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Acta Physica Polonica A

Diagonalization of a Bosonic Quadratic Form Using CCM: Application on a System with Two Interpenetrating Square Lattice Antiferromagnets

Sven E. Krüger, D. Schmalfuß, Johannes Richter

Published May 1, 2000• 0 citations
Publisher

Abstract

While the diagonalization of a quadratic bosonic form can always be done using a Bogolyubov transformation, the practical implementation for systems with a large number of different bosons is a tedious analytical task.Here we use the coupled cluster method to exactly diagonalise such complicated quadratic forms.This yields to a straightforward algorithm which can easily be implemented using computer algebra even for a large number of different bosons.We apply this method on a Heisenberg system with two interpenetrating square lattice antiferromagnets, which is a model for the quasi-2D antiferromagnet Ba 2 Cu3 O 4 Cl2 .Using a four-magnon spin wave approximation we get a complicated Hamiltonian with four different bosons, which is treated with coupled cluster method.Results are presented for magnetic ground state correlations.

DOI
10.12693/aphyspola.97.951

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While the diagonalization of a quadratic bosonic form can always be done using a Bogolyubov transformation, the practical implementation for systems with a large number of different bosons is a tedious analytical task.Here we use the coupled cluster method to exactly diagonalise such complicated quadratic forms.This yields to a straightforward algorithm which can easily be implemented using computer algebra even for a large number of different bosons.We apply this method on a Heisenberg system with two interpenetrating square lattice antiferromagnets, which is a model for the quasi-2D antiferromagnet Ba 2 Cu3 O 4 Cl2 .Using a four-magnon spin wave approximation we get a complicated Hamiltonian with four different bosons, which is treated with coupled cluster method.Results are presented for magnetic ground state correlations.
While the diagonalization of a quadratic bosonic form can always be done using a Bogoliubov transformation, the practical implementation for systems with a large number of different bosons is a tedious analytical task. Here we use the coupled cluster method (CCM) to exactly diagonalise such complicated quadratic forms. This yields to a straightforward algorithm which can easily be implemented using computer algebra even for a large number of different bosons. We apply this method on a Heisenberg system with two interpenetrating square lattice antiferromagnets, which is a model for the quasi 2D antiferromagnet Ba_2Cu_3O_4Cl_2. Using a four-magnon spin wave approximation we get a complicated Hamiltonian with four different bosons, which is treated with CCM. Results are presented for magnetic ground state correlations.
The coupled-cluster method (CCM) is applied to anisotropic-quantum-antiferromagnetic models in one and two dimensions (1D and 2D). Several hierarchical approximation schemes of the CCM are defined, which are specialized for the spin-lattice models. Good results are obtained for both ground-state and excited-state energies. The spin-spin correlation functions and the staggered magnetizations are also calculated. We have found that the CCM gives a qualitatively correct description of the entire Heisenberg-Ising phase, including the critical point where a phase transition occurs in 1D. Some interesting differences between 1D and 2D models near the critical point are discussed.
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.
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.
A new approximate scheme, DSUBm, is described for the coupled cluster method.We apply it to two wellstudied (spin-1/2 Heisenberg antiferromagnet) spin-lattice models, namely: the XXZ and the XY models on the square lattice in two dimensions.Results are obtained in each case for the ground-state energy, the sublattice magnetization and the quantum critical point.They are in good agreement with those from such alternative methods as spin-wave theory, series expansions, exact diagonalization techniques, quantum Monte Carlo methods and those from the CCM using the LSUBm scheme.
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.