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Physical Review Letters

Experimental evidence for the Haldane gap in a spin-1 nearly isotropic, antiferromagnetic chain

W. J. L. Buyers, Rose M. Morra, Robin L. Armstrong, M. J. Hogan, P. N. Gerlach, and K. Hirakawa

Published Jan 27, 1986• 433 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.56.371

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Physical review. B, Condensed matter•Mar 1, 1992•58 citations
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.
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.
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.
Since Haldane conjectured that ground state of one-dimensional Heisenberg antiferromagnet has a finite spin gap for integer spins, while gapless excitations for half-odd integer spins, it has inspired lots of theoretical and experimental studies on low-dimensional quantum magnets. Several one-dimensional chain systems of integer spins, so called Haldane chain, have been discovered and studied, such as CsNiCl3, Y2BaNiO5, PbNi2V2O8, SrNi2V2O8, AgVP2S6. It has been known that most of the Haldane chain has antiferromagnetic coupling for the intrachain exchange interaction. Recently, we have found a Haldane chain compound in which antiferromagnetic order is accompanied by antiferromagnetic interchain coupling between one-dimensional ferromagnetic chains below 30.5 Kelvin. In this presentation, we present comprehensive study of its magnetic and thermal properties.
We consider a quasi-one-dimensional system of spin-1 Heisenberg antiferromagnetic chains in two-dimensional and three-dimensional hypercubic lattices with interchain coupling J and uniaxial single-ion anisotropy D. Using large-scale numerical simulations, we map out the J-D phase diagram and investigate the low-lying excitations of the Haldane phase in the J≪1 limit. We also provide direct evidence that the Haldane phase remains a nontrivial symmetry-protected topological state for small but finite J.
Two quasi-one-dimensional S=1S=1S=1 quantum antiferromagnetic materials, PbNi2V2O8{\mathrm{PbNi}}_{2}{\mathrm{V}}_{2}{\mathrm{O}}_{8}PbNi2​V2​O8​ and SrNi2V2O8,{\mathrm{SrNi}}_{2}{\mathrm{V}}_{2}{\mathrm{O}}_{8},SrNi2​V2​O8​, are studied by inelastic neutron scattering on powder samples. A simple model Hamiltonian is proposed for the two systems and used to analyze the data. Quantitative information on in-chain and interchain interactions obtained from this analysis is discussed in the context of qualitatively different ground-state properties: magnetic long-range order in SrNi2V2O8{\mathrm{SrNi}}_{2}{\mathrm{V}}_{2}{\mathrm{O}}_{8}SrNi2​V2​O8​ and disordered ``spin liquid'' Haldane-gap state in PbNi2V2O8.{\mathrm{PbNi}}_{2}{\mathrm{V}}_{2}{\mathrm{O}}_{8}.PbNi2​V2​O8​.
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.