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Physical review. B, Condensed matter

Haldane-gap modes in theS=1 antiferromagnetic chain compound CsNiCl3

Z. Tun, W. J. L. Buyers, Robin L. Armstrong, K. Hirakawa, Bernard Briat

Published Sep 1, 1990• 74 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.42.4677

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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.
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.
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 k2{\mathit{k}}^{2}k2. As a function of energy, it shows a rounded peak somewhat above the two-magnon threshold. The effects of anisotropy are also considered.
A continuum of magnetic states has been observed by neutron scattering from the spin-1 chain compound CsNiCl3 in its disordered gapped one-dimensional phase. Results using both triple-axis and time-of-flight spectrometers show that around the antiferromagnetic point Qc = pi, the continuum lies higher in energy than the Haldane gapped excitations. At 6 K the integrated intensity of the continuum is about 12(2)% of the total spectral weight. This result is considerably larger than the 1%-3% weight predicted by the nonlinear sigma model for the three-particle continuum.
Integer-spin Heisenberg antiferromagnetic chains are expected to have a singlet groundstate and a gap to the first excited state, as first argued by Haldane. We review rigorous results on this conjecture and discuss experimental realizations. Theoretical and experimental aspects of three recent developments in this field are then reviewed: chainend excitations and the non-local order parameter, a field theory mapping and behavior near zero wave-vector, high-field behavior and one-dimensional Bose condensation.