Metastable states in the random antiferromagnetic ising chain

UA 4

Physica 108B (1981) 1325-1326 North-Holland Publishing Company

METASTABLE STATES IN THE RANDOM ANTIFERROMAGNETIC

ISING CHAIN

Jorge V. Jos6 Northeastern U., Physics Department,

Boston, MA 02115

Michael J. Mehl Physics Department,

Rutgers U., Piscataway, N.J. 08854

Jeffrey B. Sokoloff Northeastern U., Physics Department,

Boston, MA 02115

We study the kinetic Ising model with random antiferromagnetic exchange, in one dimension. The probability law is suggested on physical grounds to be, P(J)~J-~(O
INTRODUCTION + b(~)

In recent years a great amount of effort has been directed at understanding the physical properties of random magnetic systems. Among the different topics of interest in the field, the Spin-Glass problem has attracted ~ iot of attention since Edwards and Anderson ( ) suggested the possibility of a phase transition for models with competing interactions. Experimentally, spin-glasses have the distinctive signature of showing remanence in their dynamic properties. This remanence in many cases shows a logarithmic type decay of, say, the magnetization, M r ( 2 ) . Different theoretical models have been suggested that have a non-exponential decay of the magnetization. The models that have been studied most extensively are Ising models with a Gaussian distribution of exchanges. These models show a remanent M r for d=2, 3 (3) and d=l (4) . The appearance of remanence in these models is related to the formation of metaestable clusters that take a long time to decay to their ground state configurations. Here we give a brief summary of our extensive Monte-Carlo and continued fraction studies of Glauber models with random exchange. In this paper we concentrate on the antiferromagnetic case with exchange obeying the probability law, P(J) = (l-~)/J ~

(I)

This probability law is obtained by assuming randomly located magnetic moments with exponentially decaying interactions. We take Jo=l. We find that there is a weak remanent magnetization in this model that deoends strongly on the initial conditions and the fact that J has a continuous distribution. Specifically, we obtain

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(Tint) 3(i-~)]

(2)

in the low temperature limit. M 0 = 0, if the initial state is totally magnetized and M 0 # 0 otherwise, a = i and b = 0 for ~ = O, whereas a = i and b = 2/3 for ~ = 0.5. The explicit form of these coefficients is complicated and will appear elsewhere. 5 This result is fully consistent with our MC simulations, of which a sample is given in Fig. i. 015

'

'

'

o.~o

IltlIIIltllIlItIt °°°.o

I

I

I

I

5.0

~oo

~5o

300

TiME (M.C.S.)

Figure i: Remanent magnetization for a i001 spin lattice, with ~=0.3 and temperature of 0.01 (J=l). The initial magnetizations at t=O are, Min = 0.506 ( ~ ) , Min = 0.504 (A) and Min = 0 . 4 9 7 (0). The runs were carried out for 20 samples (~80,000 spins), and the bars denote one standard deviation.

II. THE MODEL We consider the kinetic Ising model introduced by Glauber with random J's. (6) The equation of motion reads

1325

1326 qi = - qi + F i+ qi+l + r i- qi-i

(3)

Here qi is the thermally averaged magnetization at site i, for a given configuration of {Ji }. The F-functions are defined from the detailed balance condition and are given by Fi+ = ~[itanh(Ji+Ji_l)/T

_+ tanh(Ji-Ji_l)/T]

THE T-CLUSTERS

In the limit when T ÷ O, we notice that the F's can take values O, -i, or $1/2. In__particular, if Ji < Ji+l''" < Ji+n' (Fi+e =0' F~e=-l),~. or if Ji > Ji+l''"

> Ji+n'

(r~+e=-l'

F~+e=O)'

whereas when Ji = Ji+l = "'" = Ji+n' then r+ = r- = -1/2. the r-clusters.

This fact lead us to define We call F 1 - clusters those

with Ji+e-1 < Ji+e' F2- clusters

M(t) = ~

1

j o~+6 st f ds Z Gik(s)qi(0)e -j~+~ i,k

(4)

The result given in Eq. (2) depends strongly on the form of the T-functions. The model is completely defined by Eqs. (i), (3) and (4). Similar problems have ~ n studied extensively in the last few yearslV'Our model differs from theirs in that we have asymmetric off-diagonal terms and we do not have correlated diagonal and off-diagonal disorder. III.

Here we just give some of the highlights of the calculations and ali the details can be found in ref. 5. The magnetization is obtained from

where j = /-I and Gik is the Green's function corresponding to Equation (3). from the continued fraction expression for G we find that the denominator has the structure (i + s) - B/[(I + s) - BI/...] with the B's equal to a product of F+ and r- . If i and i _ i+l • i+l are within the same cluster, their product is very close to zero, and the pole is close to s = -I, which leads to the exponential decay for M(t). However, if i = i~1 - 1 and i + 1 = n 1 when going from a F 2 to a rl cluster, the quenched magnetization can be found to be given by fl x_~ ]2(l-el M(t) ~ (i-~) dx [x - ( T l n t ) Tlnt which gives the result of Eq. (2) valid for T l n t < I. A full discussion of these results, plus a comparison with random ferromagnets, spin-glasses, and experiments Qn(TCNQ) 2 can be found in reference 5.

if Ji+e_l>Ji+e ,

and F 3- clusters when Ji+e_l=Ji+e , with O
Acknowledgements. J.V.J. thanks the Northeastern Research and Development Fund for support. M.J.M. has been supported by NSF Grant-DMR-80-07470 and J.B.S. by NSF Grant-DMR-79-06371 and D.0.E grant

References where j = /-1 and G., is the Green's function correspondlng to Eq. (3). Here q. is the Laplace transformed qi(t) and {qi~O)} gives the initial conditions. •

I K

~

.

The full solution with the cluster can be obtained by iterating Eq. (3) for n 2 < i < n . The important point in Eq. (5) is its pole 1 structure. Notice that if we start with {qi(0)} = i, the pole at s = 0 cancels and we have only the pole at s = -2, which leads to exponential decay for qi(t) as t ~ ~. When considering the F 3- clusters we do not find the s = 0 pole and therefore the remanent is zero. Notice, however, that with a continuous probability like the one ~iven in Eq. (4), the likelihood of F 3 clusters is much smaller than that of F~i or F 2 clusters.

IV.

EQUATION OF MOTION ANALYSIS

To further the study of the F-clusters at finite temperatures we have used a continued fraction method to calculate the total magnetization.

[i] [2] [3]

[4] [5] [6]

S.F. Edwards and P.W. Anderson, J. Phys. F5, 965 (1975). J.L. Tholence and R. Tournier, Physique Colloque 5 C4-229 (1974). See for instance the review by K. Binder, in J. Physique Colloq. 39, C6-1527, 1978. J. Jose, M. Mehl and J.B. Sokoloff, (to be published). R. J. Glauber, J. Math. Phys. 4, 294 (1963). S. Alexander, J. Bernasconi, and R. Orbach, Phys. Rev. Lett. 41, 185 (1978).