The spin-half Heisenberg antiferromagnet (HAF) on the square and triangular lattices is studied using the coupled-cluster-method (CCM) technique of quantum many-body theory. The phase relations between different expansion coefficients of the ground-state wave function in an Ising basis for the square lattice HAF is exactly known via the Marshall-Peierls sign rule, although no equivalent sign rule has yet been obtained for the triangular-lattice HAF. Here the CCM is used to give accurate estimates for the Ising-expansion coefficients for these systems, and CCM results are noted to be fully consistent with the Marshall-Peierls sign rule for the square-lattice case. For the triangular-lattice HAF, a heuristic rule is presented which fits our CCM results for the Ising-expansion coefficients of states which correspond to two-body excitations with respect to the reference state. It is also seen that Ising-expansion coefficients which describe localized, m-body excitations with respect to the reference state are found to be highly converged, and from this result we infer that the nodal surface of the triangular lattice HAF is being accurately modeled. Using these results, we are able to make suggestions regarding possible extensions of existing quantum Monte Carlo simulations for the triangular-lattice HAF.
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, of the Marshall-Peierls sign rule which is known to be satisfied at the pure Heisenberg point
(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 ${J}_{2}{/J}_{1}\ensuremath{\approx}0.61.$
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≥1/2). 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,
Δ, from the trivial Ising limit (
Δ→∞) down to a critical value
Δc, 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=1, for this model.
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.
Recent developments of high-order CCM have been to extend existing formalism and codes to
s≥21 for both the ground and excited states. In this article, we describe how "generalised" expectation values for a wide range of one- and two-body spin operators may also be determined using existing the CCM code for the ground state. We present new results for the spin-spin correlation functions of the spin-half square- and triangular-lattice antiferromagnets by using the LSUB
m approximation. We show that the absolute values of the spin-spin correlation functions
∣<s(0).s(r)>∣ converge with increasing approximation level for both lattices. We believe that the LSUB
m approximation provides reasonable results for the correlation functions for lattice separations roughly of order
r≈m for the square lattice. We compare qualitatively our results for the square lattice to those results of quantum Monte Carlo (QMC) and we see that good correspondence is observed. Indeed as seen by QMC, the spin-spin correlation function initially decays strongly with
∣r∣ before becoming constant for larger values of
∣r∣. CCM results are also compared to results of exact diagonalisations for both lattices. ED results demonstrate a strong finite-size effects at lattice separations
r=L/2 (where
N=L×L) for both lattices. The behaviour of the correlation function for the triangular lattice is qualitatively similar to that of the square lattice, namely, that it decays strongly at first before becoming constant. This is in keeping with the behaviour of both models, which are believed strongly to be Néel-ordered from approximate studies.
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) 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-
m and analytically for SUB2. It is found that SUB2-
m results converge rapidly to the full SUB2 solution, including new results for the SUB2-
m 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.