Influence of spatial anisotropy on the magnetic properties of Heisenberg magnets
with complex structures
M. Pavkov,* M. S
ˇ
krinjar, D. Kapor, M. Pantic
´
, and S. Stojanovic
´
Institute of Physics, Faculty of Sciences, Novi Sad, Trg D. Obradovic ´a 4, Yugoslavia
Received 12 March 2001; revised manuscript received 19 October 2001; published 19 March 2002
We have analyzed the influence of spatial anisotropy on spin arrangements in complex Heisenberg magnets.
In the case of a four-plane unit superlattice we have demonstrated three- to -two-dimensional crossover as a
consequence of spatial anisotropy.
DOI: 10.1103/PhysRevB.65.132411 PACS numbers: 75.30.Ds, 75.50.Ee, 75.50.Gg, 75.70.-i
Heisenberg systems with complex structures are of great
theoretical interest since magnetic materials encountered in
practice possess rather complicated structures.
1
We are going
to study the systems which were discussed previously by
Zhang Zhi-dong
2
denoted here by I in order to point out
some particularities of these systems.
We first study the system with three sublattices which is
characterized by so-called ‘‘spatial anisotropy.’’ The Hamil-
tonian of the system is
H =-
li ; l ' j
J
l , i ; l
'
, j
S
i
l S
j
l '
=-
i ,
J
ab
S
i
a S
i +
b
-
j ,
J
bc
S
j
b S
j +
c
-
m,
J
ca
S
m
c S
m+
a
l , l ' =a , b , c , further on a 1, b 2, c 3 1
with the notation following I. The basic idea was to perform
a rotation of spin operators in order to determine the ground-
state configuration of the system. After the rotation for the
set of Eulerian angles (
l
,
l
, l =1,2,3) Eq. I 2.2 and in-
troducing the Bose operators a
ˆ
l
, l =1,2,3 Bloch approxima-
tion for spin operators, one arrives at the following Hamil-
tonian:
H =H
0
+
l
A
l
i
a
ˆ
i
†
l a
ˆ
i
l +
l
C
l , l +1
z
i
a
ˆ
i
l a
ˆ
i +
l +1 +a
ˆ
i
†
l a
ˆ
i +
†
l +1 +
l
D
l , l +1
z
i
a
ˆ
i
l a
ˆ
i +
†
l +1
+a
ˆ
i
†
l a
ˆ
i +
l +1 +
l
C
l , l +1
'
z
i
a
ˆ
i
l a
ˆ
i +
l +1 -a
ˆ
i
†
l a
ˆ
i +
†
l +1 +
l
D
l , l +1
'
z
i
a
ˆ
i
l a
ˆ
i +
†
l +1
-a
ˆ
i
†
l a
ˆ
i +
l +1 +
l
B
l
N
i
a
ˆ
i
l +a
ˆ
i
†
l +
l
B
l
'
N
i
a
ˆ
i
l -a
ˆ
i
†
l . 2
Let us discuss the expression for the ground-state energy,
since it is essential for the determination of the Eulerian
angles:
h
0
=
H
0
2 NzS
2
=-
l
J
l , l +1
f
l , l +1
, 3
where N =N
tot
/3 is the number of crystal unit cells and z is
the number of nearest neighbors z =6 for a simple cubic
sc lattice and we have introduced
f
l , l +1
=cos
l
cos
l +1
+sin
l
sin
l +1
cos
l
-
l +1
.
This expression obviously corresponds to the scalar product
of two unit vectors characterizing corresponding directions,
so that -1 f
l , l +1
1.
Let us first consider the situation when all J
l , l +1
0. In
this case, the minimum of H
0
corresponds to the maximum
of all three f
l , l +1
, i.e., f
l , l +1
=1 with the following solutions.
a A trivial one discussed in I:
l
=
l +1
= ,
l
=
l +1
= , which corresponds to ferromagnetic FM ordering with
the quantization axis in an arbitrary direction.
b The particular cases:
1
=
2
=
3
=0 or and
l
-
l +1
arbitrary not mentioned in I. Neither of these solu-
tions affects the FM ground-state energy. Using the expres-
sions for the coefficients I 2.5,I 2.6 – 2.11, one arrives
at the following expression for the Hamiltonian:
PHYSICAL REVIEW B, VOLUME 65, 132411
0163-1829/2002/6513/1324114/$20.00 ©2002 The American Physical Society 65 132411-1