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arXiv:2504.08130 · The Journal of Chemical Physics

Symmetry-projected spin-AGP methods applied to spin systems

Z.L. Liu, Thomas M. Henderson, Gustavo E. Scuseria

Published Jul 10, 2025• 0 citations
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Abstract

Symmetry-projected wave function methods capture static correlation by breaking and restoring the symmetries of a system. In this article, we present the symmetry-projected spin antisymmetrized geminal power (spin-AGP) state projected onto space group symmetry as well as complex conjugation, spin-flip, and time-reversal symmetries. The method is benchmarked on the 1D XXZ model and the 2D J1 - J2 model with square and triangular lattices. Our results indicate that symmetry projection methods provide a powerful tool for frustrated spin systems.

DOI
10.1063/5.0274797

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Symmetry-projected wave function methods capture static correlation by breaking and restoring the symmetries of a system. In this article, we present the symmetry-projected spin antisymmetrized geminal power (spin-AGP) state projected onto space group symmetry as well as complex conjugation, spin-flip, and time-reversal symmetries. The method is benchmarked on the 1D XXZ model and the 2D J1 - J2 model with square and triangular lattices. Our results indicate that symmetry projection methods provide a powerful tool for frustrated spin systems.
The antisymmetrized geminal power (AGP) wavefunction has proven to be an excellent reference for strong interactions in paired electron systems. In this thesis, we extend the application of AGP methods to spin systems by utilizing the spin to pairing mapping, owing to the su(2) algebra present in both paired electron and spin systems. We have investigated the spin AGP method applied to the XXZ model and J1 − J2 model, where we have identified what we refer to as the multimodal extreme AGP as the mean-field optimized solution. We have also developed correlated methods such as configuration interaction (CI) and coupled cluster (CC) atop spin AGP for these systems. In addition, we have shown that space group symmetry plays a significant role in 2D spin lattices, and have developed space group symmetry projection methods for spin AGP. While the spin AGP method associates a spin with a pair of fermions, Jordan- Wigner Hartree–Fock (JWHF) maps a spin into a single fermion and uses a string to bridge between the spin su(2) and the fermion anticommutation algebras. We have tested JWHF on various spin systems and have additionally discussed the potential application of space group symmetry projection methods to JWHF
The antisymmetrized geminal power (AGP), a wave function equivalent to number-projected Hartree-Fock-Bogoliubov (HFB), and number-projected Bardeen-Cooper-Schrieffer (BCS) when working in the paired (natural orbitals) basis, has proven to be an excellent reference for strong pairing interactions. Several correlation methods have also been applied on top of AGP. In this work, we show how AGP can also be applied to spin systems by simply basing its formulation on a spin su(2)su(2)su(2) algebra. We here implement spin AGP and spin AGP-based correlation techniques and benchmark them on the XXZ and ${\mathrm{J}}_{1}\text{\ensuremath{-}}{\mathrm{J}}_{2}$ Heisenberg models, both in one and two dimensions. Our results indicate that spin AGP is a promising starting point for modeling spin systems.
We introduce Sz spin-projection based on cluster mean-field theory and apply it to the ground state of strongly correlated spin systems. In cluster mean-fields, the ground state wavefunction is written as a factorized tensor product of optimized cluster states. In previous work, we have focused on unrestricted cluster mean-field, where each cluster is Sz symmetry adapted. We here remove this restriction by introducing a generalized cluster mean-field (GcMF) theory, where each cluster is allowed to access all Sz sectors, breaking Sz symmetry. In addition, a projection scheme is used to restore global Sz, which gives rise to the Sz spin-projected generalized cluster mean-field (SzGcMF). Both of these extensions contribute to accounting for inter-cluster correlations. We benchmark these methods on the 1D, quasi-2D, and 2D J1 - J2 and XXZ Heisenberg models. Our results indicate that the new methods (GcMF and SzGcMF) provide a qualitative and semi-quantitative description of the Heisenberg lattices in the regimes considered, suggesting them as useful references for further inter-cluster correlations, which are discussed in this work.
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.
Abstract Electronic structure methods typically benefit from symmetry breaking and restoration, specially in the strong correlation regime. The same goes for ansätze on a quantum computer. We develop a unitary coupled cluster method based on the antisymmetrized geminal power (AGP)—a state formally equivalent to the number-projected Bardeen–Cooper–Schrieffer wavefunction. We demonstrate our method for the single-band Fermi–Hubbard Hamiltonian in one and two dimensions. We also explore post-selection as a state preparation step to obtain correlated AGP and prove that it scales no worse than O ( M ) in the number of measurements, thereby making it a less expensive alternative to gauge integration to restore particle number symmetry.
We present a general review of the projective symmetry group classification of fermionic quantum spin liquids for lattice models of spin S=1/2S=1/2S=1/2. We then introduce a systematic generalization of the approach for symmetric Z2{\mathbb{Z}}_{2}Z2​ quantum spin liquids to the one of chiral phases (i.e., singlet states that break time reversal and lattice reflection, but conserve their product). We apply this framework to classify and discuss possible chiral spin liquids on triangular and kagome lattices. We give a detailed prescription on how to construct quadratic spinon Hamiltonians and microscopic wave functions for each representation class on these lattices. Among the chiral Z2{\mathbb{Z}}_{2}Z2​ states, we study the subset of U(1) phases variationally in the antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{d}$ Heisenberg model on the kagome lattice. We discuss static spin structure factors and symmetry constraints on the bulk spectra of these phases.