Small Amplitude Osillations of Sine-Gordon Vortex
States in Planar Magnets in Three Dimensions
A . R.Pereira
Departamento deFsia,UniversidadeFederaldeViosa,
Viosa,36571-000, MG,Brazil
Reeivedon10June,2002
We investigatethe spetrum of smallamplitudeosillationsof thesine-Gordon vortex-antivortex
pairinlayeredferromagnetisystemsdesribedbytheanisotropithree-dimensionalXY-model. In
theaseofasmallinterlayerouplingonstant,vortex-pairsanbeformedindependentlyineah
plane. Analytialexpressionsfortwodisretemodeswithzero frequenyareobtained. Thesezero
modesareassoiatedwiththetwo-dimensionalmotionoftheexitationonanindividualplaneand
areimportantbeausetheinterferenebetweenthemand avortex-pairgivesrisetoontributions
totheentral peak. Theontinuumstates,whihontributetoEPRlinewidth,are derivedusing
theBornapproximation.
Asitis wellknown,two-dimensionalmagneti
sys-tems having a XY-like ontinuous symmetry at nite
temperatureandforzeromagnetieldexhibita
topo-logiallong-rangeorderand aphasetransition
involv-ing the unbinding of vortex-pairs at a ritial
tem-peratureT
KT
,theKosterlitz-Thoulesstemperature[1℄.
However, eets of relevant perturbations, suh as a
magnetieldappliedalongtheplaneorthepreseneof
aninter-planeouplingJ
z
,hangethispiture
dramat-ially. Forthethree-dimensionalXY-modelwitha
lay-eredstruture, evenforaverysmall ouplingbetween
theplanes,thesystemwillshowtheonventional
long-range orderat lowtemperaturesandan usual
seond-order transition at a temperature T
whih is higher
than T
KT
of theisolated layer. Thevortex-antivortex
interation doesnotgrowlogarithmiallywiththe
dis-tane between the vortex enters asin the ideal
two-dimensional model, but beomes linear for large
dis-tanes of separation[2, 3℄. The same happens to the
two-dimensional XY-model in the presene of a
mag-netieldappliedalongtheplane[4,5℄. Infat,ithas
beenshownthat this two-dimensionalXY-model with
anin-planemagnetieldanbeviewedasamean-eld
theory of theanisotropithree-dimensional XY-model
withaveryweakouplingbetweentheplanes[4℄. This
is beausetheorrelationsof theutuationsbetween
thelayersaresmallowingtothesmallnessofthe
inter-layeroupling.
Thelinearlargedistanedependeneforthe
vortex-antivortexinteration,E /E R ,whereRis the
dis-tanebetween thevortexenters, indiates that there
are no free vorties in the XY-model in the presene
of anexternal magneti eld applied along oneof the
in-planeaxis. However,in the three-dimensionalase,
Minnhagenand Olsson[3℄ havesuggestedthat the
ef-fetiveinterplaneouplingE
2
(T),ataritial
tempera-tureT
,renormalizestozero,leadingtothe
orrespon-dene of a Kosterlitz-Thouless transition. This
riti-al temperature hasbeendetermined in reentMonte
Carlo alulations and its value, for very small
inter-layer ouplings (J
z
J, where J is the intra-plane
oupling between spins S), is estimated in the range
0:725JS 2
< T
0:90JS 2
[6℄ depending on the
val-ues of the anisotropy (0 < J
z
=J < 0:1), while for
the isotropi three-dimensional XY-model (J
z = J),
it was estimated as T
= 1:54JS 2
in Refs.[6, 7℄ and
T
=1:5518JS 2
inRef. [8℄. Ourworryiswiththease
J
z
J,in whih vortex-pairsan beformed
indepen-dentlyin eah layerandE
2
anberenormalized. This
renormalizationisduetospinwavesandvortiesinthe
planethatthevortex-antivortexpairbelongsto,aswell
astospinwavesandvortiesinthesurroundingplanes
[3,6, 9℄. Whilethe stati properties of these systems
werestudied insomedetail,thedynamialbehavioris
notwelldeveloped. Onepoint of fundamental
impor-tanetostudyinthedynamisofthisthree-dimensional
model, and whih may also be related to the
meha-nismof renormalizationof theinter-plane oupling, is
theinteration betweenspinwavesandtopologial
belongsto. These interationsanleadto relevant
in-formationabout thepropertiesof the systemand an
alsobeusedintheperturbationstheoriesinvolving
non-linearexitationsresponsetoexternalperturbationsor
fores, quantization proedures for soliton states, as
well as in statistial mehanis [10℄. The presene of
atopologialexitation inthesystemprovidesa
loal-izedpotentialseenbylinearsolutions(magnons)inthe
sense that smalldeviations from thestatitopologial
struture must satisfy a Shrodinger-like equation in
whih the potential well is due to thepresene of the
topologialexitation. Reently,loalmodesand
sat-teringstatesofvortieswerealulated[11,12℄forpure
two-dimensional magneti materials and used to
pro-posepossibleexperimentalonrmationsofsolitons[13℄
andvorties[14℄. Sinethevortex-pairsareessentialto
understand the XY-models with perturbations, then,
in this paper we onsider the vortex-pair interating
with magnons in the three-dimensional XY
ferromag-netimodelwithweakouplingbetweentheplanes. We
startdeningtheHamiltonianofthethree-dimensional
lassialferromagnetimodel with predominant
inter-ationJ intheplaneofthelayerandweakinteration
J
z
betweenlayers(J
z J)
H= J X
X
i;j [S
x
;i S
x
;j +S
y
;i S
y
;j ℄ J
z X
X
i;j [S
x
;i S
x
+1;j +S
y
;i S
y
+1;j
℄; (1)
d
wheretheindex numbersdierentplanesofthelayer
andthesumi;jisonlyovernearestneighborsofthe
lat-tiesites intheplanes. Thelassialspinvetor
loal-izedatlayer andsitei, ~
S
;i =S
x
;i ^
i+S y
;i ^
j+S z
;i ^
kan
beparametrizedbytwosalarelds
;i ,
;i
and
writ-ten as ~
S
;i
=S(os
;i os
;i ;os
;i sin
;i ;sin
;i ).
At low temperatures T T
, we an adopt a mean
eld approximationfor theinterlayeroupling
assum-ing that hÆ ~
S
Æ
~
S
+1 i h
~
Si 2
and then, Hamiltonian
(1)anbeapproximatedby[4℄
H = X
2
4
J X
i;j (S
x
;i S
x
;j +S
y
;i S
y
;j ) 2J
z jh
~
Sij X
i S
x
;i 3
5
= X
H
: (2)
InEq.(2) wedene aredued magnetield (meaneld) 2J
z jh
~
Sij withdiretion parallel to thex-axisin the
spinspae. UsingtheontinuumapproximationtoHamiltonian(2)oneahlayer
H
=
J
2 Z
d 2
r sin 2
( ~
r ) 2
+os 2
( ~
r) 2
+ 4
a 2
0 sin
2
4 J
z jh
~
Sji
J
osos )
; (3)
wherewehavetakenS=1anda
0
isthelattiespaing,wegetthefollowingsequationsofmotion
1
J
t
=osr 2
2sin
~
r ~
r 2J
z jh
~
Sij
J
sin; (4)
1
J
t
= tansinr 2
+sin[4=a 2
0 (
~
r ) 2
℄ 2J
z jh
~
Sij
J
tanos: (5)
Statiin-planesolutionsoftheseequationsarethose
whihsolvethetwo-dimensionalsine-Gordonequation
r 2
= 2J
z jh
~
Sij
J
sin; (6)
and then, below T
this system reveals a quite
inter-esting behavior, whih is due to the vortex-antivortex
pair formations in the plane of the layer, sine the
sine-Gordonequationpossessvortexsolutions. Infat,
Hikami and Tsuneto [15℄ rst showed that thevortex
pairsanalso be formedin thelayeredsystemof
XY-model and then they proved that, in ertain
irum-stanes,itisenergetiallyfavorableforthevortexpairs
in dierent layers to beome independent instead of
making a vortex ring. An exat vortex solution for
equation(6) wasobtainedby Hudak [16℄,but we
an-notbesurethatthissolutionanbeapplieddiretlyto
thedisretelattie,sinethevortexandantivortex
vor-tiitiesinapairare4and 4respetively. Cataudella
andMinnhagen[2℄haveshownthat,inthestudyof
vor-texexitations,whenweareinterestedintheanalytial
desriptionoftopologialexitations,itismore
onve-nienttoonsiderthat,forweakouplings,theoupling
between adjaent layers leads to small deviations, at
lowtemperatures,fromthevortex-pairsolutionforthe
ideal planar ferromagnet. It is equivalent to
approx-imatesin
=
in the sine-Gordon equation and the
vortex-antivortexsolutionobtainedfrom this
approxi-mationisgivenby
vv
= tan 1
y a
x
tan 1
y+a
x
J
z jh
~
Sij
2J Z
d 2
x0K
0 0
s
J
z jh
~
Sij
J p
(x x0) 2
+(y y0) 2
1
A
(7)
tan 1
y0 a
x0
tan 1
y0+a
x0
;
d
whereK
0
isthemodiedBesselfuntionandthevortex
is loated at (0;a) andthe antivortexat (0; a). The
energyofthisexitation is[2,3℄
E
vv =E
0 +E
1 ln(R =a
0 )+E
2 (R =a
0
1); (8)
where R = 2a, E
0
is the reation energy of a vortex
pair, E
1
= 2J and E
2 =
2 q
2JJ
z h
~
Si. The
oeÆ-ientof thelinearterm in theenergydereases asthe
temperatureinreases,vanishingatT =T
andhere,it
is indiatedbythemeanmagnetization h ~
Sipresentin
E
2
[6℄. Theontributionofthestrutures(7)tothe
dy-namial orrelationfuntions in theselayeredlassial
ferromagnetwasstudiedbyPireset al. [17℄.
Theobjetnowistostudywhateetsuh
topolog-ialstrutures haveontheassoiated longwavelength
ferromagnetispin waves propagating on the layer.
To this end, we assume that the deviations from the
stati onguration (7) due to the presene of spin
waves,aresmallwithharmonitimedependene
(~r;t)=
vv
(~r)+(~r)e i!t
;(~r;t)=
vv
(~r)+(~r)e i!t
(9)
with
vv
(~r)=0 foraplanarpair and; 1.
Sub-stituting Eqs. (9) into the equations of motion and
linearizing in the small quantities, we obtain for the
in-plane spin wave osillations, aShrodinger-like
dif-ferentialequationwithzeroeigenvalue
r 2
+V(~r;!)=0 (10)
andwithafrequenydependentpotentialnot
ylindri-allysymmetrigivenby
V(~r;!)= 2J
z jh
~
Sij
J
os
vv !
2
J 2
1
[4=a 2
( ~
r
vv )
2
(2J
z jh
~
Sij=J)os
vv ℄
Theoutofplanespinwaveosillationsareobtainedfrom thein-plane osillationsthrough = i! J [4=a 2 0 ( ~ r vv ) 2 (2J z jh ~ Sij=J)os vv ℄ : (12) d
Inthemeaneldapproximation,theessentialeet
of the adjaent planes on the spin wave propagation
onalayeris tointrodueaniteativation dispersion
law given by ! 2 = ! 2 0 +b 2 0 k 2
, due to the mean
mag-netizationh ~
Siwhere ! 2 0 =J z Jjh ~ Sij(4=a 2 0 J z jh ~ Sij=J), b 2 0 =J 2 (4=a 2 0 J z jh ~
Sij=J)andk 2 =k 2 x +k 2 y
isthewave
vetormodulus. Thelowest-ordereet ofan
inhomo-geneousvortex-pairbakgroundistoprodueanelasti
satteringenterforthespinwaves,sinethepotential
(11) has a nite range of the order of the size of the
pairongurationR . Thus,in polaroordinates(r;'),
asymptoti solutions to Eq. (10) for the ontinuum
states(! 2
! 2
0
),aregivenbyphase-shiftedylindrial
spinwaves k ;n (~r) r!1 = 1 2 " H (1) jnj (kr)e in' + X n0 exp[ 2i(k)℄ nn0 H (2) jn0j (kr)e in0' # (13) d where nn0
is thephase-shift matrixoupling the
dif-ferentangularmomentumhannelsnandn0andH (1) jnj , H (2) jnj
denotestheHankelfuntions. Sinethepotential
V(~r;!)isnolongerylindriallysymmetri,the
angu-lar momentum is no longer agood quantum number,
nor is the phase-shift matrix
nn0
any longer
diago-nal. Therearetransitionsbetweenthevariousangular
momentum hannels due to the oupling betweenthe
angular dependene of the spin waveand the angular
dependeneofthesatteringpotential.
Besidesthe ontinuum stateswith ! 2
! 2
0 ,
equa-tion (10) has two disrete levels with zero-frequeny.
The zero modes are assoiatedwith thefat that the
spetrumofsmall osillationsmustontainzero-mode
translationmodesto restorethetranslationinvariane
broken bythepreseneofapairinto thesystem. The
wave funtions of these translation modes, whih we
denote by
01
(~r) and
02
(~r), are proportional to the
partial derivatives
vv
=x and
vv
=y respetively,
and hene, the zero mode with ! = !
01
= 0 is
as-soiated with a translation along the x-diretion and
analogously, the zeromode with ! =!
02
= 0is
asso-iatedwiththetranslationalongthey-diretion. They
aregivenby
01 (~r)=
02
(~r)=0and
01 (~r) =
y+a
x 2
+(y+a) 2
y a
x 2
+(y a) 2 + 2 1=2 J 3=2 z jh ~ Sij 3=2 2J 3=2 Z d 2 x0K 1 0 s 2J z jh ~ Sij J p (x x0) 2
+(y y0) 2 1 A tan 1 y0 a x0 tan 1
y0+a
x0 (x x0) p (x x0) 2
+(y y0) 2
; (14)
02 (~r) =
x
x 2
+(y a) 2
x
x 2
+(y+a) 2 + 2 1=2 J 3=2 z jh ~ Sij 3=2 2J 3=2 Z d 2 x0K 1 0 s 2J z jh ~ Sij J p (x x0) 2
+(y y0) 2 1 A tan 1 y0 a x0 tan 1
y0+a
Theseloalmodesgovernthemotionofthepaironthe
layerandareoffundamentalimportaneinthestudy
of dynamis of topologial exitations in the presene
of externalperturbations. Loal modes, speially zero
modes,areof paramountimportane inthederivation
ofatopologialexitationthermodynamis[10,18℄. In
fat,thenumberofmagnonsstatesmustbedereased
byanumberequaltothenumberofboundstates. One
waytoviewthisdereaseisthattopologialexitation
traps magnonstates dueto its preseneand thus, the
degreesof freedom(or freeenergy, et)areshared
be-tweenmodesofanon-linearsystem. Besides, theloal
modesmayberelated to theresonanephenomenain
vortex-antivortexollisionsbeausethepairinternal
in-terationduetoE
/E
2
Rmayhaveenergyexhange
withthepairtranslationalmodesduringtheollisions.
Theontinuumstateswith frequenies!!
0 an
befoundusingtheBorn approximation. Basedonthe
asymptotibehaviorofthesatteringequation,wean
rewrite Eq. (10) as r 2
+k 2
= U(~r;!), where
U(~r;!)=V(~r;!)+k 2
. TheGreenfuntionassoiated
to the Helmholtzoperator r 2
+k 2
in twodimensions
is(i=4)H (1)
0 (kj~r
~
r0 j)andhene,theformalsolutionof
Eq.(10)is
(~r)=e i
~
k~r
+ i
4 Z
d 2
r0H (1)
0 (kj~r
~
r0 j)U( ~
r0 )( ~
r0 ): (16)
Using the Born approximation we an take ( ~
r0 )
=
(0)
( ~
r0 ) = e i
~
k~r
for the rst order Born term and so
on. In generalthis integralanonlybe alulated
nu-merially, but equations(10) and (12) anbewritten
as
i!
J
=r 2
2J
z jh
~
Sij
J
(os
vv
); (17)
i!
J =
"
4
a 2
0 (
~
r
vv )
2 2J
z jh
~
Sij
J
os
vv #
; (18)
andthen, weanapproximate the spatial dependene
of the osillationssubstituting (0)
(~r) into Eq.(18) to
nd
(1)
(~r)andsubstituting (1)
(~r)intoEq.(17)tond
(1)
(~r)andsoon. Assumingthatthepotentialissmall,
weneglettermsoforderhigherthan2inthepotential
toobtain
= (
1+
1
(4=a 2
0 J
z jh
~
Sij=J) "
4
a 2
0 (
~
r
vv )
2 2J
z jh
~
Sij
J
os
vv #
"
1+ k
2
+(2J
z jh
~
Sij=J)os
vv
(k 2
+2J
z jh
~
Sij=J) # )
e i
~
k ~r
: (19)
d
The out-of-plane osillations an be written from
Eq.(19)usingtherelation(12). Ofourse,asweshould
expet,the satteredstatesas well asthe loalmodes
arenotradiallysymmetri.
As it was disussed in Ref. [13℄, these sattering
states provide an indiret method to experimentally
detet topologial exitations,throughEPRlinewidth
measurements. The linewidth is the temporal
inte-gralofthefour-spinorrelationfuntionandthe
stru-ture fator used to alulate them ontains a stati
ontributionfrom thestati pair solutionand a
time-dependent ontribution from pair-magnon sattering.
Asthisthree-dimensionalsystemismorerealistithan
the idealtwo-dimensional models, we believethat our
results may also be relevant for future investigations
on the non-linear exitations ontribution to EPR
linewidthinquasi-two-dimensionalmagnetimaterials.
wavesandvortex-antivortexpairandfoundexat
solu-tionsforzeromodesandapproximatedsolutionsforthe
ontinuum states in a layered three-dimensional
XY-model. Zero modes beome apparentin inelasti
neu-tronsatteringexperimentsandessentiallyontributes
in entral peak due to interferene between ones and
vortex-pair. Thevortex-antivortexinterationislinear
forlargedistanesofseparationandtheeetive
inter-plane oupling E
2
(T) dereases with the temperature
T, vanishing at T =T
[3, 6, 9℄. This mean that the
interplane ouplingeetively eases to befelt by the
vorties forT > T
, themodel eetively reduing to
atwo-dimensionalplanarmodelandfreevortiesstart
toexistandthedisretemodesdisappeargivingriseto
zerofrequenystatesasthelowerlimitofaontinuum
offrequenies(Goldstonemodes[19℄). Webelievethat
theloalmodesandthesatteredmodesobtainedhere
in-teration. In fat, as it was shown in Ref. [20℄, the
lowestordereetofvortexspinwaveinterationsan
bedesribedasarenormalizationoftheexhange
on-stant. Besides, zeromodesmaybeoupledto internal
interation of the vortiesin apair. Our alulations
anbeapplied toanylayeredsystemwith small
inter-layerouplingin whihsmallosillationsand
indepen-dentvortexpairsarepresent,suhasin
superondu-tors[21℄.
Aknowledgement
This work was partially supported by CNPq and
FAPEMIG(Brazil).
Referenes
[1℄ J.M.KosterlitzandD.J. Thouless,J.Phys.C6,1181
(1973).
[2℄ V.CataudellaandP.Minnhagen,PhysiaC166,442
(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. Gouv^ea,A.S.T. Pires, Phys. Rev.
B57,8274(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. Nhoand E. Manousakis, Phys. Rev.B 59, 11575
(1999).
[9℄ A.R. Pereira, A.S.T. Pires,M.E. Gouv^ea, Phys.Rev.
B51,16413(1995).
[10℄ J.F. Currie, J.A. Krumhansl, A.R. Bishop, S.E.
Trullinger,Phys.Rev.B22,477(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.B10,
3649(1996).
[14℄ A.R.PereiraandA.S.T.Pires,Phys.Rev.B60,6226
(1999).
[15℄ S.HikamiandT.Tsuneto,Prog.Theor.Phys.63,387
(1980).
[16℄ O.Hudak,Phys.Lett.A89,245(1982).
[17℄ A.S.T. Pires, A.R. Pereira, M.E. Gouv^ea, Phys.Rev.
B49,9663(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. Gouv^ea, Solid St.
Commun.86,187(1993).
