Physical review. B, Condensed matter

Theory of near-zero-wave-vector neutron scattering in Haldane-gap antiferromagnets

Ian K. Affleck, Robert Weston

Published Mar 1, 1992• 58 citations

Abstract

One-dimensional integer-spin Heisenberg antiferromagnets have disordered ground states and a gap to a triplet magnon near the antiferromagnetic wave vector, k\ensuremath{\approxeq}\ensuremath{\pi}. Near a zero wave vector the lowest energy excitation is a pair of magnons. We calculate the neutron-scattering cross section near k=0, using a Landau-Ginsburg model and exact S-matrix results for the O(3) nonlinear \ensuremath{\sigma} model. The cross section is proportional to ${\mathit{k}}^{2}$. As a function of energy, it shows a rounded peak somewhat above the two-magnon threshold. The effects of anisotropy are also considered.

DOI: 10.1103/physrevb.45.4667

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Physical review. B, Condensed matter

Theory of near-zero-wave-vector neutron scattering in Haldane-gap antiferromagnets

Ian K. Affleck, Robert Weston

Published Mar 1, 1992• 58 citations

Publisher

Abstract

{\mathit{k}}^{2}

. As a function of energy, it shows a rounded peak somewhat above the two-magnon threshold. The effects of anisotropy are also considered.

DOI: 10.1103/physrevb.45.4667

Related papers

σ-model study of Haldane-gap antiferromagnets

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We use a quantum O(N) nonlinear \ensuremath{\sigma} model to describe one-dimensional quantum Heisenberg antiferromagnets with integer spin S. A large-N analytical treatment of this effective theory leads to definite predictions for thermodynamic quantities in the low-temperature regime. We obtain the Haldane gap as a function of the temperature and note a trend similar to that found in the compound NENP. The specific heat and magnetic susceptibility are also obtained and comparison with exact numerical results for the S=1 spin chain allows us to determine the domain of validity of our approximation. By a calculation at subleading 1/N order we compute the quasiparticle lifetime. Results are given on-shell, for the width observed at the quasiparticle pole as seen in neutron-scattering experiments, and also off-shell, for the NMR relaxation rate ${\mathit{T}}_{1}^{\mathrm{\ensuremath{-}}1}$. The phenomenological interacting boson model is compared with our approach. We discuss the effects of anisotropy and compare our results with present experiments on NENP.We use a quantum O(N) nonlinear \ensuremath{\sigma} model to describe one-dimensional quantum Heisenberg antiferromagnets with integer spin S. A large-N analytical treatment of this effective theory leads to definite predictions for thermodynamic quantities in the low-temperature regime. We obtain the Haldane gap as a function of the temperature and note a trend similar to that found in the compound NENP. The specific heat and magnetic susceptibility are also obtained and comparison with exact numerical results for the S=1 spin chain allows us to determine the domain of validity of our approximation. By a calculation at subleading 1/N order we compute the quasiparticle lifetime. Results are given on-shell, for the width observed at the quasiparticle pole as seen in neutron-scattering experiments, and also off-shell, for the NMR relaxation rate ${\mathit{T}}_{1}^{\mathrm{\ensuremath{-}}1}$. The phenomenological interacting boson model is compared with our approach. We discuss the effects of anisotropy and compare our results with present experiments on NENP.

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A full description of a series of spin-wave measurements in ${\mathrm{CsNiCl}}_{3}$ in the three-dimensional (3D) and one-dimensional (1D) phases and their analysis to provide the first experimental evidence for the Haldane gap is presented. Neutron scattering experiments were performed on a single crystal of ${\mathrm{CsNiCl}}_{3}$ with both (h,0,l) and (h,h,l) as the scattering plane. The order parameter in both 3D phases was measured and the spin-wave dispersion determined in the lower phase. The spin-wave spectrum calculated from a dynamic susceptibility method was compared with the experimental response in the lower 3D phase to obtain the following values for the exchange and anisotropy constants: J=0.345\ifmmode\pm\else\textpm\fi{}0.008 THz, J'=0.0060\ifmmode\pm\else\textpm\fi{}0.005 THz, and D=-0.0130\ifmmode\pm\else\textpm\fi{}0.0015 THz. These parameters confirm that ${\mathrm{CsNiCl}}_{3}$ in its disordered phase is a good approximation to a one-dimensional Heisenberg antiferromagnet. In the 1D phase the gap frequency for an isolated chain of ${\mathrm{Ni}}^{2+}$ ions is found to be 0.32 THz close to the gap estimated in finite-chain calculations. The results constitute experimental support for the Haldane conjecture that the excitations of integer-spin chains, unlike half-integer--spin chains, exhibit an apparent anisotropy that arises not from the underlying isotropic Hamiltonian, but from many-body effects.A full description of a series of spin-wave measurements in ${\mathrm{CsNiCl}}_{3}$ in the three-dimensional (3D) and one-dimensional (1D) phases and their analysis to provide the first experimental evidence for the Haldane gap is presented. Neutron scattering experiments were performed on a single crystal of ${\mathrm{CsNiCl}}_{3}$ with both (h,0,l) and (h,h,l) as the scattering plane. The order parameter in both 3D phases was measured and the spin-wave dispersion determined in the lower phase. The spin-wave spectrum calculated from a dynamic susceptibility method was compared with the experimental response in the lower 3D phase to obtain the following values for the exchange and anisotropy constants: J=0.345\ifmmode\pm\else\textpm\fi{}0.008 THz, J'=0.0060\ifmmode\pm\else\textpm\fi{}0.005 THz, and D=-0.0130\ifmmode\pm\else\textpm\fi{}0.0015 THz. These parameters confirm that ${\mathrm{CsNiCl}}_{3}$ in its disordered phase is a good approximation to a one-dimensional Heisenberg antiferromagnet. In the 1D phase the gap frequency for an isolated chain of ${\mathrm{Ni}}^{2+}$ ions is found to be 0.32 THz close to the gap estimated in finite-chain calculations. The results constitute experimental support for the Haldane conjecture that the excitations of integer-spin chains, unlike half-integer--spin chains, exhibit an apparent anisotropy that arises not from the underlying isotropic Hamiltonian, but from many-body effects.

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Physical Review LettersJan 27, 1986433 citations

Dynamic structure factor [S(Q,ω)] of theS=1 quasi-one-dimensional Heisenberg antiferromagnet: Neutron-scattering study on AgVP2 S6

H. Mutka, C. Payen, P. Molinié +3 more

The spin correlations of the S=1 Heisenberg antiferromagnetic chains in ${\mathrm{AgVP}}_{2}$ ${\mathrm{S}}_{6}$ have been studied by neutron scattering at temperatures of the order of T\ensuremath{\sim}${10}^{\mathrm{\ensuremath{-}}2}$ J/ ${\mathit{k}}_{\mathit{B}}$ . On the polycrystalline samples, with a careful choice of experimental conditions, we have observed a Haldane gap of ${\mathit{E}}_{\mathit{g}}$ =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/ ${\mathit{E}}_{\mathit{g}}$ . These results are in quantitative agreement with the predictions and numerical results on the Haldane state.The spin correlations of the S=1 Heisenberg antiferromagnetic chains in ${\mathrm{AgVP}}_{2}$${\mathrm{S}}_{6}$ have been studied by neutron scattering at temperatures of the order of T\ensuremath{\sim}${10}^{\mathrm{\ensuremath{-}}2}$ J/${\mathit{k}}_{\mathit{B}}$. On the polycrystalline samples, with a careful choice of experimental conditions, we have observed a Haldane gap of ${\mathit{E}}_{\mathit{g}}$=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/${\mathit{E}}_{\mathit{g}}$. These results are in quantitative agreement with the predictions and numerical results on the Haldane state.

Physical Review LettersJul 22, 199175 citations

{\mathit{k}}^{2}

. As a function of energy, it shows a rounded peak somewhat above the two-magnon threshold. The effects of anisotropy are also considered.

For the antiferromagnetically coupled integer-spin

{\mathrm{Ni}}^{2+}

chains of

{\mathrm{CsNiCl}}_{3}

the Haldane gap is found by neutron scattering to be a singlet-to-triplet spin excitation. It has a 2\ensuremath{\pi} periodicity in momentum space with a dispersion relation that falls to a value of order twice the gap energy as Q\ensuremath{\rightarrow}2\ensuremath{\pi}. When the coupled

{\mathrm{Ni}}^{2+}

chains undergo the transition to three-dimensional order, a set of longitudinal modes is found to occur that are not present in conventional theory where the excitations are transverse spin waves. Our calculations of the intensities show that near the ordering wave vector (1/3, 1)/3 ,1) the new modes agree with the field theory of Affleck but not with spin-wave theory. At larger interchain wave vectors both field theory and spin-wave theory break down.

{\mathrm{CsNiCl}}_{3}.

A full description of a series of spin-wave measurements in

{\mathrm{CsNiCl}}_{3}

in the three-dimensional (3D) and one-dimensional (1D) phases and their analysis to provide the first experimental evidence for the Haldane gap is presented. Neutron scattering experiments were performed on a single crystal of

{\mathrm{CsNiCl}}_{3}

with both (h,0,l) and (h,h,l) as the scattering plane. The order parameter in both 3D phases was measured and the spin-wave dispersion determined in the lower phase. The spin-wave spectrum calculated from a dynamic susceptibility method was compared with the experimental response in the lower 3D phase to obtain the following values for the exchange and anisotropy constants: J=0.345\ifmmode\pm\else\textpm\fi{}0.008 THz, J'=0.0060\ifmmode\pm\else\textpm\fi{}0.005 THz, and D=-0.0130\ifmmode\pm\else\textpm\fi{}0.0015 THz. These parameters confirm that

{\mathrm{CsNiCl}}_{3}

in its disordered phase is a good approximation to a one-dimensional Heisenberg antiferromagnet. In the 1D phase the gap frequency for an isolated chain of

{\mathrm{Ni}}^{2+}

ions is found to be 0.32 THz close to the gap estimated in finite-chain calculations. The results constitute experimental support for the Haldane conjecture that the excitations of integer-spin chains, unlike half-integer--spin chains, exhibit an apparent anisotropy that arises not from the underlying isotropic Hamiltonian, but from many-body effects.

Neutron scattering has shown that the lowest spin excitations in the spin-1 antiferromagnet

{\mathrm{CsNiCl}}_{3}

in its one-dimensional phase occur at finite frequency. After allowance for the known weak interchain coupling the gap is found to be in good agreement with recent numerical calculations. The results support the Haldane conjecture that in integral-spin chains with isotropic interactions a gap separates the ground state from the excitations.

The spin correlations of the S=1 Heisenberg antiferromagnetic chains in

{\mathrm{AgVP}}_{2}

{\mathrm{S}}_{6}

have been studied by neutron scattering at temperatures of the order of T\ensuremath{\sim}${10}^{\mathrm{\ensuremath{-}}2}$ J/

{\mathit{k}}_{\mathit{B}}

. On the polycrystalline samples, with a careful choice of experimental conditions, we have observed a Haldane gap of

{\mathit{E}}_{\mathit{g}}

=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/

{\mathit{E}}_{\mathit{g}}

. These results are in quantitative agreement with the predictions and numerical results on the Haldane state.