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Physical Review Letters

Dynamics of an S=12 , One-Dimensional Heisenberg Antiferromagnet

Y. Endoh, Gen Shirane, R. J. Birgeneau, Peter M. Richards, S. L. Holt

Published Jan 28, 1974• 182 citations
Publisher

Abstract

We report a detailed neutron-scattering study of the spin dynamics in Cu${\mathrm{Cl}}_{2}$\ifmmode\cdot\else\textperiodcentered\fi{}2N(${\mathrm{C}}_{5}$${\mathrm{D}}_{5}$), a physical realization of the one-dimensional $S=\frac{1}{2}$ Heisenberg antiferromagnet. At $T=1.3$ K well-defined excitations are observed over the whole zone with energies given by $E(q)=\ensuremath{\pi}{J}_{\mathrm{nn}}sin(\mathrm{qc})$, the celebrated des Cloizeaux-Pearson exact solution for the spectrum of first excited states, but with intensities approximately those expected from classical spin-wave theory. At $T=8$ K the excitations are broad and ill defined.

DOI
10.1103/physrevlett.32.170

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We report a detailed neutron-scattering study of the spin dynamics in CuCl2{\mathrm{Cl}}_{2}Cl2​\ifmmode\cdot\else\textperiodcentered\fi{}2N({\mathrm{C}}_{5}$${\mathrm{D}}_{5}), a physical realization of the one-dimensional S=12S=\frac{1}{2}S=21​ Heisenberg antiferromagnet. At T=1.3T=1.3T=1.3 K well-defined excitations are observed over the whole zone with energies given by E(q)=\ensuremathπJnnsin(qc)E(q)=\ensuremath{\pi}{J}_{\mathrm{nn}}sin(\mathrm{qc})E(q)=\ensuremathπJnn​sin(qc), the celebrated des Cloizeaux-Pearson exact solution for the spectrum of first excited states, but with intensities approximately those expected from classical spin-wave theory. At T=8T=8T=8 K the excitations are broad and ill defined.
Well-defined spin-wave excitations were observed by using pulsed neutrons from CsVCl3{\mathrm{l}}_{3}l3​, an S=3/2S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3/2S=3/2 Heisenberg antiferromagnetic chain. The observed spin-wave dispersion relation shows an enhancement of the excitation energy by a factor of 1.26, which must be compared with the quantum renormalization factor of $\ensuremath{\pi}/2$ for an S=1/2S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2S=1/2 system. The linewidth of the spectrum at 40 K was broad and independent of qqq. The qqq-independent linewidth is the first experimental observation, which was predicted by rigorous theories for a classical chain at sufficiently low temperature.
The dynamic-spin-correlation function S(q,\ensuremath{\omega}) is computed for the one-dimensional antiferromagnetic Heisenberg model for S=1/2 and S=1. Imaginary-time correlation functions are evaluated with the quantum Monte Carlo method and the maximum-entropy method is used to continue these results to real frequencies. This technique produces a value for the S=1 Haldane gap that agrees with previous work and also gives a good description of inelastic neutron scatter- ing data for the one-dimensional S=1/2 antiferromagnet CuCl2{\mathrm{CuCl}}_{2}CuCl2​\ensuremath{\cdot}2N(C5{\mathrm{C}}_{5}C5​D5{\mathrm{D}}_{5}D5​).
The one-dimensional Heisenberg antiferromagnet with 8=1/2 is treated with the use of the Fermi operators and an additional condition that the long-range order is absent.Our spin wave spectrum of a one-dimensional Heisenberg antiferromagnet without longrange order is expressed by andThis is in good agreement with the exact spin wave spectrum role =1!'/2lsin kl which is obtained by des Cloizeaux and Pearson at T=O.Correlation functions <S1S2> and <SlSs> are in good agreement with the exact value at T=O, also.Thermodynamic properties of this system agree with those of Bulaevskii and those of Katsura and Inawashiro. § I. IntroductionThe one-dimensional Heisenberg antiferromagnet has been investigated by many authors.Most of previous theories have been for either zero or infinite temperatures.Recently various experiment on (CD3) 4NMnCl 3 (TMMC)ll requires further theoretical work of a Heisenberg linear-chain magnet for finite temperatures.The experimental results of linear antiferromagnet show that at low temperatures there are long-lived spin waves in spite of the absence of long-range order.At T= 0, the spin wave spectrum, wqoc I sin ql, is gained approximately by Anderson 2 l and exactly by des Cloizeaux and Pearson.8 l Recently Richards, 4 l Scales and Gersh, 5 l Kondo and Yamaji 6 l obtained the spin wave spectrum of linear antiferromagnet at finite temperatures with the use of the second-order Green's function theory.The spin wave spectrum, however, does not show exactly wqoc I sin ql at finite temperatures.
Recent low-temperature inelastic neutron scattering data on the quasi-one-dimensional (1D) Heisenberg-Ising antiferromagnet (AFM) CsCoCl3{\mathrm{Cl}}_{3}Cl3​ call for new theoretical efforts, for the available theories fail to explain the observed line shapes. In this paper we initiate a detailed study of the dynamical properties of the 1D spin-\textonehalf{} anisotropic XYZ\mathrm{XYZ}XYZ AFM with emphasis on the dynamic structure factor ${S}_{\ensuremath{\mu}\ensuremath{\mu}}(q,\ensuremath{\omega})$ at T=0T=0T=0. The structure of ${S}_{\ensuremath{\mu}\ensuremath{\mu}}(q,\ensuremath{\omega})$ is determined by the combined effects of the density of states and the matrix elements. Both effects are discussed for selected cases: the XY\mathrm{XY}XY-Heisenberg AFM (planar system), the Heisenberg-Ising AFM (uniaxial system), and the XY\mathrm{XY}XY-Ising AFM (biaxial system). The exact solution for ${S}_{\mathrm{zz}}(q,\ensuremath{\omega})$ of the XY\mathrm{XY}XY-Ising AFM has already many features in common with the experimental data on CsCoCl3{\mathrm{Cl}}_{3}Cl3​. It is an ideal starting point for a perturbation approach to the Heisenberg-Ising AFM.
Inelastic neutron scattering experiments investigating the spin dynamics along the chain direction of an S = 3 / 2 quasi-one-dimensional Heisenberg antiferromagnet CsVCl 3 were performed up to the zone boundary energy. The authors observed the quantum renormalization in the exchange constant and found that the scattering function cannot be described by either the classical delta -function or the quantum Heisenberg model for S = 1 / 2 . The spin excitations have apparent linewidths and they are compared with the theory of the classical isotropic Heisenberg system at finite temperatures.
We have examined numerical calculations on a spin 1/2 one‐dimensional (1D) Heisenberg antiferromagnet for N=8 spins, both for the states and their spectral weights. This work is of interest because of the recent neutron scattering experiments in CuCl2⋅2NC5D5 which show the spectral weight to be concentrated at the des Cloizeaux and Pearson (dCP) spin wave frequencies. The states form a two‐parameter continuum between the dCp value E1(k) =π‖J‖ ‖sin k‖ and the upper limit E2(k) =2π‖J‖ ‖ sin(k/2) ‖. Though most of the spectral weight is at the dCP frequency for N=8, about 10% of it is found near E2 for k=π. There is no numerical evidence for the existence of singlet bound states as predicted by Ovchinnikov.