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arXiv:2601.13078

SCF framework, HF stability and RPA correlation for Jordan-Wigner-transformed spin Hamiltonians on arbitrary coupling topologies

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

Published Jan 19, 2026• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2601.13078

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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.
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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 .
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.
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 ground-state energy and static and dynamic correlation functions are investigated in the inhomogeneous-Hartree-Fock (IHF) plus random-phase-approximation (RPA) approach applied to a one-dimensional spinless-fermion model showing self-trapped doping states at the mean-field level. Results are compared with those obtained using homogeneous HF and exact diagonalization. RPA fluctuations added to the generally inhomogeneous-HF ground state allow the computation of dynamical correlation functions that compare well with exact diagonalization results. The RPA correction to the ground-state energy agrees well with the exact results at strong- and weak-coupling limits. We also compare it with a related quasiboson approach. The instability towards self-trapped behavior is signaled by a RPA mode with frequency approaching zero.