geometric spin frustration.
The quantum Monte Carlo algorithm is arguably one of the most powerful computational many-body methods, enabling accurate calculation of many properties in interacting quantum systems. In the presence of the so-called sign problem, the algorithm typically relies on trial wave functions to eliminate the exponential decay of signal-to-noise ratio, usually at the expense of a bias in the result. The quality of the trial state therefore is critical for accurate simulations. In this work, benchmark results of the ground state auxiliary-field quantum Monte Carlo method are reported for the Hubbard model on several geometries. We demonstrate that when multi-determinant generalized Hartree-Fock states are used as trial wave functions, the systematic errors can be systematically reduced to a low level and the results compare favorably with the exact diagonalization data.