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Journal of Applied Physics

Unusual critical behavior in a bilinear-biquadratic exchange Hamiltonian

J. C. Bonner, John B. Parkinson, J. Oitmaa, Henk W. J. Blöte

Published Apr 15, 1987• 9 citations
Publisher

Abstract

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.

DOI
10.1063/1.338400

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

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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.

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Affleck has made predictions for the critical exponents of a class of integrable spin chains of general spin $s$. The expressions are $s$ dependent. Our interest here is a finite-size scaling analysis of spin-1 systems to determine the validity of the prediction for the alternation critical exponent. We discuss these results in the light of our numerical results for the conformal anomaly $c$, the exponent $\ensuremath{\eta}$, and the mass-gap exponent associated with biquadratic exchange. The numerical results are not generally in good agreement with theory, and hence the possible presence of logarithmic corrections is discussed.

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Spin-1 biquadratic chain: Magnetization curve (abstract)

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The antiferromagnetic (AFM) s=1 chain with biquadratic exchange coupling has recently been the subject of considerable attention, and some controversy. Theoretical arguments by Affleck that the model has dimerized character associated with a small spectral excitation gap have not received unanimous support from numerical calculations. For finite systems the field-dependent dispersion spectra are interesting and informative concerning the nature of the model. The excitations divide into two classes having very different character, corresponding to SzT even and SzT odd. Reflection symmetry about k=π/2 is apparent if states with a given total spin ST, rather than SzT, are plotted as a function of wavevector k, consistent with the Affleck prediction of dimerized character. The high degeneracy of the eigenstates reflects the existence of new symmetry operators which (a) shift the SzT value, (b) change wave vector k to π/2−k, and (c) shift the total spin quantum ST, of the energy eigenstates. Parkinson has recently observed that finite N states with SzT even and SzT≥2 map into states of the finite N s= 1/2 XXZ model at a special value of uniaxial anisotropy. Barber and Batchelor (unpublished) have shown the existence of an analytic mapping between states of the s=1 biquadratic chain and the 2D 9-state Potts model. They further show that all states of the Parkinson s= 1/2 XXZ model map into states of the s=1 biquadratic model. Hence, they find the ground state and first excited state energies exactly, and verify the Affleck picture. Their approach falls short of establishing complete integrability, however. Numerical studies on finite chains indicate that the T=0 magnetization curve is determined by the lowest-energy states for given SzT even, which are states of the s= 1/2 XXZ chain. Hence the magnetization curve of the s=1 AFM biquadratic chain is the same as the analytically known magnetization curve of the s= 1/2 XXZ chain!

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Since the Heisenberg spin chain can be considered the simplest realistic model of magnetism, surprise and some degree of controversy have resulted from recent work of Haldane. The prediction is that quantum spin chains with half-integer spin should all display T=0 phase behavior equivalent to that of the Bethe Ansatz integrable (solvable) spin-1/2 quantum chain. More remarkably, the class of integer spin chains is predicted to show very different phase behavior. In particular, a gap should be present in the spectrum of a Heisenberg antiferromagnetic chain. This remarkable feature is counterintuitive in terms of accepted wisdom in magnetism (spin-wave theory, spin-Peierls theory) and critical phenomena. Consequently the vertification of the prediction is of great interest. A considerable amount of numerical work has been done, involving finite-chain, finite-size scaling, variational, Monte Carlo and other calculations, which will be reviewed here. The present consensus is that the weight of numerical evidence supports the prediction, although puzzling features still remain. Adding additional interactions to the basic Heisenberg Hamiltonian such as spin (XXZ) anisotropy, single-ion anisotropy, biquadratic exchange, and an applied magnetic field, generates a rich and complicated phase diagram for chains with spin >1/2, particularly for the case of integer spin. The s=1 phase diagram seems to display critical behavior of a type not previously encountered. A theoretical appraisal of the Haldane phenomenon will include a discussion of the possible role of nonintegrability. Mention will also be made of current progress in experimental investigation of the phenomenon, including problems that might be encountered. More recent work of Affleck has greatly generalized the field-theoretic mappings which underlay the original work of Haldane. A number of interesting problems have been mapped into quantum spin chains of various types, including field theoretic phenomena and the localization problem of the quantum Hall effect.

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S=12 chain-boundary excitations in the Haldane phase of one-dimensional S=1 systems

Eric Polizzi, Frédéric Mila, Erik Schwartz Sørensen

The $S=\frac{1}{2}$ chain-boundary excitations occurring in the Haldane phase of $S=1$ antiferromagnetic spin chains are investigated. The bilinear-biquadratic Hamiltonian is used to study these excitations as a function of the strength of the biquadratic term, \ensuremath{\beta}, between $\ensuremath{-}1<~\ensuremath{\beta}<~1.$ At the Affleck-Kennedy-Lieb-Tasaki point, $\ensuremath{\beta}=\ensuremath{-}\frac{1}{3},$ we show explicitly that these excitations are localized at the boundaries of the chain on a length scale equal to the correlation length $\ensuremath{\xi}=1/\mathrm{ln}3$, and that the on-site magnetization for the first site is $〈{S}_{1}^{z}〉=\frac{2}{3}.$ Applying the density matrix renormalization group we show that the chain-boundary excitations remain localized at the boundaries for $\ensuremath{-}1<\ensuremath{\beta}<1.$ As the two critical points \ensuremath{\beta}=\ifmmode\pm\else\textpm\fi{}1 are approached the size of the $S=\frac{1}{2}$ objects diverges and their amplitude appears to vanish.

arXiv:cond-mat/9802163•Physical review. B, Condensed matter•Aug 1, 1998•39 citations
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.
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.
Affleck has made predictions for the critical exponents of a class of integrable spin chains of general spin sss. The expressions are sss dependent. Our interest here is a finite-size scaling analysis of spin-1 systems to determine the validity of the prediction for the alternation critical exponent. We discuss these results in the light of our numerical results for the conformal anomaly ccc, the exponent $\ensuremath{\eta}$, and the mass-gap exponent associated with biquadratic exchange. The numerical results are not generally in good agreement with theory, and hence the possible presence of logarithmic corrections is discussed.
The antiferromagnetic (AFM) s=1 chain with biquadratic exchange coupling has recently been the subject of considerable attention, and some controversy. Theoretical arguments by Affleck that the model has dimerized character associated with a small spectral excitation gap have not received unanimous support from numerical calculations. For finite systems the field-dependent dispersion spectra are interesting and informative concerning the nature of the model. The excitations divide into two classes having very different character, corresponding to SzT even and SzT odd. Reflection symmetry about k=π/2 is apparent if states with a given total spin ST, rather than SzT, are plotted as a function of wavevector k, consistent with the Affleck prediction of dimerized character. The high degeneracy of the eigenstates reflects the existence of new symmetry operators which (a) shift the SzT value, (b) change wave vector k to π/2−k, and (c) shift the total spin quantum ST, of the energy eigenstates. Parkinson has recently observed that finite N states with SzT even and SzT≥2 map into states of the finite N s= 1/2 XXZ model at a special value of uniaxial anisotropy. Barber and Batchelor (unpublished) have shown the existence of an analytic mapping between states of the s=1 biquadratic chain and the 2D 9-state Potts model. They further show that all states of the Parkinson s= 1/2 XXZ model map into states of the s=1 biquadratic model. Hence, they find the ground state and first excited state energies exactly, and verify the Affleck picture. Their approach falls short of establishing complete integrability, however. Numerical studies on finite chains indicate that the T=0 magnetization curve is determined by the lowest-energy states for given SzT even, which are states of the s= 1/2 XXZ chain. Hence the magnetization curve of the s=1 AFM biquadratic chain is the same as the analytically known magnetization curve of the s= 1/2 XXZ chain!
Since the Heisenberg spin chain can be considered the simplest realistic model of magnetism, surprise and some degree of controversy have resulted from recent work of Haldane. The prediction is that quantum spin chains with half-integer spin should all display T=0 phase behavior equivalent to that of the Bethe Ansatz integrable (solvable) spin-1/2 quantum chain. More remarkably, the class of integer spin chains is predicted to show very different phase behavior. In particular, a gap should be present in the spectrum of a Heisenberg antiferromagnetic chain. This remarkable feature is counterintuitive in terms of accepted wisdom in magnetism (spin-wave theory, spin-Peierls theory) and critical phenomena. Consequently the vertification of the prediction is of great interest. A considerable amount of numerical work has been done, involving finite-chain, finite-size scaling, variational, Monte Carlo and other calculations, which will be reviewed here. The present consensus is that the weight of numerical evidence supports the prediction, although puzzling features still remain. Adding additional interactions to the basic Heisenberg Hamiltonian such as spin (XXZ) anisotropy, single-ion anisotropy, biquadratic exchange, and an applied magnetic field, generates a rich and complicated phase diagram for chains with spin >1/2, particularly for the case of integer spin. The s=1 phase diagram seems to display critical behavior of a type not previously encountered. A theoretical appraisal of the Haldane phenomenon will include a discussion of the possible role of nonintegrability. Mention will also be made of current progress in experimental investigation of the phenomenon, including problems that might be encountered. More recent work of Affleck has greatly generalized the field-theoretic mappings which underlay the original work of Haldane. A number of interesting problems have been mapped into quantum spin chains of various types, including field theoretic phenomena and the localization problem of the quantum Hall effect.
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 S=12S=\frac{1}{2}S=21​ chain-boundary excitations occurring in the Haldane phase of S=1S=1S=1 antiferromagnetic spin chains are investigated. The bilinear-biquadratic Hamiltonian is used to study these excitations as a function of the strength of the biquadratic term, \ensuremath{\beta}, between $\ensuremath{-}1<~\ensuremath{\beta}<~1.$ At the Affleck-Kennedy-Lieb-Tasaki point, $\ensuremath{\beta}=\ensuremath{-}\frac{1}{3},$ we show explicitly that these excitations are localized at the boundaries of the chain on a length scale equal to the correlation length $\ensuremath{\xi}=1/\mathrm{ln}3$, and that the on-site magnetization for the first site is 〈S1z〉=23.〈{S}_{1}^{z}〉=\frac{2}{3}.〈S1z​〉=32​. Applying the density matrix renormalization group we show that the chain-boundary excitations remain localized at the boundaries for $\ensuremath{-}1<\ensuremath{\beta}<1.$ As the two critical points \ensuremath{\beta}=\ifmmode\pm\else\textpm\fi{}1 are approached the size of the S=12S=\frac{1}{2}S=21​ objects diverges and their amplitude appears to vanish.