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Journal of Physics C Solid State Physics

A finite-lattice study of the spin-1 Heisenberg model with biquartic exchange

Phillip M. Duxbury, J. Oitmaa

Published Jul 30, 1983• 2 citations
Publisher

Abstract

The spin-1 Heisenberg model with bilinear and biquadratic exchange is studied by means of exact calculations on finite cells of up to ten sites, in one and two dimensions. The authors investigate the nature of the ground state for different values and signs of the exchange constants and obtain estimates of the ground-state energy. Anomalous ground states are found for some, but not all, of the cases studied. The specific heat is also calculated.

DOI
10.1088/0022-3719/16/21/021

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The spin-1 Heisenberg model with bilinear and biquadratic exchange is studied by means of exact calculations on finite cells of up to ten sites, in one and two dimensions. The authors investigate the nature of the ground state for different values and signs of the exchange constants and obtain estimates of the ground-state energy. Anomalous ground states are found for some, but not all, of the cases studied. The specific heat is also calculated.
The magnetic susceptibility and specific heat of the one-dimensional S = 1 bilinear-biquadratic Heisenberg model are calculated using the transfer matrix renormalization group. By comparing the results with the experimental data of LiVGe2O6 measured by Millet et al. [Phys. Rev. Lett. 83, 4176 (1999)], we find that the susceptibility data of this material, after subtracting the impurity contribution, can be quantitatively explained with this model. The biquadratic exchange interaction in this material is found to be ferromagnetic, i.e., with a positive coupling constant.
We discuss the ground state of the Heisenberg spin-1/2 chain in one dimension, considering nearest neighbor interactions. In the thermodynamic limit the energy is calculated analytically in several ways. We start by applying the Bethe ansatz to calculate the exact ground state energy of the XXX Heisenberg model. Next we study small perturbations of the spin anisotropy, the parameter for the interaction in the z-direction for the XXZ model, by applying time-independent perturbation theory. In addition we expand an exact integral expression of the energy of the XXZ model around the spin anisotropy in the thermodynamic limit. The results are compared to numerical solutions, which are obtained by means of exact diagonalization up to twelve lattice sites. We find that for ten or more lattice sites the numerics agree well with the expectation from the analytical calculations. Finally the concept of integrability is discussed in the context of the Heisenberg model.
This is a study of finite lattice methods applied to quantum spin models on one and two dimensional lattices.The motivation for the use of these techniques is provided by placing them in the context of the wider field of approximation methods in the statistical mechanics of spin models.After reviewing previous finite lattice calculations on quantum spin models, the theory needed to efficiently generate and analyse finite lattice data is given.Finite lattice calculations on three quantum spin models are then presented.The first calculation provides an estimate of the transverse susceptibility of the one dimensional antiferromagnetic XY model, by extrapolation from calculations on rings of up to 10 sites.Good convergence is found for KB T ~ 0•4 and these results are compared with recent c -axis susceptibility measurements on the compound Cs 2 CoC1 4 .Chapters 5 and 6 study the quantum hamiltonian analog of the two dimensional axial next nearest neighbour Ising model, A perturbation analysis using non degenerate Rayleigh-Schrodingerperturbation theory is followed by a mass gap scaling analysis.These analyses provide accurate phase boundaries for, and ellucidate several interesting features of, the phase diagram of this model.In particular, the model is shown to exhibit a phase whose modulation varies continuously with a change in the interaction parameters in the hamiltonian.This "modulated" phase region lies between a normal ordered phase and a paramagnetic phase.Evid~nce is given that the transition from the paramagnetic to modulated phase is exponential in nature.The third model studied is a spin 1 biquadratic Heisenberg model with hamiltonian,<ij> (Jl s .. s. +J2(~ . .~.)2). 1 J 1 J Accurate estimates of the ground state energy in both one and two dimensions are found by calculating this quantity on 4,6,8 and 10 site lattices.Comment is made about the ground state phase diagram, and about the existence of intermediate spin states and states with a complicated magnetic ordering.The thesis concludes by commenting on the performance of finite lattice methods in the three models studied and by outlining certain directions for future advance., ACKNOWLEVGEMENTS I would like to express