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International Journal of Modern Physics B

The Coupled Cluster Method Applied to the Spin-Half XXZ Model on the Honeycomb Lattice

R. F. Bishop, J. Rosenfeld

Published Sep 20, 1998• 5 citations
Publisher

Abstract

The coupled cluster method (CCM) is applied to the spin-half XXZ model on the honeycomb lattice. Hierarchical CCM approximation schemes are applied in order to obtain the ground-state properties. The possible critical behavior of this model is also examined. Reasonably good results for the ground-state energy and the staggered magnetization are obtained, in comparison to the results from Monte Carlo simulations, spin-wave techniques, series expansions, and finite lattice diagonalizations, that have already been performed on this model.

DOI
10.1142/s0217979298001381

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The coupled cluster method (CCM) is applied to the spin-half XXZ model on the honeycomb lattice. Hierarchical CCM approximation schemes are applied in order to obtain the ground-state properties. The possible critical behavior of this model is also examined. Reasonably good results for the ground-state energy and the staggered magnetization are obtained, in comparison to the results from Monte Carlo simulations, spin-wave techniques, series expansions, and finite lattice diagonalizations, that have already been performed on this model.
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.
A new approximate scheme, DSUBm, is described for the coupled cluster method.We apply it to two wellstudied (spin-1/2 Heisenberg antiferromagnet) spin-lattice models, namely: the XXZ and the XY models on the square lattice in two dimensions.Results are obtained in each case for the ground-state energy, the sublattice magnetization and the quantum critical point.They are in good agreement with those from such alternative methods as spin-wave theory, series expansions, exact diagonalization techniques, quantum Monte Carlo methods and those from the CCM using the LSUBm scheme.
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 is applied to the spin-1/2 antiferromagnetic XXZ model on a square lattice by employing an approximation which contains two-body long-range correlations and high-order four-body local correlations. Improvement is found for the ground-state energy, the sublattice magnetization and the critical anisotropy when comparing with the approximation including the two-body correlations alone. We also obtain the full excitation spectrum which is in good agreement with the quantum Monte Carlo results and the high-order spin-wave theory.
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.
In this article, we present new results of high-order coupled cluster method (CCM) calculations, based on a Néel model state with spins aligned in the zzz-direction, for both the ground- and excited-state properties of the spin-half {\it XXZ} model on the linear chain, the square lattice, and the simple cubic lattice. In particular, the high-order CCM formalism is extended to treat the excited states of lattice quantum spin systems for the first time. Completely new results for the excitation energy gap of the spin-half {\it XXZ} model for these lattices are thus determined. These high-order calculations are based on a localised approximation scheme called the LSUBmmm scheme in which we retain all kkk-body correlations defined on all possible locales of mmm adjacent lattice sites (k≤mk \le mk≤m). The ``raw'' CCM LSUBmmm results are seen to provide very good results for the ground-state energy, sublattice magnetisation, and the value of the lowest-lying excitation energy for each of these systems. However, in order to obtain even better results, two types of extrapolation scheme of the LSUBmmm results to the limit m→∞m \to \inftym→∞ (i.e., the exact solution in the thermodynamic limit) are presented. The extrapolated results provide extremely accurate results for the ground- and excited-state properties of these systems across a wide range of values of the anisotropy parameter.