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Journal of the Physical Society of Japan

Haldane Gap and the Spin 1 Antiferromagnetic Integrable Model

Kazuo Saitoh, Satoshi Takada, Kenn Kubo

Published Nov 15, 1987• 13 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.56.3755

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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.
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.
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.
We present evidence for the absence of a gap in a class of S = 1 antiferromagnetic exchange models. The spin exchange is long ranged, of the type where . We have shown previously that without the alternating factor the model for (the S = 1 Haldane - Shastry model) has a gap, exponentially decaying correlation functions and exponentially small susceptibility at very low temperatures. In the case of the alternating interaction the stabilizing next-nearest-neighbour ferromagnetic interaction changes the behaviour of the system qualitatively. We have studied the ground state and first excited state using a modified Lanczos algorithm for system sizes up to 16 sites. Also, we performed exact diagonalization for systems up to 8 sites and obtained the thermodynamics. The correlation functions decay with distance like a power law. These models define a new class of integer spin chains that do not show a Haldane gap. The results may be relevant for describing impurity spins coupled by a RKKY interaction through a half-filled conduction electron band.
In a previous paper (Liu et al 2012 Phys. Rev. B 85 195144 ), a variational Monte Carlo method (based on Gutzwiller projected states) was generalized to S = 1 systems. This method provided very good trial ground states for the gapped phases of an S = 1 bilinear-biquadratic (BLBQ) Heisenberg chain. In this paper, we extend the approach to study the low-lying elementary excitations in S = 1 chains. We calculate the one-magnon and two-magnon excitation spectra of the BLBQ Heisenberg chain and the results agree very well with recent data in the literature. In our approach, the difference of the excitation spectrum between the Haldane phase and the dimer phase (such as the even/odd size effect) can be understood from their different topologies of the corresponding mean field theory. We especially study the Takhtajan–Babujian critical point. Despite the fact that the 'elementary excitations' are spin-1 magnons, which are different from the spin-1/2 spinons in Bethe solution, we show that the excitation spectrum, critical exponent ( ) and central charge ( c = 1.45) calculated from our theory agree well with the Bethe ansatz solution and conformal field theory predictions.
We have studied anisotropic Heisenberg antiferromagnets by using a real space renormalization-group analysis and exact diagonalization approach. For the spin-1 case, the ground-state energy and Haldane gap at the Heisenberg point are estimated, and they are in good agreement with that obtained by other methods. The critical point is identified where the gap phase changes into a gapless phase. For spin-2 case, the calculated Haldane gap supports a recent prediction suggested by a quantum O(N) nonlinear sigma model study.