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Journal of Physics Condensed Matter

The spin-one Heisenberg-biquadratic quantum spin chain treated by the coupled-cluster method

R. F. Bishop, John B. Parkinson, Yang Xian

Published Dec 6, 1993• 21 citations
Publisher

Abstract

The coupled-cluster method is used to obtain the ground-state energy of the isotropic Heisenberg-biquadratic quantum spin-one chain as a function of the ratio of the magnitudes of the two terms in the Hamiltonian. Two different model states are used which are expected to be valid in different regimes. In both cases we use simple approximation schemes to obtain numerical results for the ground-state energy which are compared with results of exact diagonalizations of short chains. For both cases we are able to incorporate some of the long-range correlations explicitly, using the so-called full-SUB2 approximation schemes, and this leads to evidence of phase changes at certain points. These are discussed in the light of known and conjectured phase transitions in this system.

DOI
10.1088/0953-8984/5/49/019

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arXiv:cond-mat/0606060•Jun 2, 2006•1 citation
The coupled-cluster method is used to obtain the ground-state energy of the isotropic Heisenberg-biquadratic quantum spin-one chain as a function of the ratio of the magnitudes of the two terms in the Hamiltonian. Two different model states are used which are expected to be valid in different regimes. In both cases we use simple approximation schemes to obtain numerical results for the ground-state energy which are compared with results of exact diagonalizations of short chains. For both cases we are able to incorporate some of the long-range correlations explicitly, using the so-called full-SUB2 approximation schemes, and this leads to evidence of phase changes at certain points. These are discussed in the light of known and conjectured phase transitions in this system.
The coupled-cluster method (CCM) is applied to anisotropic-quantum-antiferromagnetic models in one and two dimensions (1D and 2D). Several hierarchical approximation schemes of the CCM are defined, which are specialized for the spin-lattice models. Good results are obtained for both ground-state and excited-state energies. The spin-spin correlation functions and the staggered magnetizations are also calculated. We have found that the CCM gives a qualitatively correct description of the entire Heisenberg-Ising phase, including the critical point where a phase transition occurs in 1D. Some interesting differences between 1D and 2D models near the critical point are discussed.
The spin-half XXZ\mathrm{XXZ}XXZ model on the linear chain and the square lattice are examined with the extended coupled-cluster method (ECCM) of quantum many-body theory. We are able to describe both the Ising-Heisenberg phase and the XY\mathrm{XY}XY-Heisenberg phase, starting from known wave functions in the Ising limit and at the phase transition point between the XY\mathrm{XY}XY-Heisenberg and ferromagnetic phases, respectively, and by systematically incorporating correlations on top of them. The ECCM yields good numerical results via a diagrammatic approach, which makes the numerical implementation of higher-order truncation schemes feasible. In particular, the best nonextrapolated coupled-cluster result for the sublattice magnetization is obtained, which indicates the employment of an improved wave function. Furthermore, the ECCM finds the expected qualitatively different behaviors of the linear-chain and square-lattice cases.
The coupled cluster method (CCM) is applied to the spin-one anisotropic Heisenberg antiferromagnet (HAF) on the square lattice at zero temperature using a new high-order CCM ground-state formalism for general quantum spin number (s≥1/2s \ge 1/2s≥1/2). The results presented constitute the first such application of this new formalism, and they are shown to be among the most accurate results for the ground-state energy and sublattice magnetisation of this model as yet determined. We ``track'' the solution to the CCM equations at a given level of approximation with respect to the anisotropy parameter, ΔΔΔ, from the trivial Ising limit (Δ→∞Δ\to \inftyΔ→∞) down to a critical value ΔcΔ_cΔc​, at which point this solution terminates. This behaviour is associated with a phase transition in the system, and hence a primary result of these high-order CCM calculations is that they provide an accurate and unbiased (i.e., {\it ab initio}) estimation of the position of the quantum phase transition point as a function of the anisotropy parameter. Our result is, namely, that this point occurs at (or slightly below) the isotropic Heisenberg point at Δ=1Δ=1Δ=1, for this model.
The coupled-cluster method (CCM) has already been successfully applied by us to anisotropic quantum antiferromagnetic models (XXZ model) with spin s=1/2 in one and two dimensions. This approach is now generalized to the same models but with spin s>1/2. We mainly consider the one-dimensional system in this article. Several systematic approximation schemes developed previously are employed. The ground-state energy is obtained as a function of the spin s and anisotropy. It is found that, for a given approximation that contains long-range contributions, the critical value of the anisotropy for the Heisenberg-Ising phase transition rapidly approaches the classical value of 1 as the spin s increases. We also generalize one of the schemes to a hypercubic lattice of arbitrary dimensionality. By making a detailed comparison with spin-wave theory, perturbation theory, and other variational approaches, some additional intrinsic features of the CCM treatments are emphasized.
A review of the coupled cluster method (CCM) applied to lattice quantum spin systems is presented here. The CCM formalism is explained and an application to the spin-half {\it XXZ} model on the square lattice is presented. Low orders of approximation are carried out analytically and a high-order CCM formulation is presented. Results for the SUB2 approximation are carried out numerically for SUB2-mmm and analytically for SUB2. It is found that SUB2-mmm results converge rapidly to the full SUB2 solution, including new results for the SUB2-mmm limiting points compared to the SUB2 critical point. Results for the ground-state energy and the sublattice magnetisation are presented. A study of the excitation spectrum of this model at the SUB2 critical point is given. Indeed, the shape of the excitation spectrum at the SUB2 critical point is identical to that predicted by spin-wave theory for the isotropic model, albeit with a multiplicative factor of 1.1672. This result compares very well to results of cumulant series expansions and Monte Carlo simulation that again predict a similar shape for the excitation spectrum, but with multiplicative factors of 1.18 and 1.21$\pm$0.03, respectively. Results for the isotropic Heisenberg model on the square lattice for the spin-one antiferromagnet and the spin-one/spin-half ferrimagnet are also given.