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Physical review. B, Condensed matter

Coupled-cluster treatments of correlations in quantum antiferromagnets

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

Published Nov 1, 1991• 88 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.44.9425

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Journal of Physics Condensed Matter•Dec 9, 1996•13 citations
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 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 high-order coupled cluster method (CCM) for the zero-temperature ground states of quantum lattice spin systems of general spin quantum number (s> 1/2)(s>~1/2)(s> 1/2) is described and illustrated via the spin-1 anisotropic Heisenberg antiferromagnet on the square lattice. Results for the ground-state energy and sublattice magnetization are seen to be in excellent agreement with the best of other approximate methods. Furthermore, we are able to follow the solution to the CCM equations at a given level of approximation with respect to the anisotropy parameter $\ensuremath{\Delta}$ from the trivial Ising limit $(\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Delta}}\ensuremath{\infty})$ down to a critical value ${\ensuremath{\Delta}}_{c},$ at which point the physical solution terminates. The results for the ground-state energy, sublattice magnetization, and phase transition point thus support our choice of this model as a good nontrivial test case of our CCM formalism.
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.
We extend our own previous applications of the microscopic coupled-cluster method (CCM) to quantum antiferromagnets. In particular, we carry out a systematic calculation involving high-order multispin correlations for the spin-1/2 anisotropic Heisenberg models on the one-dimensional chain and the two-dimensional square lattice. Their ground-state properties are obtained as functions of the anisotropy parameter. Our CCM analysis not only produces accurate results for such physical quantities as the ground-state energy which are comparable to the best results from other techniques, but it also enables us to study the quantum phase transitions of the spin models in a systematic and unbiased manner. © 1996 John Wiley & Sons, Inc.
The coupled-cluster method (CCM) is applied to the spin- model for the 1D chain and the 2D square lattice in the region . A CCM model state is chosen in which all spins lie in the xy plane with nearest-neighbour spins anti-parallel. Very good agreement with the exactly known ground-state energies in 1D is obtained, and good agreement with Monte Carlo results is obtained for the 2D case. The sublattice magnetization is obtained, and evidence of phase transition points in both 1D and 2D is presented. The critical behaviour of the 1D ground-state energy near the ferromagnetic phase transition point is investigated.