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Journal of Physics C Solid State Physics

Crossover effects in a general spin-1-bilinear-biquadratic exchange Hamiltonian

J Otimaa, John B. Parkinson, J. C. Bonner

Published Sep 20, 1986• 37 citations
Publisher

Abstract

The authors have investigated the general 1D spin-1 bilinear-biquadratic exchange Hamiltonian by analytic and numerical techniques over the range 0<or= beta infinity . The limit beta =0 corresponds to the 1Dd Heisenberg antiferromagnet, the limit beta =1 corresponds to a Bethe ansatz integrable model and beta = infinity corresponds to pure antiferromagnetic biquadratic exchange. Haldane has predicted the existence of a singlet-triplet excitation gap in the thermodynamic limit for beta =0, whereas the beta =1 model is gapless. The authors have studied the singlet-triplet gap by scaled-gap (finite-size scaling) techniques, and it appears to vanish at an intermediate value of beta , with large uncertainty. Finite-system analytic studies show that a fundamental 'two-string' excitation picture describes the spectral excitations in the beta =1 limit, whereas a very different 'single-spin-deviation' picture characterises the excitations in the Heisenberg limit. The crossover between the two pictures shows interesting features and appears to be related to the disappearance of the singlet-triplet gap. The implication of an extended crossover region ( 1 / 3 < beta <0.62) is discussed in the context of non-integrability effects. Finally, they find the singlet-triplet gap reappears for beta >1.

DOI
10.1088/0022-3719/19/26/005

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The authors have investigated the general 1D spin-1 bilinear-biquadratic exchange Hamiltonian by analytic and numerical techniques over the range 0<or= beta infinity . The limit beta =0 corresponds to the 1Dd Heisenberg antiferromagnet, the limit beta =1 corresponds to a Bethe ansatz integrable model and beta = infinity corresponds to pure antiferromagnetic biquadratic exchange. Haldane has predicted the existence of a singlet-triplet excitation gap in the thermodynamic limit for beta =0, whereas the beta =1 model is gapless. The authors have studied the singlet-triplet gap by scaled-gap (finite-size scaling) techniques, and it appears to vanish at an intermediate value of beta , with large uncertainty. Finite-system analytic studies show that a fundamental 'two-string' excitation picture describes the spectral excitations in the beta =1 limit, whereas a very different 'single-spin-deviation' picture characterises the excitations in the Heisenberg limit. The crossover between the two pictures shows interesting features and appears to be related to the disappearance of the singlet-triplet gap. The implication of an extended crossover region ( 1 / 3 < beta <0.62) is discussed in the context of non-integrability effects. Finally, they find the singlet-triplet gap reappears for beta >1.
We have performed a variety of numerical studies on the general bilinear-biquadratic spin-1 Hamiltonian H/J=∑Ni=1[Si⋅Si+1 −β(Si⋅Si+1)2], over the range 0≤β≤∞. The model is Bethe Ansatz integrable at the special point β=1, where the spectrum is gapless, but is otherwise believed to be nonintegrable. Affleck has predicted that an excitation gap opens up linearly in the vicinity of β=1. Our studies involving spectral excitations (dispersion spectra), scaled-gap, and finite-size scaling calculations are not consistent with the Affleck prediction. The situation appears complex, with novel crossover effects occurring in both regimes, β<1 and β>1, complicating the analysis.
An efficient valence-bond technique is introduced for the numerical study of singlet states of spin-1 antiferromagnetic chains and is applied to the general bilinear-biquadratic Hamiltonian. The results support the hypothesis that the gap first vanishes, with increasing biquadratic exchange, at the Bethe ansatz integrable point, which separates regular and dimerised regimes. The authors distinguish between real gaps and the mixing of degenerate ground states due to quantum tunnelling in small systems.
A comprehensive investigation has been made of the spectral excitations and static properties of Heisenberg antiferromagnetic chains of spin 1/2, 1, 3/2, and 2, using Lanczös, Bethe Ansatz, and Monte Carlo techniques. An unusual and unanticipated crossover mechanism for spin chains with 1/2≤S≤∞ has been discovered. The validity of the Haldane conjecture concerning the presence of a spectral excitation gap for integer-spin chains has been investigated by exact finite chains calculations of (a) the primary singlet-triplet excitation gap, (b) higher excitation gaps, and (c) the Fourier transform of the ground state correlation functions. A new Monte Carlo method has extended the spin-1 gap calculations to N=32.
The ground-state properties of the spin-1/2 Heisenberg chain, which has two exchange couplings (J>J') alternatingly, are studied. This model interpolates the spin-1 antiferromagnetic Heisenberg chain (AFHC) (J'\ensuremath{\rightarrow}-\ensuremath{\infty}, J>0), the dimerized spin-1/2 AFHC (J'\ensuremath{\simeq}J>0), and the uniform spin-1/2 AFHC (J'=J>0) continuously. The string order introduced by den Nijs--Rommelse and Tasaki changes continuously through the Haldane phase and the dimer phase to vanish at J=J'. It is at a maximum for J'=0. This implies that the string order is related to the local singlet correlation in terms of the spin-1/2 model. The lowest excitation can be regarded as the renormalized triplet wave. The reversal of the anisotropy effect on the excitation spectrum of the spin-1 AFHC is also explained in a intuitive way based on this picture.
Recently there has been extensive theoretical interest in the remarkable Haldane prediction that spin-1 antiferromagnetic Heisenberg-like chains should display an energy gap between singlet ground state and excited continua. This behavior has been confirmed numerically, and it has been shown that the energy gap is of a magnitude to be observable experimentally. The phenomenon has captured the interest of experimental groups in neutron scattering and magnetism. A recent experimental study obtains very good experimental agreement with the theoretical T=0 magnetization curve obtained numerically by Parkinson and Bonner.1,2 An explanation will be provided for the unusual phase diagram and other anomalous features observed by the authors of Ref. 2. Numerical susceptibility studies of Heisenberg antiferromagnetic and general XXZ spin-1 chains will be presented, and discussed in terms of a novel quantum-classical cross-over phenomenon.2 Finally, a comparison will be made of T=0 magnetization isotherms for several spin-1 chains, including the Takhtajan–Babujian chain and the chain with pure biquadratic interactions.
States of the pure biquadratic quantum spin chain with up to four deviations are calculated exactly for arbitrary chain length. The four-deviation states have a Bethe ansatz form and consist of two interacting two-strings. They have the same form as two-deviation states of an integrable S=1/2 XXZ model. It is found that states with more than four deviations also map onto corresponding S=1/2 states. The ground states of the two systems are not the same for finite N, but appear to become the same in the limit N to infinity .