Ab Initio Simulation of the Nodal Surfaces of Heisenberg Antiferromagnets

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 localised, $m$-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.Comment: 24 pages, Latex, 3 postscript figure

High-Order Coupled Cluster Calculations Via Parallel Processing: An Illustration For CaV4_4O9_9

The coupled cluster method (CCM) is a method of quantum many-body theory that may provide accurate results for the ground-state properties of lattice quantum spin systems even in the presence of strong frustration and for lattices of arbitrary spatial dimensionality. Here we present a significant extension of the method by introducing a new approach that allows an efficient parallelization of computer codes that carry out ``high-order'' CCM calculations. We find that we are able to extend such CCM calculations by an order of magnitude higher than ever before utilized in a high-order CCM calculation for an antiferromagnet. Furthermore, we use only a relatively modest number of processors, namely, eight. Such very high-order CCM calculations are possible {\it only} by using such a parallelized approach. An illustration of the new approach is presented for the ground-state properties of a highly frustrated two-dimensional magnetic material, CaV$_4$O$_9$. Our best results for the ground-state energy and sublattice magnetization for the pure nearest-neighbor model are given by $E_g/N=-0.5534$ and $M=0.19$, respectively, and we predict that there is no N\'eel ordering in the region $0.2 \le J_2/J_1 \le 0.7$. These results are shown to be in excellent agreement with the best results of other approximate methods.Comment: 4 page

Magnetic order in a spin-1/2 interpolating kagome-square Heisenberg antiferromagnet

The coupled cluster method is applied to a spin-half model at zero temperature ($T=0$), which interpolates between Heisenberg antiferromagnets (HAF's) on a kagome and a square lattice. With respect to an underlying triangular lattice the strengths of the Heisenberg bonds joining the nearest-neighbor (NN) kagome sites are $J_{1} \geq 0$ along two of the equivalent directions and $J_{2} \geq 0$ along the third. Sites connected by $J_{2}$ bonds are themselves connected to the missing NN non-kagome sites of the triangular lattice by bonds of strength $J_{1}' \geq 0$. When $J_{1}'=J_{1}$ and $J_{2}=0$ the model reduces to the square-lattice HAF. The magnetic ordering of the system is investigated and its $T=0$ phase diagram discussed. Results for the kagome HAF limit are among the best available.Comment: 21 pages, 8 figure

Coupled Cluster Method Calculations Of Quantum Magnets With Spins Of General Spin Quantum Number

We present a new high-order coupled cluster method (CCM) formalism for the ground states of lattice quantum spin systems for general spin quantum number, $s$. This new ``general-$s$'' formalism is found to be highly suitable for a computational implementation, and the technical details of this implementation are given. To illustrate our new formalism we perform high-order CCM calculations for the one-dimensional spin-half and spin-one antiferromagnetic {\it XXZ} models and for the one-dimensional spin-half/spin-one ferrimagnetic {\it XXZ} model. The results for the ground-state properties of the isotropic points of these systems are seen to be in excellent quantitative agreement with exact results for the special case of the spin-half antiferromagnet and results of density matrix renormalisation group (DMRG) calculations for the other systems. Extrapolated CCM results for the sublattice magnetisation of the spin-half antiferromagnet closely follow the exact Bethe Ansatz solution, which contains an infinite-order phase transition at $\Delta=1$. By contrast, extrapolated CCM results for the sublattice magnetisation of the spin-one antiferromagnet using this same scheme are seen to go to zero at $\Delta \approx 1.2$, which is in excellent agreement with the value for the onset of the Haldane phase for this model. Results for sublattice magnetisations of the ferrimagnet for both the spin-half and spin-one spins are non-zero and finite across a wide range of $\Delta$, up to and including the Heisenberg point at $\Delta=1$.Comment: 5 Figures. J. Stat. Phys. 108, p. 401 (2002

Coupled Cluster Treatment of the Shastry-Sutherland Antiferromagnet

We consider the zero-temperature properties of the spin-half two-dimensional Shastry-Sutherland antiferromagnet by using a high-order coupled cluster method (CCM) treatment. We find that this model demonstrates various groundstate phases (N\'{e}el, magnetically disordered, orthogonal dimer), and we make predictions for the positions of the phase transition points. In particular, we find that orthogonal-dimer state becomes the groundstate at ${J}^{d}_2/J_1 \sim 1.477$. For the critical point $J_2^{c}/J_1$ where the semi-classical N\'eel order disappears we obtain a significantly lower value than $J_2^{d}/J_1$, namely, ${J}^{c}_2/J_1$ in the range $[1.14, 1.39]$. We therefore conclude that an intermediate phase exists between the \Neel and the dimer phases. An analysis of the energy of a competing spiral phase yields clear evidence that the spiral phase does not become the groundstate for any value of $J_2$. The intermediate phase is therefore magnetically disordered but may exhibit plaquette or columnar dimer ordering.Comment: 6 pages, 5 figure

Quantum Phase Transitions in Spin Systems

We discuss the influence of strong quantum fluctuations on zero-temperature phase transitions in a two-dimensional spin-half Heisenberg system. Using a high-order coupled cluster treatment, we study competition of magnetic bonds with and without frustration. We find that the coupled cluster treatment is able to describe the zero-temperature transitions in a qualitatively correct way, even if frustration is present and other methods such as quantum Monte Carlo fail.Comment: 8 pages, 12 Postscipt figures; Accepted for publication in World Scientifi

High-Order Coupled Cluster Method Calculations for the Ground- and Excited-State Properties of the Spin-Half XXZ Model

In this article, we present new results of high-order coupled cluster method (CCM) calculations, based on a N\'eel model state with spins aligned in the $z$-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 LSUB$m$ scheme in which we retain all $k$-body correlations defined on all possible locales of $m$ adjacent lattice sites ($k \le m$). The ``raw'' CCM LSUB$m$ 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 LSUB$m$ results to the limit $m \to \infty$ (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.Comment: 31 Pages, 5 Figure