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Physical review. B, Condensed matter

Spin dynamics and the Haldane gap in the spin-1 quasi-one-dimensional antiferromagnet CsNiCl3

Rose M. Morra, W. J. L. Buyers, Robin L. Armstrong, K. Hirakawa

Published Jul 1, 1988• 189 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.38.543

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A full description of a series of spin-wave measurements in CsNiCl3{\mathrm{CsNiCl}}_{3}CsNiCl3​ 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 CsNiCl3{\mathrm{CsNiCl}}_{3}CsNiCl3​ 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 CsNiCl3{\mathrm{CsNiCl}}_{3}CsNiCl3​ 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 Ni2+{\mathrm{Ni}}^{2+}Ni2+ 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 CsNiCl3{\mathrm{CsNiCl}}_{3}CsNiCl3​ 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.
We study a Heisenberg antiferromagnetic chain of spin S=1 with nearest-neighbor exchange interaction J and easy-plane single-ion anisotropy D. We follow the splitting of the Haldane gap into two components as a function of the anisotropy. Chains of length up to 16 sites are studied by a Lancz\"os method and longer chains up to 32 sites have been investigated by a quantum Monte Carlo simulation for D/J=0.10, 0.15, 0.20, and 0.25. A careful finite-size scaling treatment of our data leads to values that are directly relevant to the interpretation of experiments on Ni(C2{\mathrm{C}}_{2}C2​H8{\mathrm{H}}_{8}H8​N2{\mathrm{N}}_{2}N2​)2{)}_{2})2​NO2{\mathrm{NO}}_{2}NO2​ClO4{\mathrm{ClO}}_{4}ClO4​ (NENP). We discuss the relationship between our results and the magnetic-susceptibility measurements as well as the inelastic-neutron-scattering data and on the Haldane conjecture. Values are proposed for the parameters D and J in the case of NENP.
For the antiferromagnetically coupled integer-spin Ni2+{\mathrm{Ni}}^{2+}Ni2+ chains of CsNiCl3{\mathrm{CsNiCl}}_{3}CsNiCl3​ 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 Ni2+{\mathrm{Ni}}^{2+}Ni2+ 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.
We report inelastic time-of-flight and triple-axis neutron scattering measurements of the excitation spectrum of the coupled antiferromagnetic spin-1 Heisenberg chain system CsNiCl3.{\mathrm{CsNiCl}}_{3}.CsNiCl3​. Measurements over a wide range of wave-vector transfers along the chain confirm that above TNCsNiCl3{T}_{N} {\mathrm{CsNiCl}}_{3}TN​CsNiCl3​ is in a quantum-disordered phase with an energy gap in the excitation spectrum. The spin correlations fall off exponentially with increasing distance with a correlation length $\ensuremath{\xi}=4.0(2)$ sites at T=6.2K.T=6.2\mathrm{K}.T=6.2K. This is shorter than the correlation length for an antiferromagnetic spin-1 Heisenberg chain at this temperature, suggesting that the correlations perpendicular to the chain direction and associated with the interchain coupling lower the single-chain correlation length. A multiparticle continuum is observed in the quantum-disordered phase in the region in reciprocal space where antiferromagnetic fluctuations are strongest, extending in energy up to twice the maximum of the dispersion of the well-defined triplet excitations. We show that the continuum satisfies the Hohenberg-Brinkman sum rule. The dependence of the multiparticle continuum on the chain wave vector resembles that of the two-spinon continuum in antiferromagnetic spin-1/2 Heisenberg chains. This suggests the presence of spin-1/2 degrees of freedom in CsNiCl3{\mathrm{CsNiCl}}_{3}CsNiCl3​ for T< 12K,T<~12 \mathrm{K},T< 12K, possibly caused by multiply frustrated interchain interactions.
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.
The authors study the dynamic spin correlation function of a spin-1 antiferromagnetic chain with easy-plane single-ion anisotropy. They use exact diagonalization by the Lanczos method for chains of length up to N=16 spins. They show that a single-mode approximation is an excellent description of the dynamical properties. A variational calculation allows them to clarify the nature of the excitations. The existence of a two-particle continuum near-zero wavevector is clearly seen, both in finite-size effects and in the dynamical structure factor. The recent neutron scattering experiments on the quasi-one-dimensional antiferromagnet NENP are fully explained by the results.