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A cluster-based mean-field, perturbative and coupled-cluster theory description of strongly correlated systems

Athanasios Papastathopoulos-Katsaros, Carlos A. Jiménez-Hoyos, Thomas M. Henderson, Gustavo E. Scuseria

Published Feb 19, 2021• 1 citation
OpenAlex

Abstract

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.

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Cluster-based methods for strongly-correlated systems

Athanasios Papastathopoulos-Katsaros

We introduce three novel cluster-based methods to describe the ground states of strongly-correlated systems such as iron-sulfur clusters, conjugated hydrocarbons, and superconductors. These methods utilize a spatial tiling of sites as their core principle. The first approach employs unrestricted cluster mean-field theory (UcMF), with clusters of Sz eigenstates. Correlations between tiles are accounted for using perturbation theory (cPT2) and coupled-cluster (cCCSD). The second approach, generalized cluster mean-field theory (GcMF), allows Sz to break in each cluster, partially including missing intercluster correlations. A projection scheme, Sz GcMF, restores global Sz symmetry for further improvement. The third approach, a non-orthogonal configuration interaction-based theory (LC-cMF) which is still in development, is based on linear combinations of different system tilings. Various criteria, such as translational symmetry and spatial proximity, guide the selection of these tilings. Benchmark calculations on one- and two-dimensional spin models show the promise of these methods. GcMF and Sz GcMF provide a qualitative improvement over UcMF, while cPT2, cCCSD, and LC-cMF can quantitatively capture inter-cluster interactions in some systems. Overall, cluster-based methods offer valuable tools for investigating strongly-correlated spin systems with potential for further advancement.

•Dec 1, 2023•0 citations
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 J1−J2J_1-J_2J1​−J2​ 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.
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 a mean-field and a perturbative approach, based on clusters, to describe the ground state of fermionic strongly correlated systems. In the cluster mean-field approach, the ground-state wave function is written as a simple tensor product over optimized cluster states. The optimization of the single-particle basis where the cluster mean field is expressed is crucial in order to obtain high-quality results. The mean-field nature of the Ansatz allows us to formulate a perturbative approach to account for intercluster correlations; other traditional many-body strategies can be easily devised in terms of the cluster states. We present benchmark calculations on the half-filled 1D and (square) 2D Hubbard model, as well as the lightly doped regime in 2D, using cluster mean-field and second-order perturbation theory. Our results indicate that, with sufficiently large clusters or to second-order in perturbation theory, a cluster-based approach can provide an accurate description of the Hubbard model in the considered regimes. Several avenues to improve upon the results presented in this work are discussed.
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 present an innovative cluster-based method employing linear combinations of diverse cluster mean-field states and apply it to describe the ground state of strongly correlated spin systems. In cluster mean-field theory, the ground state wave function is expressed as a factorized tensor product of optimized cluster states. While our prior work concentrated on a single cMF tiling, this study removes that constraint by combining different tilings of cMF states. Selection criteria, including translational symmetry and spatial proximity, guide this process. We present benchmark calculations for the one- and two-dimensional J 1 – J 2 and XXZ Heisenberg models. Our findings highlight two key aspects. First, the method offers a semiquantitative description of the 0.4 ≲ J 2 / J 1 ≲ 0.6 regime of the J 1 – J 2 model─a particularly challenging regime for existing methods. Second, our results demonstrate the capability of our method to provide qualitative descriptions for all the models and regimes considered, establishing it as a valuable reference. However, the inclusion of additional (weak) correlations is necessary for quantitative agreement, and we explore methods to incorporate these extra correlations.
In previous work, it was shown that using the cluster mean field(cMF) wavefunction as the zeroth order wavefunction for perturbation theory and coupled cluster produces a reasonable approximation for the J 1-J 2 Heisenberg model in the thermodynamic limit(TDL). However, this work was limited to cPT4 and cCCSD. The central claim of cMF is that it takes strongly correlated systems and makes them weakly correlated so that PT weakly correlated methods can be used. To confirm that this claim is true, it is important to show that the PT and CC series converge. In this paper, we extend the previous calculations to cPT7 and cCCSDTQ5 where we show that the energy is converging. We also discuss how to perform cluster based calculations efficiently in the thermodynamic limit, which is crucial for these more expensive calculations.
We introduce three novel cluster-based methods to describe the ground states of strongly-correlated systems such as iron-sulfur clusters, conjugated hydrocarbons, and superconductors. These methods utilize a spatial tiling of sites as their core principle. The first approach employs unrestricted cluster mean-field theory (UcMF), with clusters of Sz eigenstates. Correlations between tiles are accounted for using perturbation theory (cPT2) and coupled-cluster (cCCSD). The second approach, generalized cluster mean-field theory (GcMF), allows Sz to break in each cluster, partially including missing intercluster correlations. A projection scheme, Sz GcMF, restores global Sz symmetry for further improvement. The third approach, a non-orthogonal configuration interaction-based theory (LC-cMF) which is still in development, is based on linear combinations of different system tilings. Various criteria, such as translational symmetry and spatial proximity, guide the selection of these tilings. Benchmark calculations on one- and two-dimensional spin models show the promise of these methods. GcMF and Sz GcMF provide a qualitative improvement over UcMF, while cPT2, cCCSD, and LC-cMF can quantitatively capture inter-cluster interactions in some systems. Overall, cluster-based methods offer valuable tools for investigating strongly-correlated spin systems with potential for further advancement.