Topological excitations in 2D spin system with high spin $s>= 1$

0.17 MB·English

by  Julia Bernatska

0
Topological excitations in 2D spin system with
1
0 high spin s > 1
2

n
a Julia Bernatska, Petro Holod
J

3

] Abstract
l
l
a Weconstructaclassoftopologicalexcitationsofameanfieldinatwo-
h dimensionalspinsystemrepresentedbyaquantumHeisenbergmodelwith
- high powers of exchange interaction. The quantum model is associated
s
e with a classical one (the continuous classical analogue) that is based on
m a Landau-Lifshitz like equation, and describes large-scale fluctuations of
the mean field. On the other hand, the classical model is a Hamiltonian
.
t system on a coadjoint orbit of the unitary group SU(2s+1) in the case
a
m of spin s. We have found a class of mean field configurations that can be
interpretedastopologicalexcitations,becausetheyhavefixedtopological
-
d charges. Such excitations change their shapes and grow preserving an
n energy.
o
c
[ 1 Introduction

1
AccordingtoMerminandWagner[1]thereisnoferromagneticorantiferromag-
v
5 netic order in the one- and two-dimensional isotropic Heisenberg models with
7 interactions of finite range at nonzero temperature. This statement is proven
3 duetoBogoliubov’sinequalityinthegeneralcase. Hereweconstructexcitations
0
thatcauseadestructionofalong-rangenematicormixedferromagnetic-nematic
.
1 order. This is an extension of the results of Belavin and Polyakov [2].
0
We model a planar magnet by a square atomic lattice with the same spin s
0
at each site. We describe this two-dimensional spin system by a generalized
1
: HeisenbergHamiltonian,takingintoaccounthighpowersoftheexchangeinter-
v action (Sˆ ,Sˆ ), where Sˆ is a vector of spin operators at site n. By a mean
i n n+δ n
X fieldapproximationweobtainaclassicallong-rangeequationfromthequantum
r Heisenberg one.
a
An equation for a mean field (the field of magnetization and multipole mo-
ments) is a Hamiltonian equation on a coadjoint orbit of Lie group. At the
same time, this is a generalization of the well-known Landau-Lifshitz equation
for a magnetization field. In this context we obtain effective Hamiltonians for
themagnetic systeminquestion. UsingKa¨hlerian structureofcoadjoint orbits,
we construct effective Hamiltonians such that their minimums are proportional
to topological charges of excitations. In addition, we produce these mean field
configurations that give minimums to the Hamiltonians.
1
2 Quantum and classical models
As mentioned above, we represent the spin system by a planar atomic lattice
withthesamespinsatallsites. Weassignthreespinoperators(Sˆ1, Sˆ2, Sˆ3)=
n n n
Sˆ to each atom n; the operators obey the standard commutation relations
n
[Sˆa, Sˆb ]=iε Sˆcδ ,
n m abc n nm
where a, b, c run over the values 1, 2, 3 , and δ is the Kronecker symbol.
nm
{ }
2.1 Generalized Heisenberg Hamiltonians
We are interested in so called high spins s>1. In this case, we can describe the
system by the following bilinear-biquadratic Hamiltonian
ˆ2 = J(Sˆ ,Sˆ )+K(Sˆ ,Sˆ )2 .
n n+δ n n+δ
H −Xn,δ(cid:16) (cid:17)
Here δ runs over the nearest-neighbour sites, n runs over all sites of the lattice,
constants J and K denote exchange integrals. In the case of spin s>3/2, the
above Hamiltonian can include also the bicubic exchange, namely
ˆ3 = J(Sˆ ,Sˆ )+K(Sˆ ,Sˆ )2+L(Sˆ ,Sˆ )3 ,
n n+δ n n+δ n n+δ
H −Xn,δ(cid:16) (cid:17)
where L denotes the corresponding exchange integral.
OnecaneasilywriteageneralizedHeisenbergHamiltonianforthesystemof
anarbitraryspinsorgreaterthans. ThisHamiltoniancontainsallpowersofthe
exchange interaction up to 2s. It can be reduced to a bilinear form if one takes
the 2s+1-dimensional space of irreducible representation of the group SU(2).
The spin operators Sˆa over this space generate a complete associative matrix
{ n}
algebra, whichhasasufficientnumberofoperatorstoreducethecorresponding
Hamiltonian to a bilinear form.
Forexample,inthecaseofspins=1theappropriatespaceofrepresentation
is3-dimensional,andwechooseacanonicalbasisintheform: +1 , 1 , 0 .
The spin operators Sˆa generate the algebra Mat . In orde{r|toifo|r−m ia b|ais}is
{ n} 3×3
in the algebra we take the tensor operators of weight 2
Qˆab =SˆaSˆb +SˆbSˆa, a=b,
n n n n n 6 (1)
Qˆ22 =(Sˆ1)2 (Sˆ2)2, Qˆ20 =√3 (Sˆ3)2 2
n n − n n n − 3
(cid:0) (cid:1)
inadditiontothespinoperators. Theintroducedoperatorsarecalledquadrupole
operators.
In the case of spin s=3/2 the appropriate space of representation is 4-
dimensional,and +3 , +1 , 1 , 3 isacanonicalbasis. Wecompletethe
{| 2i | 2i |−2i |−2i}
associative matrix algebra Mat of Sˆa by the tensor operators of weights 2
4×4 { n}
2
and 3, defining them by the following formulas:
Qˆab = √5 SˆaSˆb +SˆbSˆa , a=b
n 2√3(cid:16) n n n n(cid:17) 6 (2)
Qˆ22 = √5 (Sˆ1)2 (Sˆ2)2 , Qˆ20 = √5 (Sˆ3)2 5 ,
n 2√3(cid:16) n − n (cid:17) n 2 (cid:0) n − 4(cid:1)
Tˆa3 =(Qˆa2Sˆ3 +Sˆ3Qˆa2), a,b 1,2 , a=b,
n n n n n ∈{ } 6
Tˆab = 1 (Sˆa)2Sˆb +Sˆb(Sˆa)2+SˆaSˆbSˆa (Sˆb)3 ,
n √6(cid:16) n n n n n n n− n (cid:17) (3)
Tˆ3a = 1 Qˆa3Sˆ3 +Sˆ3Qˆa3+√3(Qˆ20Sˆa+SˆaQˆ20) ,
n √10 n n n n n n n n
(cid:16) (cid:17)
Tˆ30 = 1 41Sˆ3 20(Sˆ3)3 .
n 12 n− n
(cid:0) (cid:1)
Wecallthetensoroperatorsofweight3sextupole operators. Inwhatfollowswe
denote all tensor operators over the chosen space of representation by Pˆa .
{ n}
In terms of representation operators a generalized Heisenberg Hamiltonian
gets a bilinear form. For the Hamiltonians considered above we have:
ˆ2 = (J 1K) SˆbSˆb 1K QˆαQˆα 4KN;
H − −2 n n+δ− 2 n n+δ− 3
Xn,δ Xb Xn,δ Xα
ˆ3 = (J 1K+587L) SˆbSˆb 75(4K L)N
H − −2 80 n n+δ− 32 − −
Xn,δ Xb
6(K 2L) QˆαQˆα 9 L TˆβTˆβ ,
− 5 − n n+δ− 10 n n+δ
Xn,δ Xα Xn,δ Xβ
where N is the overall number of sites in the lattice. Obviously, the obtained
bilinear Hamiltonians are SU(2)-invariant, because they are constructed from
representation operators of the group SU(2).
2.2 Mean field approximation
Here a mean field is a field of expectation values for the operators Pˆa cal-
{ n}
culated after spontaneous breaking of symmetry. The breaking of symmetry is
performed by switching on an external magnetic field, which specifies an order
in the system; then the external field vanishes. Such kind of averages is also
called quasiaverages [3].
We denote a mean field averaging by , and components of the mean field
by µ (x )= Pˆa . A mean field apprho·xiimation of the bilinear-biquadratic
{ a n h ni}
Hamiltonian has the form
3 8
ˆ2 = (J 1K)z Pˆaµ (x ) 1Kz Pˆaµ (x ) 4KzN,
HMF − −2 n a n − 2 n a n − 3
Xn aX=1 Xn aX=4
where z is a number of the nearest-neighbour sites.
Evidently, a mean field Hamiltonian remains SU(2)-invariant. Then by an
action of the group SU(2) it can be reduced to a diagonal form. Of course,
3
this reduction is possible only in the case of thermodynamical equilibrium and
an infinite lattice, when the mean field becomes constant and the dependance
on site n can be omitted. Moreover, almost all components of the mean field
vanish, except the components corresponding to diagonal operators of Pˆa .
{ n}
For the bilinear-biquadratic Hamiltonian a diagonal form is the following:
ˆ2 = zN (J 1K)Sˆ3µ + 1KQˆ20µ + 4K .
HMF − (cid:16) −2 3 2 8 3 (cid:17)
Theremainingcomponentsaresuitabletoserveasorder parameters. Thecom-
ponent µ describes a normalized magnetization (a ratio of the z-projection of
3
magnetic moment to a saturation magnetization). The components µ and µ
8 15
are normalized projections of quadrupole and sextupole moments, respectively.
Possible values of order parameters can be obtained from the self-consistent
equations
µ = Pˆa =TrPˆae−hˆkMTF
a h iMF hˆMF
Tre− kT
for all diagonal operators Pˆa. Here we use the density matrix with the one-site
Hamiltonian hˆ = ˆ /N. Note, that for the standard Heisenberg Hamilto-
MF MF
H
nian, when only the spin operators are considered, a self-consistent equation
turns into the well-known Weiss equation.
Inthecaseofbilinear-biquadraticHamiltonian,ananalysisofself-consistent
equations gives the following. We adduce probable values of order parameters
inthelimitT →0asJ, K>0: 1)|µ3|=1, µ8=2√J3−KK; 2)|µ3|=12, µ8=J√−3KK/2;
3) µ =0, µ = 2 ; 4) µ =0, µ = 1 . Two of the solutions correspond
3 | 8| √3 3 | 8| √3
tostateswith theferromagnetic order, becausetheyhave anonzeromagnetiza-
tionµ . Theothertwosolutionscorrespondtostateswiththequadrupoleorder
3
(so called nematic states), which are states with zero magnetization. Solutions
2) and 4) are unstable and called partly ordered. If K becomes negative, only
solution 1) remains. These results accord with the results of [4, 5] and with the
phase diagram of ordered states in a one-dimensional spin system from [6].
In what follows we consider an SU(3)-invariant system with the bilinear-bi-
quadratic Hamiltonian. The SU(3)-invariance is reached by assigning J=K;
this is the boundary line between the ferromagnetic and the nematic regions
(see [6]). It means the system can appear in the both states. In the case of
Hamiltonian ˆ3,themaximalSU(4)-invarianceisreachedasJ= 81K= 81L,
H −44 −16
that is located within the ferromagnetic region.
NowweapplythemeanfieldaveragingtothequantumHeisenbergequation
dPˆa
i~ n =[Pˆa, ˆ]. (4)
dt n H
The averaging is performed with the assumption of zero correlations between
fluctuations of Pˆa at distinct sites: PˆaPˆb = Pˆa Pˆb . Then we take a
{ n} h n mi h nih mi
large-scale limit and obtain the Landau-Lifshitz like equation
∂µ
~ a =2Jl2C µ (µ +µ ), (5)
abc b c,xx c,yy
∂t
4
where C are structure constants of the Lie algebra of Pˆa with the commu-
tation realbactions [Pˆa,Pˆb]=iC Pˆcδ . Equation (5) is{anne}quation of motion
n m abc n nm
for the mean field µ (x) over the plain x = (x,y) x, y R that replaces
a
{ } { | ∈ }
the lattice.
InthecaseofstandardHeisenbergHamiltonian(5)coincideswiththeLandau-
Lifshitz equation for an isotropic magnet. Therefore, in the general case we call
(5) a generalized Landau-Lifshitz equation for the vector field µ . The vector
a
{ }
fieldhas8componentsifoneexploitsthebilinear-biquadraticHamiltonian,and
15 components for the Hamiltonian with the bicubic exchange.
2.3 Effective Hamiltonians on coadjoint orbits
The generalized Landau-Lifshitz equation (5) can be interpreted as a Hamilto-
nian equation on a coadjoint orbit of Lie group. Inthecaseofspins=1wedeal
with the group SU(3), in the case of an arbitrary spin s the group is SU(2s+1).
Note, that the matrices Pˆa serve as a basis in the corresponding Lie algebra
su(2s+1), and componen{ts o}f the mean field µ serve as coordinates in the
a
dual space to su(2s+1). { }
We start with brief description of the groups SU(3) and SU(4). For more
material see, in particular, [7, 8].
The group SU(3) has two types of orbits: the generic SU(3) of dimen-
U(1) U(1)
×
sion 6, and the degenerate SU(3) of dimension 4. Each orbit of SU(3) is
SU(2) U(1)
defined by two numbers m and q×, which are values of the coordinates µ and
3
µ at an initial point (a point in the positive Weyl chamber). Simultaneously,
8
these numbers are limiting values of the mean field components µ and µ at
3 8
zero temperature. For a degenerate orbit one has to assign m=0, or m=√3q.
Evidentlythedegenerateorbitswithm=0aredomainsofmeanfieldconfigura-
tions that realize nematic states. Ferromagnetic states are realized on all other
orbits of SU(3).
The group SU(4) has four types of orbits: the generic SU(4) of
U(1) U(1) U(1)
× ×
dimension 12, the degenerate SU(4) of dimension 10, the degenerate
SU(2) U(1) U(1)
× ×
SU(4) of dimension 8, and the maximal degenerate SU(4) of dimen-
S(U(2) U(2)) SU(3) U(1)
sion6×. Eachorbitisdefinedbynumbersm,q,p,whicharelimiti×ngvaluesofthe
mean field components µ , µ , µ . Almost all orbits are domains of ferromag-
3 8 15
netic mean field configurations. Nematic states are realized on the degenerate
orbits of dimension 8 as m=p=0 and q is arbitrary. So it is probable to reveal
a nematic state even in the ferromagnetic region of the phase diagram [6].
Asshownabove,limitingvaluesofdiagonalcomponentsofameanfield serve
as order parameters. Simultaneously, they define a coadjoint orbit where the
corresponding mean filed configuration lives.
Each orbit possesses a Hamiltonian system with an equation of motion and
a group-invariant Hamiltonian. For a degenerate orbit of SU(3) the equation is
∂µ
a = 4 C µ (µ +µ ), (6)
∂t 3(m2A+q2) abc b c,xx c,yy
5
where denotes a dimensional constant. The values m, q of order parameters
A
(a magnetization and a projection of quadrupole moment) define an orbit via
the following equations, which we call constraints:
δ µ µ =m2+q2
ab a b
d µ µ = m2+q2 µ .
abc b c ±q 5 a
Hered = √3 Tr(Pˆ Pˆ Pˆ +Pˆ Pˆ Pˆ )isasymmetrictensor. Thecorresponding
abc 4√5 a b c b a c
SU(3)-invariant Hamiltonian is the following
8
deg = 2 (µ )2+(µ )2 dxdy, (7)
Heff 3(m2J+q2)Z aX=1(cid:16) a,x a,y (cid:17)
where the dimensional constant has a meaning of exchange integral.
J
Obviously, equations (6) and (5) coincide. It is easy to show, that (5) coin-
cides with the equation of motion on a maximal degenerate orbit. Recall, that
(5) is obtained when correlations between fluctuations of Pˆa are neglected.
{ n}
Presumably, equations of motion on other orbits are derived from (4) via the
mean field averaging with more complicate correlation rules.
A generic orbit of SU(3) is determined by the following equations:
δ µ µ =m2+q2
ab a b
d µ µ µ = 1 q(3m2 q2).
abc a b c √5 −
The SU(3)-invariant Hamiltonian on this orbit has the form
8
gen = (m2+q2)2(µ )2+(m2+q2)(η )2
Heff 2m2(mJ2−3q2)2 aX=1(cid:16) a,x a,x −
2√3q(3m2 q2)µ η , (8)
a,x a,x
− − (cid:17)
whereη isaquadraticformin µ : η =√5d µ µ . Formoredetailssee[9].
a a a abc b c
{ }
TheHamiltoniansystemsoncoadjointorbitsofSU(3)serveasclassicaleffec-
tive models for the spin system of s>1 with biquadratic exchange. Evidently,
these models describe large-scale (or slow) fluctuations of the mean field. In
this paper we suppose that the order parameters m, q are fixed numbers. But
generally speaking, they depend on a temperature T and the interaction con-
stant J. Taking into account these dependencies, one can consider small-scale
(or quick) fluctuations of the mean field.
2.4 Geometrical properties of effective Hamitonians
EachcoadjointorbitofasemisimpleLiegroupisahomogeneousspacethatad-
mitsaKa¨hlerianstructure. Thusonecanintroduceacomplexparameterization
of an orbit. For this purpose we use a generalized stereographic projection (for
6
more details see [8]). In the case of group SU(3), the projection is represented
by the following formulas:
µa =−m−2√3q ζa+mξa, ηa = √3(m2−2q2)−2mq ζa+2mqξa,
w2+w¯2+w3+w¯3 (1 w¯1)(w3 w1w2)+(1 w1)(w¯3 w¯1w¯2)
ζ1=−√2(1+ w22+ w32) ξ1=− − √2(1−+ w12+ w3−w1w22)−)
| | | | | | | − |
w3 w¯3 w2+w¯2 (1+w¯1)(w3 w1w2) (1+w1)(w¯3 w¯1w¯2)
ζ2=i√2(1−+ w−22+ w32) ξ2=i √2(1−+ w12+−w3 w1w22−)
| | | | | | | − |
w22 w32 1 w12
ζ3= 1+| w|2−2+| w|32 ξ3= 1+ w12+−|w3| w1w22
| | | | | | | − |
w¯2w3 w2w¯3 w1 w¯1
ζ4=i1+ w2−2+ w32 ξ4=i1+ w12+−w3 w1w22 (9)
| | | | | | | − |
w2+w¯2 w3 w¯3 (1+w¯1)(w3 w1w2)+(1+w1)(w¯3 w¯1w¯2)
ζ5= √2(1+ w−22+−w32) ξ5=− √2(1−+ w12+ w3 w1w22−)
| | | | | | | − |
w2 w¯2+w3 w¯3 (1 w¯1)(w3 w1w2) (1 w1)(w¯3 w¯1w¯2)
ζ6=i√2(1−+ w22+−w32) ξ6=i − √2(1−+ w12+−w3− w1w22−)
| | | | | | | − |
w¯2w3+w2w¯3 w1+w¯1
ζ7=−1+ w22+ w32 ξ7=−1+ w12+ w3 w1w22
| | | | | | | − |
2 w22 w32 1+ w12 2w3 w1w22
ζ8= √3(1−+| w|22−+| w|32) ξ8= √3(1+| w|1−2+| w3− w1w2| 2).
| | | | | | | − |
Thecoordinates w , w , w (Bruhatcoordinatesaccordingto[7])parameter-
1 2 3
{ }
ize a generic orbit of SU(3). In the case of a degenerate orbit, one has to assign
m=0 and w =0, or m=√3q and w =0.
1 2
In terms of w the effective Hamiltonians (7) and (8) have the form
α
{ }
∂w ∂w¯ ∂w ∂w¯
α β α β
Heff =J Z Xα,β gαβ¯(cid:16) ∂z ∂z¯ + ∂z¯ ∂z (cid:17)dzdz¯, α, β =1, 2, 3, (10)
wherez =x+iy isacomplexcoordinateontheplaneobtainedfromtheatomic
latticeafteralarge-scalelimitingprocess(seeSection2.2). Thetensorg isnon-
degenerate and positively defined, thus it can serve as a metrics on an orbit.
Its components {gαβ¯} come from (7) for a degenerate orbit, and from (8) for a
generic one. In terms of the auxiliary vector fields ζ and ξ we have
a a
{ } { }
∂ζ ∂ζ ∂ξ ∂ξ
gdeg1 = 1 a a , gdeg2 = 1 a a ,
αβ¯ 2 ∂w ∂w¯ αβ¯ 2 ∂w ∂w¯ (cid:12)
Xa α β Xa α β(cid:12)w2=0
(cid:12)
∂ζ ∂ζ ∂ζ ∂ξ ∂ξ ∂ξ (cid:12)
ggen = 1 a a a a + a a .
αβ¯ 2 (cid:18)∂w ∂w¯ − ∂w ∂w¯ ∂w ∂w¯ (cid:19)
Xa α β α β α β
Note, that in terms of w the tensor g does not depend on a particular orbit.
α
{ }
BeingaKa¨hlerianmanifoldanorbitofSU(3)possessesaKa¨hlerianpotential.
For this purpose we use a potential Φ of the Kirillov-Kostant-Suoriau form:
Φ=mΦ1− m−2√3q Φ2,
Φ =ln(1+ w 2+ w w w 2), Φ =ln(1+ w 2+ w 2),
1 1 3 1 2 2 2 3
| | | − | | | | |
7
A topological structure of the orbit is characterized by the second cohomology
groupH2 ofdimension2. Thatiswhythereexisttwobasis2-forms,forexample
generated by the potentials Φ , Φ . Each of them defines a topological charge
1 2
1 i∂2Φ ∂w ∂w¯ ∂w ∂w¯
= k α β α β dz dz¯, k =1, 2.
k
Q 4π Z Xα,β ∂wα∂w¯β(cid:16) ∂z ∂z¯ − ∂z¯ ∂z (cid:17) ∧
On a degenerate orbit only one potential is governing, and only one topolo-
gicalchargeexists. Thentheexpressionsfor and degk differonlyinasign.
Qk Heff
Evidently,
degk >4π . (11)
Heff J|Qk|
Hence, on a degenerate orbit a minimum of is realized if the equality holds,
eff
H
that takes place if w are holomorphic or antiholomorphic functions. Herewe
α
{ }
use an idea of Belavin and Polyakov [2].
For a generic orbit we define a topological charge by = + . In order
1 2
Q Q Q
to extend inequality (11) to generic orbits with the topological charge , we
Q
construct an effective Hamiltonian by the following formula:
8
gen = C (µ )2+C (η )2+C µ η , (12)
Heff 2m2(mJ2−3q2)2 aX=1(cid:16) 1 a,x 2 a,x 3 a,x a,x(cid:17)
C =m4+q4 4 mq(q2 m2)+14m2q2,
1 − √3 −
C = 5m2+q2 2 mq,
2 3 − √3
C = 2 m3+2q3 26m2q+ 2 mq2.
3 √3 − 3 √3
In terms of w it is reduced to the form (10) with the metrics
α
{ }
∂ζ ∂ζ ∂ξ ∂ξ
ggen = 1 a a + a a .
αβ¯ 2 (cid:18)∂w ∂w¯ ∂w ∂w¯ (cid:19)
Xa α β α β
Then we get
gen >4π ,
Heff J|Q|
anda minimum of gen is realized if the equality holds, that takes place if w
Heff { α}
are holomorphic or antiholomorphic functions.
3 Large-scale topological excitations
Now we construct a particular class of topological excitations that give min-
imums to the effective Hamiltonians deg1, and gen defined by (12). We
Heff Heff
describe these excitations by holomorphic functions w (z) . Each set w (z),
α 1
{ } {
w (z), w (z) represents a mean field configuration of the system in question.
2 3
}
First, we consider a degenerate orbit of SU(3) with m=0, q= 2 , where
−√3
a nematic state is realized. In order to satisfy the limiting conditions: µ m,
3
→
µ q, the functions w (z) have to vanish as z . Let
8 α
→ { } →∞
w (z)=0, w (z)= a2 , w (z)= a3 , a , z , a , z C, (13)
1 2 z z2 3 z z3 2 2 3 3 ∈
− −
8
be a large-scale excitation in the magnet in question. The corresponding mean
field configuration is obtained by substitution of (13) into (9). A behavior of
the components µ and µ is represented on the Fig.1. As z the values of
3 8
→∞
µ , µ tend to m, q.
3 8
m (w ,w)
3 2 3 m(w ,w)
8 2 3
a 1 1
2 z a Ö3 a z
3 21 3
z2 z z2 z3
-1 a3
2
-
Ö3
Fig. 1. Profiles of the mean field components µ , µ in the case of
3 8
configuration (13).
Suppose z =z . By shifting of the coordinate z one can easily reduce (13)
2 3
to the configuration
w (z)=0, w (z)=a2, w (z)=a3, a , a C,
1 2 z 3 z 2 3 ∈
and calculate its topological charge:
2i (a2+a2)
= 2 3 dz dz¯=1. (14)
Q 4π ZZC (|z|2+a22+a23)2 ∧
In the case of z =z the topological charge equals 2.
2 3
6
Itcanbeinterpretedasfollows. Eachpoleofameanfieldconfigurationrep-
resents a kind of Belavin-Plyakov soliton. Each soliton gives a unit topological
charge. Thus,twodistinctsolitonshavethetopologicalcharge2. Nocontinuous
deformation take a configuration of topological charge 2 to a configuration of
topological charge 1. If we allow noncontinuous deformation, then two solitons
canmeetatanypointandjoinintoone, atthesametimeanenergyisreleased.
From (11), it follows that the released energy equals 4π per one pole.
J
Note, that the energy of configuration (13) does not depend on parameters
of solitons: a , z , a , z . It means that the excitation can grow (when a and
2 2 3 3 2
| |
a grow) preserving an energy. Such growth immediately leads to destruction
3
| |
of an order in the system.
One can construct a configuration with more than two solitons:
n m
w (z)=0, w (z)=a / (z z ), w (z)=a / (z z ), (15)
1 2 2 2k 3 3 3k
− −
kY=1 kY=1
here a , a , z n , and z m are fixed complex numbers. If all values
2 3 { 2k}k=1 { 3k}k=1
z n and z m are distinct, a topological charge equals n+m. When a
{ 2k}k=1 { 3k}k=1
poleofthefunctionw (z)coincideswithapoleofw (z), thetopologicalcharge
2 3
decreasesby1. Butacoincidenceoftwopolesofthesamefunction(forexample
9
w (z)) does not lead to a decrease of the topological charge. It is easy to see,
2
that the minimal energy of configuration (15) equals 4π min(n,m).
J ·
Now we consider a generic orbit, where a ferromagnetic state is realized.
Suppose m=1 and q= 2 . In this case, we describe a mean field in the
−√3
magnet by the effective Hamiltonian gen, defined by (12). Let
Heff
w (z)= a1 , w (z)= a2 , w (z)= a3 , a ,z C, k =1,2,3. (16)
1 z z1 2 z z2 3 z z3 k k ∈
− − −
bealarge-scaleexcitationofthemeanfield. Acalculationoftopologicalcharges
gives: 1) =3, =2, if a =a =a or a =a =a , 2) =2, =2, if
1 2 1 2 3 1 2 3 1 2
Q Q 6 6 6 Q Q
a =a =a , 3) =2, =1, if a =a =a or a =a =a .
1 3 2 1 2 1 2 3 1 2 3
6 Q Q 6
4 Conclusion and discussion
Each generalized Heisenberg Hamiltonian with high powers of the exchange in-
teraction can be reduced to a bilinear form. By a mean field averaging we
obtain a classical system from the original quantum one. An averaging of the
Heisenberg equation gives a Landau-Lifshitz like equation for a mean field. Us-
ing Lie group apparatus, we construct effective Hamiltonians for the classical
system with SU(3) symmetry. One of them deg is an SU(3)-analogue of the
Heff
Hamiltonian commonly used in theory of magnetism. In addition, we propose
anotherone gen, whichisbiquadraticinthemeanfield. Further,weconstruct
Heff
examples of topological excitations that give minimums to the Hamiltonians.
Such excitations can change their shapes and grow preserving an energy. This
is a probable scenario for the destruction of an ordered state in a 2D magnet at
nonzero temperature, that agrees with the Mermin-Wagner theorem.
5 Acknowledgments
ThisworkispartlysupportedbythegrantoftheInternationalCharitableFund
for Renaissance of Kyiv-Mohyla Academy.
References
[1] MerminN.D.,WagnerH.Phys. Rev. Lett171133(1966).
[2] BelavinA.A.,PolyakovA.M.JETP Letters22245(1975).
[3] Bogolyubov N N and Bogolyubov N N (Jr.) Introduction to Quantum Statistical Me-
chanics [in Russian](Moscow: Nauka)1984p384
[4] MatveevVMZh. Eksp. Teor. Fiz.651626(1973)
[5] Nauciel-BlochM,SarmaGandCastetsAPhys. Rev. B54603(1972)
[6] BuchtaK.,FathG.,LegezaO¨.,SolyomJ.Phys. Rev. B72054433(2005)
[7] PickenR.F.J. Math. Phys.31616(1990).
[8] BernatskaJ.,HolodP.Proceedings of the IXth International Conference on Geometry,
Integrability and Quantization(Sofia)2008,p.146–166;arXiv: math.RT/0801.2913
[9] BernatskaJ.,HolodP.J. Phys. A42(tobepublishedin2009)
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