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Exact nonlinear spin waves in some models of interacting classical spins on a one-dimensional lattice
29 December
1997
PHYSICS
ELSEVIER
LETTERS
A
Physics Letters A 237 (1997) 73-79
Exact nonlinear spin waves in some models of interacting classical spins on a one-dimensional lattice Radha Balakrishnan a,‘, Rohit Dhamankar b a Institute h Department
of Mathemaiical
of Physics.
Sciences. C.I.T. Campus,
Indian
Institute
of Technology.
Madras
600 113, India
Kanpur
Received 3 March 1997; revised manuscript
208016.
India
received 19 August 1997; accepted for publication Communicated by A.R. Bishop
26 August 1997
Abstract We present exact finite-amplitude spin wave solutions for the nonlinear differential-difference equations describing the dynamics of spin vectors in a ferromagnetic (FM) and an antiferromagnetic (AFM) classical Heisenberg model of interacting spins on a one-dimensional lattice. We find that the dispersion relation between the frequency and wave vector of the FM nonlinear spin wave has the same form as that of the well-known low-amplitude spin wave, but with a prefactor which is explicitly dependent on its amplitude. This is in contrast with the AFM nonlinear spin wave, where we obtain the following intriguing results: Borh the frequency and the wave vector acquire complicated dependences on the sublattice spin wave amplitudes. However, the dispersion relation between them is identical to that of the usual low-amplitude AFM spin wave, with no explicit dependence on the amplitudes. We outline how such solutions can also be supported in certain variants of the above models. 0 1997 Elsevier Science B.V.
1. Introduction
The nonlinear dynamics of one-dimensional magnetic systems has attracted a lot of attention in recent years [I]. This is essentially because experimentalists have been successful in synthesizing crystals with regular chains of magnetic ions such that the interchain interaction is much weaker than the intrachain interaction, by substituting non-magnetic ions or large organic complexes between chains [2]. Consider a magnetic chain modeled by the nearest-neighbour Heisenberg exchange Hamiltonian
H=
- Sn+,_
-JcS,
(1.1)
n
’ Senior Associate
of the ICTP, Trieste, Italy.
0375-9601/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved PII SO37S-9601(97)00686-S
Here S, denotes the spin at the nth site, with Si = S’. Further, J > 0 and J < 0 correspond, respectively, to ferromagnetic (FM) and antiferromagnetic (AFM) coupling. Strictly speaking, the S, appearing in Eq. (1.1) are quantum spin operators. But in experimental situations [2], it turns out that they can be regarded as classical vectors for S r 1. The classical spin evolution equation is found from Eq. (1.1) by using [3]
~={H,S,}.
( *.2)
where (,) denotes the Poisson bracket, and the Cartesian components Sz satisfy the angular momentum algebra
R. Balakrishnan.
74
R. Dhamunkar/
obtained by cal variables component in Eq. (1.2)
recognizing that the appropriate canoniare the azimuthal angle (pn of S, and its S,Z, i.e. {cp,, S$ = S,,,. Using Eq. (1.3) yields
-
x (Sn+* +s,_,).
=JS,
dt
(1.4)
then obtains linear differential-difference equations which support (small-amplitude) spin wave solutions of the form S,’ = u exp[i(kna - ot>], where ST = Si + iSi, a is the lattice separation and u is a small constant. Further, the dispersion relation can be shown to be 0=2JS(l
It must also be noted that the analysis of the dynamics will be different for the FM (J > 0) and the AFM (J < 0) chains essentially because in the ground state, the nearest-neighbour vectors are parallel and antiparallel, respectively, in these cases. Thus in the FM case, Eq. (1.4) can be used directly. For the AFM case, it is more appropriate to write down the equations for the spin vectors at the even and odd sites (regarded as two sublattices) separately as follows,
ds,, - dt = JL dS -
Physics Letters A 237 (I 997) 73-79
2n+1
dt
=
x CL,-, +%n+,)T J&n+, x
(%l
+S2,+2).
( 1.5a)
(1.5b)
Eqs. (1.4) and (1.5) are coupled nonlinear differential-difference equations. The natural question that arises is whether they can be solved exactly. In this Letter, we obtain a class of exact solutions of these equations with physical relevance. This is of interest because, when confronted with such nonlinear equations in physics, one usually analyses either their nonlinear continuum approximation or their linear discrete approximation. The former has been discussed for the Heisenberg chain and many of its variants [I]. In particular, it is we11 known that the continuum version of Eq. (1.4) is the Landau-Lifshitz equation [4], which has been shown [5] to be completely integrable using the inverse scattering transform, leading to strict soliton solutions. Finding the continuum version of the AFM equations (1.5) is less straightforward and has been considered by several authors [6,1]. But the question of its complete integrability is still unclear. The latter, i.e. the linear discrete approximation of Eqs. (1.4) and (1.5), can be found in any standard solid state physics textbook [7], wherein one considers small-amplitude modes Sz, Si < S around the respective ground states S,Z= S for the FM and S,L= (- 1)“s for the AFM. One
( 1.6a)
-coska)
for the FM chain, and ( I .6b)
w=2JSsinka
for the AFM chain. Our new results are as follows: We obtain a class of exact solutions for Eqs. (1.4) and (1.5) in the form of finite-amplitude FM and AFM spin waves respectively. We call these solutions nonlinear spin waves. We find that the frequency of the FM nonlinear spin wave becomes amplitude dependent, but its wave vector is unconstrained, so that its dispersion relation has the same functional form as Eq. (1.6a), but with a prefactor which is explicitly dependent on the amplitude. For the AFM nonlinear wave, in contrast, both the frequency and the wave vector acquire complicated dependences on the sublattice spin wave amplitudes. But we find the surprising result that its dispersion relation is identical to (1.6b), the usual low-amplitude AFM spin wave, with no explicit dependence on the amplitudes. Extensions of these results to an alternating chain and a chain with staggered magnetic field term are also outlined. In the analysis that follows, we will find it convenient to represent the Cartesian components of S, in terms of the polar and azimuthal angles 0, and d,,, i.e. S” = (S;,S,\,S,z) = S(sin 0, cos 4,) sin 0, sin 4,) cos 0,))
(1.7)
where 0 2 0, I rr and 0 I 4, < 2~. Thus S,+ = Ssin 0, exp(i$,).
2. Nonlinear ferromagnetic
(1.8)
spin waves
For the FM case, the ground state corresponds to all the spin vectors aligned in the same direction,
R. Balakrishnan,
say, S; = S. Writing ponents, we obtain dS,+ = -i dt
R. Dhamankm/Physics
out Eq. (1.4) in terms of com-
-
dt
-J[S&q+, -s&s;+,
J>O.
As B0 increases from 0 to n/2, the total energy increases by an amount proportional to sin%,.
(2.2)
3. Nonlinear antiferromagnetic
+.si_,)
+.sd‘_,)].
= S sin 19~exp[i( kna - cot)],
(2.3)
where 8, represents the finite constant polar angle of all the spin vectors on the chain. Firstly, since S,’ = S’, a solution such as Eq. (2.3) imposes the condition Si = S cos B0 = const., and hence dS5 /dt must vanish for such solutions. Indeed, by using Eq. (2.3) in Eq. (2.2), we can verify that this is identically satisfied. Next, substituting Eq. (2.3) in Eq. (2.11, we find the amplitude-dependent dispersion relation
where 018,I
-JiVS*[l--sin20,(l-cosku)],
(2.3) in
(2.1)
S,+ = S sin 8, exp( i 4,)
=2.lscos8,(l
by substituting
(2.5)
We now show that Eqs. (2.1) and (2.2) support finite-amplitude spin wave solutions of the form
w(k)
can be found
+Si_,)
-.s$q+, +.y_,)],
w
total energy (1.1): E=
J[S,‘(S,Z+,
7s
Letters A 237 (1997) 73-79
a is the nearest-neighbour 7r.
Since the nearest-neighbour spin directions are antiparallel to each other in the ground state, it is more convenient to study the dynamics of the spins at the even and odd sites separately. Writing out Eqs. (1.5) in terms of the spin components, we get (with J < 0) -=
-i
dt
J[STn(Si,+,
-S&(Szf,+, dS;,,+, ~ dt
= -i
+Sin-,)
(2.4) separation,
and
J[S2+,+,(S&+2
dS&, = -J[S,\‘,(S;,+, dt -S;,(S;n+,
2.1. Linear approximation As already mentioned, linear approximations discussed in textbooks correspond to taking S:= u exp[i(knu - ot)] with u small, and thus approximating Sz = (S’ - u’)‘/’ by S. Thus Eq. (2.1) is = - i 6S(2ST - Sz+ , approximated by dSz/dt .SJ_ ,), and Eq. (2.2) gives dS,L/dt = 0. Substituting the above expression of Sz leads to the well-known linear spin wave dispersion relation (1.6a). Comparing this relation with Eq. (2.4) shows that for a fixed value of k, the nonlinear spin wave frequency depends on the amplitude (u = sin 0,) according to w = 2./S(l - u’)‘/‘(l - cos ku). (This exact result goes over to the linear spin wave result on setting u = 0.) Moreover we see that, even for a fixed value of k, there is a nonlinear spin wave band of w values, corresponding to different values of Be. The
d%+, dt
(3.la)
+Sl:,_,)]. +.Sj,)
%!)I~
-SZ;n+,(SZ+n+2 +
-coska),
spin waves
(3.lb)
+$-,)
(3.lc)
+Sl’“_,)].
= -J[SZ\n+,(S;n+z -S;n+,(S;‘n+?
We show that Eqs. (3.1) spin waves of the form
+ S;,,)
(3.ld)
+.K,)]. support
finite-amplitude
S: = S sin B0 exp(i 4,) = Ssin 8, exp[i( kju - wt)],
(3.2a) (3.2b)
s; = scos 8, for j even, and ST =Ssin(r-
e,)exp(i+,)
= Ssin 8, exp[i( kju - wt + a)],
(3.3a)
SCOS(~- e,) = -~COS 8,
(3.3b)
siJ =
for j odd. Here
f9,, and (rr-
0,)
represent
finite
76
R. Balakrishnan,
R. Dhamankar/
(constant) polar angles at the even and odd sites respectively, and CY is a constant relative phase factor to be determined. We have sought solutions of the form given above so that in the limit of very small 19~ and el, the spin vectors are almost along the _+z axis and we can thus read off the corresponding linear spin wave solutions from our finite-amplitude analysis. First, it can be verified that Eqs. (3.2a) and (3.3a) when used in Eq. (3.1~) indeed yield dS,‘/dr = 0 for all even j as required by Eq. (3.2b). On the other hand, using them in Eq. (3.ld), we obtain dS’ / = 2 JS’ sin 8, sin 8, cos ka sin (Y at for j odd. Since this must vanish (see Eq. (3.3b)), we obtain (Y= 0 or 7r. Hence (Y cannot take on arbitrary values in Eq. (3.3a). Next, substituting Eqs. (3.2a) and (3.3a) in Eq. (3.1 a> and Eq. (3.1 b) respectively, we get o + 2 JScos8,
= f2 JScos
sin 0, cos ka BO-_ sin 0,
= f2JScos
e,-_
(3.4a)
and w-
2JScos8,
sin B0 sm 8,
cos ka, (3.4b)
where the upper sign corresponds lower to cr = rr.
to (Y= 0 and the
3.1. Linear approximation Before proceeding further, we study Eqs. (3.4) in the linear approximation by considering the amplitudes of the spin vectors around Sj = +S, that is, when 8, and 8, are very small. We obtain (o+2JS)e0i(2JScoska)e,=0,
(3.5a)
+(ZJScoska)e,+(w-2JS)e,=O.
(3.5b)
Physics Letters A 237 (I 9971 73-79
relation for a (linearized) AFM spin wave (Eq. (1.6b)). Returning to the finite-amplitude relations (3.4), we note that they are two simultaneous equations for w and cos ka, which can be solved to yield
w= +2JS
cos 8, - cos 8, i - cos eO cos 8,
(3.7)
and sin eO sin 8,
cos k4l= _t
i - cos eO cos e1
(3.8)
Thus the situation for the AFM chain is different from the FM case where one could obtain the dispersion relation (2.4) directly by substituting the spin wave solution (2.3) in the evolution equations (2.1) and (2.2). For the AFM case however, both w and k are functions dependent on the amplitudes (B,, 8,) as seen from Eqs. (3.7) and (3.8). Although these are complicated functions, it turns out that we can actually eliminate 8, and 0, from them to obtain w as a function of k alone. What is more surprising is the fact that the dispersion relation thus obtained for the AFM nonlinear spin wave is identical to that for the linear spin wave (Eq. (3.6)). This can be shown by first finding the expressions for w2 and cos’ ka from Eqs. (3.7) and (3.8) and using them to verify that Eq. (3.6) is identically satisfied ‘. This is in contrast to the FM case where the nonlinear spin wave dispersion relation (2.4) depends on the amplitude 8, and therefore renormalizes the prefactor of the linear spin wave dispersion relation (1.6a). Some special features of these AFM nonlinear modes can be understood by studying the expressions (3.7) and (3.8). Let us first consider LY= 0, which corresponds to the positive sign in (3.8) and therefore implies 0 I k 5 7r/2a. Let us fix 0, at some constant value (less than n/2, say) and study the modes as 0, is varied from 0 to 7r. When 0, =O, w= -2JS=21JIS (since J
Eqs. (3Sa) and (3.5b) represent two linear homogeneous equations in B0 and 0, which have a nontrivial solution only if ’ A geometrical way to verify this is to recognize that the factor
w2 = 4 J'S'
sin' ka,
or w = 2 JS sin ka, the well-known
(3.6) [7] dispersion
1-cos B(,cos 0,
occurring in the denominator of o and cos kcu
(see Eqs. (3.7) and (3.8)) is just the hypotenuse of a right-angled triangle with base sin B,, sin 0, and altitude cos f3,, -cos
0,.
R. Bulukrishnon.
Qa
R. Dhamunkur/
IT
Fig. I. Contour plot of kc@,,, 0,) (Eq. (3.8)). The arrows indicate the directlon of increasing k values for (Y = 0 and decreasing k values for u = 7r (see text).
rr/2u.
As 0, is increased from 0, both w and k decrease, and they both vanish when 0, becomes equal to BO. As 8, is increased beyond f3,,, w takes on negative values and k also increases. When 8, = n-/2, i.e. the odd spin vectors lie in the xv plane, the frequency w = - 21 J].S cos 0, and cos ka = sin Be. Finally, for 0, = rr, w= -2]JIS, the maximum negative value possible, but k = 7r/2a once again. Next consider (Y= rr which corresponds to the negative sign in (3.81, implying rr/2a I k 5 m,/ cr. An analogous discussion can be given for this case. It is instructive to study the contour plot of k(fl,, 0,) obtained from Eq. (3.8). This is given in Fig. 1. The two arrows indicate the directions of increasing k from 0 to n-/2 cr for CK= 0. (On the other hand, they indicate the directions of decreasing k from rr/ CYto 7r/2 cy for cy = n.) We find that there is a symmetry about the 8, = 0, line, which divides the 0,-8, plane into two halves corresponding to w < 0 and w > 0 respectively. The above analysis generalises to all finite amplitudes, the results of Keffer et al. [8] for linearized AFM spin waves of very small amplitudes 8, and 8,, where two types of modes w < 0 and w > 0 were found to appear for 8, > 8, and 0, < 0, resoectivelv. 1
”
1
J
II
Physrcs Letters A 237 (1997) 73-79
--~_i_
n
”
^0
Oo
Fig. 2. Contour plot of E(H,,, 0,) (E7.q.(3.9)). The arrows indmte the direction of increasing E values.
The total energy of the AFM chain is expressible in terms of B0 and 0, as E=J(NX
1)s’
I
sin’ 19~sin’ 0,
1 - cos t), cos 0,
.
+cosCI,cosH,
(3.9)
1
Note that E is the same for both values of cy. The contour plot of E(B,, 0,) obtained from Eq. (3.9) is presented in Fig. 2. The two arrows indicate the direction of increasing energy from the ground state value - 1JI(N - 1)S2 to the maximum value 1J I( N - I)S’. This plot clearly shows that as the energy is increased, the set of allowed values for 8, and 0,
Fig. 3. E versus ka
78
R. Bulakrishnun.
R. Dhumankar/
decreases. This in turn restricts the allowed values of k for larger energies. This can also be seen from the E versus ka plot given in Fig. 3, which shows a triangular-shaped allowed region, with the base corresponding to the ground state energy. From this figure, we find that although the number of allowed k modes does decrease as the energy increases, the number of spin wave modes that can be excited is quite large, even at intermediate energies. This can have observable consequences, as will be pointed out at the end. We have also made some preliminary studies on nonlinear spin waves in the case of the following variants of the Heisenberg spin chain. (a) The alternating chain with exchange energies J and J’ described by the Hamiltonian * (JSZ, + J’&n+2).
H = CSzn+, n
For the FM case we find that the dispersion becomes o=Scos0,
I
J+J’-
(J-J’)2+4JJfcos2ka
relation
I
,
which reduces to (2.4) when J = J’. For the AFM case we obtain for w and k. certain generalizations of the expressions (3.7) and (3.8). Interestingly, we find that once again it is possible to eliminate 8, and 8, from these expressions to get w=2mSsinka, where sinka=
(J+J’)(co~~o+co~~~) 2LZ(
1 + cos e. cos 6,)
The important difference is that those values of 0, and 8, are (sinka] I 1. (b) A Heisenberg chain with a field B along the z-axis modeled
.
for this case, only possible for which staggered magnetic by
H=J~(S;S,+,+(-l)“BG,). n This destroys the preferred parallel orientations in ferromagnets and now, for both the cases one can write equations of motion for odd and even sites. We see that o as well as k are functions of &,, 8, and B. But the elimination of these parameters is not possible and hence o is a multivalued function of k.
Physics Letters A 237 (1997) 73-79
The higher the staggered field, the more it tries to align the spins along it. Therefore, as the value of the staggered field is increased, the range of allowed 8, and 8, values becomes smaller.
4. Discussion Nonlinear spin waves have been found in the continuum version of certain anisotropic magnetic chains [9]. However, it must be mentioned that exact solutions of nonlinear discrete evolution equations are rare. The Toda chain [lo] of particles with exponential nearest-neighbor interactions is one of the few examples where a class of exact dynamical solutions for the displacement of the particle at the nth site is a cnoidal wave. This transforms into a sinusoidal wave in the small-amplitude approximation. However, in systems such as the interacting nearest-neighbor spin vectors on a discrete chain considered in this work, one must solve coupled equations for the spin components. While in the existing literature for the isotropic FM and AFM chains, the analysis of these discrete equations has been carried out only in the small-amplitude approximation to obtain (sinusoidal) spin waves, our work shows that these systems in fact support finite-amplitude nonlinear spin waves as exact solutions. For the FM case, the functional form of the w versus k dispersion relation of these exact nonlinear modes is the same as that for the linear modes except for an amplitude-dependent prefactor cos BO, (One could regard this as a kind of “renormalization” induced by the nonlinearity.) Intuitively, one might expect an analogous renormalization factor to appear in the AFM case as well. But this does not happen. We find that there is no such explicit amplitude-dependent factor in the dispersion relation for the AFM nonlinear modes, in spite of the fact that both w and k are nontrivial functions of the sublattice amplitudes 8, and 8, (see Eqs. (3.7) and (3.8)). Intriguingly, their functional dependences are such that the dispersion relation of a nonlinear AFM spin wave is identical to that of the well-known linear spin wave. We conclude with a remark. Some early experiments 17,111 on certain AFM systems reported the following striking observation: Well-resolved spin
R. Balakrishnan.
R. Dhamankar/Physics
wave modes with finite k were seen to persist for fairly high temperatures (i.e. well up to the NCel ordering temperature) and their dispersion relation could be fitted well with that of the small-amplitude linear modes. Obviously, at such temperatures, the amplitude of the modes will be large, suggesting that nonlinear effects should be included in any satisfactory theoretical justification of this observation. In this connection, our nonlinear analysis of the AFM chain is useful and shows the following: In spite of the fact that the range of allowed k values does decrease as the energy increases, Fig. 3 indicates that for a range of energies well above the ground state, quite a large number of spin wave modes are still available. This could be a possible reason for the observed persistence of finite K modes even when the temperature is increased. In addition, it is clear that it should also be possible to solve for the values of the two unknown amplitudes 8, and 0,. by substituting the experimental values of the spin wave excitation energy o, the corresponding wave vector k and the lattice spacing c( in the two equations (3.7) and (3.8).
providing excellent faciIities during the preparation of this manuscript. RD thanks the Institute of Mathematical Sciences for summer-studentships during 1995 and 1996.
References [ll
H.J. Mikeska,
M. Steiner. Adv.
Phys. 40 (1991)
191, and
references therein.
[21 L.J.
de Jongh, A.R. Miedema.
Adv. Phys. 23 (1974)
I.
131 N.D. Mermin, J. Math. Phys. 8 (1967) 106 141 L. Landau, E. Lifshitz. Phys. Z. Sowjetunion 8 (1935) M. Lakshmanan, Th.W. 84 (1976)
iSI
[61 I.V.
Phys. Len. A 64 (1977)
235.
Baryakhtar. B.A. Ivanov, Fiz. Niz. Temp. 5 (1979)
R. Balakrishnan, A.R. Bishop. R. Dandoloff, 64 (1990)
[71 C.
Kittel.
2107: Phys. Rev. B 47 (1993) Introduction
Wiley, New York,
[81 F.
153;
Ruijgrok, C.J. Thompson. Physica A
577.
L.A. Takhtajan,
to Solid
759;
Phys. Rev. Lett.
3108.
State Physics,
5th edn..
1976.
Keffer, H. Kaplan. Y. Yafet. Am. J. Phys. 71 (1953)
250.
and references therein.
191 A.
Chubukov, Fiz. Niz. Temp.
K.O. Abdulloev,
Acknowledgement
79
Letters A 237 f 1997) 73-79
10 (1984)
K.K. Muminov.
3450 [Phys. Solid State 37 (1995)
[IO1 M.
Toda.
Theory
of Nonlinear
1166;
Fiz. Tverd. Tela 37 (1995) 18951. Lattices,
Springer.
Berlin.
1981.
RB would like to thank ICTP, Trieste, hospitality through a Senior Associateship,
Italy for and for
[III
M.T. Hutchings, G. Shirane, R.J. Birgeneau, S.L. Holt, Phys. Rev. 5 (1972)
1999.
1997
PHYSICS
ELSEVIER
LETTERS
A
Physics Letters A 237 (1997) 73-79
Exact nonlinear spin waves in some models of interacting classical spins on a one-dimensional lattice Radha Balakrishnan a,‘, Rohit Dhamankar b a Institute h Department
of Mathemaiical
of Physics.
Sciences. C.I.T. Campus,
Indian
Institute
of Technology.
Madras
600 113, India
Kanpur
Received 3 March 1997; revised manuscript
208016.
India
received 19 August 1997; accepted for publication Communicated by A.R. Bishop
26 August 1997
Abstract We present exact finite-amplitude spin wave solutions for the nonlinear differential-difference equations describing the dynamics of spin vectors in a ferromagnetic (FM) and an antiferromagnetic (AFM) classical Heisenberg model of interacting spins on a one-dimensional lattice. We find that the dispersion relation between the frequency and wave vector of the FM nonlinear spin wave has the same form as that of the well-known low-amplitude spin wave, but with a prefactor which is explicitly dependent on its amplitude. This is in contrast with the AFM nonlinear spin wave, where we obtain the following intriguing results: Borh the frequency and the wave vector acquire complicated dependences on the sublattice spin wave amplitudes. However, the dispersion relation between them is identical to that of the usual low-amplitude AFM spin wave, with no explicit dependence on the amplitudes. We outline how such solutions can also be supported in certain variants of the above models. 0 1997 Elsevier Science B.V.
1. Introduction
The nonlinear dynamics of one-dimensional magnetic systems has attracted a lot of attention in recent years [I]. This is essentially because experimentalists have been successful in synthesizing crystals with regular chains of magnetic ions such that the interchain interaction is much weaker than the intrachain interaction, by substituting non-magnetic ions or large organic complexes between chains [2]. Consider a magnetic chain modeled by the nearest-neighbour Heisenberg exchange Hamiltonian
H=
- Sn+,_
-JcS,
(1.1)
n
’ Senior Associate
of the ICTP, Trieste, Italy.
0375-9601/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved PII SO37S-9601(97)00686-S
Here S, denotes the spin at the nth site, with Si = S’. Further, J > 0 and J < 0 correspond, respectively, to ferromagnetic (FM) and antiferromagnetic (AFM) coupling. Strictly speaking, the S, appearing in Eq. (1.1) are quantum spin operators. But in experimental situations [2], it turns out that they can be regarded as classical vectors for S r 1. The classical spin evolution equation is found from Eq. (1.1) by using [3]
~={H,S,}.
( *.2)
where (,) denotes the Poisson bracket, and the Cartesian components Sz satisfy the angular momentum algebra
R. Balakrishnan.
74
R. Dhamunkar/
obtained by cal variables component in Eq. (1.2)
recognizing that the appropriate canoniare the azimuthal angle (pn of S, and its S,Z, i.e. {cp,, S$ = S,,,. Using Eq. (1.3) yields
-
x (Sn+* +s,_,).
=JS,
dt
(1.4)
then obtains linear differential-difference equations which support (small-amplitude) spin wave solutions of the form S,’ = u exp[i(kna - ot>], where ST = Si + iSi, a is the lattice separation and u is a small constant. Further, the dispersion relation can be shown to be 0=2JS(l
It must also be noted that the analysis of the dynamics will be different for the FM (J > 0) and the AFM (J < 0) chains essentially because in the ground state, the nearest-neighbour vectors are parallel and antiparallel, respectively, in these cases. Thus in the FM case, Eq. (1.4) can be used directly. For the AFM case, it is more appropriate to write down the equations for the spin vectors at the even and odd sites (regarded as two sublattices) separately as follows,
ds,, - dt = JL dS -
Physics Letters A 237 (I 997) 73-79
2n+1
dt
=
x CL,-, +%n+,)T J&n+, x
(%l
+S2,+2).
( 1.5a)
(1.5b)
Eqs. (1.4) and (1.5) are coupled nonlinear differential-difference equations. The natural question that arises is whether they can be solved exactly. In this Letter, we obtain a class of exact solutions of these equations with physical relevance. This is of interest because, when confronted with such nonlinear equations in physics, one usually analyses either their nonlinear continuum approximation or their linear discrete approximation. The former has been discussed for the Heisenberg chain and many of its variants [I]. In particular, it is we11 known that the continuum version of Eq. (1.4) is the Landau-Lifshitz equation [4], which has been shown [5] to be completely integrable using the inverse scattering transform, leading to strict soliton solutions. Finding the continuum version of the AFM equations (1.5) is less straightforward and has been considered by several authors [6,1]. But the question of its complete integrability is still unclear. The latter, i.e. the linear discrete approximation of Eqs. (1.4) and (1.5), can be found in any standard solid state physics textbook [7], wherein one considers small-amplitude modes Sz, Si < S around the respective ground states S,Z= S for the FM and S,L= (- 1)“s for the AFM. One
( 1.6a)
-coska)
for the FM chain, and ( I .6b)
w=2JSsinka
for the AFM chain. Our new results are as follows: We obtain a class of exact solutions for Eqs. (1.4) and (1.5) in the form of finite-amplitude FM and AFM spin waves respectively. We call these solutions nonlinear spin waves. We find that the frequency of the FM nonlinear spin wave becomes amplitude dependent, but its wave vector is unconstrained, so that its dispersion relation has the same functional form as Eq. (1.6a), but with a prefactor which is explicitly dependent on the amplitude. For the AFM nonlinear wave, in contrast, both the frequency and the wave vector acquire complicated dependences on the sublattice spin wave amplitudes. But we find the surprising result that its dispersion relation is identical to (1.6b), the usual low-amplitude AFM spin wave, with no explicit dependence on the amplitudes. Extensions of these results to an alternating chain and a chain with staggered magnetic field term are also outlined. In the analysis that follows, we will find it convenient to represent the Cartesian components of S, in terms of the polar and azimuthal angles 0, and d,,, i.e. S” = (S;,S,\,S,z) = S(sin 0, cos 4,) sin 0, sin 4,) cos 0,))
(1.7)
where 0 2 0, I rr and 0 I 4, < 2~. Thus S,+ = Ssin 0, exp(i$,).
2. Nonlinear ferromagnetic
(1.8)
spin waves
For the FM case, the ground state corresponds to all the spin vectors aligned in the same direction,
R. Balakrishnan,
say, S; = S. Writing ponents, we obtain dS,+ = -i dt
R. Dhamankm/Physics
out Eq. (1.4) in terms of com-
-
dt
-J[S&q+, -s&s;+,
J>O.
As B0 increases from 0 to n/2, the total energy increases by an amount proportional to sin%,.
(2.2)
3. Nonlinear antiferromagnetic
+.si_,)
+.sd‘_,)].
= S sin 19~exp[i( kna - cot)],
(2.3)
where 8, represents the finite constant polar angle of all the spin vectors on the chain. Firstly, since S,’ = S’, a solution such as Eq. (2.3) imposes the condition Si = S cos B0 = const., and hence dS5 /dt must vanish for such solutions. Indeed, by using Eq. (2.3) in Eq. (2.2), we can verify that this is identically satisfied. Next, substituting Eq. (2.3) in Eq. (2.11, we find the amplitude-dependent dispersion relation
where 018,I
-JiVS*[l--sin20,(l-cosku)],
(2.3) in
(2.1)
S,+ = S sin 8, exp( i 4,)
=2.lscos8,(l
by substituting
(2.5)
We now show that Eqs. (2.1) and (2.2) support finite-amplitude spin wave solutions of the form
w(k)
can be found
+Si_,)
-.s$q+, +.y_,)],
w
total energy (1.1): E=
J[S,‘(S,Z+,
7s
Letters A 237 (1997) 73-79
a is the nearest-neighbour 7r.
Since the nearest-neighbour spin directions are antiparallel to each other in the ground state, it is more convenient to study the dynamics of the spins at the even and odd sites separately. Writing out Eqs. (1.5) in terms of the spin components, we get (with J < 0) -=
-i
dt
J[STn(Si,+,
-S&(Szf,+, dS;,,+, ~ dt
= -i
+Sin-,)
(2.4) separation,
and
J[S2+,+,(S&+2
dS&, = -J[S,\‘,(S;,+, dt -S;,(S;n+,
2.1. Linear approximation As already mentioned, linear approximations discussed in textbooks correspond to taking S:= u exp[i(knu - ot)] with u small, and thus approximating Sz = (S’ - u’)‘/’ by S. Thus Eq. (2.1) is = - i 6S(2ST - Sz+ , approximated by dSz/dt .SJ_ ,), and Eq. (2.2) gives dS,L/dt = 0. Substituting the above expression of Sz leads to the well-known linear spin wave dispersion relation (1.6a). Comparing this relation with Eq. (2.4) shows that for a fixed value of k, the nonlinear spin wave frequency depends on the amplitude (u = sin 0,) according to w = 2./S(l - u’)‘/‘(l - cos ku). (This exact result goes over to the linear spin wave result on setting u = 0.) Moreover we see that, even for a fixed value of k, there is a nonlinear spin wave band of w values, corresponding to different values of Be. The
d%+, dt
(3.la)
+Sl:,_,)]. +.Sj,)
%!)I~
-SZ;n+,(SZ+n+2 +
-coska),
spin waves
(3.lb)
+$-,)
(3.lc)
+Sl’“_,)].
= -J[SZ\n+,(S;n+z -S;n+,(S;‘n+?
We show that Eqs. (3.1) spin waves of the form
+ S;,,)
(3.ld)
+.K,)]. support
finite-amplitude
S: = S sin B0 exp(i 4,) = Ssin 8, exp[i( kju - wt)],
(3.2a) (3.2b)
s; = scos 8, for j even, and ST =Ssin(r-
e,)exp(i+,)
= Ssin 8, exp[i( kju - wt + a)],
(3.3a)
SCOS(~- e,) = -~COS 8,
(3.3b)
siJ =
for j odd. Here
f9,, and (rr-
0,)
represent
finite
76
R. Balakrishnan,
R. Dhamankar/
(constant) polar angles at the even and odd sites respectively, and CY is a constant relative phase factor to be determined. We have sought solutions of the form given above so that in the limit of very small 19~ and el, the spin vectors are almost along the _+z axis and we can thus read off the corresponding linear spin wave solutions from our finite-amplitude analysis. First, it can be verified that Eqs. (3.2a) and (3.3a) when used in Eq. (3.1~) indeed yield dS,‘/dr = 0 for all even j as required by Eq. (3.2b). On the other hand, using them in Eq. (3.ld), we obtain dS’ / = 2 JS’ sin 8, sin 8, cos ka sin (Y at for j odd. Since this must vanish (see Eq. (3.3b)), we obtain (Y= 0 or 7r. Hence (Y cannot take on arbitrary values in Eq. (3.3a). Next, substituting Eqs. (3.2a) and (3.3a) in Eq. (3.1 a> and Eq. (3.1 b) respectively, we get o + 2 JScos8,
= f2 JScos
sin 0, cos ka BO-_ sin 0,
= f2JScos
e,-_
(3.4a)
and w-
2JScos8,
sin B0 sm 8,
cos ka, (3.4b)
where the upper sign corresponds lower to cr = rr.
to (Y= 0 and the
3.1. Linear approximation Before proceeding further, we study Eqs. (3.4) in the linear approximation by considering the amplitudes of the spin vectors around Sj = +S, that is, when 8, and 8, are very small. We obtain (o+2JS)e0i(2JScoska)e,=0,
(3.5a)
+(ZJScoska)e,+(w-2JS)e,=O.
(3.5b)
Physics Letters A 237 (I 9971 73-79
relation for a (linearized) AFM spin wave (Eq. (1.6b)). Returning to the finite-amplitude relations (3.4), we note that they are two simultaneous equations for w and cos ka, which can be solved to yield
w= +2JS
cos 8, - cos 8, i - cos eO cos 8,
(3.7)
and sin eO sin 8,
cos k4l= _t
i - cos eO cos e1
(3.8)
Thus the situation for the AFM chain is different from the FM case where one could obtain the dispersion relation (2.4) directly by substituting the spin wave solution (2.3) in the evolution equations (2.1) and (2.2). For the AFM case however, both w and k are functions dependent on the amplitudes (B,, 8,) as seen from Eqs. (3.7) and (3.8). Although these are complicated functions, it turns out that we can actually eliminate 8, and 0, from them to obtain w as a function of k alone. What is more surprising is the fact that the dispersion relation thus obtained for the AFM nonlinear spin wave is identical to that for the linear spin wave (Eq. (3.6)). This can be shown by first finding the expressions for w2 and cos’ ka from Eqs. (3.7) and (3.8) and using them to verify that Eq. (3.6) is identically satisfied ‘. This is in contrast to the FM case where the nonlinear spin wave dispersion relation (2.4) depends on the amplitude 8, and therefore renormalizes the prefactor of the linear spin wave dispersion relation (1.6a). Some special features of these AFM nonlinear modes can be understood by studying the expressions (3.7) and (3.8). Let us first consider LY= 0, which corresponds to the positive sign in (3.8) and therefore implies 0 I k 5 7r/2a. Let us fix 0, at some constant value (less than n/2, say) and study the modes as 0, is varied from 0 to 7r. When 0, =O, w= -2JS=21JIS (since J
Eqs. (3Sa) and (3.5b) represent two linear homogeneous equations in B0 and 0, which have a nontrivial solution only if ’ A geometrical way to verify this is to recognize that the factor
w2 = 4 J'S'
sin' ka,
or w = 2 JS sin ka, the well-known
(3.6) [7] dispersion
1-cos B(,cos 0,
occurring in the denominator of o and cos kcu
(see Eqs. (3.7) and (3.8)) is just the hypotenuse of a right-angled triangle with base sin B,, sin 0, and altitude cos f3,, -cos
0,.
R. Bulukrishnon.
Qa
R. Dhamunkur/
IT
Fig. I. Contour plot of kc@,,, 0,) (Eq. (3.8)). The arrows indicate the directlon of increasing k values for (Y = 0 and decreasing k values for u = 7r (see text).
rr/2u.
As 0, is increased from 0, both w and k decrease, and they both vanish when 0, becomes equal to BO. As 8, is increased beyond f3,,, w takes on negative values and k also increases. When 8, = n-/2, i.e. the odd spin vectors lie in the xv plane, the frequency w = - 21 J].S cos 0, and cos ka = sin Be. Finally, for 0, = rr, w= -2]JIS, the maximum negative value possible, but k = 7r/2a once again. Next consider (Y= rr which corresponds to the negative sign in (3.81, implying rr/2a I k 5 m,/ cr. An analogous discussion can be given for this case. It is instructive to study the contour plot of k(fl,, 0,) obtained from Eq. (3.8). This is given in Fig. 1. The two arrows indicate the directions of increasing k from 0 to n-/2 cr for CK= 0. (On the other hand, they indicate the directions of decreasing k from rr/ CYto 7r/2 cy for cy = n.) We find that there is a symmetry about the 8, = 0, line, which divides the 0,-8, plane into two halves corresponding to w < 0 and w > 0 respectively. The above analysis generalises to all finite amplitudes, the results of Keffer et al. [8] for linearized AFM spin waves of very small amplitudes 8, and 8,, where two types of modes w < 0 and w > 0 were found to appear for 8, > 8, and 0, < 0, resoectivelv. 1
”
1
J
II
Physrcs Letters A 237 (1997) 73-79
--~_i_
n
”
^0
Oo
Fig. 2. Contour plot of E(H,,, 0,) (E7.q.(3.9)). The arrows indmte the direction of increasing E values.
The total energy of the AFM chain is expressible in terms of B0 and 0, as E=J(NX
1)s’
I
sin’ 19~sin’ 0,
1 - cos t), cos 0,
.
+cosCI,cosH,
(3.9)
1
Note that E is the same for both values of cy. The contour plot of E(B,, 0,) obtained from Eq. (3.9) is presented in Fig. 2. The two arrows indicate the direction of increasing energy from the ground state value - 1JI(N - 1)S2 to the maximum value 1J I( N - I)S’. This plot clearly shows that as the energy is increased, the set of allowed values for 8, and 0,
Fig. 3. E versus ka
78
R. Bulakrishnun.
R. Dhumankar/
decreases. This in turn restricts the allowed values of k for larger energies. This can also be seen from the E versus ka plot given in Fig. 3, which shows a triangular-shaped allowed region, with the base corresponding to the ground state energy. From this figure, we find that although the number of allowed k modes does decrease as the energy increases, the number of spin wave modes that can be excited is quite large, even at intermediate energies. This can have observable consequences, as will be pointed out at the end. We have also made some preliminary studies on nonlinear spin waves in the case of the following variants of the Heisenberg spin chain. (a) The alternating chain with exchange energies J and J’ described by the Hamiltonian * (JSZ, + J’&n+2).
H = CSzn+, n
For the FM case we find that the dispersion becomes o=Scos0,
I
J+J’-
(J-J’)2+4JJfcos2ka
relation
I
,
which reduces to (2.4) when J = J’. For the AFM case we obtain for w and k. certain generalizations of the expressions (3.7) and (3.8). Interestingly, we find that once again it is possible to eliminate 8, and 8, from these expressions to get w=2mSsinka, where sinka=
(J+J’)(co~~o+co~~~) 2LZ(
1 + cos e. cos 6,)
The important difference is that those values of 0, and 8, are (sinka] I 1. (b) A Heisenberg chain with a field B along the z-axis modeled
.
for this case, only possible for which staggered magnetic by
H=J~(S;S,+,+(-l)“BG,). n This destroys the preferred parallel orientations in ferromagnets and now, for both the cases one can write equations of motion for odd and even sites. We see that o as well as k are functions of &,, 8, and B. But the elimination of these parameters is not possible and hence o is a multivalued function of k.
Physics Letters A 237 (1997) 73-79
The higher the staggered field, the more it tries to align the spins along it. Therefore, as the value of the staggered field is increased, the range of allowed 8, and 8, values becomes smaller.
4. Discussion Nonlinear spin waves have been found in the continuum version of certain anisotropic magnetic chains [9]. However, it must be mentioned that exact solutions of nonlinear discrete evolution equations are rare. The Toda chain [lo] of particles with exponential nearest-neighbor interactions is one of the few examples where a class of exact dynamical solutions for the displacement of the particle at the nth site is a cnoidal wave. This transforms into a sinusoidal wave in the small-amplitude approximation. However, in systems such as the interacting nearest-neighbor spin vectors on a discrete chain considered in this work, one must solve coupled equations for the spin components. While in the existing literature for the isotropic FM and AFM chains, the analysis of these discrete equations has been carried out only in the small-amplitude approximation to obtain (sinusoidal) spin waves, our work shows that these systems in fact support finite-amplitude nonlinear spin waves as exact solutions. For the FM case, the functional form of the w versus k dispersion relation of these exact nonlinear modes is the same as that for the linear modes except for an amplitude-dependent prefactor cos BO, (One could regard this as a kind of “renormalization” induced by the nonlinearity.) Intuitively, one might expect an analogous renormalization factor to appear in the AFM case as well. But this does not happen. We find that there is no such explicit amplitude-dependent factor in the dispersion relation for the AFM nonlinear modes, in spite of the fact that both w and k are nontrivial functions of the sublattice amplitudes 8, and 8, (see Eqs. (3.7) and (3.8)). Intriguingly, their functional dependences are such that the dispersion relation of a nonlinear AFM spin wave is identical to that of the well-known linear spin wave. We conclude with a remark. Some early experiments 17,111 on certain AFM systems reported the following striking observation: Well-resolved spin
R. Balakrishnan.
R. Dhamankar/Physics
wave modes with finite k were seen to persist for fairly high temperatures (i.e. well up to the NCel ordering temperature) and their dispersion relation could be fitted well with that of the small-amplitude linear modes. Obviously, at such temperatures, the amplitude of the modes will be large, suggesting that nonlinear effects should be included in any satisfactory theoretical justification of this observation. In this connection, our nonlinear analysis of the AFM chain is useful and shows the following: In spite of the fact that the range of allowed k values does decrease as the energy increases, Fig. 3 indicates that for a range of energies well above the ground state, quite a large number of spin wave modes are still available. This could be a possible reason for the observed persistence of finite K modes even when the temperature is increased. In addition, it is clear that it should also be possible to solve for the values of the two unknown amplitudes 8, and 0,. by substituting the experimental values of the spin wave excitation energy o, the corresponding wave vector k and the lattice spacing c( in the two equations (3.7) and (3.8).
providing excellent faciIities during the preparation of this manuscript. RD thanks the Institute of Mathematical Sciences for summer-studentships during 1995 and 1996.
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