Calculation of the out-of-plane dynamical correlation for CsNiF3

S. A. LeonelDepartamento de Fisica, Universidade Federal de Juiz de Fora, Juiz de Fora, Minas Gerais, Brazil

A. S. T. PiresDepartamento de Fisica, Universidade Federal de Minas Gerais, CP 702, Belo Horizonte, 30161970 Minas Gerais, Brazil

~Received 9 July 1996!

We calculate the out-of-plane dynamical spin correlation function for the one-dimensional easy-plane fer-romagnet CsNiF3 at low temperatures using a diagrammatic expansion for the temperature-dependent Greenfunction. We compare our theoretical calculation for the spin wave linewidth with experimental data ofKakurai, Steiner, and Dorner.@S0163-1829~96!02842-1#

In the past years the spin dynamics of one-dimensional~1D! magnetic systems have been studied extensively boththeoretically and experimentally.1–3 The Hamiltonian

H522J(iSW i•SW i111A(

i~Si

z!2, ~1!

with S51, has been found to describe the easy-plane ferro-magnet CsNiF3 quite well. The values of the parameters forthis compound were obtained by Steiner, Dorner, andVillain4 from the low temperature classical spin wave disper-sion relation measured in neutron scattering experiments.5

This procedure yieldsJ511.5 K andA54.5 K. A quantumrenormalization of the Hamiltonian6 ~1! however reducesAto A „121/(2S)… which yieldsA59.0 K. We have foundthat using the valueA59.0 K ~bare parameter! in our theo-retical calculation we could fit the experimental data for thespin wave peak position quite well. In zero field, the mostimportant excitation in this system is the spin-wave-likepropagating modes for wave vectorsq larger than the inversecorrelation lengthk(T). Villain7 using a self-consistent har-monic approximation predicted the existence of two charac-teristic linewidths, due to in-plane~IP! and out-of-plane~OP!spin fluctuations. The IP component of the dynamic structurefactorSaa(q,v)(a5x,y) has been quite well studied show-ing a well behaved structure.8 However the linewidth of theOP componentSzz(q,v) displays an anomalous wave vectordependence due to a singularity in the three spin wave den-sity of states. This anomalous behavior was predicted byReiter9 using a classical zero-temperature spin wave theory,and observed in CsNiF3 by Kakurai, Steiner, and Dorner

8 bymeans of inelastic polarized neutron scattering.

In this Brief Report we will calculate the in-plane dy-namic structure factor for the quantum Hamiltonian~1!. Wewill take the classical easy-plane ferromagnetic state as thezeroth approximation, and expand in powers of the ampli-tude of the out-of-plane fluctuations^(Sn

z)2&/@S(S11)#, andin-plane fluctuations(fn2fn11)

2&. In a diagrammatic ex-pansion of the perturbation series for semiclassical spin theleading term is the classical system, one-loop graphs are oforder \, two-loop graphs of order\2, and so on.10 Moreaccurately, the graphs are of order 1/S, 1/S2, etc., and for asemiclassical system\S is of order unity. We will carry out

the expansion to the two-loop level, which involves compu-tation of 46 different diagrams. The multiplicity of diagramsand their greater complexity make it impractical to go be-yond two loops.

We start by introducing the Villian’s representation7

Sn15eifn@S~S11!2Sn

z~Snz11!#1/2,

Sn25@S~S11!2Sn

z~Snz11!#1/2eifn, ~2!

wheref is the angular variable in thexy plane canonicallyconjugated toSn

z . Substitution of~2! into the Hamiltonian~1!, expanding to second order inSn

z/AS(S11) and(fn2fn11), and Fourier transforming gives

H5E01H21H41•••, ~3!

where E0522JNS(S11) is the energy of the classicalground state,H2 is the harmonic Hamiltonian given by

H252JS~S11!(q

~12cosq!fqf2q

12J(q

~12cosq1d!SqzS2q

z , ~4!

with d5A/2J, and

H45J

N (q1 ,q2 ,q3 ,q4

dq11q21q31q4 ,0H 2S~S11!

6@12cosq1

2cosq22cosq32cosq41cos~q11q2!1cos~q11q3!

1cos~q11q4!#fq1fq2

fq3fq4

2@12cosq22cosq3

1cos~q11q4!#Sq1z fq2

fq3Sq4z

11

6S~S11!@32cos~q11q2!2cos~q11q3!

2cos~q11q4!#Sq1z Sq2

z Sq3z Sq4

z J . ~5!

Hamiltonian~4! can be diagonalized by using the canonicaltransformation

PHYSICAL REVIEW B 1 NOVEMBER 1996-IVOLUME 54, NUMBER 17

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fq5aq~aq11a2q!, Sq

z5 iba~aq12a2q!, ~6!

whereaqbq 5 1/2, and

aq5S 1

4S~S11!

12cosq1d

12cosq D 1/4. ~7!

We obtain

H25(q

\vq~aq1aq11/2!, ~8!

where

\vq54J@S~S11!~12cosq!~12cosq1d!#1/2. ~9!

As we can see from~2! and ~4! a measurement ofSaa(q,v) (a5x,y) does not reduce to the observation ofone single magnon of wave vectorq, but involves the obser-vation of all magnons with wave vectors between, roughly,q2k and q1k, wherek is the inverse correlation length.1

On the contrary, a measurement ofSzz(q,v) does in factcorrespond to the observation of a single magnon with amuch smaller linewidth. In the model discussed here, at lowtemperatures, the linewidth of the out-of-plane componentSzz(q,v) is produced by the decay of the magnon into othermagnons, by the scattering by other magnons, or, in math-ematical terms, by the anharmonic terms in~4!. Since theseterms are functions of bothf andSz we use a matrix formfor the temperature-dependent Green function

D5FDff DfS

DSf DSSG , ~10!

where

Dma~q,t2t8!5^T$cqm~t!c2q

a ~t8!%&, ~11!

wherem,a5f or S, cqs5Sq

z ,cqf5fq , and 0,t,b, with

b51/kBT. T in ~11! means that the operators are arranged sothatt is decreasing from left to right. The nonperturbed~barepropagator! Green function is

D0~q,ivn!5F 2aq2vq i ~ ivn!

2 i ~ ivn! 2bq2vq

G 1

~ ivn!22vq

2 , ~12!

where

D~q,ivn!5E0

b

dt eivntD~q,t!, vn52npT. ~13!

The temperature-dependent Green function for the interact-ing system obeys a matrix form of Dyson’s equation and canbe written as

D~q,ivn!5F 2aq2vq1(

SS2vn2(

fS

vn2(Sf

2bq2vq1(

ff

G3

1

@~ ivn!22vq

2~11D!#,~14!

where

D52

vqS aq

2(ff

1bq2(SS

D 1vn~(fS2(Sf!

vq2

1(ff(SS2(fS(fS

vq2 , ~15!

and the(ma are elements of a self-energy matrix. The cal-culation of(ma follows the procedure given in Ref. 10 bymaking the following replacement11 i t→t. The remainingtask is the computation of the self-energy~irreducible con-nected! graphs which sum to give the( function. We havedone this calculation up to the two loop level; this is theentire second order expansion. Since the calculation is stan-dard we quote only our results for the dynamic relaxationfunction Szz(q,v) which is related to the temperature-dependent Green function by11

Szz~q,v!524p

vImDSS~q,v1 id!. ~16!

We have calculatedSzz(q,v) for the compound CsNiF3 as afunction of v using Eq.~16! and in Fig. 1 we show someexamples of our calculations. In Fig. 2 we show the peakposition ~magnon dispersion relation! as a function of thewave vectorq at a temperatureT54.2 K. The experimentaldata is from Ref. 5. As we said before we fitted the experi-mental data by adjusting the anisotropy parameterA toA59.0 K, in agreement with previous estimates.6,8 For thepeak position the second-order correction~two-loop graphsplus the second-order contribution from one-loop graphs!adds, at most, 2% to the one-loop result, and this leads us tobelieve that our calculation, by just going to the two-looplevel, is accurate. Of course the first-order correction leads toa null linewidth. In Fig.3 we show our theoretical predictionfor the OP linewidthG at T54.7 K compared with experi-mental results of Kakurai, Steiner, and Dorner.8 As pointed

FIG. 1. The out-of-plan dynamical relaxation function in arbi-trary units, as a function ofv for T54.7 K andq/p 5 0.2 ~solidline!; q/p50.3 ~dashed line!.

54 11 945BRIEF REPORTS

out by Reiter9 the anomalous behavior in the OP linewidth isdue to a singularity in the three spin wave density of states,which are the major channels for the decay processes of theOP spin wave.

In conclusion, we have calculated the OP dynamical spincorrelation function for the easy-plane one-dimensional fer-romagnet, to the two-loop level. We have applied this calcu-

lation to CsNiF3, and obtained the anomalous wave vectordependence in the spin wave linewidth, observed experimen-tally by Kakurai, Steiner, and Dorner.8

This work was partially supported by Conselho Nacionalde Desenvolvimento Cientifico e Technologico~Brazil!, andCapacitac¸ao e Aperfeicoamento de Pessoal de Nivel Superior~Brazil!.

1M. Steiner, J. Villain, and C. G. Windsor, Adv. Phys.25, 87~1976!.

2H. J. Mikeska, Adv. Phys.40, 191 ~1991!.3T. Schneider and E. Stoll, inSolitons, edited by S. E. Trullinger,V. E. Zakharov, and V. L. Pokrovsky~North-Holland, Amster-dam, 1986!.

4M. Steiner, B. Dorner, and J. Villain, J. Phys. C8, 165 ~1975!.5M. Steiner and J. K. Kjems, J. Phys. C10, 2665~1977!.6E. Rastelli and P. A. Lindgaard, J. Phys. C12, 1899~1979!.

7J. Villain, J. Phys.~Paris! 35, 27 ~1974!.8K. Kakurai, M. Steiner, and B. Dorner, Europhys. Lett.12, 653

~1990!.9G. Reiter, Phys. Rev. Lett.60, 2214~1988!.10N. F. Wright, M. D. Johnson, and M. Foweler, Phys. Rev. B32,

3169 ~1985!.11A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinskii,Methods

of Quantum Field Theory in Statistical Physics~Prentice-Hall,Englewood Cliffs, NJ, 1963!.

FIG. 2. Peak position as a function of the wave vector forT54.2 K. The solid line is our theoretical prediction. The filledcircles are experimental data from Ref. 9.

FIG. 3. q-dependence of the OP linewidthG at T54.7 K. Thesolid line is our theoretical prediction. The filled circles are experi-mental data from Ref. 5.

11 946 54BRIEF REPORTS