Mixed spins Ising model on the honeycomb lattice

Journal of Magnetism and Magnetic Materials 104-107 (1992) 261-263 North-Holland

AI I=

Mixed spins Ising model on the honeycomb lattice * J.R. Gon~alves and L.L. Gon~alves Departamento de F(sica da UFCe, Campus do Pici, Caixa Postal 6030, Fortaleza, Ceard, Brazil This Ising model on the honeycomb lattice with spins of magnitudes ½ and 1 is studied. A random uniaxial anisotropy is considered, and the exact solution is obtained for binary annealed distributions of this anisotropy. The transition temperature is explicitly determined as a function of the anisotropy strength for two different distributions and it is shown that the system presents a reentrant behaviour for some range of values of the parameters. The study of the Ising model with mixed spins of magnitudes ~1 and s has attracted recently considerable attention. In particular the homogeneous model on the honeycomb lattice has been studied by Gon~alves [1] who was able to obtain an exact solution. For the random uniaxial anisotropy model approximate results, for s = 1, have been obtained by Kaneyosni [2] for quenched disorder and various two-dimensional lattices. In this paper we will consider this disordered model, also for s = 1, on the honeycomb lattice for a n n e a l e d distributions of the r a n d o m uniaxial anisotropy. In this case each set of spins occupies one of the equivalent sublattices of the honeycomb lattice (denoted by A A and A B) so that the nearest neighbours of a given spin belong to the other spin set. We will only consider binary distributions of the random uniaxial anisotropy such that the Hamiltonian is given by:

~=-

~., J % . r , (jl)

E I~AB

D, r 2 -

E

/'zSt,

(1)

/~AB

j ~ A A, l ~ A B

Dl(1 +51) D2(1 - - ~ / ) 2 + 2 '

(2)

with 5 = _+ 1. For the distribution of random uniaxial anisotropy given by

P ( D I ) = p 6 ( D , - D , ) + (1 - p ) a ( D t -

D2),

(3)

the pseudo-chemical potential /~ is determined from the equation 1+(5)

P = --5---

t

[zZG(3J' DI)+G(3J'

/3J'= aln / _ 2 7 , ,

[ Z (1(J,

D1) +

G(J,

D2)]

D2)

],

(S)

where z = e/3~'

fl = 1 / k B T ,

(6)

and 1

G(x, y) =

~

e ~ymz cosh(/3mx).

(7)

m=--I

It should be noted that J ' > 0 and is an even function of J. By a similar procedure we can show that ( Y ) is given by ( 5 ) = 3X1(O'1O'2)t +X2,

where ~r = + 1, r = O, + 1, and tz is the pseudo-chemical potential which controls the distribution of the random parameter D t which is given by

DI =

The partition function is obtained by considering a partial trace on the spins r and on the variables 5, so that we can map the model onto the spin-½ Ising model on the triangular lattice with an effective exchange constant J ' given by

(8)

where (~rlO-e)t denotes the nearest-neighbour correlation function of the effective Ising model on the triangular lattice, and x I and x 2 are given by

x, = V l / U l, x 2 = V 2 / U ) ,

(9)

with U~ = 2 [ G ( J , D 2 ) G ( 3 J , O l ) - G ( J , D 1 ) G ( 3 J , O 2 ) ] ,

II1 = G ( J , D 1 ) G ( 3 J , D 2 ) + G ( J ,

D2)G(3J , Dl)

_ e4C~g'G(g, D 1 ) G ( J , D2) _ e-4t~J'G(3J ' D 1 ) G ( 3 J , D2), Vz = 3 e 4 ~ J ' G ( j , D 1 ) G ( J , D2)

(4)

* Work partially financed by the Brazilian Agencies CNPq and Finep.

_ e-4t3J'G(3J ' D 1 ) G ( 3 J , D2) -G(J,

D a ) G ( 3 J , D2) - G ( J ,

0312-8853/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved

D e ) G ( 3 J , D1).

(lo)

J.R. and L.L. Gongah,es / Mixed spins Ising model on the honeycomb lattice

262

The transition temperature is obtained by considering the exact critical value o f / 3 J ' for the triangular lattice [3], namely exp(4/3Jc') = 3, and the critical value of the nearest neighbour correlation [3] (o'l~r2)t = ~, in eqs. (4), (5) and (7). From this, after eliminating the fugacity z, we obtain a final equation for the transition temperature which is given by

3G( J, D1)G( J, D2) + ( 4 p -

T'20

15

1 ) G ( J , D , ) G ( 3 J , D2)

IC

+ (3 - 4 p ) G ( J, D z ) G ( 3J, D , ) -G(3J,

D,)G(3J, D2)=0

(11)

_

We will consider two different distributions: one where D 2 = 0 and Dj = D , and one where Dj = D and D 2 = - D . In the first case eq. (11) is explicitly given by

~=-Z8

0.5

=- . ---

e/31)[(3 cosh [3J - cosh 3/3J)(cosh 3flJ+ cosh flJ) + ( 2 p + 1) cosh +(3

flJ-

I

I

2p) cosh~J+(2p-1)cosh3[

I

0.2

( 2 p - 1) cosh 3/3J]

I

[

0.4

I.

06

08

1.0

p

3J+l=O,

(l~)

Fig. 2. Transition temperature T* (T* =-kBT:,/J) as a function of p for the distribution P ( D ) = p ~ ( D I - D ) + ( I p)~(D) and various a(~ ~ D / J ) .

and in the second case by 1 + et~t~[(2p + 1) cosh /3J - ( 2 p - 1) cosh 3/3J] +e

t~t)[(3 - 2 p )

cosh / 3 J + ( 2 p -

1) cosh 3/3J]

+ (3 cosh /3J - cosh 3/3J)(cosh 3,/3J + cosh /3J ) = 0.

(t3)

,=oo

'4

,=o.2

i

In figs. 1 and 2 we present the solution of eq. (12), for the dilute case. From fig. 1 we conclude that for p > 0.5 a reentrant behaviour is present and this is a consequence of the competition between the exchange interaction and the uniaxial anisotropy. There is a critical ac (a =- D / J ) equal to - 3 . 0 such that for c~ <(~c the sytem does not present an ordered state.

1 p=O.O

1.6

p=l.O

p=0.8

/

-p=l.o 0 6

p=0.4

-6

-4

-2

0

I

I

I

2

4

6 0(:

Fig. 1. Transition temperature T* (T* =-kBTc/J) as a function of a(a = D / J ) for the distribution P(D t) = P6(Dt D) + ( l - p ) g ( D t) and various p.

-I0

-8

-6

-4

-2

0

2

4

6

8

I0 O(

Fig. 3. The same as in fig. 1 for the distribution P ( D ) = p 6 ( D / - D ) + ( 1 - p ) 6 ( D t + D).

J.R. and L.L. Gon~ahres / Mixed spins lsing model on the honeycomb lattice T * 2.0

I.=~

1.0

0.5

I

0

I

0.2

I

I

0.4

0.6

0.8

1.0

p

Fig. 4. The same as in fig. 2 for the distribution P ( D / ) = p 6 ( D t - D ) + ( 1 - p ) ~ ( D I + D).

263

function of p. T h e i m p o r t a n t point we conclude is t h a t we have two percolation thresholds, namely Plc = 0.5 for a < - 3 . 0 a n d P2c = 0.75 for c~ = - 3 . 0 . A r e e n t r a n t b e h a v i o u r is also p r e s e n t for a * < a < - 3.0, w h e r e a * can be d e t e r m i n e d by imposing on eq. (12) the condition t h a t for this a t h e r e are two d e g e n e r a t e roots. In figs. 3 and 4 we p r e s e n t the solution of eq. (13). As we can see, the results are essentially the same as in the previous case. As excepted, we have the same percolation thresholds a n d critical c~, since they are universal characteristics of the system. R e e n t r a n t behaviour is p r e s e n t a n d the main new feature is the symmetry property a -~ - a , p ~ 1 - p which is a consequence of the choice of the p a r a m e t e r s . In passing we would like to m e n t i o n that eqs. (12) and (13) reduce to the o n e o b t a i n e d for the homogen e o u s model [1], a n d that the r e e n t r a n t b e h a v i o u r has also b e e n found for q u e n c h e d distributions of the r a n d o m uniaxial anisotropy [2]. It also should be n o t e d t h a t for p = 1, D large and positive, the system behaves as an s = ~I system.

References F o r p = 0.5, a c is equal to - 5.0, a n d for p < 0.5 t h e r e will always be an o r d e r e d state irrespective of the value of a. Fig. 2 shows the transition t e m p e r a t u r e as a

[1] L.L. Gon~alves, Phys. Scripta 32 (1985) 248, 33 (1986) 192. [2] T. Kaneyoshi, Physica A 153 (1983) 556. [3] J. Stephenson, J. Math. Phys. 5 (1964) 1009.