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Journal of Applied Physics

Critical behavior of the S=1 antiferromagnetic Heisenberg chain (abstract)

Jill C. Bonner

Published Apr 15, 1991• 1 citation
Publisher

Abstract

The spectral properties, and associated critical singularities, of spin-1 Heisenberg antiferromagnetic chains are of high current interest, both theoretical and experimental. Theoretical studies using the single mode approximation and also a Monte Carlo calculation indicate that the spectrum has a gap at k=π, as predicted by Haldane. They also predict the unusual feature that the excited state continuum is asymmetric about the mid-point of the Brillouin zone, k=π/2. However, neither approach is sufficiently accurate to predict reliably the magnitude of the gap at k=0 which was not discussed by Haldane. Our approach is to use static (spectral) calculations of excited states of chains of N=4 through 14 spins for all values of k, a novel feature not associated with previous finite-size calculations, to discuss the dispersion spectra of the S=1 Heisenberg chain. In addition, we have calculated dynamical properties, in particular S(q,ω), exactly for finite chains up to N=14 spins. This allows us to estimate the gap at k=0 to good accuracy, and we find that to within this accuracy the gap at k=0 is twice the Haldane gap at k=π. This result is important in connection with very recent neutron scattering and magnetization measurements. In addition, we have calculated the correlation length and critical singularities associated with dynamical quantities.

DOI
10.1063/1.347889

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Physical review. B, Condensed matter•May 1, 1994•7 citations
The spectral properties, and associated critical singularities, of spin-1 Heisenberg antiferromagnetic chains are of high current interest, both theoretical and experimental. Theoretical studies using the single mode approximation and also a Monte Carlo calculation indicate that the spectrum has a gap at k=π, as predicted by Haldane. They also predict the unusual feature that the excited state continuum is asymmetric about the mid-point of the Brillouin zone, k=π/2. However, neither approach is sufficiently accurate to predict reliably the magnitude of the gap at k=0 which was not discussed by Haldane. Our approach is to use static (spectral) calculations of excited states of chains of N=4 through 14 spins for all values of k, a novel feature not associated with previous finite-size calculations, to discuss the dispersion spectra of the S=1 Heisenberg chain. In addition, we have calculated dynamical properties, in particular S(q,ω), exactly for finite chains up to N=14 spins. This allows us to estimate the gap at k=0 to good accuracy, and we find that to within this accuracy the gap at k=0 is twice the Haldane gap at k=π. This result is important in connection with very recent neutron scattering and magnetization measurements. In addition, we have calculated the correlation length and critical singularities associated with dynamical quantities.
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.
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.
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.
The spin correlations of the S=1 Heisenberg antiferromagnetic chains in AgVP2{\mathrm{AgVP}}_{2}AgVP2​S6{\mathrm{S}}_{6}S6​ have been studied by neutron scattering at temperatures of the order of T\ensuremath{\sim}${10}^{\mathrm{\ensuremath{-}}2}$ J/kB{\mathit{k}}_{\mathit{B}}kB​. On the polycrystalline samples, with a careful choice of experimental conditions, we have observed a Haldane gap of Eg{\mathit{E}}_{\mathit{g}}Eg​=26 meV at the antiferromagnetic point at q=\ensuremath{\pi}. The excitation spectra are consistent with a mode that has a spin-wave velocity C=150 meV and a correlation length \ensuremath{\xi}\ensuremath{\approxeq}5.5, close to C/Eg{\mathit{E}}_{\mathit{g}}Eg​. These results are in quantitative agreement with the predictions and numerical results on the Haldane state.
We develop an extended self-consistent mean-field approach to study a spin-1 antiferromagnetic (AF) chain with the Hamiltonian H=tsumi{\mathit{tsum}}_{\mathit{i}}tsumi​ [Six{\mathit{S}}_{\mathit{i}}^{\mathit{x}}Six​Si+1x{\mathit{S}}_{\mathit{i}+1}^{\mathit{x}}Si+1x​+Siy{\mathit{S}}_{\mathit{i}}^{\mathit{y}}Siy​Si+1y{\mathit{S}}_{\mathit{i}+1}^{\mathit{y}}Si+1y​ +\ensuremath{\lambda}Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​Si+1z{\mathit{S}}_{\mathit{i}+1}^{\mathit{z}}Si+1z​+D(Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​)2{)}^{2})2] based on Anderson's spin-wave theory. The ground-state energy changes little compared with the traditional spin-wave theory and the excitation spectrum shows a gap which lends support to Haldane's conjecture. At the Heisenberg point (\ensuremath{\lambda}=1,D=0), the ground-state energy and the gap are -1.3505 and 0.3914, respectively. In the Heisenberg-Ising case (\ensuremath{\lambda}>0,D=0), the gap decreases when \ensuremath{\lambda} goes down from 1 and at a critical ${\ensuremath{\lambda}}_{\mathit{c}}$>0 the gap goes to 0, while for the single-ion anisotropy (\ensuremath{\lambda}=1,D>0) the gap decreases with the increasing of D and at a critical value Dc{\mathit{D}}_{\mathit{c}}Dc​ the gap goes to 0. These results agree well with those of other theoretical methods and experiments. We study the one-magnon excitation spectrum carefully and suggest an explanation for its crossing with the two-magnon excitation spectrum at low momentum. We estimate the crossing occurring at k\ensuremath{\simeq}0.3\ensuremath{\pi}. The results agree well with the very recent theoretical and experimental results. The picture of the Haldane gap is discussed from our extended self-consistent mean-field ground state and many interesting properties of the spin-1 AF chain are discussed.