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arXiv:2607.21929

Phaseless auxiliary-field quantum Monte Carlo in the Hartree-Fock-Bogoliubov manifold

Zhaozhan Zhang

Published Jul 24, 2026• 0 citations
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Abstract

We formulate and implement the phaseless auxiliary-field quantum Monte Carlo (AFQMC) method in the Hartree-Fock-Bogoliubov (HFB) manifold for systems with strong pairing correlations. Our formulation introduces a more flexible representation of HFB walkers with an explicit analytical expression for the normalization factor derived from the representation matrix with respect to the reference vacuum. We also extend our previously developed stochastic gauge formalism from the Slater determinant manifold to the HFB framework, providing additional flexibility for controlling stochastic fluctuations. Benchmark calculations for the Richardson model demonstrate the accuracy and numerical stability of the method.

DOI
10.48550/arxiv.2607.21929

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We formulate and implement the phaseless auxiliary-field quantum Monte Carlo (AFQMC) method in the Hartree-Fock-Bogoliubov (HFB) manifold for systems with strong pairing correlations. Our formulation introduces a more flexible representation of HFB walkers with an explicit analytical expression for the normalization factor derived from the representation matrix with respect to the reference vacuum. We also extend our previously developed stochastic gauge formalism from the Slater determinant manifold to the HFB framework, providing additional flexibility for controlling stochastic fluctuations. Benchmark calculations for the Richardson model demonstrate the accuracy and numerical stability of the method.
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We explore the possibility of implementing random walks in the manifold of Hartree-Fock-Bogoliubov wave functions. The goal is to extend state-of-the-art quantum Monte Carlo approaches, in particular the constrained-path auxiliary-field quantum Monte Carlo technique, to systems where finite pairing order parameters or complex pairing mechanisms, e.g., Fulde-Ferrell-Larkin-Ovchinnikov pairing or triplet pairing, may be expected. Leveraging the flexibility to define a vacuum state tailored to the physical problem, we discuss a method to use imaginary-time evolution of Hartree-Fock-Bogoliubov states to compute ground state correlations, extending beyond situations spanned by current formalisms. Illustrative examples are provided.
Abstract The so-called phaseless quantum Monte-Carlo method currently offers one of the best performing theoretical framework to investigate interacting Fermi systems. It allows to extract an approximate ground-state wavefunction by averaging independent-particle states undergoing a Brownian motion in imaginary-time. Here, we extend the approach to a random walk in the space of Hartree–Fock–Bogoliubov (HFB) vacua that are better suited for superconducting or superfluid systems. Well-controlled statistical errors are ensured by constraining stochastic paths with the help of a trial wavefunction. It also guides the dynamics and takes the form of a linear combination of HFB ansätze. Estimates for the observables are reconstructed through an extension of Wick’s theorem to matrix elements between HFB product states. The usual combinatory complexity associated to the application of this theorem for four- and more- body operators is bypassed with a compact expression in terms of Pfaffians. The limiting case of a stochastic motion within Slater determinants but guided with HFB trial wavefunctions is also considered. Finally, exploratory results for the spin-polarized Hubbard model in the attractive regime are presented.
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.
High Resolution Image Download MS PowerPoint Slide We report a scalable Fortran implementation of the phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) and demonstrate its excellent performance and beneficial scaling with respect to system size. Furthermore, we investigate modifications of the phaseless approximation that can help to reduce the overcorrelation problems common to the ph-AFQMC. We apply the method to the 26 molecules in the HEAT set, the benzene molecule, and water clusters. We observe a mean absolute deviation of the total energy of 1.15 kcal/mol for the molecules in the HEAT set, close to chemical accuracy. For the benzene molecule, the modified algorithm despite using a single-Slater-determinant trial wavefunction yields the same accuracy as the original phaseless scheme with 400 Slater determinants. Despite these improvements, we find systematic errors for the CN, CO 2, and O 2 molecules that need to be addressed with more accurate trial wavefunctions. For water clusters, we find that the ph-AFQMC yields excellent binding energies that differ from CCSD(T) by typically less than 0.5 kcal/mol.
We describe the computational ingredients for an approach to treat interacting fermion systems with pairing fields, based on path integrals in the space of Hartree-Fock-Bogoliubov (HFB) wave functions. The path integrals can be evaluated by Monte Carlo, via random walks of HFB wave functions whose orbitals evolve stochastically. The approach combines the advantage of HFB theory in paired fermion systems and many-body quantum Monte Carlo techniques. The properties of HFB states, written in the form of either product states or Thouless states, are discussed. The states preserve forms when propagated by generalized one-body operators. They can be stabilized for numerical iteration. Overlaps and one-body Green's functions between two such states can be computed. A constrained-path or phaseless approximation can be applied to the random walks of the HFB states if a sign problem or phase problem is present. The method is illustrated with an exact numerical projection in the Kitaev model, and in the Hubbard model with attractive interaction under an external pairing field.