Abstract
We investigate the spectrum of small amplitude oscillations of the sine-Gordon vortex-antivortex pair in layered ferromagnetic systems described by the anisotropic three-dimensional XY-model. In the case of a small interlayer coupling constant, vortex-pairs can be formed independently in each plane. Analytical expressions for two discrete modes with zero frequency are obtained. These zero modes are associated with the two-dimensional motion of the excitation on an individual plane and are important because the interference between them and a vortex-pair gives rise to contributions to the central peak. The continuum states, which contribute to EPR linewidth, are derived using the Born approximation.
a25v32n3
Small Amplitude Oscillations of Sine-Gordon Vortex States in Planar Magnets in Three Dimensions
A . R. Pereira
Departamento de Física, Universidade Federal de Viçosa,
Viçosa, 36571-000, MG, Brazil
Received on 10 June, 2002
We investigate the spectrum of small amplitude oscillations of the sine-Gordon vortex-antivortex pair in layered ferromagnetic systems described by the anisotropic three-dimensional XY-model. In the case of a small interlayer coupling constant, vortex-pairs can be formed independently in each plane. Analytical expressions for two discrete modes with zero frequency are obtained. These zero modes are associated with the two-dimensional motion of the excitation on an individual plane and are important because the interference between them and a vortex-pair gives rise to contributions to the central peak. The continuum states, which contribute to EPR linewidth, are derived using the Born approximation.
As it is well known, two-dimensional magnetic systems having a XY-like continuous symmetry at finite temperature and for zero magnetic field exhibit a topological long-range order and a phase transition involving the unbinding of vortex-pairs at a critical temperature TKT, the Kosterlitz-Thouless temperature [1]. However, effects of relevant perturbations, such as a magnetic field applied along the plane or the presence of an inter-plane coupling Jz , change this picture dramatically. For the three-dimensional XY-model with a layered structure, even for a very small coupling between the planes, the system will show the conventional long-range order at low temperatures and an usual second-order transition at a temperature Tc which is higher than TKT of the isolated layer. The vortex-antivortex interaction does not grow logarithmically with the distance between the vortex centers as in the ideal two-dimensional model, but becomes linear for large distances of separation [2,3]. The same happens to the two-dimensional XY-model in the presence of a magnetic field applied along the plane [4,5]. In fact, it has been shown that this two-dimensional XY-model with an in-plane magnetic field can be viewed as a mean-field theory of the anisotropic three-dimensional XY-model with a very weak coupling between the planes [4]. This is because the correlations of the fluctuations between the layers are small owing to the smallness of the interlayer coupling.
The linear large distance dependence for the vortex-antivortex interaction, Evµ E2R, where R is the distance between the vortex centers, indicates that there are no free vortices in the XY-model in the presence of an external magnetic field applied along one of the in-plane axis. However, in the three-dimensional case, Minnhagen and Olsson [3] have suggested that the effective interplane coupling E2(T), at a critical temperature Tc, renormalizes to zero, leading to the correspondence of a Kosterlitz-Thouless transition. This critical temperature has been determined in recent Monte Carlo calculations and its value, for very small interlayer couplings (Jz
where the index n numbers different planes of the layer and the sum i,j is only over nearest neighbors of the lattice sites in the planes. The classical spin vector localized at layer n and site i, n,i =
can be parametrized by two scalar fields qn,i, fn,i and written as
,i = S(cosqn,icosfn,i,cosqn,isinfn,i, sinqn,i). At low temperatures T
Tc, we can adopt a mean field approximation for the interlayer coupling assuming that ád
n·d
n+1ñ
á
ñ2 and then, Hamiltonian (1) can be approximated by [4]
In Eq.(2) we define a reduced magnetic field (mean field) 2Jz|áñ| with direction parallel to the x-axis in the spin space. Using the continuum approximation to Hamiltonian (2) on each layer
where we have taken S = 1 and a0 is the lattice spacing, we get the followings equations of motion
Static in-plane solutions of these equations are those which solve the two-dimensional sine-Gordon equation
and then, below Tc this system reveals a quite interesting behavior, which is due to the vortex-antivortex pair formations in the plane of the layer, since the sine-Gordon equation possess vortex solutions. In fact, Hikami and Tsuneto [15] first showed that the vortex pairs can also be formed in the layered system of XY-model and then they proved that, in certain circumstances, it is energetically favorable for the vortex pairs in different layers to become independent instead of making a vortex ring. An exact vortex solution for equation (6) was obtained by Hudák [16], but we cannot be sure that this solution can be applied directly to the discrete lattice, since the vortex and antivortex vorticities in a pair are 4 and -4 respectively. Cataudella and Minnhagen [2] have shown that, in the study of vortex excitations, when we are interested in the analytical description of topological excitations, it is more convenient to consider that, for weak couplings, the coupling between adjacent layers leads to small deviations, at low temperatures, from the vortex-pair solution for the ideal planar ferromagnet. It is equivalent to approximate sin f f in the sine-Gordon equation and the vortex-antivortex solution obtained from this approximation is given by
where K0 is the modified Bessel function and the vortex is located at (0,a) and the antivortex at (0,-a). The energy of this excitation is [2,3]
where R = 2a, E0 is the creation energy of a vortex pair, E1 = 2pJ and E2 = p2. The coefficient of the linear term in the energy decreases as the temperature increases, vanishing at T = Tc and here, it is indicated by the mean magnetization á
ñ present in E2 [6]. The contribution of the structures (7) to the dynamical correlation functions in these layered classical ferromagnet was studied by Pires et al. [17].
The object now is to study what effect such topological structures have on the associated long wavelength ferromagnetic spin waves propagating on the layer n. To this end, we assume that the deviations from the static configuration (7) due to the presence of spin waves, are small with harmonic time dependence
with qv(
) = 0 for a planar pair and a ,b
1. Substituting Eqs. (9) into the equations of motion and linearizing in the small quantities, we obtain for the in-plane spin wave oscillations, a Schrödinger-like differential equation with zero eigenvalue
and with a frequency dependent potential not cylindrically symmetric given by
The out of plane spin wave oscillations are obtained from the in-plane oscillations through
In the mean field approximation, the essential effect of the adjacent planes on the spin wave propagation on a layer is to introduce a finite activation dispersion law given by 2 =
+
k2, due to the mean magnetization á
ñ where
= Jz J | á
ñ | (4/
-Jz | á
ñ |/J),
= J2 (4/
-Jz | á
ñ |/ J ) and k2 =
+
is the wave vector modulus. The lowest-order effect of an inhomogeneous vortex-pair background is to produce an elastic scattering center for the spin waves, since the potential (11) has a finite range of the order of the size of the pair configuration R. Thus, in polar coordinates (r,j), asymptotic solutions to Eq. (10) for the continuum states (
2³
), are given by phase-shifted cylindrical spin waves
where Dn n¢ is the phase-shift matrix coupling the different angular momentum channels n and n¢ and ,
denotes the Hanckel functions. Since the potential V(
,
) is no longer cylindrically symmetric, the angular momentum is no longer a good quantum number, nor is the phase-shift matrix Dn n¢ any longer diagonal. There are transitions between the various angular momentum channels due to the coupling between the angular dependence of the spin wave and the angular dependence of the scattering potential.
Besides the continuum states with
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These local modes govern the motion of the pair on the layer n and are of fundamental importance in the study of dynamics of topological excitations in the presence of external perturbations. Local modes, specially zero modes, are of paramount importance in the derivation of a topological excitation thermodynamics [10,18]. In fact, the number of magnons states must be decreased by a number equal to the number of bound states. One way to view this decrease is that topological excitation traps magnon states due to its presence and thus, the degrees of freedom (or free energy, etc) are shared between modes of a non-linear system. Besides, the local modes may be related to the resonance phenomena in vortex-antivortex collisions because the pair internal interaction due to Enµ E2R may have energy exchange with the pair translational modes during the collisions.
The continuum states with frequencies ³
0 can be found using the Born approximation. Based on the asymptotic behavior of the scattering equation, we can rewrite Eq. (10) as Ñ2b+k2b = U(
,
)b, where U(
,
) = V(
,
)+k2. The Green function associated to the Helmholtz operator Ñ2+k2 in two dimensions is (i/4)H0(1)(k|
-
|) and hence, the formal solution of Eq.(10) is
Using the Born approximation we can take b()
b(0)(
) = e
for the first order Born term and so on. In general this integral can only be calculated numerically, but equations (10) and (12) can be written as
and then, we can approximate the spatial dependence of the oscillations substituting b(0)() into Eq.(18) to find a(1)(
) and substituting a(1)(
) into Eq.(17) to find b(1)(
) and so on. Assuming that the potential is small, we neglect terms of order higher than 2 in the potential to obtain
The out-of-plane oscillations can be written from Eq.(19) using the relation (12). Of course, as we should expect, the scattered states as well as the local modes are not radially symmetric.
As it was discussed in Ref. [13], these scattering states provide an indirect method to experimentally detect topological excitations, through EPR linewidth measurements. The linewidth is the temporal integral of the four-spin correlation function and the structure factor used to calculate them contains a static contribution from the static pair solution and a time-dependent contribution from pair-magnon scattering. As this three-dimensional system is more realistic than the ideal two-dimensional models, we believe that our results may also be relevant for future investigations on the non-linear excitations contribution to EPR linewidth in quasi-two-dimensional magnetic materials.
We have investigated the interactions between spin waves and vortex-antivortex pair and found exact solutions for zero modes and approximated solutions for the continuum states in a layered three-dimensional XY-model. Zero modes become apparent in inelastic neutron scattering experiments and essentially contributes in central peak due to interference between ones and vortex-pair. The vortex-antivortex interaction is linear for large distances of separation and the effective interplane coupling E2(T) decreases with the temperature T, vanishing at T = Tc [3,6,9]. This mean that the interplane coupling effectively ceases to be felt by the vortices for T > Tc, the model effectively reducing to a two-dimensional planar model and free vortices start to exist and the discrete modes disappear giving rise to zero frequency states as the lower limit of a continuum of frequencies (Goldstone modes [19]). We believe that the local modes and the scattered modes obtained here may contribute to the renormalization of the vortex interaction. In fact, as it was shown in Ref. [20], the lowest order effect of vortex spin wave interactions can be described as a renormalization of the exchange constant. Besides, zero modes may be coupled to internal interaction of the vortices in a pair. Our calculations can be applied to any layered system with small interlayer coupling in which small oscillations and independent vortex pairs are present, such as in superconductors [21].
Acknowledgement
This work was partially supported by CNPq and FAPEMIG (Brazil).
- [1] J.M. Kosterlitz and D.J. Thouless, J. Phys. C 6 ,1181 (1973).
- [2] V. Cataudella and P. Minnhagen, Physica C 166, 442 (1990).
- [3] P. Minnhagen and P. Olsson, Phys. Rev. B 44, 4503 (1991).
- [4] M. Ito, Prog. Theor. Phys.66, (1981) 1129 (1981).
- [5] G.M. Wysin, M.E. Gouvęa, A.S.T. Pires, Phys. Rev. B 57, 8274 (1998).
- [6] B.V. Costa, A.R. Pereira, A.S.T. Pires, Phys. Rev. B 54, 3019 (1996).
- [7] S.K. Oh, C.N. Yoon, J.S. Chung, Phys. Rev. B 56 ,13677 (1997).
- [8] K. Nho and E. Manousakis, Phys. Rev. B 59, 11575 (1999).
- [9] A.R. Pereira, A.S.T. Pires, M.E. Gouvęa, Phys. Rev. B 51, 16413 (1995).
- [10] J.F. Currie, J.A. Krumhansl, A.R. Bishop, S.E. Trullinger, Phys. Rev. B 22, 477 (1980).
- [11] B.A. Ivanov, A.K. Kolezhuk, G.M. Wysin, Phys. Rev. Lett. 76, 511 (1996).
- [12] A.R. Pereira, F.O. Coelho, A.S.T. Pires, Phys. Rev. B 54, 6084 (1996).
- [13] C.E. Zaspel, J.E. Drumheller, Int. J. Mod. Phys. B 10, 3649 (1996).
- [14] A.R. Pereira and A.S.T. Pires, Phys. Rev. B 60, 6226 (1999).
- [15] S. Hikami and T. Tsuneto, Prog. Theor. Phys. 63, 387 (1980).
- [16] O. Hudák, Phys. Lett. A 89, 245 (1982).
- [17] A.S.T. Pires, A.R. Pereira, M.E. Gouvęa, Phys. Rev. B 49, 9663 (1994).
- [18] S.E. Trullinger and R.M. Deleonardis, Phys. Rev. A 20, 2225 (1979).
- [19] R. Rajaraman, Solitons and Instantons (North-Holland, Amsterdam, 1984).
- [20] A.R. Pereira, A.S.T. Pires, M.E. Gouvęa, Solid St. Commun. 86, 187 (1993).
- [21] O. Hudák, Phil. Magazine B 81, 1533 (2001).
Publication Dates
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Publication in this collection
04 Dec 2002 -
Date of issue
Sept 2002
History
-
Received
10 June 2002