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DYNAMICS OF UNBOUND VORTICES IN THE
2-DIMENSIONAL XYAND ANISOTROPIC
HEISENBERG MODELS
F. Mertens, Adrian Bishop, M. Gouvea, G. Wysin
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C8, Supplement au no 12, Tome 49, dkembre 1988
DYNAMICS OF UNBOUND VORTICES IN THE 2-DIMENSIONAL X Y A N D
ANISOTROPIC HEISENBERG MODELS
F. G. Mertens l , A. R. Bishop, M. E. Gouvea and G. M. Wysin
Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, U.S.A.
Abstract. - Assuming an ideal gas of vortices above the Kosterlitz-Thouless transition temperature, the dynamic form factors are calculated. For the in-plane correlations a Lorentzian central peak is predicted which is independent of the vortex size and shape. However, for the out-of-plane correlations the velocity dependence of the vortex structure is decisive for the occurrence of a Gaussian central peak. Both results are in good agreement with combined Monte Carlo-molecular dynamics simulations.
1. Introduction
Quasi-two-dimensional magnetic materials with easy-plane symmetry, e.g. RbzCrCl4 or BaCoz (AsOs),
,
have been studied recently both by inelastic neutron scattering experiments 11, 2, 87and by a phenomenological theory for the dynamic corre- lations [3]. In this theory the anisotropic Heisenberg model with nearest-neighbor interactionsis considered, where Sm is a classical spin vector and 0
5
X<
1; X = 0 corresponds to the XY-model.At a critical temperature
T,
(A) Monte Carlo (MC) data [5] show a Kosterlitz-Thouless phase transition. Above T, a part of the vortex-antivortex pairs unbind and the unbound vortices are in motion due to their in- teractions. Assuming that the positions are random lo- cally, the velocity distribution is Gaussian [4], therefore the u n b o p d vortices can be treated phenomenologi- cally as an ideal gas, in the same spirit as the soliton- gas approach for 1-d magnets.The correlations for the in-plane components S, or Sy are quite distinct from those for the out-of-plane component S,. We show here that the velocity depen- dence of the vortex structure is decisive for the out- of-plane correlations, in contrast to reference [3] where only the static structure has been considered.
S, and Sy are not localized, i.e. they have no spatial Fourier transform. Therefore the in-plane correlation function Sx, (r, t) = (S, (r, t) S, (0,O)) is only globally sensitive to the presence of vortices. Thus the char- acteristic length is the average vortex-vortex separa- tion 2
c,
where is the Kosterlitz-Thouless correlation length.When a planar vortex starts moving it develops an out-of-plane structure (see next section). However, for S,, (r, t) this is not important because the dom- inant effect of moving vortices is to act like "2-d sign functions" or "2-d kinks"
,
i.e; every vortex that passes with its center between 0 and r in time t diminishes the correlations, changing cos4
by a factor of (-I), inde- pendent of the direction of movement and independent of the internal structure of the vortex [3].The detailed calculation of S,, (r, t ) is published elsewhere 141 and gives a (squared) Lorentzian central peak for the dynamic form factor
with y =
6%
/
(2E). Here is the rms velocity of the vortices which can be taken from Huber [7] who calcu- lated the velocity auto-correlation function. The cen- tral peak (2.2) is in excellent agreement with data ob- tained from combined MC-MD simulations [4]. More- over there is a qualitative agreement with the above mentioned neutron scattering experiments 1, 23. We use a continuum description and spherical coor- 3. Out-of-plane correlationsdinates for the spin configuration
S, (r, t) is localized for a single vortex, therefore cor- (r, t, = (cOs
4
sin e, sin4
sin e, e) (2.1) relations are sensitive to the vortex size and structure. where = (z, y ) . T - , ~ equations of motion have We assume a dilute gas of N, unbound vortices with Static vortex or antivortex solutions [6] 4(,,) = positions I% and velocities ui and consider the inco-f tan-' (ylz)
.
Molecular dynamics (MD) simulations herenthave shown [6] that for 0
5
X5
0.7 only a planar NVsolution 6 (r)
=
a/ 2 is stable, whereas for
X& 0.8
~ , ( r , t ) = ~ ~ c o s ~ ( r - I % - u i t ) . (3.1) only a solution which has an out-of-plane structure i=l6 (r) # T
/
2 is stable; only the former case is consid-ered here. The thermal average in S,, (r, t) = (S, (r, t) S, (O,0))
' p e r m a n e n t address: Physics Institute, University of Bayreuth, D-8580 Bayreuth, F. R. G .
C8 - 1386 JOURNAL DE PHYSIQUE
is evaluated by integration over R and u
n V s 2
1
/
d 2 m a u P (u) cos B ( r-
R
-
ut)x
where n, is the vortex density and P ( u ) is the ve- locity distribution. Introducing the vortex form factor
f (q) = Fourier transform of cos % (r) , we get S2
s z z (q, t) = m n v
x
P (u) e-jqeut. (3.3) This can be evaluated easily if the static vortex solu- tions are inserted [3]. However, for X 0.7 only the planar solution turns out to be stable [6] and S,, would then vanish, in contradiction t o the MC-MD simula- tion [3].Therefore the velocity dependence of 0 (r) must be taken into account. For X
5
0.7 and small velocity u the equations of motion yield the asymptotic solution (in the moving frame, with time unit h l J S )-1 u.e,
-
cos % =--
, r + co46 r
which has been checked by MD-simulations; S = 1 -A, and e, is the azimuthal unit vector in the xy-plane. The solution for r -t 0 can be obtained also, but we are interested here only in the correlations for small q
where the asymptotic solution should be a good ap- proximation. This leads t o a velocity dependent form factor and eventually t o
This is a Gaussian central peak which reflects the ve- locity distribution. The width I', = aq has a linear q-dependence, which is very well supported by the MC- MD data [3]. The integrated intensity is
Here the divergence for q -+ 0 results from the infi- nite range of the structure (3.4). However, the actual radius of a vortex must be on the order of
<
(see intro- duction), which can be taken into account e.g. by an ad-hoc cut-off function exp ( - ~ r1
J) with a free pa- rameter E. This gives an extra factor of re2 in (3.6), withre= 1-11 W a n d W = [ l + ( < q / ~ ) ~ ] ~ ' ~ . he final result for I, (q) is consistent with our MC-MD data for small q (Fig. 1). Note that absolute intensi- ties are compared here; we have chosen E such that I,is smaller than the data because other effects can also contribute t o the central peak, e.g. 2-magnon differ- ence processes and vortex-magnon interactions which will be treated in future publications.
0.0 0.l 0.2 0.3 '0.4 0.5
q/r
Fig. 1. - Intensity L of central peak for a temperature
T
>
T, 1:0.8. Data points result from M G M D simulations on a 50x50 lattice (circles) and 100 x 100 lattice (crosses). Solid line from (3.6) including the cut-ofl;, with E and from reference [4].Acknowledgments
The research of FGM and MEG was supported by DFG (Germany) and CNPQ (Brazil), respectively.
[I] Hutchings, M. T., Day, P., Janke, E. and Pynn, R., J. Magn. Magn. Mater. 54-57 (1986) 673. [2] Regnault, L. P., Boucher, J. P., Rossat-Mignod,
J., Bouillot, J., Pynn, R., Henry J. Y. and Renard, J. P., Physica B 136 (1986) 329.
[3] Mertens, F. G., Bishop, A. R., TNysin, G. M. and Kawabata, C., Phys. Rev. Lett. 59 (1987) 117. [4] Mertens, F. G., Bishop, A. R., TVysin, G. M. and
Kawabata, C., Phys. Rev. B, in press.
[5] Kawabata, C. and Bishop, A. R., Solid State Commun. 42 (1982) 595; J. Phys. Soc. Jpn 52
(1983) 27; Solid State Commun. 60 (1986) 169. [6] Wysin, G. M., Gouvea, M. E., Elishop, A. R. and
Mertens, F. G., Computer Simt~lation Studies in Condensed Matter Physics: Recent Developments (Springer, Berlin) 1988, in press.
[7] Huber, D. L., Phys. Rev. Lett.
