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Journal of Physics Condensed Matter

A study of the bilinear-biquadratic spin-1 antiferromagnetic chain using the valence-bond basis

K Chang, Ian K. Affleck, Geoffrey W. Hayden, Z. G. Soos

Published Jan 9, 1989• 34 citations
Publisher

Abstract

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.

DOI
10.1088/0953-8984/1/1/014

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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.
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.
A Monte Carlo simulation study based on the heat-bath algorithm on a checkerboard arrangement is applied to the spin-1 antiferromagnetic chain model with both bilinear and biquadratic exchange interactions. Changing the parameter \ensuremath{\lambda} which denotes strength of the biquadratic interaction against the bilinear one from -1.0 to 2.0, we obtain the following results according to the measurements of energy and two-point correlation function. At \ensuremath{\lambda}=1 and -(1/3 where some physical quantities are derived rigorously, our Monte Carlo simulation results agree well with the rigorous ones. For ${\ensuremath{\lambda}}_{c}$\ensuremath{\le}\ensuremath{\lambda}\ensuremath{\le}${\ensuremath{\lambda}}_{1}$ with taking 0.25\ensuremath{\le}${\ensuremath{\lambda}}_{c}$\ensuremath{\le}0.5 and ${\ensuremath{\lambda}}_{1}$\ensuremath{\ge}2.0, its model has the gapless ground state accompanied by a power-law decaying correlation function. This implies the existence of a critical line. For -1.0${\ensuremath{\lambda}}_{c}$, on the other hand, its model has the ground state with energy gap accompanied with exponentially decaying correlation function. This indicates that the valence-bond-solid (VBS) state asserted by Affleck et al. is stabilized there. As the point of \ensuremath{\lambda}=0 comes to the standard Heisenberg antiferromagnetic model and is in the region of -1.0${\ensuremath{\lambda}}_{c}$, the Haldane's conjecture is supported that this ground state is described by the VBS state for the case of integer spin, e.g., S=1.
We have studied the Spin 1 antiferromagnetic chain with both bilinear and biquadratic exchange interactions. The spin correlations have been calculated by means of the transfer matrix method based on the Trotter formula. The mass gap ( m (0)) at the ground state has been investigated as a function of the ratio λ between bilinear and biquadratic interactions. The result suggests that the gapless ground state persists in a finite range of λ, i.e., λ c < λ< λ 1 , with λ c = 0.5 and λ 1 > 1 which is consistent with the exact solution at the completely integrable point (λ= 1). The calculated mass gap ( m (0)) has been fitted to a Kosterlitz-Thouless (KT)-type for λ< λ c ( m (0)= A exp [- B /(λ c - λ) σ ]) producing the results σ= 0.49±0.04 and λ c =0.49±0.03.
We study numerically the elementary excitation spectrum of the most general isotropic spin-1 chain with bilinear-biquadratic nearest-neighbor coupling. Using finite-size scaling, a massless phase with a period tripling in the ground state is found to exist in an extended region around the Lai-Sutherland point. The location of the transition from the known valence-bond-like phase to the trimerized phase cannot, however, be given precisely. We also present results on the ground-state two-point correlation function.
An exact correspondence between the staggered biquadratic spin-1 antiferromagnetic chain and the quantum Hamiltonian limit of the nine-state Potts model is established for finite chains with free ends. For uniform interactions this equivalence is used (via a further exact mapping to a Bethe ansatz soluble XXZ model) to calculate exactly the infinite lattice values of the ground-state energy per site and the (nonzero) gap to the lowest-energy excited state. Periodic boundary conditions and the nature of the ground state as a function of bond alternation are also discussed.
The bond operator representation of quantum S=12S=\frac{1}{2}S=21​ spins with the singlet state being projected out is applied to investigate the isotropic spin-1 Heisenberg antiferromagnetic chain $H={\ensuremath{\sum}}_{i}[{\mathbf{S}}_{i}\ensuremath{\cdot}{\mathbf{S}}_{i+1}\ensuremath{-}\ensuremath{\beta}({\mathbf{S}}_{i}\ensuremath{\cdot}{\mathbf{S}}_{i+1}{)}^{2}]$. In the approximation of Bose condensation, the ground-state properties of the system and their $\ensuremath{\beta}$ dependences are studied. The results at Heisenberg point $\ensuremath{\beta}=0$ and at valence-bond-solid (VBS) point $\ensuremath{\beta}=\ensuremath{-}\frac{1}{3}$ agree quite well with the analytical, numerical, and experimental studies. The excitation spectrum is nonsymmetric with respect to $k=\ensuremath{\pi}/2$ and shows a minimum at $k=\ensuremath{\pi}$. The $\ensuremath{\beta}$ dependences of the ground-state energy and the excitation gap show the similarities between the Heisenberg point and VBS point. The related properties of the spin ladders are also discussed.