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Journal of Chemical Theory and Computation

Scalable Implementation of Mean-Field and Correlation Methods Based on Lie-Algebraic Similarity Transformation of Spin Hamiltonians in the Jordan–Wigner Representation

Shadan Ghassemi Tabrizi, Thomas M. Henderson, Thomas D. Kühne, Gustavo E. Scuseria

Published Oct 31, 2025• 2 citations
Publisher

Abstract

High Resolution Image Download MS PowerPoint Slide Recent work has highlighted that the strong correlation inherent in spin Hamiltonians can be effectively reduced by mapping spins to Fermions via the Jordan–Wigner transformation (JW). The Hartree–Fock method is straightforward in the Fermionic domain and may provide a reasonable approximation to the ground state. Correlation with respect to the Fermionic mean field can be recovered based on Lie-algebraic similarity transformation (LAST) with two-body correlators. Specifically, a unitary LAST variant eliminates the dependence on site ordering, while a nonunitary LAST yields size-extensive correlation energies. Whereas the first recent demonstration of such methods was restricted to small spin systems, we present efficient implementations using analytical gradients for the optimization with respect to the mean-field reference and the LAST parameters, thereby enabling the treatment of larger clusters, including systems with local spins s > 1 2 .

DOI
10.1021/acs.jctc.5c01412

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High Resolution Image Download MS PowerPoint Slide Recent work has highlighted that the strong correlation inherent in spin Hamiltonians can be effectively reduced by mapping spins to Fermions via the Jordan–Wigner transformation (JW). The Hartree–Fock method is straightforward in the Fermionic domain and may provide a reasonable approximation to the ground state. Correlation with respect to the Fermionic mean field can be recovered based on Lie-algebraic similarity transformation (LAST) with two-body correlators. Specifically, a unitary LAST variant eliminates the dependence on site ordering, while a nonunitary LAST yields size-extensive correlation energies. Whereas the first recent demonstration of such methods was restricted to small spin systems, we present efficient implementations using analytical gradients for the optimization with respect to the mean-field reference and the LAST parameters, thereby enabling the treatment of larger clusters, including systems with local spins s > 1 2 .
High Resolution Image Download MS PowerPoint Slide Mapping spins to fermions via the Jordan–Wigner (JW) transformation can render mean-field (Hartree–Fock, HF) descriptions effective for strongly correlated spin systems. As established in recent work, the application of such approaches is not limited by the nonlocal structure of JW strings or by site ordering because string operators can be absorbed into Thouless rotations of a Slater determinant, and the variational optimization of a unitary Lie-algebraic similarity transformation removes any ordering dependence. Leveraging these ideas, we develop a self-consistent field (SCF) scheme that expresses the mean-field energy as a functional of the single-particle density matrix, providing an alternative to gradient-based optimization of Thouless parameters. We derive the analytical orbital Hessian to diagnose HF stability and compute the ground-state correlation energy through the random-phase approximation (RPA). Benchmark results for the XXZ and J 1 – J 2 model on one- and two-dimensional lattices demonstrate that RPA significantly improves mean-field accuracy.
Mapping spins to fermions via the Jordan-Wigner (JW) transformation can render mean-field (Hartree-Fock, HF) descriptions effective for strongly correlated spin systems. As established in recent work, the application of such approaches is not limited by the nonlocal structure of JW strings or by site ordering, because string operators can be absorbed into Thouless rotations of a Slater determinant, and the variational optimization of a unitary Lie-Algebraic similarity transformation removes any ordering dependence. Leveraging these ideas, we develop a self-consistent field (SCF) scheme that expresses the mean-field energy as a functional of the single-particle density matrix, providing an alternative to gradient-based optimization of Thouless parameters. We derive the analytic orbital Hessian to diagnose HF stability and compute ground-state correlation energy through the random-phase approximation (RPA). Benchmark results for the XXZ and J1-J2 model on one- and two-dimensional lattices demonstrate that RPA significantly improves mean-field accuracy.
The Jordan–Wigner transformation establishes a duality between su(2) and fermionic algebras. We present qualitative arguments and numerical evidence that when mapping spins to fermions, the transformation makes strong correlation weaker, as demonstrated by the Hartree–Fock approximation to the transformed Hamiltonian. This result can be rationalized in terms of rank reduction of spin shift terms when transformed to fermions. Conversely, the mapping of fermions to qubits makes strong correlation stronger, complicating its solution when one uses qubit-based correlators. The presence of string operators poses challenges to the implementation of quantum chemistry methods on classical computers, but these can be dealt with using established techniques of low computational cost. Our proof of principle results for XXZ and J1-J2 Heisenberg (in 1D and 2D) indicates that the JW transformed fermionic Hamiltonian has reduced complexity in key regions of their phase diagrams and provides a better starting point for addressing challenging spin problems.
The Jordan–Wigner transformation is a powerful tool for converting systems of spins into systems of fermions, or vice versa. While this mapping is exact, the transformation itself depends on the labelling of the spins. One consequence of this dependence is that approximate solutions of a Jordan–Wigner-transformed Hamiltonian may depend on the (physically inconsequential) labelling of the spins. In this work, we turn to an extended Jordan–Wigner transformation which remedies this problem and which may also introduce some correlation atop the Hartree–Fock solution of a transformed spin Hamiltonian. We demonstrate that this extended Jordan–Wigner transformation can be thought of as arising from a unitary version of the Lie algebraic similarity transformation (LAST) theory. We show how these ideas, particularly in combination with the standard (non-unitary) version of LAST, can provide a potentially powerful tool for the treatment of the XXZ and J1−J2 Heisenberg Hamiltonians.
We present a class of correlated wavefunctions generated by exponentials of two-body on-site Hermitian operators that can be evaluated with polynomial computational cost via a Hamiltonian similarity transformation. Wavefunctions of this form have been studied with variational Monte Carlo methods, but we present a formalism to perform non-stochastic calculations. The Hausdorff series generated by these Jastrow factors can be summed exactly without truncation resulting in a set of equations with polynomial computational cost. The correlators include the density-density, collinear spin-spin, spin-density cross terms, and on-site double occupancy operators. The resulting non-Hermitian many-body Hamiltonian can be solved in a biorthogonal mean-field approach with only a small set of correlation terms required for accurate calculations in systems with local interactions. Although the energy of the model is unbound, projective equations in the spirit of coupled cluster theory lead to well-defined solutions. The theory is tested on the one and two-dimensional repulsive Hubbard model where it yields accurate results for large systems with low computational cost. Symmetry projection methods are included to further improve the reference wavefunction and results under strong correlation without sacrificing good quantum numbers resulting in very accurate energies for small systems and producing a better ground state for the calculation of other properties.
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