Vol.2, No.11, 1287-1291 (2010) Natural Science
http://dx.doi.org/10.4236/ns.2010.211155
Excitations for the one-dimensional S = 1
pseudo-Heisenberg antiferromagnetic chain
Elena V. Orlenko1*, Fedor E. Orlenko1, George G. Zegrya2
1Theoretical Physics Department St.Petersburg State Polytechnic University, St.Petersburg, Russia; *Corresponding author:
2Solid State Electronic Department. A. F. Ioffe Physical-technical Inst. RAS, St.Petersburg, Russia
Received 19 September 2010; revised 22 October 2010; accepted 25 October 2010.
ABSTRACT
We are interested in the anisotropic S=1 anti-
ferromagnetic chain. System of particles with an
arbitrary spin is described directly from the first
principles, based on the symmetry law. The
ground state of the one-dimensional S=1
pseudo-Heisenberg antiferromagnet with sin-
gle-ion anisotropy is calculated. Excitations of
the chain in the form of nonlinear spin waves
and, in particular, the possibility of a soliton
solution is considered.
Keywords: Pseudo-Heisenberg Hamiltonian;
Antiferromagnetic Spin-1 Chain; Soliton Excitations
1. INTRODUCTION
We investigate an anisotropic S=1 antiferromagnetic
chain. Interest in one-dimensional S=1 antiferromagnets
is traced back to the works concerned with Bose
–Einstein condensates of alkali atomic system with arbi-
trary spin [1] from one hand and to the original work by
Haldane [2,3] from other hand. In the case of spin-1
Bose gas with antiferromagnetic interaction like 23 Na ,
it has been pointed out [4] that as the magnetic field gra-
dient is reduced the single condensate involves toward an
angular momentum eigenstate, which becomes a spin
singlet as the magnetic field is reduced to zero. The
singlet state has a “fragmented” structure which bears no
resemblance to single condensate state. The part of the
Hamiltonian that describes the interaction between the
boson particles with angular momentum s=1 [5] has a
Heisenberg form, but in general case assumes to the
Hamiltonian has a polynomial form:12
0
=()
ns n
n
n
VCSS





,
where the operators 1 S

and 2 S


the spins of atom 1 and
2, respectively, Cn is a linear combination of the
s-scattering interaction constants, S – the maximal
total spin of two particles. It was emphasized there that
“for bosons (or fermions), the symmetry implies that
only even (or odd) S terms appear in V.” Contrary to this
statement, it is shown in [6] that both the even and the
odd S-values in the interaction Hamiltonian are possible
for the both fermionic and bosonic systems in the spin
representation, concerned to the symmetry of the coor-
dinate part of wave function. The polynomial form of the
interaction is very useful for the description of the sym-
metric (antisymmetric) states in the spin representation,
but the general form of the interaction of Hamiltonian
was not found in reference [5].
By analyzing the presence of topological terms in ef-
fective-field theories for one-dimensional antiferromag-
nets, Haldane conjectured that integer-spin chains dis-
play a ground state with exponentially decaying
spin-spin correlations and a gapped excitation spec-
trum—properties markedly different from those dis-
played by the exactly solvable S=1/2 chain. Despite ear-
ly controversy, this so-called “Haldane conjecture” is
now supported by solid numerical and experimental
evidences [7]. The single-ion anisotropy is relevant in
accounting for the magnetic properties of a number of
compounds: CsNiCl3 (weak axial anisotropy), NENP
[Ni(C2 H
8N 2)2 NO2 )ClO4] (weak axial anisotropy),
CsFeBr3, NENC [Ni(C2H8 N
2)2 Ni(CN)4 ], and DTN
[NiCl2−4SC(NH2)2 ] (strong planar anisotropy) [7].
Haldane3, 4 has analyzed the isotropic point. The inter-
mediate phase was investigated by studying an extended
S=1 model with biquadratic interactions,
2
11
()
ii ii
AKLT i
HJSS SS 









(1)
Affleck, Kennedy, Lieb, and Tasaki (AKLT) [8]
showed that this Hamiltonian is exactly solvable at

=1/3, where it displays a simple valence-bond solid
(VBS) ground state with gapped excitations. Since the
ground state at

=0 has been shown [3] to exhibit
long-ranged string correlations and is adiabatically con-
nected to the ground state at

=1/3 (see, e.g., [9]), one

E. V. Orlenko et al. / Natural Science 2 (2010) 1287-1291 1288
concludes that the Haldane phase has a VBS character.
In contrary to this position, it was shown by E. Orlenko
[10], that neither

=1/3, nor -

=0 could appear in
the Eq.1, the constant only possible here for such system
is

=1. The interest in Haldane-type phases exhibiting
long-range string correlations has been renewed and
proposals for investigating string order in cold atomic
systems have recently been made [11,12]. The investiga-
tion of string order in cold atomic system has been made
in reference [10]. Here the ground state of a set of parti-
cles with spins j= 1 and j= 2 were obtained and nonlinear
magnetic phenomena in the set of particles with j = 1
were considered. The presence of nonlinear terms in the
Hamiltonian of a system may give rise to nonlinear spin
waves in atomic systems with the spin j= 1and, in par-
ticular, to a soliton.
Analyzing excited states above the Néel ground state,
following den Nijs and Rommelse13, it is interpreted the
spin S=1 chain as a diluted system of a pseudospin S˜
=1/2 pseudoparticles; sites with spin projections Siz=1

as being occupied by spin-half particles with pseudospin
components S˜iz=±1/2 and sites with Siz=0 as being
empty (occupied by “holes”). Using this language, the
Néel ground state is equivalent to an “undoped” anti-
ferromagnet, and for small positive values of D/J, the
low-lying excited states lie in the “one-hole sector”
(containing one site with Siz=0). The situation is remi-
niscent of spin-charge separation in one-dimensional
fermionic systems where a hole doped into the system
fractionalizes into “holon” and “spinon” constitu-
ents.Unlike den Nijs and Rommelse’s artificial interpre-
tation the S=1 chain as a diluted system of S =1/2 pseu-
doparticles; we show that the system of particles with an
arbitrary spin can be described directly from the first
principles, based on symmetry law.
The theoretical description of the spin ordering in the
antiferromagnetic S1- chain requires a new form of the
model Hamiltonian. The universal Hamiltonian of the
exchange interaction for the system of particles with an
arbitrary spin is developed here from the first principles.
In this paper, we are interested in improving on previous
estimates for ground state of the one-dimensional S=1
pseudo-Heisenberg antiferromagnet with single-ion ani-
sotropy. Additionally, we obtain results for the excita-
tions in the form of nonlinear spin waves and, in par-
ticular, the possibility of a soliton solution is considered.
We will show that the soliton is stable if the nonpoint
potential varies slowly within the soliton length. For the
one –dimensional S=1spin system, we will arrive at the
pattern of nonlinear magnetic vortices that transform
into macroscopic vortices of a magnetic field in a system
and bring the phase separation.
2. MAGNETIC ORDERING IN THE
ONEDIMENSIONAL CHAIN OF IONS
WITH S=1 SPINS
2.1. Pseudo-Heisenberg Hamiltonian of the
Exchange Interaction for the Chain of
Particles with s= 1
In the simplest case of the chain of ions with
two-particle interaction, the Hamiltonian in the coordi-
nate representation can be written as follows:
(1,...,)( ,1)
i
NHii





(2)
where the numbers of particles are 1, … i, i+1, …N.
The Hamiltonian (, 1)Hii


describing pair interaction
can be represented as a sum of noninteracting particle
Hamiltonians 0
()( )(,)hihjHi j



and the pair inter-
action operator(,)Vij

.
The first order correction of the total energy of a
two-particle system is
(1) ,EKA

 (3)
where K is the direct input and A is the exchange input
to the energy correction, and sign is corresponded
with the symmetry of coordinate part of wave function .
The total wave function of two bosons is a product of
the coordinate-dependent part and /(, 1)
sa iia spin
part of the same permutation symmetry.
//
(, 1)(, 1)(, 1)
ssasa
iiii ii

   . (4)
In other words, the factor 1 for the exchange input
Eq.3 depended of the symmetry of the spatial part cor-
responds to the determined spin part symmetry of the
wave function. The symmetry of the spin part of the two
particle system is connected with the total spin S be-
cause of the symmetry of the Clebsch-Gordan coeffi-
cients. In the case s1 =s2 =s is:

12 21
2
1,
zz
z
zzz
sS
SS SS
ssssssss
CC

 (5)
where the two particle spin function is represented in the
form
112 2
12
1211 22
,;,, ,
z
zz
zzz
SS
z
zz sss s
ss S
SSs sCs sss



(6)
The vectors of the spin states of particles number 1
and number 2 are 11
,
z
s
s and 22 ,
z
s
s.
The permutation operator ,
s
s
P

acting on the spin state
Eq.6 gives the eigenvalue(1)
S

, because of the two par-
ticle spin functions are symmetric or antisymmetric with
respect to particle permutation. It can be seen from Eq.6
for Clebsch-Gordan coefficients.
Then we will change the factor 1 in Eq.4 by the
permutation operator 1, 2
s
s
P

in the spin representa-
tion:
(1)
1,2 ,
s
s
EKAP


(7)
Then the Hamiltonian of the chain taken into account

E. V. Orlenko et al. / Natural Science 2 (2010) 1287-1291 1289
pair interactions and acting on spin functions, can be
presented as
1
int, 1, 1,
()
ii
iiiis s
i
HKAP






. (8)
For obtaining the permutation operator in the spin
representation the following condition is used:
121 21 2
,;,(1) ,;,
S
ss zz
P SSssSSss

. (9)
The permutation operator can be presented in the form
of a polynomial expansion as
 
221
1,22 1221121120
... .
ss
ss ss
Pcss csscssc


 

 


(10)
Here the number of free coefficients cn is equal to the
number of possible eigenvalues of the scalar product
operator. The eigenvalue of the scalar product 12
s
s





operator depends on the value of total spin S:


______________
_______
22
12
11
2(1)2(1)
22
ssS sSSss 



 . (11)
In the case s1 =s2 =1, the total spin of the two particle
system has three eigenvalues S=2, 1, 0.
The free coefficients cn can be determined from the
system of three linear equations resulting from Eq.10.
This system has only one, unique solution with determi-
nate coefficients:
c0 =-1, c1=c2=1. Then, the permutation operator for the
two particle system in the case s1 =s2 =1 has the form:
2
1,21212
()()1
ss
Pssss 



 , (12)
and the Hamiltonian of N particles with s = 1 interacting
in pairs can be represented in the form:


2
int, 1, 1111.
iiiii ii i
i
HKAssss
 








  (13)
This Hamiltonian of the exchange interaction has a
biquadratic term, which gives rice for the nonlinear phe-
nomena in the S=1 antiferromagnetic chain.
2.2. Ground State of the S=1 Chain
Unlike the s=1/2 system of particle, the total spin Σ of
the system with s=1 spins is not a good quantum number
because it does not commutate with the Hamiltonian
Eq.13 int
2
,0H





. The Hamiltonian Eq.14 can be
rewritten in the form, where

2
2
,1 1iii i
Sss





 is the
two-particles spin operator, with indices i and i+1 de-
noting the particles numbers:

2
22 22
11 1
2
11
221
22
int
i ,ii,iii ,ii
i
N
HK
A
Ss Ss.
 














(14)
First of all we calculate the exchange energy exc
i,i+1
Eof
ions couple as a function of couple spin S. It will be
equal to


2
1
112 1
2
1121 1
2
exc
i,i+1 i,i
EA S(S)s(s)
S( S)s( s).




 








(15)
Let us consider the spin of ion’s couple as an integer
variable, then the new integer variable x can be written
as

112 1
2
x
S( S)s( s), (16)
where x varies in the [-2, 1].
Then the exchange energy exc
i,i+1
Eof ions couple can be
presented as a function of x
2
1
() (1).
exc
i,i+1 i,i
ExAxx


 (17)
Here the lowest energy value is achieved at x=-0.5,
which corresponds the spin S=1.3, close to the physi-
cally available total spin of couple S=1. This state is
anti-symmetric and stable with respect to the small fluc-
tuations from the equilibrium. Then we come to conclu-
sion, that the most preferable value of the total couple
spin is S=1. Then we can present the Hamiltonian of
whole chain as the perturbation expansion in the follow-
ing schema, which conserves the spin anti-symmetry of
whole chain:
11
12
1
11
14
2
,
inti,ii i
i,N/
()
q,q qq
q,N/
N
HKA
A
...






 



(18)
where symbolic expressions mean for neighbour ions



2
11 111
1
i,i iii,iiiii
AA ssss
 






and


1
11
2
1
11 1
1
()
q,q qq
()
q,qq qq q
A
Ass ss

 
 

 
 
 
for the neighbour couple of ions with the first renormal-
ized constant of the exchange interaction Aq,q+1(1) and the
total couple’s spin
1
2
,
iii
q
s
ss






223
12
iii
q
s
ss.

 




Because of these constants Aq,q+1 (ν) are equal for each
couple in the chain, it is possible to find the energy of the
system in the representation of S spin of couples. Because
of each term of this expansion corresponds to the most
preferable spin Sq,q+1=1 anti-symmetric state, then the
total spin of whole chain Σ will be also equal to 1, (Σ=1)

E. V. Orlenko et al. / Natural Science 2 (2010) 1287-1291 1290
which corresponds to anti-symmetric state. It is a anti-
ferromagnetic state, because of the averaged spin pro ion
is reduced to zero as Σ/N. In this case the total energy of
the chain can be written as follows:
1
0
1
22 222
()( )
()
int
NNA A
EKA ...









. (19)
We assume, that the renormalized exchange parameter
of the k step A(k) is connected with the k-1 approximation
exchange parameter A(k-1 ) as follows:
1, (k)(k )
B
b
AAexp a




(20)
where aB – is Bohr’s radius, b- is the constant of lattice.
Then the total energy of the basic anti-symmetric state of
infinite length chain for each ion will be
0
0
11 1
22 22 22
11
22
B
b
a () ()
int
()
B
EKA e
KA
N
KA b
exp( )
a




 


 
 
 










(21)
It is easy to see that the exchange interaction plays more
important role for the long chains than for the short and
brings to the system a strong long order correlation.
3. SPIN DARK-BRIGHT SOLITON
Let us consider the anisotropic spin-1 antiferromag-
netic chain that preserves ground state Eq.21 of the sys-
tem. The Hamiltonian of the system that takes into ac-
count only exchange interaction is taken in the form of
Eq.14. Then, assuming that the interaction takes place
with the nearest neighbors in the chain, we can set the
spin of a pair of atoms to be equal to S=1 and the eigen-
value of the operator 1ii
s
s 




, to 1ii
s
s 




= –1. If a spin
k in a chain of atoms is flipped, the operator of the exci-
tation energy can be written in the form:

2
01 11
2224
1
{()
()2(1)(1)},
intk kkkk k
kkk k
V HEAssssss
sss s


 


 



 


(22)
where (considering the flip of a single spin k)
24 2
0(( 1)( 1)1) ii
i
EAss
.
In the semi-classical continuous approximation of
magnetic moment eigenvalues, the magnetic moment of
an atom may be represented by a function that smoothly
varies with distance. Then, the spin k 1 can be ex-
panded as
2 2
12,
2
kk
kk
s
s b
ssb
x
x


 




(23)
where b is the constant of the lattice. After the substitu-
tion of the above expansion into Eq.22, the excitation
energy becomes
2
22
4
22
2
2 2
2
2
22 4
12
2
kk
k
k
kk
ss
b
VAs bxx
s b(s)s.
x



 

 
 
 
 













(24)
On the other hand, in the approximation of an effec-
tive field H* produced by all spins of the system, the
excitation energy of the system can be defined as the
energy of interaction of each of the spins with this field:
0
*
k
VgsH.  


(25)
Then, each of the spins precesses (in accordance with
the Bloch theorem) in the effective magnetic field
*
me
H
m
tMc












2 4
*222
2
0
2
2
2
2
{(2( )())
4
(1 2)}.
2
kk
k
k
k
sS Ab
Hsb
gx x
s
bSx



 






 (26)
Applying Eq.24 to the components of the magnetic
moment vector m = sgµ0 and assuming mz >> mx, my ,
one can rewrite Eq.26 in the cyclic coordinate system:
222 2
2222222
00
222 2
2222222
00
() 4()
[( )
4
[( )];
mm m
Ab
mi m];
xbgbg
mm m Ab
mi m
xbgbg



 




 








(27)
where 2
0
()
xy
A
mmim, b.
g

 
The equation obtained is similar to that for a
dark–bright soliton [14-16]. Following [14], we repre-
sent a solution in the form
2
0{(( ))}
2
i tixktg
mka
meesechkxqt

,
3({[()]})mAaisincostanhkxqt

  (28)
Here, 1/k is the soliton length, mz = m0 ,
222
22 000
2222 22
000
4
1{()},
44
mmm
kcos
aggg




E. V. Orlenko et al. / Natural Science 2 (2010) 1287-1291 1291
2
2
(1)()
2
Atgka



 

is the precession frequency with regard to the shift, and
2
() (0)Aa qt qkttg



is the soliton coordinate.
Such integrable systems in which the free energy is
conserved are known as Manakov equations [17]. The
higher the velocity of a soliton, the lower the free energy
of system [17]; therefore, the soliton is formally unstable
(it is accelerated). However, the effect of other excita-
tions (solitary or ordinary spin waves), which is disre-
garded here, does not cause dissipation [17]. With an
additional inhomogeneous potential added to Eq.26, the
system becomes nonintegrable, allowing for the nontriv-
ial interaction of a soliton with the environment. Never-
theless, if the interaction potential varies slowly within
the scale of the soliton length k-1 , the variations (propa-
gations) in the soliton and potential correlate and the free
energy can be treated as an adiabatic invariant in this
case [15,16]. Thus, for the spin system considered above,
we will arrive at the pattern of nonlinear magnetic vor-
tices that transform into macroscopic vortices of a mag-
netic field in a antiferromagnetic chain system. It is im-
portant to note that we analyzed the case when only one
spin was flipped. Considering the flip of two, three, etc.
spins, we will generate a set of solitons with different
frequencies 1, 2, …,

n that differ in the ex-
change interaction constants.
4. CONCLUSIONS
The universal form of the exchange interaction Ham-
iltonian of the system of particles with an arbitrary spin
in the spin representation is developed from the first
principles. The Hamiltonian described the antiferro-
magnetic S1-chain contains the biquadratic term with
determined coefficient. The ground state energy of the
chain is more dependent from the exchange interaction
in the case of a long chain. The exchange interaction
makes such system antiferromagnetic and strong
long-order correlated. The presence of biquadratic term
in the Hamiltonian of the system gives rice to nonlinear
spin waves in the chain and, in particular, to a soliton.
The soliton is stable if the nonpoint potential varies
slowly within the soliton length. For the one-dimen-
sional S=1spin system, we will arrive at the pattern of
nonlinear magnetic vortices that bring the phase separa-
tion.
5. ACKNOWLEDGEMENTS
This work was supported by the Ministry of Education and Science
of the Russian Federation (state contract no. P2326, federal program
“Scientific and Pedagogical Staff of Innovative Russia for
2009–2013").
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