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arXiv:cond-mat/0511544 · Physical Review B

High-order coupled cluster calculations via parallel processing: An illustration for CaV4O9

DAMIAN J. J. FARNELL, J. Schulenburg, Johannes Richter, Klaus A. Gernoth

Published Nov 15, 2005• 22 citations
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Abstract

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, ${\mathrm{CaV}}_{4}{\mathrm{O}}_{9}$.

DOI
10.1103/physrevb.72.172408

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High-Order Coupled Cluster Method Calculations using Three-Dimensional Model States: An Illustration for the Triangular-Lattice Antiferromagnet in an External Field

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arXiv:1312.0785•Dec 3, 2013•1 citation
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 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.
The coupled cluster method (CCM) of microscopic quantum many-body theory has become an ab initio method of first choice in quantum chemistry and many fields of nuclear, subnuclear and condensed matter physics, when results of high accuracy are required. In recent years it has begun to be applied with equal success to strongly correlated systems of electrons or quantum spins defined on a regular spatial lattice. One regularly finds that the CCM is able to describe accurately the various zero-temperature phases and the quantum phase transitions between them, even when frustration is present and other methods such as quantum Monte Carlo often fail. We illustrate the use and powerfulness of the method here by applying it to a square-lattice spin-half Heisenberg model where frustration is introduced by competing nearest neighbour bonds. The model exhibits the physically interesting phenomenon of competition between magnetic order and dimerization. Results obtained for the model with the CCM are compared with those found from spin-wave theory and from extrapolating the results of exact diagonalizations of small lattices. We show that the CCM is essentially unique among available methods in being able both to describe accurately all phases of this complex model and to provide accurate predictions of the various phase boundaries and the order of the corresponding transitions.
High-Order Coupled Cluster Method (CCM) Formalism 1: Ground- and Excited-State Properties of Lattice Quantum Spin Systems with s>=1/2s >= 1/2s>=1/2
The coupled cluster method (CCM) is a powerful and widely applied technique of modern-day quantum many-body theory. It has been used with great success in order to understand the properties of quantum magnets at zero temperature. This is due largely to the application of computational techniques that allow the method to be applied to high orders of approximation using localised approximation schemes, e.g., such as the LSUBmmm scheme. In this article, the high-order CCM formalism for the ground and excited states of quantum magnetic systems are extended to those with spin quantum number s≥12s \ge \frac 12s≥21​. Solution strategies for the ket- and bra-state equations are also considered. Aspects of extrapolation of CCM expectation values are discussed and future topics regarding extrapolations are presented.
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.
We report on recent results for strongly frustrated quantum J1 - J2 antiferromagnets in dimensionality d = 1, 2, 3 obtained by the coupled cluster method (CCM). We demonstrate that the CCM in high orders of approximation allows us to investigate quantum phase transitions driven by frustration and to discuss novel quantum ground states. In detail we consider the ground-state properties of (i) the Heisenberg spin-1/2 antiferromagnet on the cubic lattice in d = 1, 2, 3, and use the results for the energy, the sublattice magnetization and the spin stiffness as a benchmark test for the precision of the method; (ii) coupled frustrated spin chains (the quasi-one-dimensional J1 - J2 model) and discuss the influence of the quantum fluctuations and the interchain coupling on the incommensurate spiral state present in the classical model; (iii) the Shastry-Sutherland antiferromagnet on the square lattice; and (iv) a stacked frustrated square-lattice Heisenberg antiferromagnet (the quasi-two-dimensional J1 - J2 model), and discuss the influence of the interlayer coupling on the quantum paramagnetic ground-state phase that is present for the strictly two-dimensional model.
The coupled cluster method (CCM) has previously been applied to study the ground- and excited-state properties of many different types of frustrated and unfrustrated quantum spin systems. A common feature in the application of the CCM is to rotate the local spin axes of the (often classical) model state so that (notationally only) the spins all appear to point in the downwards zzz-direction. Hitherto, we remark that only coplanar model states have been used because they do not lead to imaginary terms in the new Hamiltonian. By contrast, non-coplanar "three-dimensional" (3D) model states can lead to imaginary terms in the new Hamiltonian after rotation. In principle, however, macroscopic quantities predicted by the CCM (such as the ground-state energy and order parameter) should still be real (even though the Hamiltonian may be complex) because the transformations of local spin axes are unitary. Here we explain how we may use such 3D model states for the CCM and how we may solve for the (now possibly complex) CCM correlation coefficients. We present results for the spin-half triangular-lattice antiferromagnet in an external magnetic field. We use both coplanar model states and a single 3D non-coplanar model state. We calculate the ground-state energy and the lattice magnetisation as a function of the magnetic field strength. We find that the energies of the coplanar states lie lower than that of the non-coplanar state for all values of the external field, as seen in other approximate studies for this model. We find that the non-coplanar state does not detect the well-known spin plateau occurring in this model, although (as seen before) it is clearly observed for the coplanar model states using the CCM. These results are an excellent initial validation of the new approach for the application of the CCM using 3D model states.