Disordered magnetic systems in two dimensions

Journal

of Magnetism

and Magnetic

Materials

96 (1991) 77-81

77

North-Holland

Disordered magnetic systems in two dimensions K. Ziegler Institut

ftir Theorie der Kondensierten

Received

5 August

Materie,

Universitiit Karlsmhe,

1990; in revised form 1 October

Physikhochhaus,

W-7500 Karlsruhe,

Germany

1990

Recent theoretical results for the two-dimensional Ising model with random bonds are applied to dilute and mixed magnetic systems, The rounding effect on the quenched specific heat peak, observed in experiments with strong disorder, is explained by the the appearance of a new thermodynamic phase between the ferro- and paramagnetic phase. The relations of this result with computer simulations are discussed. In particular, finite-size effects on the disorder-induced phase are estimated.

1. Introduction In this article we want to discuss some recent developments in the theory of the two-dimensional Ising spin systems with random bonds. In particular, we will comment on recent extensive computer simulations [l] and give a perspective of related experiments. Magnetic alloys or dilute magnetic systems have been an object of investigation for several decades. Among other questions the effect of disorder (due to impurities) on the transition from the magnetically-ordered phase to the paramagnetic phase has been studied [2-41. Another question is the generation of new thermodynamical phases due to disorder (e.g. spin glass phase). Concerning the first equation, a number of experiments indicate a strong effect on the specific heat peak at the critical point at least when disorder is strong [2,3]. On the other hand, the specific heat peak does not seem to be affected when disorder is weak [3]. There is no clear observation for the generation of a new phase between the magnetically ordered and the paramagnetic phase. Beside these experiments there have been extensive numerical simulations using, e.g., Monte Carlo methods [l] or transfer matrix calculations [6]. These investigations of the two-dimensional Ising model indicate at most a weakening of the loga0304-8853/91/$03.50

0 1991 - Elsevier Science Publishers

rithmic singularity of the specific heat to a double logarithmic singularity but not a rounding of the specific heat as observed in several experiments. We will concentrate the following discussion on the thermodynamic properties of the Ising model (i.e. the specific heat) and not on the magnetization or spin-spin correlation functions. From the theoretical side there are also a number of investigations mainly based on perturbation theory, self-consistent methods [7] (like CPA) and renormalization group calculations [8]. The twodimensional Ising model is a good candidate with which to study the effect of disorder, since the model can be solved exactly in the absence of disorder (i.e. no random bonds). Thus, it seems to be natural to expand the model in powers of the random fluctuations of the bonds. Such an expansion is infrared divergent near the critical point of the pure system. Therefore, one has to apply a renormalization procedure in order to control these singularities. This renormalization has been performed by several groups [8-lo]. All these perturbative approaches lead to a single logarithmic, i.e. it is weakened but not rounded. This result is in contradiction to the experimental observation at least for moderate disorder. In order to understand this discrepancy we have studied the two-dimensional Ising model with random bonds by

B.V. (North-Holland)

78

K. Ziegler / Disordered magnetic systems rn IWODimensions

using a different perturbation theory [ll]. The idea is not to restrict the randomness to a very weak one because such a restriction could cover some important properties if the model is nonanalytic in the perturbation by disorder (which turns out to be indeed the case). For this purpose we have constructed a replica model. The idea for this construction is based on the fact that there is a fermion representation of the Ising model on the square lattice [12]. The fermion model can be generalized by introducing N different colors for the fermions; i.e., we replicate the original model N times. Then we introduce random bonds which also couple the different replicas. Although there is not an obvious interpretation for the interaction of the different colors in terms of the Ising spins, we can study the limit N --f cc and a l/N-expansion. The limit N + cc can be solved. There is no indication from the l/N-expansion [13] that the case N = 1 is qualitatively different from that with a large value of N. Therefore, we may apply the l/N expansion in order to get results which are relevant for the model with N = 1. Fortunately, this expansion is not plagued with infrared singularities except for the new critical temperatures at T = T,. These temperatures mark two phase boundaries which enclose a new phase between the magnetic and the paramagnetic phase. Thus the critical temperature T, of the pure system is split into two new critical points T+. The results of refs. [g-10] can also be understood from this model: the perturbation with respect to disorder is correct if disorder is not too large compared to the distance from the critical point T, of the pure system (an explicit bound has been given in refs. [ 11,131). In the opposite case, where disorder is strong compared to the distance from T,, the perturbation theory of refs. [g-10] is not valid. This is related to the fact that the limits T + T, and vanishing disorder cannot be interchanged. In the following we will evaluate the quenched specific heat for two types of disorder: the dilute system and the antiferro-ferromagnetically mixed system. Our calculation is based on the results found from the replica model in refs. [11,13]. In particular, the width of a new thermodynamic phase between the ferro- and paramagnetic phase is calculated explicitly. Furthermore, in section 3

we estimate the finite-size effects on the existence of this new phase of the two-dimensional random bond Ising model.

2. Quenched

specific heat

The Ising model describes the statistics of discrete classical spins {S,.} with the two orientations S, = +l. It is assumed that the spins are on a square lattice A where nearest neighbouting spins interact with the energy Z=

-

c J,,,S,.S/. r,r’t,\

(1)

The coupling J,.,., is zero if Y. r’ are not nearest neighbours and random otherwise. In order to discuss magnetic systems we will distinguish two different situations. First, a dilute magnetic system, where the impurities are non-magnetic atoms. can be described by a distribution of J,,, as Prob[(J,,,)

=O] =p,

Prob[( J,.,,) = J] = 1 -p.

(2)

i.e., p is the concentration of non-magnetic atoms. Secondly, there are magnetic alloys, where two different kinds of magnetic atoms are mixed. This can be described, for instance, by introducing ferro- and antiferromagnetic couplings in the Ising model : Prob[(J,.,,)

= -J]

=p.

Prob[(J,,,)

= J] = 1 -p.

(3)

Under the assumption that there is a unique critical point T,(p) for the quenched system, the phase diagram of both cases can be evaluated by a self-consistent method [14]. Thermodynamic properties of the Ising spin system at the inverse temperature /3 = l/R,T can be evaluated from the quenched energy per spin

where 1A 1 is the volume of the lattice A and ( . . . )J is the average with respect to the distribution of the couplings { Jr,, }. In particular, we

K. Ziegler / Disordered magnetic systems in two dimensions

obtain the specific heat by differentiation respect to the temperature as (w/

with

a = ~WJ.

(ii) is stable for n2 > 0 and vice versa. With these findings we can calculate the quenched energy as

(9

with the following omitted): g = 4( (Y$)

(c?)~ the result of the replica in the Introduction. We will of calculations here but refer [11,13]. There we obtained in with

-‘log[(m+mJ2]

Iml >mo,

-i-

Iml Imo,

I

2

(QJZ

Now we can use for model we mentioned not give the details to the original work thelimit N+cc

mTyc

04)

2e-“y:/g

2e-0.54’R. The quenched specific heat reads according to eq. (5) as

quantities

(the index

m.

-

I

(7) - l),

dk, dk,,

(10)

K

=yC(i sin k, - sin k2), (11)

&=a-1. The quantities the equations

m,

and n in (10) are determined by

17= rig-l,,

(12a)

ms= -ml+gI.

gI

(12’4

:<2,=(i) (ii)

-2ny,2/g

q2 - e

withgI=l*m,=--.

_

gI

n

m

2

epTY2/g,

(16)

1 1+gz

Il.

(17)

W) W)

4

*2

where the qualitative behaviour of the specific heat changes. There is an order parameter 77which characterizes the new phase between the ferromagnetic phase for 0 < m < m, and the paramagnetic phase for m-c m -c0. An interpretation of this order parameter can be given in terms of dynamical properties of the Ising models [13]. The multiplicative ‘renormalization’ m + m, = mZ, of the temperature dependent term m is restricted by

Eq. (12a) has two solutions: ~=O=Im,=-ml+gI,

lmllm,.

In the limit g --, 0 we recover the well-known logarithmic singularity of the pure two-dimensional Ising model at the critical point m = 0. If disorder is present (g > 0), however, this divergence is cut-off. Moreover, there are two critical points

m+/- -

with

$

(15)

(8)

(9)

+]K]~]-~

z

J is

Y,,,’ = ta~(~J,,,~L

- (~4)~)~

m = (Y) - (fi

19

(13b)

The validity of the solutions has been determined by a stability analysis in the l/N-expansion [13].

For small g and sufficiently large m (i.e. gZ small), the renormalization factor is Z, = 1 such that we obtain the result of the pure Ising model. All the results of the replica model have been derived for weak disorder, i.e., g is small. Now we can use the distributions of y given in (2) and (3) to evaluate m and g in the specific

K. Ziegler

80

/ Disordered

magnerrc systems in two drmensions

where g was obtained for m = 0. For the mixed magnetic system one obtains

1.10

m=(l-2p) 0.90 g_o

0

F \0.70 z vO.60

0.40

m.

OXJ 0.00

0.10

0.20

IJ

Fig. 1. The critical point T, (p)/T,(O) as a function of the impurity concentration p. The splitting is less than 10m4 for the dilute (upper curve) system (see eq. (16)).

heat of (15). One system m=(l-p)

finds

for the dilute

PC1 -p)

(21)

(l-2p)"

Then we can evaluate the critical temperatures T,(p), i.e., the temperatures with m(T+) = m *. The graphs of m( ri) are shown in fig. 1. We notice that the splitting 1T, - T_ 1 is extremely small for the dilute system such that it is not visible in the graph. Furthermore, we can evaluate the quenched specific heat according to eq. (15). For 1m ( > m,, the specific heat grows logarithmically as in the pure system. However, for 1m 1 < m,, it is cut-off (i.e., there is a maximum of the quenched specific heat) with some smoothing. The value of the cut-off is shown in fig. 2 for both systems.

magnetic

(18)

P _p

(191

80.00

Since we are interested in comparing the results given in the previous section with those obtained by Monte Carlo simulations [l], we must estimate the effect of a finite lattice. According to eq. (16) the new phase is very narrow because we consider weak disorder where g is small. The new phase is characterized by a solution 17# 0 of eq. (12a). Therefore, it is necessary that the integral I in (10) satisfies

I(

a z

m = q = 0) 2

l/g.

(22)

For a finite lattice the integral goes over to a sum. However, for small g only the contributions from small wave vectors k, are relevant for the evaluation of I. Therefore, we may approximate this sum by an integral with a cut-off for small wavevectors. Suppose, we consider an LX L lattice. Eq. (22) then reads approximately

60.00 :: E 40.00

20.00

0.00

(20)

3. Finite size effects

tanh(PJ)-(a-l),

g=o.0861

_

34

.

-0.80

tanh(PJ)-(a--l).

?,...,,,,,,,.,...,.,,,,,,,,,,,,,,,,,,,,,, 0.00 0.10

0.20

0.30

0.40

sin’ k, + sin’ k2)-’

dk,

dk,

P Fig. 2. The maximum of the quenched specific heat as a function of the impurity concentration p (in arbitrary units). The upper (lower) curve is for the dilute (mixed) systems.

(23)

K. Ziegler / Disordered magnetic systems in two dimensions

which implies for L za 1: Llog TY,'

L2$.

(24)

Thus we will not see the new phase if the linear length of the system is smaller than L, with L, Z

eTY2/n z eo.wg_ (25)

When we assume that our approximations are valid for g < 0.1 then finite size effects are irrelevant for L 2 Lo = 220. Systems larger than this have been studied by Monte Carlo simulations (L I 600 in ref. [l]) such that it should be possible to detect the new phase by simulations. In the transfer matrix calculations of ref. [6] the width of the system is 10. In this case we cannot expect to find deviations from the behaviour of the pure system. From these observations we believe that high accuracy experiments would provide a very reliable test for the results of the specific heat singularity.

4. Conclusion We have evaluated the quenched specific heat of the two-dimensional random bond Ising model. This calculation was based on results derived from the replica model of ref. [ll]. Here we considered two special cases of random bonds which present a dilute magnetic system and a ferromagnetic system with antiferromagnetic impurities. Our results may explain the experimental observation that the specific heat peak at a critical temperature seems to be rounded only if the disorder is strong; the rounding effect appears only on a temperature interval AT = 1T, - T_ 1 - T,c,exp( - c/p) where p is the concentration of impurities, T, the critical temperature of the pure system and co, ci are some numbers depending on the type of dis-

81

order (see section 2). The rounding is related to the existence of a new phase between the ferroand the paramagnetic phase. This phase is characterized by a slow dynamics [13] and has no evident static interpretation. Since the width of this phase is very small, it is difficult to observe it in an experiment. On the other hand, even the rounding of the specific heat has not been observed in computer simulations. This can be explained by the fact that the new phase (and therefore the rounding) is completely covered by finite size effects: if the system is not very large only the logarithmic specific heat of the pure Ising model or the double logarithmic cross-over [S-lo] is indicated. Therefore, experiments on real materials would be an interesting alternative to computer simulations for studying the effect of disorder in low-dimensional magnetic systems.

References PI J.S.

Wang, W. Selke, V1.S. Dotsenko and V.B. Andreichenko, Europhys. Lett. 11 (1990) 301; Physica A 164 (1990) 221. 121K. Yamada, Y. Ishikawa, Y. Endoh and T. Masumoto, Solid State Connnun. 16 (1975) 1335. 131 M. Suzuki and H. Ikeda, J. Phys. C 11 (1978) 3679. 141 H. Ikeda, I. Hatta and M. Tanaka, J. Phys. Sot. Japan 40 (1976) 334. PI H. Ikeda, T. Abe and I. Hatta, J. Phys. Sot. Japan 50 (1981) 1488. [61 B. Derrida, B.W. Southern and D. Stauffer, J. de Phys. 48 (1987) 335. [71 J.A. Blackman and J. Poulter, J. Phys. C 17 (1984) 107. VI V. Dotsenko and V. Dotsenko, Adv. Phys. 32 (1983) 129. [91 G. Jug, Phys. Rev. Lett. 53 (1984) 9. WI R. Shanker, Phys. Rev. Lett. 58 (1987) 2466. 1111 K. Ziegler, Europhys. Lett. (in press). WI T. Schultz, D. Mattis and E. Lieb, Rev. Mod. Phys. 36 (1964) 856. u31 K. Ziegler, Nucl. Phys. B344 (1990) 499. u41 K. Ziegler, J. Magn. Magn. Mat. 38 (1984) 239.