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INTERNAL SPIN PRECESSIONS IN MAGNETIC
SOLITONS
J. Boucher
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C8, SupplCment au no 12, Tome 49, dkembre 1988
INTERNAL SPIN PRECESSIONS IN MAGNETIC SOLITONS
Centre d7Etudes Nucltaires de Grenoble, DRF/SPh/DSPE, 85X, 38041 Grenoble Cedex, France
Abstract. - In antiferromagnetic Ising-like chains, internal modes are expected to occur in the soliton fluctuations. For the case S
=
1/2, it is shown that they correspond to precessions of the spins inside the solitons. Fkcent Soliton Magnetic Resonance experiments performed on the compound CsCoC13 agree with this description.In 1983, Haldane [I] has established that, in anti- ferromagnetic (AF) Ising-like classical spin chains, in- ternal modes should be associated with the soliton ex- citations. More recently, Affleck [2] has related these predictions t o the "Dyon" concept which is used in par- ticle physics. As discussed in [I] and [2] these internal modes depend on whether the spin value is integer or half integer. For integer spin the lowest energy soliton state is predicted to be non magnetic with quantum number m = 0. However, for half integer spin the low- est energy soliton state is predicted t o be magnetic with m =
f
112. Accordingly, the application of an ex- ternal magnetic field H should generate internal spin precessions. This problem is considered here in the case of AF Ising-like quantum spin (S = 1/2) chains where LLexact" derivations can b e performed. Recent Soliton Magnetic Resonance (SMR) experiments with CsCoC13 support this description.The Hamiltonian for AF Ising-like quantum spin chains is:
dent of the wave vector k : it corresponds to a Zeeman effect. For HL ( ~ 1 1 = 0) the effect of the field is t o change both the soliton energy (hwk) and the soliton velocity (dWk/dk)
.
Exact analytical expressions can be also derived for the dynamical structure factors [5]:sa
(q, w)-
J
dt e-iut(s:
(t) SO.,) where S," are the Fourier components (a = x, y, z) of the spin operators SE. For HIl (HL = 0),
while the fluctuations along the chainsSi
(q, w) are practically not affected by the external field, the transverse com- ponents Sf (q, w) with a! = z , y undergo a doublingof the soliton modes ?cording t o (H, EJ
<<
T) :St (q, w) = S; ( q , w)
=
n, (1-n,) N-'x x q f cos2(k+q/2)x (6 [w - 0 , s i n ( 2 k - q ) + ~ ~ ~ ] +6 [w
-
0, sin (2k-
q)-
HI(])
where n.= exp (- J/T) is the soliton density,
H=
~JS:S:+~+
2eJ (SESz+l+
SgS:+l) - 0,= 4eJ sin q,41
= exp [-2sJ cos (2k/T)] /Zllwith S = 1/2 and E, HII/J, HL/J
<<
1. The first term ( l o ( X ) is the zer~order modified Bessel function). where J is the ex&ange coupling describes the hing Within the same approximations, one can write forenergy. At low temperature, T --+ 0, the spins tend t o HL :
align along the z direction, which coincides with the ST (q, w) ^. ns N-I
x
P! Cos2(h
+
:)
chain axis. In (I), HII = g I I p ~ H and HL = ~ I ~ B H
define the Zeeman energy when H is applied parallel 6
-
4
.in (2k-
q) - 2~~ cos(k
-f )
cos!!] and perpendicular to the chains. The soliton energy isfirst evaluated in the subspace of states which contain n s ( l - n ~ ) ~ - i ~ ~ one soliton in the chain. Such a diagonalization of 7.1 S1(9, W) ^. 4 c0s2 q/2
has been given by Villain for H = 0 [3] and by Shiba
and Adachi for H
#
0 [4]:-
0, sin (2k-
q)-
2 H l cos (1:-
i)
cosf] with -T5
k5
+T. The application of an external andfield yields a splitting of the soliton energy into two
N - ~ C P ~
= I . branches. For HII (HL = 0),
the splitting is indepen-In the limit q -+ 0, these equations agree with the re- sults of [4]. Interactions between solitons, between soli- l ~ e r n b e r of Equipe de Recherche, CNRS no 216. tons and magnons and with impurities are expected .to
C8 - 1440 JOURNAL DE PHYSIQUE
round off the square root singularities predicted by the above one-soliton model. TO account for such round- ing effects, we introduce a "soliton damping" in the S" (q, w) by the following substitution
6 [w
-
w (9, H, k)l -+ A/{a2
+
[w-
w (9, H, Ic)I2).
Examples of such "Realistic" soliton modes are shown in figure 1 for HII and HL. In particular, one ob- serves that a t q = 0, the soliton modes Sf (q = 0, w) and S i (q = 0, w) are shifted by f Hll (Zeeman effect), while the modes ST (q = 0, w) and S i (q = 0, w) ex- tend continuously from zero to the characteristic fre- quencies
f
2HL, where we expect to observe a maxi- mum. The Electron Spin Resonance (ESR) technique, as it probes the uniform (q = 0) dynamics at a finite frequency (w#
0) should allow one to observe the soli- ton fluctuations.Fig. 1.
-
Doublings of soliton modes in AF Ising-like quan- tum spin chains for H applied parallel and perpendicular t o the chains.The compound CsCoC13 is a good realization of AF Ising-like quantum spin chains with J = 75 K and
E = 0.12, and solitons fluctuations have been observed
by neutron inelastic scattering measurements [6] above the magnetic ordering temperature T~11121 K. Previ- ous ESR measurements were performed by Adachi [7] at the frequency of 9 GHz but below
T N ~
in an or- dered phase where the one-dimensional soliton concept is questionable. The ESR measurements presented in figure 2 were performed a t the frequency w E 35 GHzby varying the field from 0 to 16 kOe and in the tem- perature range 10 _< T
5
60 K, which covers the one- dimensional phase ( T>
21 K) [8]. For H applied per- pendicular t o the chains, the fluctuations were probed along the two axes cu = y and cu = z , perpendicular t o the field. For H parallel, the fluctuations were probed in a direction ( a = x, y) perpendicular to the chains.In these measurements, the modula.tion field procedure was used and the ESR signal is given by
it reproduces essentially the derivative of the soliton mode with respect to H. The dotted lines in figure 2 are the resulting theoretical curves obtained by assum- ing the damping A t o be independlent of Ic, H and w. The present soliton model explains remarkably well the asynimetric lineshapes observed experimentally above and below T N ~ . While above T N ~ , the damping de- creases slowly with T (A ci 1 K)
,
iL drastic narrowingof A is observed at the transition. This result suggests that the damping is of dynamical ,origin: the narrow- ing at TN1 would correspond to it "freezing" of the
solitons, which, however, persist in the ordered phase. We may ask if this result is general for solitons or if it is specific to CsCoC13, which, below Tpql, presents strong interchain frustrations.
Fig. 2.
-
Examples of Soliton Magne6ic Resonance signals observed in CsCoC13 as a function of the field (0-16 kOe) applied parallel and perpendicular to the chains, at the fre- quency of 35 GHz. The dotted lines are theoretical curves. The theoretical derivation and the experimental data presented above have been obtained in the case of half integer quantum spin (S = I/:!) chains. However, the results agree with the semi-classical descriptions given by Haldane [I] and Affleck [2]. In the context of the Haldane's conjecture for integer spins [I], similar investigations for S = 1 spin chains, are now highly desirable.[l] Haldane, F. D. M., Phys. Rev. Lett. 50 (1983) 1153.
[2] Affleck, I., Phys. Rev. Lett. 5'7 (1986) 1048. [3] Villain, J., Physica B 79 (197!j) 1.
[4] Shiba, H. and Adachi, K., J . Rhys. Soc. Jpn 50
(1981) 3278.
[5] Devreux, F. and Boucher, J. P., J . Phys. 48
(1987) 1663.
[6] Yoshizawa, H., Hirakawa, K., Satija, S. K. and Shirane, G., Phys. Rev. B 23 (1981) 2298. Boucher, J. P., Regnault, L. I?., Rossat-Mignod, J., Henry, Y., Bouillot, J. and Stirling, W. G., Phys. Rev. B 31 (1985) 3015.
[7] Adwhi, K., J. Phys. Soc. Jpn 50 (1981) 3904. [8] Boucher, J. P., Rius, G. and Henry, Y., Europhys.
