Multicritical behaviour of the antiferromagnetic spin-32 Blume-Capel model

Physica A 210 (1994) 415-423

Multicritical behaviour of the antiferromagnetic spin-; Blume-Cape1 model

A. Bakchich”‘“, S. BekhechiaSb, A. Benyoussef”

“Laboratoire de Magnttisme et Hautes l%aergies, Dkpartement de Physique, Fact&e des Sciences, B.P. 1014, Rabat, Maroc

bLaboratoire de Physique ThCorique, Dipartement de Physique, Facultk des Sciences, B.P. 1014, Rabat, Maroc

‘Groupe de Physique de la Matitre Condenske, DZpartement de Physique, Facultk des Sciences, B.P. 20, El-Jadida, Maroc

Received 14 March 1994

Abstract

The multicritical behaviour of the spin-$ king model on the square lattice with nearest-neighbour antiferromagnetic exchange interaction (J < 0), a crystal-field interaction (D) and an external magnetic field (H) is studied within mean field approximation (M.F.A.). The phase diagram exhibits a rich variety of behaviour: second order, first order and critical points of different order.

1. Introduction

The Blume-Cape1 model [1,2] and its generalisation, the Blume-Emery- Griffiths (B.E.G.) model or S = 1 Ising model, which presents a rich variety of critical and multicritical phenomena, have been extensively studied. The B.E.G. model was initially introduced in connection with phase separation and superfluid ordering He3-He4 mixtures 131. As discussed in Ref. [4], the strong interest in these models arises partly from the unusually rich phase-transition behaviour they display as their interaction parameters are varied, and partly from their many possible applications.

In most of the cases considered so far the bilinear interaction is ferromagnetic. In the antiferromagnetic case, the spin-l Ising systems are used to describe both the order-disorder transition and the crystallization of the binary alloy, and it was solved in the M.F.A. [5]. One of the most interesting and elusive features of the mean field phase diagram for the antiferromagnetic spin-l Blume-Cape1 model in an external magnetic field is the decomposition of a line of tricritical points into a

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416 A. Bakchich et al. I Physica A 210 (1994) 415-423

line of critical end points and one of double critical points. (A number of different

names are used in the literature to describe the “double critical point”, including

“double critical end point” or “bicritical end point” and “ordered critical point”

[6].) This model was also studied by transfer-matrix and Monte-Carlo finite-size-

scaling methods [7], but such decomposition does not occur in this two-dimension-

al model.

An extension of the B.E.G. model is the possibility of inclusion of higher spin

values. The spin-s B.E.G. model with dipolar and quadrupolar interactions was

introduced to explain phase transitions in DyVO, [8,9] and its phase diagram was

obtained within M.F.A. [lo]. Another spin-; model was later introduced to study

tricritical properties in ternary fluid mixtures [ll], which was also solved in the

M.F.A. Recently the phase transition in the spin-; B.E.G. model with nearest

neighbor interactions, both bilinear and biquadratic, and with a crystal-field

interaction has been studied within M.F.A. and Monte-Carlo simulation [12] and

by renormalisation group method [13].

Our aim in this paper is to extend the model [6] to the spin-5 where the spins at

sites i and j interact with a direct exchange J, < 0 and at each site exists a single

ion anisotropy of strength D and an external magnetic field H. The structure of

this paper is as follows: In Section 2 we give the Hamiltonian which defines the

model, discuss the ground state diagram and briefly present the M.F.A. In

Section 3, we present the results and discuss the different phase diagrams

obtained. In Section 4 we draw our final conclusion.

2. Ground state and mean field treatment

The model Hamiltonian is:

H = -J c S,S; + D c S’ - H c S, . (1) (i.j) ’ I I

Here the local spin variable can take the values ?$ and ? i. The first term

describes the antiferromagnetic coupling (J < 0) between the spins at sites i and j,

this interaction is restricted to the z nearest neighbour pairs of spins. The second

term describes the single ion anisotropy and the last term represents the effects of

an external magnetic field. The Hamiltonian and phase diagrams are invariant

under the transformation (H-t -H, S+ -S).

In order to calculate the ground state energy, we divide the lattice into two

equivalent sublattices a and b and express the Hamiltonian as a sum of the

contributions of the pairs of nearest neighbours. So, the contribution of a pair S,

and S, is:

E, = IJIS,S, +; (S; + St) - 5 (S, + S,) . (2)

A. Bakchich et al. I Physica A 210 (1994) 415-423 417

DAJI

HAJI

Fig. 1. Ground state phase diagram.

By comparing the values of E, for different configurations we obtain the ground state phase diagram (Fig. 1).

For Hl IJ~s 0, there are three ordered ground states with antiferromagnetic symmetry referring to the sublattice magnetization (M,, Mb), we denote these states by (5, -$), (5, -+) and (3, -t), respectively. The three uniform ground states are analogously termed as (5, 5), (+, 3) and (:, i).

To study the multicritical properties.of this Hamiltonian, we used the model, of two equivalent sublattices (a, b) and within the M.F.A. we have the self- consistent equations:

3e Ma=

-9pD’4 sinh[-$P(/JlzM, -H)] + eepDi4 sinh[-+P(]JjzM, -H)]

-G >

(3)

3eP M, = 9PD’4 sinh[-$P(IJlzM, -H)] + e-pD’4 sinh[-+/3(\JlzM, -H)]

G >

(4) i’

where M,, M, denote the aimantations of each sublattice. The free energy per site is given by:

F iJk N = - & log(z;z;) - - 2 MaM, > (5)

with:


-G = 2e-9pD’4 cosh[-$P(]J]zM, -H)] + 2e-pD’4 cosh[-+P((J]zM, -H)] ,

(6)

z; = 2e-%‘D/4 cosh[-:p(/J]zM, -H)] + 2e-PD’4 cosh[-$p(]J]zM, -H)] ,

(7)

where p = l/k,T is the inverse temperature and z is the coordination number. The self-consistent equations (3) and (4) are solved numerically. The solution

which minimizes the free energy (5) represents the stable equilibrium phase, and the others correspond to metastable states. If there are two solutions which have the same minimum free energie, these phases coexist and the system has a first order phase transition. We introduce the order parameter M which is defined as the difference between the two aimantations M, and M,. The disordered state corresponds to M = 0, while the ordered state corresponds to a non-zero value of M. We can now obtain the phase diagrams for the temperature as a function of DIIJI and HIIJI, which will be presented in the next section.

3. Results and discussions

Following Griffiths [ 141, we use the following symbols to denote the various entities on the phase diagram: A - one-phase point (disordered phase); A” - two- phase point (ordered phase); A3 -three-phase point (coexistence of ordered A” and disordered A phases); A” - four-phase point (coexistence of two ordered A’ phases); B - critical point; BA2 - critical end point (coexistence of critical B and ordered A’ phases); B2 - isolated critical point in the midst of an ordered phase (coexistence of two critical B phases), which we shall call double critical point; C - tricritical point.

(i) In the absence of the external magnetic field, H = 0, the behaviour of the antiferromagnet (Fig. 2a) is similar to the ferromagnet [12]. There is a second order transition line separating the disordered phase from the two antiferromagnetic phases (s, - $) and (+, - 3) which are separated by a first order transition line. This line terminates at an end point, as shown by the behaviour of the magnetization M (Fig. 2b), instead of the tricritical point obtained by Barreto et al. in the ferromagnetic case [12].

(ii) In the presence of the magnetic field, H # 0, the behaviour of the antiferromagnet is completely different from the ferromagnet. Typical phase diagrams in the (H/(./I, D/lJI) pl ane for various values of D/IJI are shown in Figs. 3-6.

For negative values of DI j.lI, DIIJI 6 0, the transition line between the antiferromagnetic phase (s, - +) and the disordered phase is always of second order (Fig. 3).

For 0 s D/IJI < 1.94, the disordered phase is separated from the three ordered


1.6

1

F 0.8 -

0.6

0.4

0.2

0

” 0.5 1 1.5 D&

2.5 3 3.5 4

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

D/IJi

Fig. 2. (a) Phase diagram for H/I./( = 0: solid line, critical points; dashed line, first order transition.

An end point occurs. (b) Behaviour of the order parameter IM( for TIJJ( = 1.1 and TI\J[ = 1.25,

showing the existence of an end point.


0 2 3 Ei?;, 5 6 7

Fig. 3. Phase diagram for D/lJl = -2.0: solid line, critical points.

phases (3, -$), (5, -i) and ($,+) by a 1 ine of critical points B (Fig. 4a). In Fig.

4b we show the sublattice magnetizations M,, M, as functions of HIIJI at finite

low temperature, and we find that there is no transition between these ordered

phases, results confirmed by using the elementary excitations and low tempera-

ture development.

For 1.94 G DIIJ( c 2.3, see Fig. 5, the disordered phase is separated from the

ordered phases by a line of critical points B and a first order line of A” points,

joining at a tricritical point C and at a critical end point BA’. The critical end

point BA* occurs at the intersection of a line of critical points B and a line of first

order transition points A” and A”. In the ordered phase there is a first order line

of A” points, separating the phases ($, -3), ($, 5) at high temperature and the

phases (5, -$), (C, - +) at low temperature, this line ends at a double critical

point B*. The double critical point B2 terminates a line of first order transition

points A” in the midst of the ordered phase A2.

As DI1.l is increased, DilJl > 2.3, the phase diagram is divided into two blocks

of critical points B separating the disordered phase from the ordered phases

(+, i) and (+, - +) at high and low magnetic field, respectively.

4. Conclusion

We have dealt with the phase transition of an antiferromagnetic spin-3 Blume-

Cape1 model with a bilinear exchange interaction (J < 0), a single-ion anisotropy


2

-1.5

-i

i

1

0 2 4 8' 10 12

/ / /

(b)

0 2 4 8 10 12

Fig. 4. (a) Phase diagram for D/(J\ = 1.90: solid line, critical points. (b) Plot of the magnetizations M, and M, for DIIJI = 1.9 and TIIJI = 0.13 showing the no transition between the ordered phases.

(D) and an external magnetic field (H). In this simple model we find that the phase diagram exhibits, in the absence of the magnetic field, a line of critical points separating the disordered phase from the two antiferromagnetic phases which are separated by a first order transition at low temperature and merge at


5 _, E-

2.5

2

1.5

1

0.5

0 0 2 4

H/;iJ, 8 10 12

Fig. 5. Phase diagram for DIjJ( = 1.98: solid lines, critical points; dashed lines, first order transition

Two double critical points B’, a critical end point BA’ and a tricritical point C occur.

1.2

0.8

0.2

0 0 2 4 6 i0 12 14 16

Fig. 6. Phase diagram for D//J1 = 4.0: solid line, critical points.


the end point. In the presence of the magnetic field, the phase diagram in the

WlJI~ WJI, ~$w P s ace consists of surfaces of critical and first order transition. These surfaces are variously bounded by an ordinary tricritical line, isolated lines of double critical points and a line of critical end points. The decomposition of the line of first order in the antiferromagnetic spin-5 model is qualitatively similar to that obtained for the antiferromagnetic spin-l Blume- Cape1 model [6].

References

[l] M. Blume, Phys. Rev. 141 (1966) 517. [2] H.W. Cape& Physica 32 (1966) 966; 33 (1967) 295. [3] M. Blume, V.J. Emery and R.B. Grifiths, Phys. Rev. B 4 (1971) 1071. [4] J.B. Collins, P.A. Rikvold and E.T. Gawlinski, Phys. Rev. B 38 (1988) 6741. [5] Y. Saito, J. Chem. Phys. 74 (1981) 713. [6] Y.L. Wang and K. Rauchwarger, Phys. Lett. A 59 (1976) 73. [7] J.D. Kimel, Per-Arne Rikvold and Y.L. Wang, Phys. Rev. B 45 (1992) 7237. [8] A.H. Cooke, D.M. Martin and M.R. Wells, J. Phys. (Paris) C 1 (1971) 488. [9] A.H. Cooke, D.M. Martin and M.R. Wells, Solid State Commun. 9 (1971) 519.

[lo] J. Sivardiire and M. Blume, Phys. Rev. B 5 (1972) 1126. [ll] S. Krinsky and D. Mukamel, Phys. Rev. B 11 (1975) 399. [12] F.C. Barreto, O.F. De Alcantara Bonfim, Physica A 172 (1991) 378. [13] A. Bakchich, A. Bassir and A. Benyoussef, Physica A 195 (1993) 188. [14] R.B. Griffiths, Phys. Rev. B 12 (1975) 345.

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Multicritical behaviour of the antiferromagnetic spin-32 Blume-Capel model