The Ising spin glass phase transitions and phase space geometry

The Ising spin glass phase transitions and phase space geometry

322 Journal of Magnetism and Magnetic Materials 90 & 91 (1990) 322-325 N orth-Holland Invited paper The Ising spin glass phase transitions and phas...

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322

Journal of Magnetism and Magnetic Materials 90 & 91 (1990) 322-325 N orth-Holland

Invited paper

The Ising spin glass phase transitions and phase space geometry LA. Campbell a

b

a

and L. de Arcangelis

b

Physique des Solides, Unicersite Paris Sud, 91405 Orsay, France SPilT, CEN Saclay, 91191 Gil sur Ycette, France

The 3D ± J Ising spin glass has been shown in massive simulations by Ogielski to have strongly non-exponential relaxation between Tg and the ferromagnetic ordering temperature To. "Damage spreading" numerical data by Derrida and collaborators have indicated anomalous behaviour.in a similar temperature range. The results from the two different types of simulation are compared and discussed in terms of the geometry of the phase space of the system; conclusions are drawn on the phase transitions in spin glasses.

We want to discuss simulation results on the 3D ± J Ising spin glass (lSG) in order to see what physical picture can be obtained from them and to what extent the picture can be extended to other systems so as to throw light on phase transitions in disordered materials in general. The most massive numerical simulations on this problem were performed by Ogielski [1] on a taylor made computer. He obtained two main classes of results: on the relaxation and on the critical exponents. In the relaxation measurements Ogielski observed the evolution in time of q, the average distance in phase space between an initial configuration A(O) of the spins and the configuration A(t) after a relaxation time t. This distance is equal to the fraction of spins which are differently oriented in the two configurations. The individual spins were allowed to relax at each time step according to Glauber dynamics. The results were parametrised using the phenomenological expression

where A, x, T and f3 depend on the temperature (we will quote all temperatures in units of J). Ogielski defined the spin glass temperature Tg by the divergence of T at T= 1.18, and found that f3 = 1 (i.e. long time relaxation becomes exponential) above the Griffiths temperature [2], To = 4.51, which is the temperature at which the equivalent ferromagnet has its Curie point. The non-exponential form of the relaxation in a spin glass above Tg and the special behaviour related to the Griffiths temperature have been widely discussed without clear predictions leading to the type of behaviour observed by Ogielski [3]. A quite independent approach to the ISG was initiated by Derrida and coworkers [4] and by De Arcangelis et al. [5] who exploited the possibilities of the "damage spreading" method [6]. Suppose an ISG with a given set of interactions; starting from two

different initial configurations of the spins A(O) and B(O), one imposes identical dynamics. At any time step the same set of spins i are chosen in the two configurations and the same random numbers x are used in the heat bath dynamics. At sufficiently high temperatures where the molecular field is small compared to kT, after and SiB each time step the two individual spin states will tend to become the same, so after a few time steps per spin the two configurations A(t) and B(t) will have become identical - each spin i in B will have the same orientation as in A. As the temperature is lowered, this is no longer necessarily true. It remains true only so long as the accessible phase space remains" simple"; it ceases to be true for instance below ~ in a ferromagnet where the phase space splits into two, the spin up set of configurations and the spin down set of configurations [4]. Derrida et al. [4] studied the 3D ISG with three choices of initial conditions

st

(a) A(O) and B(O) chosen independently and at random, (b) B(O) the mirror image of A(O) (all spins inverted), (c) B(O) differs from A(O) by one single inverted spin. They measured the distance D(t) between the configurations A(t) and B(t) after a long time t and found the following remarkable behaviour. Above T"" 4, D(t) fell to zero. Below T"" 4 and whatever the initial conditions (a, b or c), D(t) remained at a temperature dependent non-zero value, D(T). Below T"" 1.8 the values of D(T) began to depend on the initial conditions. As pointed out by De Arcangelis et al. [5], these results imply fundamental changes in the phase space structure of the ISG near these two temperatures. However, the two critical temperatures given by these data are tantalizingly close to but not the same as the To and Tg of Ogilski's simulations. We suggest that this is a technical problem related to the choice of time scale on which D(t) is measured in the simulation. If the time scale chosen is too short, the

0304-8853/90/$03,50 V 1990 - Elsevier Science Publishers B.V. (North-Holland) and Yamada Science Foundation

J.A. Campbell. L de Arcangelis / Ising spin glass phase transitions

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2

3

4

T Fig. 1. D(T) as defined in the text as a function of temperature. The curve is extrapolated to low temperatures (see discussion); below T,. the three branches are schema tic indications of the expected behaviour correspond ing to the three initial conditions. Dat a points are from refs. [4,5] and from present result s.

system will not be able to adequately sample the available phase space; on the other hand if the time scale chosen is too long, all parameters including D(l) will average to zero because one is dealing with a finite sized sample of spins . The criteria "too long" and "too short" will depend on the natural relaxation rate of the system , hence on the temperature. Derrida et al. chose to use a fixed time at all temperatures for the measure-

ment of the distance D(l) between A(t) and B(l). We have repeated the same type of simulation as their s but with a slightly modified protocol. First, following De Arcangelis et al. (5) the system was annealed at temperature T to produce the initial configurations A(O) and B(O). Then the "damage spreading" procedure was followed at the same T with a temperature dependent time scale obtained by .consulting Ogielski 's results [1). The

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I.A. Campbell, L de Ar cangelis / Ising spill glass phase transitions

"measuring time" at each T was chosen as that for which q(t) became negligible (less than 10- 3 ) in Ogielski's data. This time goes from about 10 MCS (Monte Carlo Steps per spin) at T = 5 to about 5000 MCS at T = 1.8 and diverges rapidly at lower T. This criterion gives an estimate of the value of D(T) at a time such that phase space has been fully explored (this is what q(t) z 0 means for a system for which q(cc) is zero) but the system is not overannealed. The curve for D(T) defined in this way is in excellent agreement with the previous work for the intermediate range of temperature where our time scales are reasonably close to those used before; our D(T) however tends to zero close to TG (rather than at T z 4) in satisfactory agreement with Ogielski's data. In the earlier calculations [4,5] it was found that for temperatures below T z 1.8 the values of D(T) depended on the starting conditions (a, b or c). We think that this is due to an insufficient time scale (time scales of 500 MCS were used, while on our criterion 5000 MCS are necessary by this temperature). We surmise that if appropriately long time scales were used the values of D(T) would become independent of the starting conditions for temperatures right down to Tl'; In practice, at temperatures below T= 2 the computer time imposed by the present protocol made calculations prohibitively long; however we can extrapolate the D(T) curve from points above this temperature. D(T) extrapolated in this way intersects the line D = 0.5 at a temperature very close to Tg, fig. 1. If we now plot D(T) from fig. 1 against f3(T) values from Ogielski, we get a respectable straight line, fig. 2. This is remarkable as the values a re obtained for two completely different parameters from two quite distinct simulations. We will now attempt to interpret these results using . the phase space approach already outlined in ref. [7J. The phase space of an N spin Ising system is an N dimensional hypercube; the available space at temperature T consists of those of the 2N possible configurations which have energy E z VeT). In ref. [7] it was suggested by analogy with the percolation transition in a randomly occupied hypercube that the phase space of an lSG was simple or "compact" at high temperatures, became "complex" but still fully connected (a giant spanning cluster) below a first transition (identified with TG ) and before finally splitting up into disconnected clusters of configurations for temperatures below Tg • The curve for D(T) shown in fig. 1 follows just the pattern expected on this picture. In terms of the familiar "valleys in phase space" language, the fact that D(T) is zero at T above TG means that all configurations are in the same simple valley so that A(t) and B(t) rapidly collide in the "damage spreading" procedure. For T between TG and Tg, D(T) goes to the same non-zero

value whether initially A(O) and B(O) are close together in phase space (starting condition c) or far apart (conditions a and b); this implies th at all configurations occupied at temperature T lie in one complex valley [8] where complex means maze like so that the two configurations take an extremely long time (possibly infinite in the thermodynamic limit) to collide under the damage spreading procedure. The value of D(T) is a measure of the " complexity" ; the maximum po ssible value of D for a complex spanning cluster is 0.5 (no correlation between A(t) and B(t». As the temperature is lowered, complexity increases; the condition D(Tg) = 0.5 suggested by the results in fig. 1 means th at Tg corresponds physically to the lowest temperature at which the spanning cluster can exist, and below this the thermodynamically available phase space must shatter into small clusters [9]. On a purely practical level, these results suggest that simulation data giving D(T) might well provide a reliable and reasonably economical method of estimating Tg and TG in other Ising spin glasses, with dimensions other than 3, distributions of interactions, etc. The simple relation, fig. 2, between f3 and D implies that the form of the relaxation of a single configuration q(t) reflects directly the complexity of the accessible phase space. The "damage spreading" procedure used in the simulations is very instructive and gives vital insight into the phase space structure, but it would seem to have no conceivable experimental analogue; f3 on the other hand is a parameter whieh can be extracted from experimental data and its value could be considered a direct observational measure of the complexity of real life systems. In the preceding discussion we have made a number of assertions which lack rigour; in particular we have not said clearly what "complexity" means. \\le imagine this as a concept which would play an analogous role in a closed space (such as the hypercube) to that played by fractality in Euclidean space, but we do not know how to define it. Nevertheless, we feel that the results summarised above provide us with a concrete physical model for the 3D ISG . If we define a phase transition as a qualitative chan ge in the morphology of available phase space with temperature, then the lSG presents intrinsically two successive transitions, at TG and Tg. The first is a change from a simple to a complex phase space structure and its signature is the onset of non-exponential relaxation; the second is the sha ttering of the complex spanning cluster of configurations and is the conventional spin glass transition. . This picture may well be generalizable; the 3D ±J ISG lend s itself admirably to computer simulations which would be much less easy to carry out in other cases, but we suggest that the overall picture of the

LA . Campbell. L de Arcangelis / Ising spin glass phase transitions

sequence of transitions in phase space obtained in this special case could be applicable mutatis mutandi to other complex systems such as structural glasses [10]. References [1) A.T. Ogielski, Phys. Rev. B 32 (1986) 7384.

(2) R.D. Griffiths, Phys. Rev. Lett. 23 (1969) 17. (3) A.l . Bray, Phys. Rev. Lett. 60 (1988) 720. [4) B. Derrida and G. Weisbuch , Europhys. Leu . 4 (1987) 657. A.U. Neumann and B. Derrida, 1. de Phys. 49 (1988) C8·1647. [5) L. de Arcangelis, A. Coniglio and H.l. Herrmann, Europhys . Lett. 9 (1989) 749. (6) H.E. Stanley, D. Stauffer, D. Kertesz and H.l . Herrmann, Phys. Rev. Lett. 59 (1987) 2326.

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(7) I.A. Campbell, J.M. Flesselles, R. Jullien and R. Botet. J.

Phys. C 20 (1987) L47. I.A. Campbell, J .M. Flesselles, R. Jullien and R. Botet, Phys. Rev. B 37 (1988) 3825. [8) The results are also compatible with a situation where there are many interpenetrating spanning clusters ; we will assume that there is only one spanning cluster as is the case of the randomly occupied hypercube, P. Erdos and J. Spencer, Comput. Math . Appl. 5 (1979) 33. M. Ajtai, J. Komlos and E. Szemerdi, Cornbinatorica 2 (1982) 1. K. Weber , J. Inr. Proc. Cyber. 22 (1986) 601. (9) R.G. Palmer, Adv, Phys. 31 (1982) 669. (lO) K..L. Ngai and U. Strom, Phys. Rev. D 27 (1983) 6031. CA. Angell and L.M. Torell, J. Chern. Phys. 78 (1983) 937. M. Adam, M. Delsanti, J.P. Munch and D. Durand, Phys. Rev. Lett. 61 (1988) 706.