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arXiv:cond-mat/0311126

High-Order Coupled Cluster Method (CCM) Calculations Via Parallel Processing: An Application To The Kagome Antiferromagnet

DAMIAN J. J. FARNELL, R. F. Bishop

Published Nov 6, 2003• 1 citation
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Abstract

A simple ``brute-force'' parallelisation procedure for the computational implementation of high-order coupled cluster method (CCM) calculations is presented here. This approach is investigated and illustrated by an application of high-order CCM to the Heisenberg antiferromagnet on the Kagomé lattice with nearest- and next-nearest-neighbour bonds. Ferromagnetic next-nearest-neighbour bonds are used to stabilise a model state which contains three sublattices in which the spins make angles of 120$^{\circ}$ to each other. Ground-state results for up to approximately 10000 fundamental clusters are presented, and our best estimate for the ground-state energy per spin of the spin-half Kagomé lattice antiferromagnet with only nearest-neighbour bonds is $E_g/N = -0.43104$. We believe that further increases (of at least another order of magnitude) in the number of fundamental clusters might be possible in future by using parallel processing techniques. The extension of high-order CCM calculations in order to consider non-Néel (e.g., dimer solid) model states, simulation of excitation spectra, lattice boson and fermion models, and finite-sized systems is very briefly considered.

DOI
10.48550/arxiv.cond-mat/0311126

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•Jul 1, 2019•1 citation
A simple ``brute-force'' parallelisation procedure for the computational implementation of high-order coupled cluster method (CCM) calculations is presented here. This approach is investigated and illustrated by an application of high-order CCM to the Heisenberg antiferromagnet on the Kagomé lattice with nearest- and next-nearest-neighbour bonds. Ferromagnetic next-nearest-neighbour bonds are used to stabilise a model state which contains three sublattices in which the spins make angles of 120∘^{\circ}∘ to each other. Ground-state results for up to approximately 10000 fundamental clusters are presented, and our best estimate for the ground-state energy per spin of the spin-half Kagomé lattice antiferromagnet with only nearest-neighbour bonds is Eg/N=−0.43104E_g/N = -0.43104Eg​/N=−0.43104. We believe that further increases (of at least another order of magnitude) in the number of fundamental clusters might be possible in future by using parallel processing techniques. The extension of high-order CCM calculations in order to consider non-Néel (e.g., dimer solid) model states, simulation of excitation spectra, lattice boson and fermion models, and finite-sized systems is very briefly considered.
Recent developments of high-order CCM have been to extend existing formalism and codes to s≥12s \ge \frac 12s≥21​ for both the ground and excited states, and independently to "generalised" expectation values for a wide range of one- and two-body spin operators. An advantage of the CCM is that the Goldstone linked-cluster theorem is obeyed at all levels of approximation and so it provides results in the infinite lattice limit N→∞N \to \inftyN→∞ from the outset. However, recent results have also shown that the CCM can provide exact (symmetry-breaking) results for the spin-half linear-chain J1J_1J1​--J2J_2J2​ at the Majumdar-Ghosh point J2/J1=0.5J_2/J_1=0.5J2​/J1​=0.5 by identifying special solutions of the CCM equations for the usual Néel model state. Interestingly, the CCM provides exact (non-symmetry-breaking) results for systems in which small magnetic clusters become de-coupled from each other when the bonds connecting them tend to zero. These exact results involve the identification of "special solutions" of the CCM equations for the Néel state. An example of this is given by a spin-half system with nearest-neighbour bonds for an underlying lattice corresponding to the magnetic material CaV4_44​O9_99​ (CAVO) in which one of the two different types of bonds on the lattice tend to zero. Larger finite-sized systems may be considered by appropriate choice of the unit cell and the bonds on it. We show here that exact diagonalisation results for ground-state energy and excitation energy gap for the spin-half and spin-one linear Heisenberg model on chains of length up to N=12 sites for s=1/2 and N=6 sites for s=1 with periodic boundary conditions are reproduced exactly using high-order CCM via this "brute-force" approach; i.e., one in which none of the translational or point-group symmetries of the finite lattice are used.
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 an 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 only by using such a parallelized approach. An illustration of the new approach is presented for the ground-state properties of a magnetic material, CaV4O9{\mathrm{CaV}}_{4}{\mathrm{O}}_{9}CaV4​O9​.
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.
Coplanar model states for applications of the coupled cluster method (CCM) to problems in quantum magnetism are those in which all spins lie in a plane, whereas three-dimensional (3D) model states are, by contrast, non-coplanar ones in which all the spins do not lie in any single plane. A crucial first step in applying the CCM to any such lattice quantum spin system is to perform a passive rotation of the local spin axes so that all spins in the model state appear mathematically to point in the same (say, downwards z-)direction. Whereas this process leads to terms with only real coefficients in the rotated Hamiltonian for coplanar model states, an additional complication arises for 3D model states where the corresponding coefficients can become complex-valued. We show here for the first time how high-order implementations of the CCM can be performed for such Hamiltonians. We explain in detail why the extension of the computational implementation of the CCM when going from coplanar to 3D model states is a non-trivial task that has not hitherto been undertaken. To illustrate these new developments, we present results for three cases: (a) the spin-half one-dimensional Ising ferromagnet in an applied transverse magnetic field (as an exactly solvable test model to use as a yardstick for the viability and accuracy of our new methodology); (b) the spin-half triangular-lattice Heisenberg antiferromagnet in the presence of an external magnetic field; and (c) the spin-S triangular-lattice XXZ antiferromagnet in the presence of an external magnetic field, for the cases 12≤S≤5\frac{1}{2} \le S \le 5 21​≤S≤5 . For 3D model states the sets of algebraic CCM equations for the ket- and bra-state correlation coefficients become complex-valued, but ground-state expectation values of all physical observables are manifestly real numbers, as required, and as we explicitly demonstrate in all three applications. Indeed, excellent correspondence is seen with the results of other methods, where they exist, for these systems. In particular, our CCM results demonstrate explicitly that coplanar ordering is favoured over non-coplanar ordering for the triangular-lattice spin-half Heisenberg antiferromagnet at all values of the applied external magnetic field, whereas for the anisotropic XXZ model non-coplanar ordering can be favoured in some regions of the parameter space. Specifically, we present a precise determination of the boundary (i.e., the critical value of the XXZ anisotropy parameter Δ\Delta Δ ) between a 3D ground state and a coplanar ground state for the XXZ model for values for the external magnetic field near to saturation, for values of the spin quantum number S≤5S \le 5S≤5 . Although the CCM calculations are computationally intensive for this frustrated model, especially for high spin quantum numbers, our accurate new results certainly improve our understanding of it.
In this article we present new formalism for high-order coupled cluster method (CCM) calculations for "generalized" ground-state expectation values and the excited states of quantum magnetic systems with spin quantum number s ≥ 1/2. We use high-order CCM to demonstrate spontaneous symmetry breaking in the spin-half J1-J2 model for the linear chain using the coupled cluster method (CCM). We show that we are able to reproduce exactly the dimerized ground (ket) state at the Majumdar-Ghosh point (J2/J1= 1/2) using a Néel model state. We show that the onset of dimerized phase is indicated by a bifurcation of the nearest-neighbour ket- and bra-state correlation coefficients for the nearest-neighbour Néel model state. We show that ground-state energies are in good agreement with the results of exact diagonalizations of finite-length chains across this entire regime (i. e., J1>0 and J2≤1/2). The effects of the bifurcation point are also observed for the sublattice magnetization for the nearest-neighbour model state. Finally, we use the new formalism for the excited state in order to obtain the excitation energy as a function of J2/J1 for the nearest-neighbour model state by solving up to the LSUB14 level of approximation. We obtain an extrapolated value for the excited-state energy gap of -0.0036 at J2/J1=0.0 and of 0.2310 at J2/J1=0.5. We show that an excitation energy gap opens up at J2/J1 ≈ 0.24, although the gap only becomes large at J2/J1≈ 0.4.
Coplanar model states for applications of the coupled cluster method (CCM) to problems in quantum magnetism are those in which all spins lie in a plane, whereas three-dimensional (3D) model states are, by contrast, non-coplanar ones in which all the spins do not lie in any single plane. A crucial first step in applying the CCM to any such lattice quantum spin system is to perform a passive rotation of the local spin axes so that all spins in the model state appear mathematically to point in the same (say, downwards z-)direction. Whereas this process leads to terms with only real coefficients in the rotated Hamiltonian for coplanar model states, an additional complication arises for 3D model states where the corresponding coefficients can become complex-valued. We show here for the first time how high-order implementations of the CCM can be performed for such Hamiltonians. We explain in detail why the extension of the computational implementation of the CCM when going from coplanar to 3D model states is a non-trivial task that has not hitherto been undertaken. To illustrate these new developments, we present results for three cases: (a) the spin-half one-dimensional Ising ferromagnet in an applied transverse magnetic field (as an exactly solvable test model to use as a yardstick for the viability and accuracy of our new methodology); (b) the spin-half triangular-lattice Heisenberg antiferromagnet in the presence of an external magnetic field; and (c) the spin-S triangular-lattice XXZ antiferromagnet in the presence of an external magnetic field, for the cases <= S <= 5. For 3D model states the sets of algebraic CCM equations for the ket- and bra-state correlation coefficients become complex-valued, but ground-state expectation values of all physical observables are manifestly real numbers, as required, and as we explicitly demonstrate in all three applications. Indeed, excellent correspondence is seen with the results of other methods, where they exist, for these systems. In particular, our CCM results demonstrate explicitly that coplanar ordering is favoured over non-coplanar ordering for the triangular-lattice spin-half Heisenberg antiferromagnet at all values of the applied external magnetic field, whereas for the anisotropic XXZ model non-coplanar ordering can be favoured in some regions of the parameter space. Specifically, we present a precise determination of the boundary (i.e., the critical value of the XXZ anisotropy parameter Delta) between a 3D ground state and a coplanar ground state for the XXZ model for values for the external magnetic field near to saturation, for values of the spin quantum number S <= 5. Although the CCM calculations are computationally intensive for this frustrated model, especially for high spin quantum numbers, our accurate new results certainly improve our understanding of it.