Journal of Magnetism and Magnetic Materials 60 (1986) 311-313 North-Holland, Amsterdam

311

ON THE REENTRANT BEHAVIOUR OF THE TWO-DIMENSIONAL ISING MODEL

Klaus ZIEGLER

Fachbereich Physik, Gesamthochschule Essen, 4300 Essen, Fed. Rep. Germany

Received 20 November 1985

The phase diagram of the two-dimensional Ising model with binary disorder is evaluated (in the region - 1~ J1/J, < 0) using a pertubative approach.

Classical spin models with a two-spin interac- tion have been introduced to describe the phase transition behaviour of ferro- or antiferromagnetic systems. The interaction energy of a system on a square lattice A is then represented by the Ham- iltonian

H= - c Jijsisi, i,jSA

where qj=O for Ii-j]+1 and Jii=.$. The coupling between two adjacent spin Si, Sj is called ferro- (antiferro-) magnetic if Jij > 0, since positive J prefer parallel (i.e. ferromagnetic) ordering of the spins for sufficiently low temperatures of the statistical model and vice versa. An interesting situation appears experimentally as well as theoret- ically if the sign of the couplings are chosen ran- domly. Then we expect a competetion of ferro- and antiferromagnetic domains on the lattice for temperatures below the critical temperatures of the pure systems.

Eu,Sr,_,S, for instance, is believed to be an example representing such a system [I]. Measure- ments on this material have shown that two differ- ent phase transitions occur at temperatures Tc,, T,, for certain concentrations p[2]. The phenomenon was explained as a reentrant behaviour of the phase boundary between ordered and disordered regions. To investigate this behaviour by means of the model defined in (1) we introduce a discrete distribution of random couplings { Jij} :

Prob(Jij=J,>O)=l-p,

Prob(J,,=J,<:O)=p. (2)

Now we consider the p-T-phase diagram for the model with temperature T. We observed that the coupling scales with the temperature in the Gibbs measure. Therefore, the limit T --, 0 depends only on the sign of Jlj providing it exists; i.e. we expect in general a discontinuous change of the proper- ties of the system when we go from J, < 0 (com- peting interaction) to J, = 0 (diluted system) or J1 > 0 (ferromagnetic interactions) at T = 0.

For the Ising model, this observation was com- bined in a recent paper with the assumption that the critical temperatures T,(p) depends continu- ously on the parameter

a = J,/J,

as long as T(p) is positive [3]. The authors con- clude then for a two-dimensional system from their knowledge of the phase boundary for a = 0 and a = - 1 a reentrant behaviour between these two values and, therefore, two phase transitions with a fixed concentration p.

We will present here the phase diagram in the discussed parameter region - 1 < a < 0 for the two-dimensional Ising model. Our approach is based on the belief that the critical temperature can be determined reliably by perturbative meth- ods.

0304-8853/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

312 K. Ziegler / Reentrant behaviour of the 2-D Ising model

Let us consider the average internal energy per Ising spin

where /3 is the inverse temperature. The critical temperatures of the pure model (p = 0, 1) are defined here as the singularities of U(T). Then we introduce narrow distributed disorder in (2) by choosing 1 J1 - Jo ) small. The disorder will change the location of the singularities and perhaps smoothes them out [4]. A pertubative method was developed in ref. [5] to study the first effect (the second effect is probably not observable using pertubative methods).

We describe it here briefly. The expansion parameter is

A = tanh /3Jo - tanh /3J, (4)

and singularities of the internal energy are related to the singularities of a Green’s function

gk,,k, = [ m2 + 2.dY2 - ‘1

x (cos k, + cos k, - 2)] -l,

Y(A) = (tanh@(A) Jb

m(A) = I- b0) - [y(A)]‘,

(5)

(6) at m, k,, k, =0 or m = 4(& - l), k,, k, = +T.

The A-dependence of m is important (“mass- renormalization”), since m = m(0) would generate expansion terms for U(T) which are divergent when m(0) + 0. However, the average internal en- ergy of the king model

]A]-’ C (+ C AjSisje-BH)J, i,jCA tsk)

with Z = c ewBH, tsk)

(7)

is always finite. Therefore, we obtain the expan- sion of m(A) in powers of A by the requirement that the divergent expansion terms of U(T) cancel against each other. Thus we find

m(A) = a,p(l -p)A’+ 0(A3),

a4 = 1.32, (8)

T@)/T(O)

L--.x2_ .2 4 p

Fig. 1. Phase diagram of the 2D Ising model with disordered

bonds. The curves show the phase boundary for a = J,/J, = -0.5, -0.3, -0.2 and -0.05 (from left to right). The arrows indicate T,(l)/T,(O) = 1 a 1, i.e. the ratio of the critical temper- atures of the antiferromagnetic and the ferromagnetic system

with coupling J1 and J,-,, respectively.

at the critical temperature T,(p). The evaluation of T,(p) was given explicitly in ref. [5] for a = - 1, 0 using (6) and (8). However, for other values of a is only a numerical solution of these equations available. Some results are shown in fig. 1. The phase diagram could be interpreted as a transition from a ferromagnetic to nonferromagnetic states. On the left-hand-side of the phase boundary exists an infinite cluster which is ferromagneticly ordered. Thermodynamic fluctuations and higher con- centrations of antiferromagnetic couplings destroy this cluster.

The effect of the latter decreases with decreas- ing temperature down to a certain temperature T = T,(a). But for temperatures below T,(a) the tendency of destroying the infinite ferromagnetic cluster increases surprisingly. To(a) is almost inde- pendent of Tc,antirerro(a) ( = - C,,,,) in our calculation: T,(a) 1 Tc,antifcrro(a) for a > -0.3 whereas T,(a) < Tc,antiferr,, for a < -0.3. Further- more, the system leaves the ferromagnetic region for a fixed concentration of antiferromagnetic bonds p at temperatures depending on a which are not obviously related to the critical tempera- tures of the antiferromagnetic clusters. Therefore,

K. Ziegler / Reentrant behaviour of the 2-D Ising model 313

the phenomenon is probably not caused only by the formation of large antiferromagnetic clusters as soon as T is below the critical temperature of a pure antiferromagnetic system with coupling J1. An alternative explanation is the occurrence of a new phase below the backbending part of the phase boundary which is neither ferro- nor para-

References

[l] P.J. Ford, Contemp. Phys. 23 (1982) 141. [2] H. Maletta, Lecture Notes in Physics, vol. 192 (Springer,

Berlin, Heidelberg, New York, 1983). [3] W.F. Wolff and J. Zittartz, Z. Phys. B 60 (1985) 185.

[4] K. Ziegler, J. Phys. A 18 (1985) L 801. [S] K. Ziegler, J. Magn. Magn. Mat. 45 (1984) 239.

magnetic.