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Journal of Physics Condensed Matter

The coupled-cluster method applied to theXXZmodel using a planar model state

R. F. Bishop, DAMIAN J. J. FARNELL, John B. Parkinson

Published Dec 9, 1996• 13 citations
Publisher

Abstract

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.

DOI
10.1088/0953-8984/8/50/038

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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.
We study quantum spin systems in the 1D, 2D square and 3D cubic lattices with nearest-neighbour XY-exchange. We use the coupled-cluster method (CCM) to calculate the ground-state energy, the T = 0 sublattice magnetization and the excited-state energies, all as functions of the anisotropy parameter . We consider the case with S = 1/2 in detail and give some results for higher S. In 1D these results are compared with the exact S = 1/2 results and in 2D with Monte Carlo and series expansions. We obtain critical points close to the expected value and our extrapolated LSUBn results for the ground-state energy are well converged for all except very close to the critical point.
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.
Interest in lattice quantum spin systems as models of quantum magnets has increased with the discovery of new and interesting magnetic materials. Here we use a well-known technique of quantum many-body theory, namely the coupled-cluster method (CCM), to investigate the nearest-neighbour, spin-½, anisotropic Heisenberg model on the square lattice. Ground-state expectation values for quantities such as the ground-state energy and the sublattice magnetisation are determined to an accuracy comparable with that of the best of other available techniques including Monte Carlo methods. In order to demonstrate this point we present results for various values of the anisotropy parameter, including those for the isotropic Heisenberg model and the isotropic XY model. We show that it is now possible to determine the presence and position of the quantum phase transitions using ab initio CCM calculations, and furthermore that we can accurately predict the critical behaviour at these points.
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.
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.