Table Of ContentQuantum effects for the 2D soliton in isotropic ferromagnets
Boris A. Ivanov,1,2 Denis D. Sheka,2,∗ Vasilii V. Krivonos,2 and Franz G. Mertens3
1Institute of Magnetism, NASU, 03142 Kiev, Ukraine
2National Taras Shevchenko University of Kiev, 03127 Kiev, Ukraine
3Physikalisches Institut, Universit¨at Bayreuth, D–95440 Bayreuth, Germany
(Dated: January 22, 2007)
7
We evaluate a zero–point quantum correction to a Belavin–Polyakov soliton in an isotropic 2D
0
ferromagnet. Byrevisingthescatteringproblemofquasi–particlesbyasolitonweshowthatitleads
0
to the Aharonov–Bohm type of scattering, hence the scattering data can not be obtained by the
2
Born approximation. We proof that the soliton energy with account of quantum corrections does
n nothaveaminimumasafunction ofitsradius,which isusually interpretedas asoliton instability.
a Ontheotherhand,weshowthatlonglifetimesolitonscanexistinferromagnetsduetoanadditional
J
integral of motion, which is absent for the σ–model.
2
2 PACSnumbers: 75.10.Hk,75.30.Ds,05.45.Yv
]
l
e Solitonsareknowntoplayanimportantroleinseveral theproblemsemiclassicallyusingtheone–loopcorrection
- branches of field theory and condensed matter physics, to the classical soliton energy, originally calculated by
r
t see Ref. 1 for a review. In particular, solitons treated as Dashen et al. [10, 11] for 1D solitons, see also Ref. 12.
s
. nonlinear excitations are important in 1D and 2D mag- To generalize these results to the 2D case one needs to
t
a netism [2, 3, 4]. A serious impediment in studying 2D solvethesoliton–magnonscatteringproblemfor2Dmag-
m
spinsystemsarisesduetotheabsenceofexactanalytical nets. We show a unique new feature of the 2D soliton–
- solutions for most models. Thereupon special attention magnon interaction, which is absent in 1D: the soliton
d
is deserved to models which admit an analytical treat- acts on magnons not only by some local potential, but
n
o ment. One ofthe well–knownexamples is amodelofthe alsointhesamewayasaneffectivelong–rangedmagnetic
c 2D isotropic ferromagnet(FM), which provides an exact field acts on a charged particle; this essentially changes
[ analytical soliton, the so–called Belavin–Polyakov (BP) the scattering picture, leading to the Aharonov–Bohm
2 soliton [5]. In terms of the normalized magnetization, (AB) scenario. We state that the AB type of scattering
v m = (sinθcosφ,sinθsinφ,cosθ), the soliton structure is a general consequence of 2D scattering by a topologi-
2 m is described by the formula [5] calsoliton. We calculate the Casimirenergyfor the BP–
BP
5
soliton in FMs and show that the quantum correction
1 θBP R |q| cannotprovidea fixedsize forthe soliton. Nevertheless,
2 tan = , φ =ϕ +qχ. (1)
BP 0
1 2 (cid:18)ρ(cid:19) we show that long lifetime solitons can exist in FMs due
6 to an additional integral of motion, contrary to antifer-
Here ρ and χ are the polar coordinates in the magnet
0 romagnets.
/ plane, the integer q is the π2–topological charge of the
t
a soliton, R and ϕ0 are arbitrary parameters. BP–solitons The macroscopic dynamics of the classical FM follows
m of the form (1) appear in different models of non-linear
the Landau–Lifshitz equations
- field theory and condensed matter physics [1]. In partic-
d ular,theBP–solitonsareimportantfortheferromagnetic
n
quantum Hall effect [6].
o 1
c The unique problem of the BP soliton is that its en- sinθ ∂ φ=∇2θ sinθcosθ(∇φ)2,
t
v: ergyEBP =4πJS2|q|(J istheexchangeintegral,S isthe D1 − (2)
Xi afrtoommitchsepisnc)aldeoiensvnaoritadnecpeeonfdtohneistyssrtaedmiu,swRh:icthhiissraespualrtst D sinθ ∂tθ =−∇· sin2θ∇φ ,
(cid:0) (cid:1)
r of the general conformal invariance of the model. Since
a
thesolitonradiusisnotfixed,oneisthenfreetoletitgo
up to the systemsize, andthe thermal excitationofsoli- where D is the stiffness coefficient of the spin–waves,
tons will break the long-rangeorder[5]. However,recent which are characterized by the dispersion law ω(k) =
studies have shown that a quantization of the soliton of Dk2. Toanalyzethesoliton–magnoninteraction,wecon-
the classical σ-model, which can be attributed to anti- sider small oscillations of the magnetization m on the
ferromagnets, breaks the static scale invariance [7, 8, 9]. background of the stationary BP–soliton m . These
BP
The natural question whether it works for FMs is still oscillationscanbedescribedintermsofthecomplexval-
open. ued “wave function” Ψ = θ θ +isinθ (φ φ ),
BP BP BP
− −
The purpose of our study is to examine the role of see Ref. 13. For the further analysis it is instructive to
quantumfluctuationsforthesolitonproperties. Wetreat rewrite the linearized equation for the Ψ–function in the
2
form of the Schr¨odinger equation: a magnetic field. However,due to the topologicalsoliton
properties, A is a long–rangedfield
i
HΨ= D∂tΨ, H =(−i∇−A)2+V, (3a) 1 ρ 2|q| +qe when ρ R,
V =−ρq22 sin2θBP, A=−qcosρθBPeχ. (3b) A(ρ)= 1+−(cid:16)Rρ(cid:17)2|q|ρqeχ ∼−ρρqeχχ when ρ≪≫R, (6)
R
(cid:16) (cid:17)
The Hamiltonian H has a form which is typical for whichistypicalforthe ABeffect[14]. Forsuchatypeof
a quantum–mechanical charged particle in the presence scattering, some standard scattering results fail. For ex-
of a scalar potential V and an additional magnetic field ample,the Levinsontheoremmustbe modified forlong–
withavectorpotentialA. Theappearanceofaneffective range potential systems [14, 15]. Since scattering phase
magneticfieldisanewfeatureofthe 2Dsoliton–magnon shifts are not still localized, there appears a problem of
interaction, which is always absent in 1D systems. Dis- the regularization of the scattering series like in conven-
cerning this effective magnetic field gives the possibility tional AB scattering picture [16]. As was firstly noted
to draw a number of general conclusions about soliton– by Feinberg [17], the Born approximation fails for the
magnon scattering in the 2D case, see below. AB scattering;it givesanaverageof twodifferentmodes
Forthe system(3)weapply the standardpartialwave with opposite signs of m [18]. Thus we need a more pre-
expansion, using the Ansatz: cise approach than the Born approximation. One needs
to stress that such a long–range behavior is not a result
Ψ(ρ,χ,t)= ψ (ρ)exp(imχ iω t+β ). (4) of slow algebraic decay of the out–of–plane structure of
m α α
−
α=X(k,m) the BP soliton. It is a consequence of the topology of
the BP–soliton, namely, of the relation φ=qχ, thus the
Here the integer m is the azimuthal quantum number, k
AB–type of scattering is valid also for anisotropic mag-
is the radial wave number, and β is an arbitrary initial
α nets [19, 20, 21].
phase. Each partial wave ψ is an eigenfunction of the
m Let us discuss the soliton with the topological charge
2D radial Schr¨odinger equation
q = 1, which has the lowest energy. Such a soliton has
two internal zero–frequency modes, which are the limit
2+U ψ =k2ψ ,
−∇ρ m m m of the continuum spectrum as k 0: [22]
(cid:0) m2+(cid:1)2mqcosθ +q2cos2θ (5) →
Um = ρB2P BP. ψm(k==+0)1 = ρ2+1R2, ψm(k==00) = ρ2+ρR2. (7)
Here the term linear in m reminds of an effective mag- The mode with m = +1 is a local translational mode,
netic field A. The scattering problem can be formulated which describes a soliton shift, the mode with m = 0 is
in the usual way. The eigenfunctions for free magnon the half–local rotational mode. The mode with m =+1
modes have the form ψfree J (kρ), with an asymp- has anexact analyticalsolution for any finite values ofk
m ∝ |m|
twohtiecnbkeρha≫vior|mψ|mf;reeJm∝i(skρB)e−s1s/e2lcfousn(cktρio−n.|mI|nπ/t2he−pπr/e4s)- ψm=1(ρ)=J2(kρ)− k2ρ(ρ/JR1()k2ρ+) 1, (8)
ence of a soliton the behavior of a magnon solution can
hence this mode does not scatter at all, η = 0
be analyzed at large distances, ρ R. In view of the m=+1
asymptotic behavior U m+q≫2/ρ2, in the limiting [13]. Notethatintheinterestingcaseoflong–wavelength
m
≈ | | asymptotic behavior (kρ 1) at large distances ρ R,
case kρ m one has the usual result [13]:
≪ ≫
≫| | this expression has the same form as a combination of
1 m+q π π Bessel and Neumann functions, J2(kρ) (kρ)2 and
ψm ∝ √kρcos(cid:18)kρ− | 2 | − 4 +ηm(k)(cid:19). Y2(kρ) (kρ)−2. Thus, the second term∝ in (8) imi-
∝
tates the presence of the function Y and the presence
2
The phase shift η contains all information about the of scattering, which caused a conclusion in Ref. 8 that
m
scattering process. the mode with m=+1 can be scattered. The scattering
The main features of the scattering on a topological phaseshift canbe found inboth limiting cases,for small
BP–soliton are caused by the magnetic field A. As an andlargedimensionlessradialwavenumberκ =kR[13].
analogueoftheZeemansplittingofelectronenergyterms For long–wavelengths, κ 1
≪
inanexternalmagneticfield,the presenceofaneffective
0, when m=1,
magnetic field breaks the symmetry η (k)=η (k). It
m −m π
insengSaeoctmievseestaimmry’esst.othtaekseoilnittoona–cmcaogunnotnsespcaartateterliyngpopsriotbivleemanids ηm κ∼≪1−−π2κln2(l1n/κκ1),, wwhheenn mm==−0,1, (9a)
treated perturbatively using the Born approximation [7, πκ2
9], which, in principle, can be used for the scattering in 2m(m+1)sgn m, otherwise,
3
0
and in the opposite case of short–wavelengths [13]
ηm κ∼≫1πsgn (m−1)− π(mκ−1). (9b) ates ( ) -0.5
st
Now we are able to calculate the density of magnon of
y
states. Let us generalize the main arguments of Dashen sit -1
n
et al. [10, 11] for the 2D system. The idea of the ap- de
proach is to calculate energy shifts of vacuum magnon he
states in the presence ofa soliton, whichare constructed e in t -1.5
as one-loop quantum corrections to the soliton energy. g
n
a
The energy of the vacuum comes from the zero-point h
C
fluctuations of the magnon states. Without the soliton, -2
eachvacuummagnonmakesacontributionasℏDk2 /2, 0 1 2 3
vac
where k issetofallowablewavevectors. Inthesoli- Wave number ( )
vac
{ }
ton presence the set of allowable wave vectors changes,
k . The energy of the state with the wave vector k is FIG.1: ThechangeR inthedensityofmagnonstatesdueto
ℏ{D}k2/2. Therefore the energy correctionis thesoliton vs dimensionless wavenumber(κ =kR),obtained
bynumerical integration of theSchr¨odinger equation (5).
ℏD ℏD
E1–loop = k2 k2 . (10)
2 − 2 vac
Xk kXvac
To determine the set of allowable states, we put the sys- Hereη(k)isasumofthe phaseshifts ofallpartialwaves
tem in a very large box of the size L, making all states with the fixed radial wave number k. Note that our def-
discrete. In the limiting case L the energy correc- inition of η(k), in contrast to the case of the σ-model
→ ∞
tion (10) does not depend on the form of the boundary [9], contains independent summations over positive and
conditions. For the 2D case we choose fixed boundary negative m, which reflects the breaking of the symmetry
conditions for a circular box of radius L [13]. Since the m m.
→−
free magnons are described by the Bessel function ψfree,
m
byenforcingfixedboundaryconditionsψfree(ρ=L)=0, The short–wave length behavior of the phase shift is
m
we fix the allowed values of the radial wave number, responsible for ultraviolet (UV) singularities. Follow-
kvacL = j(n), where j(n) is the n–th zero of the Bessel ing Refs. 9 in order to avoid the UV singularity one
n m m
needs to derive the short–wave length asymptotics for
function J . In the region of interest, n 1, the zeros
m
(n) ≫ the phase shift. There appears a problem to sum to in-
of the Bessel function j πn. Thus the allowed val-
m ≈ finity an alternating series, which has no absolute con-
ues of the wave numbers are kvac πn/L, similar to the
n ≈ vergence. Symmetric limits regularize this summation
1D case. However, the above–used simple equation for
similar to the exponential regularization for the original
j(n) is valid only if m is not very large. In the case of
m | | AB–scattering[16];differentwavesaretakenintoaccount
|m|≫ 1, the first zero of the Bessel function jm(1) ≈ |m|. inthe orderin whichthe poles km(p) inthe scatteringam-
Hence, in a finite system there appears a restriction for
plitudeappearask increases[24]. Finally,wefoundthat
the allowednumberofmodes, m L,andthesumrule
| |≤ η( ) = π for the soliton with q = 1[25]. Note that
for the 2D case takes the form: ∞ −
this result can not be obtained from the Born approx-
imation because of the long–range nature of the AB–
L ∞ kL
dk . (11) scattering; namely, perturbative Born calculations re-
Xk,m(cid:16)•(cid:17)−→ π Z0 m=X−kL(cid:16)•(cid:17) sultedindifferentconclusionsthatη( )=π (seeRef.7)
∞
and η( )=2π (see Ref. 9).
∞
For magnon states in the presence of the soliton there
appearsaphaseshiftη duetothesoliton–magnonscat- Tocheck ouranalyticalpredictionswe havecalculated
m
tering, therefore η(k) by the numerical integration of the Schr¨odinger
equation(5)form [ 100;100],whichgivesη( )= π
η ∈ − ∞ −
k L+η =kvacL=j(n) = k kvac = . with the precision 10−3.
n m n m ⇒ n− n −L
Finally,the one-loopcorrectionreadsE1–loop =E +
The one-loop correction to the soliton energy reads ECas. Here the counterterm E = ℏDη(k)k2 ∞CTcan
CT − 2π 0
be evaluated using a momentum cutoff Λ, which(cid:12)results
E1–loop = ℏD ∞kη(k)dk, η(k)= kL ηm(k).(12) in ECT = 21ℏDΛ2. Taking away this UV term, e(cid:12).g. by
− π Z0 m=X−kL renormalizing the exchange constant [9], we ends with
4
the finite Casimir energy: limit, one can finally obtain the soliton lifetime
ℏD ∞ 1 κL/R dη (κ) 4πℏ R 10
ECas = κ2R(κ)dκ, R(κ)= m . τ . (15)
2R2 Z π dκ ∼ JC2 (cid:18)a(cid:19)
0 m=X−κL/R
(13) Notethatthelifetimeismuchbiggerthanℏ/J forR>a.
HereR(κ)describesthechangeinthedensityofmagnon To conclude, quantum effects decrease the energy of
states due to the soliton. This expression can be ana- the BP soliton in isotropic FMs, more strongly for small
lyzed analytically in limiting cases. Using the asymp- soliton radius. Nevertheless, the original argumentation
totical behavior for the phase shift for different modes by Belavin and Polyakov [5] about the breaking of the
one can conclude that the maximum scattering in the long-range order of the system is still valid. It is based
long wavelength limit corresponds to the mode with on the fact that in isotropic FMs the energy is indepen-
m = 0, hence the density of states has a singularity: dent of the soliton radius R, and R can be comparable
R(κ) (2κ)−1ln2κ when κ 0. In the short with the system size, R L. However, in the case of
∼− →−∞ → ∼
wavelength limit all modes compensate each other and large radii the quantum correction is negligible, the en-
R(κ ) 0. In the intermediate range the change ergy has a finite limit for R , so the problem of
→ ∞ → → ∞
in the density of states can be found numerically only, long-range ordering has the classical form. Another as-
see Fig. 1. One can conclude that for all κ the change pectofthe problemisthe fate ofthe BPsolitonatsmall
in the density of states takes only negative values and finite R. As can be seen form Eq. (15), the lifetime is
∞R(κ)dκ = 1. Finally, the Casimir energy is ob- small for small R. In some respects the situation is sim-
0 −
tRained as ilar to the problem of the black hole evaporation, i.e.
∞ the large radius soliton dissipation is very slow, and can
ECas C a 2
= , C = κ2 R(κ) dκ, (14) be neglected. At the same time the dissipation of small
EBP −8πS (cid:16)R(cid:17) Z | | radius solitons is very fast. When the soliton radius is
0
small enough, the speed of dissipation increases rapidly;
where C is a constant, which we calculated numerically
inthe finalstagewithS S the solitoncancollapseby
z
tobe C 0.38. We introducedin(14)thetypicallength ∼
a quantum jump, which is accompanied by a change of
scale a =≈ ~D/JS, which is about a lattice constant.
the topological charge.
Note thatpthis parameter is absent for the static BP–
D. D. Sh. thanks the University of Bayreuth, where
soliton problem, but it naturally breaks an initial scale
partofthisworkwasperformed,forkindhospitalityand
invariance of the model in the dynamics, see Eq. (2).
acknowledgesthe supportfromthe AlexandervonHum-
The soliton energy is reduced when the soliton radius
boldt Foundation. The work in Kiev is partly supported
decreases.
by the grant INTAS Ref.05-1000008-8112.
LetusdiscussphysicalconsequencesofEq.(14). First,
the soliton energy with account of the quantum correc-
tiondoesnothaveaminimumasafunctionofthesoliton
radius. Usually this is interpretedas a solitoninstability
∗
inthecontextoftheHobart–Derricktheorem,seeRef.1. Electronic address: Denis [email protected]
We will show here that this property leads to a dissipa- [1] N. Manton and P. Sutcliffe, Topological solitons, Cam-
tion of the soliton energy caused by magnon radiation, bridge Monographs on Mathematical Physics (Cam-
bridge University Press, 2004).
common to that for 3D Hopf solitons in isotropic FMs
[2] A. M. Kosevich, B. A. Ivanov, and
[23]. As an important contrast to σ–models, a model A. S. Kovalev, Physics Reports 194, 117
of the FM has an additional integral of motion, the z– (1990), URL http://www.sciencedirect.
component of the total spin Sz 2πS(R/a)2 [2]. Even com/science/article/B6TVP-46SXP0P-19/2/
≈
for the static limit S takes a nonzero value; the energy d0a99bbbccf078602123db3e5d601202.
z
dissipationcausedbytheradiationofmagnonpairswith [3] H.J.MikeskaandM.Steiner,Adv.Phys.40,191(1991).
wavevectorskand kwillbeaccompaniedbyadecrease [4] V.G.Bar’yakhtarandB.A.Ivanov,Sov.Sci.Rev.Sec.A.
− 16, 3 (1993).
of the value of S by two. Therefore the soliton lifetime
z [5] A. A. Belavin and A. M. Polyakov, JETP Lett. 22, 245
τ =S /(dS/dt) can be sizeable, when S is large (S is
z z z z (1975).
the number of bound magnons in the soliton). [6] M. Abolfath, Phys.Rev. B 58, 2013 (1998), URLhttp:
The amplitude of the radiation process is ̟ //link.aps.org/abstract/PRB/v58/p2013.
JS(ak)2, see [23]. In accordance to Fermi’s golden rule∼, [7] J. P. Rodriguez, Phys. Rev. B 39, 2906 (1989), URL
http://link.aps.org/abstract/PRB/v39/p2906.
dSz = 2π|̟|2δ dE ℏω(k) . [8] B. Moussallam, Phys. Rev. B 43, 3325 (1991), URL
dt Xk ℏ (cid:18)dSz − (cid:19) [9] Hht.tWp:a/l/lilsienrka.nadpsG.o.rHg/oalzbwsatrrtahc,t/PPhRyBs/.vR43ev/p.3B32651., 2819
Here E is the total energy of the soliton with account of (2000),URLhttp://link.aps.org/abstract/PRB/v61/
theCasimirenergy. CalculatingdSz/dtinthecontinuum p2819.
5
[10] R.F.Dashen,B.Hasslacher,andA.Neveu,Phys.Rev.D com/science/article/B6WB1-4DDR3KP-NJ/2/
10,4114(1974),URLhttp://link.aps.org/abstract/ d8f56ee8b6ab36e8635946f3f839dfbf.
PRD/v10/p4114. [19] D. D. Sheka, B. A. Ivanov, and F. G. Mertens, Phys.
[11] R.F.Dashen,B.Hasslacher,andA.Neveu,Phys.Rev.D Rev. B 64, 024432 (2001), URL http://link.aps.org/
10,4130(1974),URLhttp://link.aps.org/abstract/ abstract/PRB/v64/e024432.
PRD/v10/p4130. [20] D. D. Sheka, I. A. Yastremsky, B. A. Ivanov, G. M.
[12] R. Rajaraman, Solitons and Instanton (North–Holland, Wysin, and F. G. Mertens, Phys. Rev. B 69,
Amsterdam, 1982). 054429 (pages 13) (2004), URL http://link.aps.org/
[13] B. A. Ivanov, V. M. Murav’ev, and D. D. Sheka, JETP abstract/PRB/v69/e054429.
89,583(1999),URLhttp://link.aip.org/link/?JET/ [21] B.A.IvanovandD.D.Sheka,JETPLett.82,436(2005),
89/583/1. URL http://link.aip.org/link/?JTP/82/436/1.
[14] D.D.ShekaandF.G.Mertens,Phys.Rev.A74,052703 [22] B. A. Ivanov, JETP Lett. 61, 917 (1995),
(pages5)(2006),URLhttp://link.aps.org/abstract/ URL http://www.jetpletters.ac.ru/ps/936/
PRA/v74/e052703. article 14292.shtml.
[15] D. Sheka, B. Ivanov, and F. G. Mertens, Phys. Rev. A [23] I.E.DzyaloshinskiiandB.A.Ivanov,JETPLett29,592
68, 012707 (pages 5) (2003), URL http://link.aps. (1979).
org/abstract/PRA/v68/e012707. [24] The pole of the scattering amplitude corresponds to the
[16] W. C. Henneberger, Phys.Rev.A 22, 1383 (1980). condition when η(km(p)) = ±π/2. Asymptotically km(p) ∼
[17] E. Feinberg, Sov. Phys. Usp. 5, 753 (1963), URL |m|/R when k>>1/R, see details in Ref. 13.
http://www.ufn.ru/archive/russian/abstracts/ [25] More general, η(∞) = −π|q| for higher topological
abst3970.html. charges [14].
[18] S. N. M. Ruijsenaars, Annals of Physics 146,
1 (1983), URL http://www.sciencedirect.
