Solitons in the one-dimensional easy-plane ferromagnet

Physica 120B (1983) 235-240 North-Holland Publishing C o m p a n y

§3.1. SOLITONS

S O L I T O N S IN T H E O N E - D I M E N S I O N A L EASY-PLANE F E R R O M A G N E T H.J. M I K E S K A

Institut fiir Theoretische Physik, Universitiit Hannover, F.R. Germany Invited paper W e present theoretical results on nonlinear excitations in the one-dimensional easy-plane ferromagnet in a s y m m e t r y breaking external field. Soliton-like solutions to the nonlinear classical equations of motion for the real magnetic chain (as opposed to an idealized planar chain, equivalent to a Sine-Gordon system) are discussed and the approach of the soliton instability in this system is described. Q u a n t u m mechanical corrections to the soliton energy in the real magnetic chain are given and shown to be substantially different from results derived in the Sine-Gordon approximation. T h e status of quantitative predictions for the strength of contributions to the dynamic structure factor from nonlinear m o d e s in thermal equilibrium and the investigation of these m o d e s by neutron scattering is discussed.

1. Introduction

where we have introduced

In recent years neutron scattering experiments on nearly one-dimensional magnetic solids have allowed us to study nonlinear excitations in these materials and to compare experimental and theoretical results. Nonlinear modes of m a n y different types are supported by one-dimensional magnets with various symmetries and couplings (for a review see [1]). A m o n g these, topological solitons in the easy-plane ferromagnetic chain in a symmetry breaking external magnetic field have been studied in greatest detail both theoretically and, using the c o m p o u n d CsNiF3, experimentally. In the following we will describe recent results for this type of topological solitons. We will concentrate on theoretical results and refer to the contribution by Steiner to this conference for an up-to-date account of the present status of experimental results. The system of interest is defined by the Hamiltonian

s. o = s # / h $ ,

H = - J ~ S.S.+, + A ~. (S.Zy- ~B ~ S. ~ n

n

(1)

and cycl. ,

(3)

The operators sn~ have a simple meaning in the classical limit 5 ~ (when they commute). In this limit they are conveniently parametrized by two angles s. = {cos 0. cos 4'., cos On sin &., sin On}.

(4)

The dynamic properties of our system are usually discussed in terms of the dynamic correlation functions

g ~ (x, t) = (s ~ (x, t)s ~ (0, 0))

(5)

and their Fourier transforms

S~(q, to)= ~ 1 f dt dx ei(qx_~Ot)gaa (x, t),

(6)

S'~"(q) = f do) S~'~'(q, w)

(7)

n

and the c o m m u t a t o r relations Is. x, s,. y] = ~i s. z 6.,.

~2= s ( s + 1).

(2)

directly related to inelastic and elastic neutron scattering cross sections. In the limit A ~ ~ (which essentially confines the dynamics to the easy plane) and S ~ o ~ (the classical limit), the long-wavelength part of the

0378-4363/83/0000-0000/$03.00 © 1983 North-Holland and Y a m a d a Science Foundation

236

H.J. Mikeska / Solitons in the ld easy-plane ferromagnet

system given by eqs. (1) and (3) can be m a p p e d exactly to the classical S i n e - G o r d o n (SG) chain, which in the c o n t i n u u m a p p r o x i m a t i o n is defined by

1 (&,b'/z + m2(1 _ cos

+ 2Jc2 \ ot /

&)}

(8)

with Poisson brackets

{4,(x), 6(x')}

i 6(x - x').

=

E q u i v a l e n c e to requires the parameters J~o = J S 2 ,

(9)

c

the magnetic chain, eq. following identification

C2 = 2 J A S 2 ,

m

2 =

~B/JS.

(1), of

(10)

T h e d y n a m i c properties of the classical S G chain are discussed in terms of (anti) solitons, breathers and m a g n o n s , see e.g. [2]. T o lowest o r d e r in t e m p e r a t u r e these m o d e s are i n d e p e n d e n t of each other, realising in particular a gas of nonoverlapping solitons with density n = 4m

e 8/3m,

(11)

w h e r e f l - I is the t e m p e r a t u r e in units of E o / k a . A t this simplest level of the description the contributions of solitons and m a g n o n s to the d y n a m i c structure functions are calculated ind e p e n d e n t l y of each o t h e r with the results [3]

cq m2( )

T h e emerging simple picture of a central soliton p e a k in addition to the conventional m a g n o n peaks at w =-+wq has been qualitatively confirmed t h r o u g h the early n e u t r o n scattering experiments by K j e m s and Steiner [4]. For a m o r e detailed discussion of topological solitons it is necessary to go b e y o n d this simple picture based exclusively on the m a p p i n g to the ideal classical S G chain and to deal with the real magnetic chain system. In the three main sections of this p a p e r we will discuss the consequences of relaxing s o m e of the simplifications used above: (i) Allowing finite values of t z B / 2 A S = h ~ (h = 5 for CsNiF3 at B = 10 k G ) requires one to consider the d y n a m i c s of the out-of-plane comp o n e n t of spin in m o r e detail. At A = 3 the static classical S G soliton b e c o m e s unstable against out-of-plane distortions [5, 6]. Precursors of this instability show up already at larger values of A and will be discussed in section 2. (ii) Finite values of S ( S = 1 for CsNiF3) require the consideration of q u a n t u m corrections. In Section 3 we will discuss the calculation of the soliton e n e r g y in a semiclassical approach. (iii) A t finite t e m p e r a t u r e interference effects between solitons and m a g n o n s as well as a n h a r m o n i c rnagnon terms contribute to the d y n a m i c structure functions. These contributions, which are essential for a quantitative c o m p a r i s o n to n e u t r o n scattering data will be discussed in section 4.

2. D y n a m i c s of the easy-plane ferromagnetic chain beyond the SG approximation

32n

S~o~,o(q, w ) - - -

X exp{--4flmoo2/c2q2}f~ (q),

(13)

f~(q) = (Q/sinh Q)2, f y ( q ) = (Q/cosh Q)2,

1

(12)

00 a 02q5 Oq~ 00 ~ - = cos 2 sin 0 - - - - - sin ~ , v 0z 2 0z 0z

Q = "rrq/2m,

1

S~,.%(q, w ) = 4 z r f l m 2 + qZ x

{a(,o - .,.)

oJq = c ( m 2 + q2)1/2 .

+

a(~

+

%)}a~.~,

T h e equations of m o t i o n for the easy-plane ferromagnetic chain as given by the Hamiltonian (1) in the classical c o n t i n u u m approximation are

(14) (15)

0~ _ 0t

(16)

1 020 - - + A sin 0 cos 0 0z 2 - sin 0 /+~/.n. /, 2¢ ' , tg 0 cos th \az /

(17)

237

H.J. Mikeska / Solitons in the ld easy-plane ferromagnet

Here, the spin vector s has been p a r a m e t r i z e d according to

F(s ; A) = 3 sech s - ~ ~ dk e iks[lx t>, + tg h2 s

(18)

s = {cos 0 cos 05, cos 0 sin 05, sin 0}

and time and distance are m e a s u r e d in units of m ~ a n d ( J S m 2 ) -1 respectively (see eq. (10) for the definition of m). T h u s the only remaining p a r a m e t e r in the classical t h e o r y is A = 2AS/tzB. For A ~ (at finite B ) o n e has 0 ~ 1, the term A sin 0 d o m i n a t e s the rhs of eq. (17) and the classical S G equation is o b t a i n e d in the f o r m 1 32&

A Ot2

_

024)

cgz2

sin qS,

0 = 1 005

(26)

(20)

F o r two limiting values of A results for F(s; A) and A(A) can be given explicitly

(21)

0=-u

d~

or d20 ds 2

- --+

(a + 1 - 6 sech 2 s)O = - 2 u sech s.

(22)

T h e solution of eq. (22) is

O(s) = ~ - 3 F(s; a)(-2u),

is

A ( A ) = ½f~ ds F(s; A) sech s.

= ~o A-\--d~-s} + c ° s & °

energy

(19)

is an exact solution to eqs. (16) and (17). Starting f r o m this observation the b e h a v i o u r of solitons moving with small velocities u for finite values of A can be investigated in successive approximations in u and 0. W e start f r o m an ansatz 05=05(s), 0 = 0 ( s ) , s=z-ut and obtain to lowest o r d e r in u and 0

-ds----~+

In this approximation the soliton d e t e r m i n e d up to terms of O(u4):

(25)

Thus, in o u r units, A v2 is the limiting velocity of the S G t h e o r y and it b e c o m e s obvious that the m a p p i n g of the magnetic chain to the S G system m a k e s sense only in the nonrelativistic limit of the S G theory. It is i m p o r t a n t to note that for arbitrary values of A the static S G soliton 0 = 0

(24)

1~ A ( A ) + O ( u 4 ) } E = 8 m { 1 + Tu2 a-----

A at

05 = 050 = 4 arctg e z ,

- ik tgh s) sech rrk/2 1 + A + k 2"

(23)

A~

F(s;A)~sechs,

A-~3

F ( s ; A ) = ~ - s e c h 2s,

37r

A(A)~I

A(A)=37r2/32.

Thus it appears that A ( A ) is varying with A within very n a r r o w limits whereas in this approximation the characteristic velocity of the s p e c t r u m is c h a n g e d f r o m A 1/2 in the S G approximation to ( A - 3) 1/2 owing to the d y n a m i c s of the out-of-plane spin c o m p o n e n t . F o r A--+3 the static soliton b e c o m e s unstable to meridional distortions as has been shown by K u m a r [5] and Magyari and T h o m a s [6]. In the present app r o a c h this instability shows up as a diverging out-of-plane spin c o m p o n e n t 0 for A--,3. For A < 3 the static soliton continues to be an exact solution but according to eq. (25) it can lower its e n e r g y by moving with an arbitrarily small velocity u and is thus unstable. This d y n a m i c source of the instability can also be shown to e m e r g e f r o m a consideration of p h o n o n frequencies in the p r e s e n c e of the static soliton [7]. T h e result eq. (25) has an immediate cons e q u e n c e for the soliton contribution to dynamic structure factor: T h e u 2 term in the e n e r g y is directly related to the Gaussian f r e q u e n c y distribution in eq. (12) and the coefficient of this term determines the f r e q u e n c y width of the central peak. C h a n g i n g A to A - 3 implies that the f r e q u e n c y width is now given by

238

n , J . M i k e s k a / Solitons in the l d e a s y - p l a n e f e r r o m a g n e t

cq ./ Aw - ~flmV2~ ~/ 1

31xB 2AS .

where we have introduced (27)

4, = g'/', Thus with increasing magnetic field the central peak is predicted to sharpen, a property which should be observable in neutron scattering experiments and which could serve to indicate the approach of the instability. The accompanying reduction of the integrated intensity will be discussed in section 4. The behaviour of the system for A < 3 is not clear at present but it appears probable that no topological stability of soliton solutions survives and that the system enters a new type of dynamic behaviour which is currently under theoretical investigation. Indications of this new phase may have been seen in c o m p u t e r experiments [8].

3. Semiclassical t r e a t m e n t of the m a g n e t i c quant u m soliton

Starting from the representation of spin operators in eqs. (2) and (3) a semiclassical approach, i.e. an expansion in 1/S can be performed. Treating a system with planar symmetry it is useful to replace s x and s y by the operator analog of the classical angle 4, s x + is y = eie~X/1

(sZ) 2 -

s~l~

sz = g

g2= [2A'~ m

rr,

\)~]

(32) In the planar limit, A ~ m , one can neglect the second line in eq. (30) and arrive at the Hamiltonian of the quantum SG chain. This has been used by Maki [9] to treat the quantum magnetic chain (with the additional replacement S ~ S ) , employing the procedure of Dashen, Hasslacher and Neveu [10] to renormalize ultraviolet divergencies. Using the fact that the static soliton is an exact solution of the equations of motion (16) and (17) (which follow from (30) and (31) in the classical limit) this approach can be extended [11] to the full Hamiltonian (30). However, ultraviolet divergencies originate from the continuum approximation, which is artificial and can be avoided completely for the magnetic chain, where the cutoff is given by the finite lattice spacing. Thus lowest order quantum corrections to the energy of the topologically stable soliton state are given by the difference in the energy of zero point vibrations in the one soliton state and the ground state, which is obtained as [11]

E~o, = E~ ~)+ ½~_. (hw o - hw~ '))

(2s)

q

= E~)(I

g2

7r2j

(33)

with c o m m u t a t o r relations i

[4,,., s. z] = ~ 6.,..

(29)

with

F ( m , t) = ~-/2 + In 2 ~ / m T o first order in 1/S the following quantum form of the theory is obtained in the continuum limit

H = j~Zg2 fj dx

= ~-/2 + 4X/t

t~ 1 t>> 1

(34)

whereas for the field theoretic SG problem one has F = 1. For CsNiF3 the value of "n'2j/2A is of the order of 20 and neglecting rr/2 in eq. (34) one obtains a quantum correction to the soliton energy which depends only on the value of S, namely

f l [ OCI)'~ 2 , ! 2 t 277"

lm 2~'(cos g O ) ~ ' ] } ,

(30)

[qO(x), 1r(x')] = iN(x - x'),

(31)

+

.

Eso, = E f t ( 1 _ l + 2S

O(1/f")).

(35)

H.J. Mikeska / Solitons in the l d easy-plane ferromagnet

Owing to the occurrence of S in eqs. (10) E ~ ) is larger than the soliton energy 8 m J used in classical calculations by a factor S/S. For CsNiF3 with S = 1 quantum corrections therefore decrease the soliton energy by a factor of V/2 - ~1- ~ 0.91 as c o m p a r e d to the conventional classical value. Even this correction, however, is practically c o m p e n s a t e d by an increase in the value of J, when this constant is determined from the spin wave spectrum including q u a n t u m effect to the same order [12]. It should, however, be emphasized that higher order terms in the expansion in 1/S are likely to be needed to obtain a quantitatively reliable result.

4. Central peak intensities at finite temperatures The simple result, eq. (12), for the soliton induced central peak, valid for the ideal SG theory has to be modified in several respects if a meaningful comparison with neutron scattering results on CsNiF3 is to be performed: (i) Soliton-magnon interferences reduce the central peak intensity by a factor [12]

1 - 2 ~ m (1

m2-(-2---~7 + q2,~] +

0(/~-2)

.

(36)

(ii) At finite temperatures even the static soliton will take advantage of the out-of-plane degree of freedom offered by the magnetic system. This can be expressed as a decrease of the soliton energy with temperature, leading to the following effective increase in soliton density [1]

tribution to the scattering intensity around zero frequency from two magnon processes. In the classical continuum approximation its intensity is given by [12]

1

1

S~X(q) = 327rm3/32 1 + ( q / 2 m ) 2"

(37)

(iii) T h e effect of the out-of-plane degree of freedom on the frequency width of the central peak as discussed in section 2 leads to a decrease in central intensity by a factor (1 - 3 1 z B / 2 A S ) 1/2 .

(39)

It has also been numerically evaluated including discreteness effects and quantum corrections (Bose factors) [13]. For a comparison with experiments all these corrections have to be taken into account and the most useful way to state the results is to give integrated intensities in units of (47rm)-l ( 2 A / J S 2 ) m = Io, which is the zero t e m p e r a t u r e magnon strength (apart from small quantum corrections discussed in [12]). Using this normalization, in addition to dependences on temperature magnetic field, wave-vector and frequency, " a b s o l u t e " soliton intensities can be c o m p a r e d to experiment. In table I we give central peak intensities in these units for q = 0, m = 0.29, corresponding approximately to B = l0 k G in CsNiF3, and two temperatures. The difference between the entries in the first two lines does not illustrate the importance of the modifications (i)-(iii) listed above since rather large corrections cancel. The twomagnon contribution, given in the last line, is seen to be of comparable size to the soliton contribution for c~ = x; for a = y, on the other hand, it is absent, whereas the soliton contribution is

Q n~exp( 3E~' )n. \8V2AJ /

239

= (co-

Q)=

2 xx

O)

- 0),

Table I Central peak intensities m = 0.29 in units (4~rrn) 1X/2A/JS 2

for of

q =(1, h~ =

T=6K

T=8K

0.08 (I. 10 (I.23

0.50 (I.47 0.40

(38)

(iv) In addition to the soliton induced central peak there exists (for a = x) a further con-

S~oXl.o(q= (I) S~,~(q = 0) S~X(q = 0)

"rrq

2m"

240

H.J. Mikeska / Solitons in the l d easy-plane ferromagnet

Both the existence of a central c o m p o n e n t for c~ = y and of central intensity in addition to the two-magnon contribution has been demonstrated experimentally [14] for T = 1 2 K . This temperature is too high to allow a reliable quantitative discussion (in particular terms of higher order in /3 -1 in the important correction factor (36) would have to be considered); the qualitative results, however, strongly support the importance of solitons for the dynamics of the easy-plane spin chain as represented by CsNiF3.

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[51 P. Kumar, Phys. Rev. B25 (1982) 483. 16] E. Magyari and H. Thomas, Phys. Rev. B25 (1982) 531 I71 H.J. Mikeska and K. Osano, to be published. [81 G. Wysin, A.R. Bishop and P. Kumar, J. Phys. C15 (1982) L337.

Acknowledgements I wish to thank Drs. K. Maki and K. Osano for useful discussions. The hospitality of Dr. S. Doniach and the D e p a r t m e n t of Applied Physics at Stanford University, where much of this work was done, as well as financial support from the Bundesministerium ffir Forschung und Technologie and from the Fulbright Commission is gratefully acknowledged.

[91 K. Maki, Phys. Rev. B24 (1981) 3991. [1(1] R.F. Dashen, B. Hasslacher and A. Neven, Phys. Rev. DI0 (19741 4114; Dll (1975) 3424.

111] H.J. Mikeska, Phys. Rev. B26 (19821 5213. 1121 E. Allroth and H.J. Mikeska, Z. Phys. B43 (19811 209. [131 G. Reiter, Phys. Rev. Lett. 46 (1981) 202: Erratum 46 (1981) 518.

I141 M. Steiner, K. Kakurai, W. Knop, R. Pynn and J.K. Kjems, Solid State Comm. 41 (1982) 329.