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arXiv:cond-mat/0105386

AB INITIO CALCULATIONS OF THE SPIN-HALF XY MODEL

DAMIAN J. J. FARNELL, M. L. RISTIG

Published Sep 1, 2001• 1 citation
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Abstract

In this article, the correlated basis-function (CBF) method is applied for the first time to the quantum spin-half {\it XY} model on the linear chain, the square lattice, and the simple cubic lattice. In this treatment of the quantum spin-half {\it XY} model a Jastrow ansatz is utilised to approximate the ground-state wave function. Results for the ground-state energy and the sublattice magnetisation are presented, and evidence that the CBF detects the quantum phase transition point in this model is also presented. The CBF results are compared to previous coupled cluster method (CCM) results for the spin-half {\it XY} model, and the two formalisms are then compared and contrasted.

DOI
10.1142/9789812799760_0020

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In this article, the correlated basis-function (CBF) method is applied for the first time to the quantum spin-half {\it XY} model on the linear chain, the square lattice, and the simple cubic lattice. In this treatment of the quantum spin-half {\it XY} model a Jastrow ansatz is utilised to approximate the ground-state wave function. Results for the ground-state energy and the sublattice magnetisation are presented, and evidence that the CBF detects the quantum phase transition point in this model is also presented. The CBF results are compared to previous coupled cluster method (CCM) results for the spin-half {\it XY} model, and the two formalisms are then compared and contrasted.
In this article, the correlated basis-function (CBF) method is applied for the first time to the quantum spin-half XY model on the linear chain, the square lattice, and the simple cubic lattice. In this treatment of the quantum spin-half XY model a Jastrow ansatz is utilised to approximate the ground-state wave function. Results for the ground-state energy and the sublattice magnetisation are presented, and evidence that the CBF detects the quantum phase transition point in this model is also presented. The CBF results are compared to previous coupled cluster method (CCM) results for the spin-half XY model, and the two formalisms are then compared and contrasted. In this paper we consider the T = 0 properties of the quantum spin system known as the spin-half XY model, described by the Hamiltonian
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.
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.
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.
The coupled cluster method (CCM) is a well-known method of quantum many-body theory, and in this article we present an application of the CCM to the spin-half ${J}_{1}{\ensuremath{-}J}_{2}$ quantum spin model with nearest- and next-nearest-neighbor interactions on the linear chain and the square lattice. We present results for ground-state expectation values of such quantities as the energy and the sublattice magnetization. The presence of critical points in the solution of the CCM equations, which are associated with phase transitions in the real system, is investigated. Completely distinct from the investigation of the critical points, we also make a link between the expansion coefficients of the ground-state wave function in terms of an Ising basis and the CCM ket-state correlation coefficients. We are thus able to present evidence of the breakdown, at a given value of J2/J1,{J}_{2}{/J}_{1},J2​/J1​, of the Marshall-Peierls sign rule which is known to be satisfied at the pure Heisenberg point (J2=0){(J}_{2}=0)(J2​=0) on any bipartite lattice. For the square lattice, our best estimates of the points at which the sign rule breaks down and at which the phase transition from the antiferromagnetic phase to the frustrated phase occurs are, respectively, given by ${J}_{2}{/J}_{1}\ensuremath{\approx}0.26$ and ${J}_{2}{/J}_{1}\ensuremath{\approx}0.61.$
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.