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arXiv:2609.34128

Bogoliubov coupled-cluster theory for su(2) Hamiltonians

Swarnamoy Ghosh, Thomas M. Henderson, Gustavo E. Scuseria

Published Sep 28, 2026• 0 citations

Abstract

Coupled-cluster theory is the method of choice for weakly correlated systems, but in strongly correlated systems where the mean-field reference is qualitatively poor, it can break down badly. These failures can be ameliorated by using symmetry-broken mean-field references. For systems that spontaneously break spin symmetry, we might favor an unrestricted reference. In systems that instead break number symmetry, we turn to Bogoliubov coupled cluster theory. In this work, we apply Bogoliubov coupled-cluster to Hamiltonians with an underlying su(2) algebra to study the pairing Hamiltonian and the spin XXZ and J1-J2 models by exploiting the correspondence between fermionic-pair operators and spin-1/2 operators. For the pairing problem, the reference is the usual Bardeen--Cooper--Schrieffer (BCS) quasiparticle vacuum, while for spin systems we use the analogous spin-BCS reference, which breaks Sz symmetry and provides a flexible starting point for strongly correlated regimes. Correlation is incorporated through a hierarchy of coupled-cluster approximations. Beyond energies, we compute the response density matrices for the evaluation of spin--spin correlation functions for the XXZ and J1-J2 Heisenberg models, and the pairing parameter and particle-number variance for the pairing Hamiltonian. We also derive the corresponding relaxed density matrices, including the effects of orbital relaxation. Benchmark calculations for the half-filled pairing Hamiltonian and spin models demonstrate that Bogoliubov coupled cluster provides a systematically improvable wave-function-based approach for su(2) Hamiltonians, including geometrically frustrated spin systems requiring complex and noncollinear BCS references.

DOI: 10.48550/arxiv.2609.34128

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Coupled-cluster theory is the method of choice for weakly correlated systems, but in strongly correlated systems where the mean-field reference is qualitatively poor, it can break down badly. These failures can be ameliorated by using symmetry-broken mean-field references. For systems that spontaneously break spin symmetry, we might favor an unrestricted reference. In systems that instead break number symmetry, we turn to Bogoliubov coupled cluster theory. In this work, we apply Bogoliubov coupled-cluster to Hamiltonians with an underlying su(2) algebra to study the pairing Hamiltonian and the spin XXZ and J1-J2 models by exploiting the correspondence between fermionic-pair operators and spin-1/2 operators. For the pairing problem, the reference is the usual Bardeen--Cooper--Schrieffer (BCS) quasiparticle vacuum, while for spin systems we use the analogous spin-BCS reference, which breaks Sz symmetry and provides a flexible starting point for strongly correlated regimes. Correlation is incorporated through a hierarchy of coupled-cluster approximations. Beyond energies, we compute the response density matrices for the evaluation of spin--spin correlation functions for the XXZ and J1-J2 Heisenberg models, and the pairing parameter and particle-number variance for the pairing Hamiltonian. We also derive the corresponding relaxed density matrices, including the effects of orbital relaxation. Benchmark calculations for the half-filled pairing Hamiltonian and spin models demonstrate that Bogoliubov coupled cluster provides a systematically improvable wave-function-based approach for su(2) Hamiltonians, including geometrically frustrated spin systems requiring complex and noncollinear BCS references.

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The ground-state pairing correlations in finite fermionic systems are described with a high degree of accuracy within a variational approach based on a combined coupled-cluster and particle-number-projected BCS ansatz. The flexibility of this symmetry-preserving wave function enables a unified picture valid from weak to strong coupling, both in small and large systems. The present variational approach consistently yields an energy upper bound while operating at the same level of precision as the nonvariational particle-number-projected Bogoliubov-coupled-cluster theory [Phys. Rev. C 99, 044301 (2019)].

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.

We introduce perturbation and coupled-cluster theories based on a cluster mean-field reference for describing the ground state of strongly correlated spin systems. In cluster mean-field, the ground state wave function is written as a simple tensor product of optimized cluster states. The cluster language and the mean-field nature of the ansatz allow for a straightforward improvement which uses perturbation theory and coupled-cluster to account for intercluster correlations. We present benchmark calculations on the 1D chain and 2D square J 1 – J 2 Heisenberg model, using cluster mean-field, perturbation theory, and coupled-cluster. We also present an extrapolation scheme that allows us to compute thermodynamic limit energies accurately. Our results indicate that, with sufficiently large clusters, the correlated methods (cPT2, cPT4, and cCCSD) can provide a relatively accurate description of the Heisenberg model in the regimes considered, which suggests that the methods presented can be used for other strongly correlated systems. Some ways to improve upon the methods presented in this work are discussed.

We introduce cluster-based mean-field, perturbation and coupled-cluster theories to describe the ground state of strongly-correlated spin systems. In cluster mean-field, the ground state wavefunction is written as a simple tensor product of optimized cluster states. The cluster-language and the mean-field nature of the ansatz allows for a straightforward improvement based on perturbation theory and coupled-cluster, to account for inter-cluster correlations. We present benchmark calculations on the 2D square

J_1-J_2

Heisenberg model, using cluster mean-field, second-order perturbation theory and coupled-cluster. We also present an extrapolation scheme that allows us to compute thermodynamic limit energies very accurately. Our results indicate that, even with relatively small clusters, the correlated methods can provide an accurate description of the Heisenberg model in the regimes considered. Some ways to improve the results presented in this work are discussed.