A STAGGERED QUADRUPOLE PHASE FOR A SPIN-ONE ISING SYSTEM WITH ANTIFERROMAGNETIC BIQUADRATIC INTERACTIONS

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A STAGGERED QUADRUPOLE PHASE FOR A

SPIN-ONE ISING SYSTEM WITH

ANTIFERROMAGNETIC BIQUADRATIC

INTERACTIONS

I. Ono

To cite this version:

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JOURNAL DE PHYSIQUE

Colloque C8, Supplement au no 12, Tome 49, d6cembre 1988

A STAGGERED QUADRUPOLE PHASE FOR A SPIN-ONE ISING SYSTEM WITH

ANTIFERROMAGNETIC BIQUADRATIC INTERACTIONS

I. Ono

Department of Physics, Tokyo Institute of Technology, Oh-Okayama, Meguro-ku, Tokyo 152, Japan

Abstract. - We have found a so-called "staggered quadrupole phase " for a simpler version of Blume, Emery and Griffiths model of spin-one Ising system, including only biquadratic interaction and anisotropic energy. The bilinear (exchange) interaction is not always necessary for the occurrence of this phase. The phase diagram obtained by the mean-field approximation has been confirmed by our Monte Carlo simulations.

Many works were devoted to the phase transition for Blume

,

Emery and Griffiths models [l] of spin-one Ising systems, where biquadratic as well as bilinear interactions were included. Most of studies on its phase diagram have been concentrated in the case of ferromagnetic interactions [2]. When the biquadratic interactions are antiferromagnetic (negative), the competi- tions with ferromagnetic bilinear interaction brought about a new ordered phase [3]. Wang and Wentworth

[4] have found a "staggered quadrupole phase7' (SQ phase) consisting of two interpenetrating sublattices by Monte Carlo simulations.

In this paper we will show that the SQ phase occurs also in a simpler model excluding bilinear interaction. Monte Carlo simulations of this model on the square and simple cubic lattice have confirmed the above pre- diction.

Our Hamiltonian can be written by

imation the coupled self-consistent equations which will determine the order parameters q ~ = and

q~ = for the A and B sublattice are easily de- rived as 9 A = 2 exP [-P (IJzl Z Q B - D)] 1

+

2 exp [-P (l Jzl Z ~ B

-

D)] 2 ~ X P [-P

(l

J2l ZQA

-

D)]

(3) q B = l

+

2 exp

[-P

(l JZ

l

z q ~ - D)]

'

where

P

= l/kBT. The magnetizations (S;) always vanishes because the Hamiltonian (1) is invariant un- der the inversion of any spin (S; -+ -S;). In the high temperature limit the q ~ and q ~ take both the value

2/3. Since the symmetry of this phase is same as the quadrupole one, this phase may falls continuously into the Q phase or NQ phase, as the temperature decreases, without a phase transition. The transition temperature can be determined from the condition of the solutions q~

#

q ~ being allowed t o occur below it.

The phase diagram obtained is shown in figure 1, where

Eg (SQ) = - (N/2) D , ( 2 ~ ) Fig. 1. - Phase diagram for the spin-one model of antiferromagnetic biquadratic interaction. Solid curve r e p

respectively. Here N is the total number of spins and resents the transition tempefatures below which a "stag- z is the number of nearest neighbor sites. When D

>

gered quadrupole" or two sublattice ordered phase ap-

2

1 J2

1 ,

the Q phase becomes the lowest and when D

<

Pears. In the region the 0

>

D / z lJ2l

>

-0.15 and

1.05 2 D / z lJzl

>

1 a re-entrant transition appears. Mar-

'

the NQ phase In the region of

'

IJ2l

>

D

>

'

_rks_{(0) or}_(A)_{denote the transition temperature estimated}

the SQ phase is the lowest. _{from Monte Carlo simulations on the simple cubic}_or_square Within the framework of the mean-field approx- lattices, respectively.

where each S; take the values 1, 0 or -1. Firstly let

us consider what is the lowest spin arrangement in the 0.3-

case of Jz

<

0. We have assumed here the three spin

k T/z JF.

arrangements as follows: 1) the quadrupole (Q) phase

in which S; on both sublattices may take 1 or -1 ran- 0 . 2 . -

domly; 2) the non-quadrupole (NQ) phase is which each Si take the value 0; 3) the staggered quadrupole (SQ) phase in which each S; on the A sublattice take

0 . 1

arbitrary value 1 or -1 and each S; on the B sublattice take the value 0 or vice versa. Thus we have

E, (Q) = (N/2) ('

I

J2l - 2 0 ) ( 2 4 0 . 0

--

- , , , - .

-0 5 0.0 0 . 5 1 . 0

Eg (NQ) = 0 (2b) D/? J,

(3)

C8

-

1542 JOURNAL DE PHYSIQUE

the transition temperature versus D / z 1521 is presented

by the solid curve. Our Monte Carlo results also are plotted in figure 1. The temperature dependence of the order parameters are shown in figure 2. In the

SQ ordered phase, the A sublattice is occupied dom- inantly by S; = 61, while the B sublattice does by S; = 0. For a special value D = 0, the A sublattice is

occupied with equal probabilities by

Si

= 1, 0 or -1 or q.~, = 213 a t the absolute zero of temperature, while

the B lattice entirely by Si = 0. This ordered phase has been also confirmed by Monte Carlo simulations on the simple cubic lattice, as shown in figure 3, but

k T / Z J z Fig. 2. - a) Temperature dependences of the two order parameters q~ = (S;) and qg = (S;) of the A and B

sublattices for D / z [J21 = 0, 0.5 or 1.0; b) The order p* rameters in the re-entrant region when D / z

I

J2/ = -0.1.

Fig. 3. - Temperature dependence of the order parameters obtained from our Monte Carlo data on the simple cubic lattice in the case of D = 0.

not found on the square lattice. In the narrow region of 0

>

D / z lJzl

2

-0.15 or 1.05

2

D/z1J21

>

1.0, a re-entrant phase transition is expected to occur in the mean-field approximation as shown in figure 2b. Our Monte Carlo studies have succeede,d in finding it for the narrower region.

As a conclusion we have shown that the bilinear interactions are not always necessary for occurrence of the staggered quadrupole phase in the antiferromagnetic BEG Model. 'Moreover the re-entrant phase transition is possible t o occur in a certain region of the anisotropic energy. This has been also confirmed by our simulations.

[l] Blume, M., Emery, V. J. and Griffiths, R. B.,

Phys.

Rev.

A 4 (1971) 1071.

[2] Lawrie, I. D. and Sarbach, S. in Phase Transition

and Critical Phenomena, Eds. (3. Domb and J. L. Lebowitz (Springer, New York) 1983 Vol. 9.

[3] Chen, H. H. and Levy, P. H., Phys. Rev. B 7

(1973) 4267.

[4] Wang, Y.-L. and Wentworth, C., J. Appl. Phys.