spinons and spin waves in one-dimensional heisenberg antiferromagnets

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Page 1: Spinons and spin waves in one-dimensional Heisenberg antiferromagnets

ELSEVIER

Journal of Magnetism and Magnetic Materials 140-144 (1995) 1651-1652

Spinons and spin waves in one-dimensional Heisenberg antiferromagnets

R.A. Cowley a,* D.A. Tennant a S.E. Nagler b, T. Perring c a Oxford Physics, Clarendon Laboratory, Parks Rd, Oxford, UK

b Department of Phystcs, University of Florida, Gainesville, Florida, USA c RutherfordAppleton Laboratory, Chilton, Berks, UK

~ Journal of am:tnetlsm magnetic ~ i materials

Abstract 1 The spin excitations of the s = ~ nearly one-dimensional Heisenberg antiferromagnet, KCuF3, have been determined

using neutron scattering techniques. The results at high energies show that the excitations are described by pairs of unbound spinons but, at low temperatures and energies, the scattering is described by spin wave theory.

One-dimensional Heisenberg antiferromagnets (HAFC) are systems with strong interactions, like fractional quan- tum Hall states and possibly, high-temperature supercon- ductors, for which the fundamental excitations, spinons [1], have fermion character as opposed to the usual picture of boson excitations from a long-range ordered ground state.

1 In the case of s = ~ one-dimensional antiferromagnets, the spinons are unbound and this leads to the dynamical correlation function, measured by neutron scattering tech- niques, having the character of a pair spectrum rather than well-defined spin wave excitations [2].

Our measurements [3] of these effects were performed using neutron scattering techniques. One set of measure- ments was performed using the MAR1 time-of-flight spec- trometer at the pulsed neutron source ISIS of the Ruther- ford Appleton Laboratory. These measurements determine the dynamical correlation function along lines in the en- ergy/momentum transfer plane, and the line varies with the incident neutron energy as shown in Fig. 1. Also shown in Fig. 1 is the area of the plane in which two-spinon scattering is expected to occur, lying between the spinon dispersion curve e L = ~rJ Isin Qland the upper bound e u = 2 7 r J I s i n ( Q / 2 ) l [2]. The experiments were per- formed with a single crystal of KCuF 3, which has effec- tively nearly one-dimensional C u - F - C u chains. Above T N = 40 K there is no long-range order and the material is in its magnetic one-dimensional phase, but below Trq the weak interchain coupling causes three-dimensional order- ing. Fig. 2 shows the scattering observed after background subtraction and it clearly shows three peaks corresponding

* Corresponding author. Fax: +44 865 272400; email: r.cow- [email protected].

120 ~

100

80

a~ zo

0 1

2 8 0 m e V /

J 181meV 140meV / ¢ 127meV

2 a 4 5

Momentum Transfer (Tr/a)

Fig. 1. Trajectories measured in energy/wavevector along the c-axis for the low-angle detector bank on MARl, when the c-axis is along the incident neutron direction and the incident energy is varied. The curves give the boundaries of the two-spinon contin- uum. The lattice parameter along the c-axis (spacing between spins) is a.

ENERGY (meV)

0 20 40 60 80

' ' ' f ' ' , i , , , i , , , 1 ,

lO

,~ T=50 K 8 :~ el= 150meV

Sum over ol 6 low-angle i 4 ~ "~ ~ 2 0 de tec to rs

MOMENTUM (Tr/c)

Fig. 2. The scattering observed for the low-angle bank on MARl from KCuF 3 with incident neutron energy 150 meV. The solid line is the ansatz of Mueller et al. [4]. The crystal is in the one-dimensional state at 50 K, and a is the lattice parameter.

0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0304-8853(94)00535-4

Page 2: Spinons and spin waves in one-dimensional Heisenberg antiferromagnets

1652 R.A. Cowley et al. /Journal of Magnetism and Magnetic Materials 140-144 (1995) 1651-1652

to the three scattering regions shown in Fig. 1. The solid line is the ansatz for the scattering proposed by Mueller et al. [4], which has recently been shown to be the exact result for non-interacting spinons and which is applicable if the interactions fall off along the chain as J / L 2 where L is the distance between the spins [5]. Since the ansatz has only two parameters, J and a scale factor, the agreement is very satisfactory as it is also for other incident energies and shows the correctness of the spinon picture.

The field theory formulation of the continuum approxi- mation to the HAFC can be used [6] to calculate the temperature dependence of the dynamical correlation func- tion exactly. In our recent papers [3], it was shown that this provides an excellent description of the low-energy data at temperatures from 40 to 250 K. In this paper we report on the results at low temperatures in the three-dimensional long-range ordered state. Fig. 3 shows the same scan as Fig. 2, but for KCuF 3 in the magnetically ordered phase at 20 K. Clearly there is little difference between the data at 50 and 20 K, and the 20 K data are very well described by the spinon picture.

In order to examine the low-energy part of the spec- trum in more detail, we have performed measurements using a triple-axis spectrometer at the Brookhaven High Flux Beam Reactor, which extend the earlier work of Satija et al. [7]. Typical results are given in Fig. 4 showing two well-defined spin wave peaks and some continuum scattering between them. Since these results were obtained in a long-range ordered phase, it might be expected that they can be explained in terms of a spin wave model and so we have calculated the one- and two-spin wave contri- butions to the dynamical correlation functions using a full three-dimensional spin wave theory. Fig. 4 shows that this gives an excellent description of the low-energy data, but Fig. 3 shows it totally fails for high-energy transfers. In contrast, the one-dimensional spinon picture works well at high energies, but is less satisfactory at low energies

Energy Transfer (meV) 20 40 60 80

lo . . . . . . . . . . . . . . . . . . . . 3 **~ 8 ~ E0= 150meV I

6 "r: zoK

~ 2

~ o - ~

] 2 3 4 Momentum Transfer (n/a)

Fig. 3. The same scan as in Fig. 2 but with T = 20 K in the three-dimensicaaal phase. The dotted line shows a one- and two- spin wave theory, and a is the lattice parameter.

20O

E tso O3

~ 100

5O

T= 1 OK i

0.3 0.4 0.5 0.6 0.7 [ 1,0.,fl

Fig. 4. The scattering observed in a constant energy scan along (10r/) at 10 K for 25 meV [7]. The solid line is given by the spin wave theory and the dashed line by the spinon theory; ~7 is the reduced wavevector in units of 2 ~ / c .

because it does not include the inter-chain coupling. The cross-over occurs at about 25 meV, which is about twice the zone boundary spin wave energy perpendicular to the chain direction. Full details of these results and the associ- ated calculations will be published elsewhere. They do, however, show that the spinon picture provides a good description of the scattering in the one-dimensional phase. In the three-dimensional ordered phase, spin wave theory works well at low energies but fails at high energies, while spinon theory is the other way round. We hope these results will lead to further work to unify these models in nearly one-dimensional systems.

Acknowledgements: We are grateful to M. Arai, Z. Bowden, G. Shirane, A.D. Taylor and A.M. Tsvelik. Fi- nancial support has been provided by SERC in the UK, a NATO travel grant, and the US Department of Energy under Awards no. DE-FGOJ-92 ER 45280 and DE-AC0276 CH 00016.

R e f e r e n c e s

[1] P.W. Anderson, Science 235 (1987) 1196. [2] L.D. Faddeev and L.A. Takhtajan, Phys. Lett 85 A (1981)

375. [3] S.E. Nagler, D.A. Tennant, R.A. Cowley, T.G. Perring and

S.K. Satija, Phys. Rev. B 44 (1991) 12361; see also Phys. Rev. Lett. 70 (1993) 4003.

[4] G. Mueller, H. Thomas, H. Beck and J.C. Bonner, Phys. Rev. B 24 (1981) 1429.

[5] F.D.M. Haldane and M.R. Zirnbauer, Phys. Rev. Lett. 71 (1993) 4055.

[6] H.J. Schulz, Phys. Rev. B 34, (1986) 6372; A. Luther and I. Peschel, Phys. Rev. B 9 (1974) 2911.

[7] S.K. Satija, J.D. Axe, G. Shirane, H. Yoshizawa and K. Hirakawa, Phys. Rev. B 21 (1980) 2001.