Study of a simple hypothesis for the mean-field theory of spin-glasses

Transcription

1 Study of a simple hypothesis for the meanfield theory of spinglasses J. Vannimenus, G. Toulouse, G. Parisi To cite this version: J. Vannimenus, G. Toulouse, G. Parisi. Study of a simple hypothesis for the meanfield theory of spinglasses. Journal de Physique, 1981, 42 (4), pp < /jphys: >. <jpa > HAL Id: jpa Submitted on 1 Jan 1981 HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 . and A This à J. Physique 42 (1981) l AVRIL 1981, 565 Classification Physics Abstracts K Study of a simple hypothesis for the meanfield theory of spinglasses J. Vannimenus, G. Toulouse Laboratoire de Physique de l Ecole Normale Supérieure, 24, rue Lhomond, Paris 05, France G. Parisi INFN, Frascati (Roma), Italy (Reçu le 30 octobre 1980, accepté le 8 décembre 1980) Résumé Nous présentons une étude détaillée des conséquences d une hypothèse simple pour le modèle de Sherrington et 2014 Kirkpatrick savoir que dans la phase verre de spin l entropie est indépendante du champ magnetique appliqué. Cette hypothèse conduit à des prédictions en excellent accord avec les résultats de simulations numériques, pour l énergie de l état de base, l entropie, et pour l énergie et l aimantation spontanée en fonction de la valeur moyenne Jo des interactions. La pertinence de la théorie de champ moyen pour les matériaux réels est discutée; les domaines où un accord existe déjà sont soulignés, et de nouvelles expériences sont suggérées Abstract. detailed study is presented of the consequences of a simple hypothesis for the SherringtonKirkpatrick model, namely that in the spinglass phase the entropy is independent of the applied magnetic field. This hypothesis leads to predictions in excellent agreement with the results of available MonteCarlo simulations, for the groundstate energy, the entropy in zero field, the energy and spontaneous magnetization as a function of the interaction mean value Jo. The relevance of meanfield theory to real spinglass materials is discussed, underlying the extant areas of agreement and suggesting new experiments. 1. Introduction. paper is the third in a series [1, 2] examining the consequences of a simple Hypothesis for the spinglass phase, of the infinité ranged SherringtonKirkpatrick (SK) model [3, 4]. Interest in this model started with the physical idea that its solution is of meanfield type and that it should provide the best definition of what meanfield means for a spinglass. We are therefore led to address two issues : i) How close to the true solution of the SK model does this hypothesis lead? This involves a comparison between the existing MonteCarlo data and our predictions. ii) What comes out of the comparison with experimental measurements on real (threedimensional) spinglass materials? 2. The SK model and the projection Hypothesis. The Hamiltonian of the SK model in a uniform magnetic field is for N Ising spins Si. The bond interactions Jij are taken as independent random variables, with a Gaussian random law of mean value Jo/N and of variance J/k (the normalizations ensure a sensible thermodynamic limit). In the following, the energy unit is fixed by the choice J 1. The phase diagram (T, H, Jo) expected for this model is depicted in figure 1 of reference [21 with a critical surface corresponding to the onset of replicasymmetry breaking. The projection (or PaT) hypothesis consists in assuming that within the spinglass phase (or replicasymmetry broken phase), restricting (for notation simplicity) to the case Jo 0, the entropy does not vary with the magnetic field : As a consequence, the magnetization is temperature independent : Besides, it is hypothesized that : Article published online by EDP Sciences and available at

3 Entropy Early Zerotemperature 566 where q is the EdwardsAnderson (EA) order parameter. Expression (2) implies that the free energy F has an additive character : 3. Fits with exiating MonteCarlo data for the SK model. MonteCarlo data have been obtained by Kirkpatrick and Sherrington [4]. It is interesting to observe that they were published early in the game, much before the advent of the Parisi replicasymmetry breaking scheme [5] and of the subsequent PaT hypothesis [1]. e Entropy as a function of temperature, Jo H 0. Within the projection hypothesis, analytical expansions at low temperatures ( T ~ 0) and around the transition temperature (T 1) can be obtained and have been derived [1]. Between these limits, the calculation can be done numerically. The results for the entropy S( T) are displayed in figure 1. They fall slightly below the MonteCarlo results (Fig. 8 of Ref. [4]), but the latter were obtained on samples of 500 spins and the difference is within the estimated size effects. Fig. 2. magnetization predicted by the projection hypothesis (full curve), compared to the magnetization computed without replicasymmetry breaking (dotdashed curve). Parisi s scheme and with the most recent computer simulations [5, 6] (Fig. 2). e Groundstate energy, Jo T H 0. The groundstate energy Eo is directly obtained from the previous function M(H1 using the relation : To achieve a good accuracy with less effort it is convenient to compute the difference between Eo and the groundstate energy without replicasymmetry breaking Es (2/n)1/2 : The sign of (M 0 S K _ M ) corresponds to the expectation that the true groundstate energy is higher than E SK (Fig. 2). One hnds with the PaT hypothesis : hence Fig. 1. of the SK model computed via the projection hypothesis (H 0). The triangles represent the MonteCarlo data of Kirkpatrick and Sherrington, obtained on 500spin samples [Ref. 4]. e Magnetization as a function of field, Jo T 0. Although this quantity was not calculated in reference [4], we present our predictions here for completeness purposes. The zerotemperature MSK (H) derived from the original SK theory without replicasymmetry breaking is also drawn for comparison. The initial slope corresponds to a susceptibility X(O) 1 in the spinglass phase, in agreement with whereas Parisi [5] obtains Eo ± , a remarkably close âgreement. [In fact, (ÊMIDT) is very small on the critical line and the result for Eo is altered by less than 104 if for instance the maximum value of MSK at given H is used in equation (7), instead of M(H) given by the projection hypothesis.] e Groundstate energy as a function of J(» H 0. It was shown in reference [2] how a simple thermodynamic argument allows to derive the general case Jo * 0 from the case Jo 0. For the groundstate energy Eo(Jo), this leads to the prédiction :

4 567 Fig. 4. Zerotemperature magnetization as a function of Jo in an applied field H 0 (full curve) and H 0.01 (dashed curve). The circles and error bars correspond to the MonteCarlo data in a held H 0.01 [Ref. 4]. Fig. 3. Groundstate energy per spin Eo as a function of Jo. The triangles with error bars represent the MonteCarlo results on 500spin samples [Ref. 4]. The complete results are plotted in figure 3, together with the MonteCarlo data of Kirkpatrick and Sherrington [4] (500spin samples). The agreement is considerably better than with the SK theory without replicasymmetry breaking ; it would still be improved (for Jo 1) by taking into account the size effects in the MonteCarlo data which are expected to become more pronounced in the spinglass phase. Spontaneous magnetization as a function of JO. A nonzero groundstate magnetization develops for Jo> 1, in zero magnetic field : This magnetization is rather strongly affected by an external field. Thus, for Jo 1, one has : In figure 4, the magnetization M(Jo) is drawn both for H 0 and H 0.01, and is compared with the MonteCarlo data [4], obtained with a field H The agreement is considerably better than with the nonreplicasymmetry breaking solution. Indeed, it is as good as it could be. This is certainly the most convincing fit in favour of the PaT hypothesis and one can only hope that a new generation of Monte Carlo calculations will reduce the error bars and therefore make the test more stringent. 4. Comparison with experiments crn real materials. Before attempting any comparison with real materials, we must first consider two caveats. First, the spins in most real spinglass materials have vector (Heisenberg) character rather than Ising character. The results or prédictions presented above pertain to the Ising case, where the number of spin components m 1. The only other firm knowledge that we have so far bears on the limit m oo, corresponding to the exactly soluble spherical model [7]. Comparing these two extreme limits, we recognize that the susceptibility is predicted to be constant in the spinglass phase for both cases. It is naturally tempting to make the guess that this behaviour is m independent and therefore valid also for Heisenberg spins. Second cornes the general question of the reliability of a meanfield theory for a real (threedimensional) material. By its définition, meanfield theory should hold in the limit of infinite space dimensionality. Deviations due to fluctuations are expected to occur in finite dimensionalities and to increase when the dimensionality goes down. These fluctuations are expected to affect mostly the critical region around the transition temperature (eventually suppressing the transition below the socalled lower critical dimension) and for instance the cusp in the specihc heat predicted for the SK model is not observed in real materials : We should therefore trust better the meanfield theory deep in the low température spinglass phase than in the critical region. Besides, the fluctuations are expected to be weaker in systems with longrange interactions : this favours the standard spinglass materials, such as CuMn and similar alloys, for a comparison with meanfield theory. Keeping these remarks in mind, let us consider the expérimental facts. Expérimental evidence for a plateau in x(t), or equivalently in M(T) at low external fields, has indeed

5 It 568 been found repeatedly on CuMn alloys of various concentrations [812], when the measurements are made with cooling under fixed applied field (fieldcooling). In these conditions, the magnetization is found to be at equilibrium i.e. timeindependent, whereas it is not when the field is applied at low temperatures and the sample is heated up thereafter. It is therefore the former quantity which should be compared with an equilibrium theory (leaving as a task for a future dynamical theory to explain why fieldcooling is reversible). Another very important prediction of the meanfield theory is the existence of a critical line H c( T), below which replicasymmetry breaking takes place. Note that this prediction goes against an established folklore, claiming that the spinglass transition disappears in a finite field. Several groups [12, 13, 14] are now trying to find experimental evidence for such a line and to draw its locus in the phase diagram. There are experimental difficulties for this determination which are best illustrated by the sketch made in figure 5. In finite field, meanfield theory predicts the existence of a critical line Hc(T). But it predicts also a crossover line Hp(T) within the paramagnetic phase, with the opening up of a critical region where Curie behaviour (M function only of H/T, for instance) does not hold. This critical region leads to a rather smooth change from the Curie regime to the spinglass regime (e.g. for the magnetization), whereas in zero field the change is sharp. Indeed it is this smooth change that led to the conviction that the transition was smeared out in a field. As a consequence, the définition of Hc(T) by the end in the spinglass plateau for M(T) is not very precise [12]. An alternative experimental definition of the instability line is through the onset of irreversibility effects [14], and it will be interesting to compare both determinations. A third domain of comparison between meanfield concerns the anomalous theory and experiments power laws predicted by the PaT hypothesis. Thus it is predicted that the equilibrium magnetization M under small field H behaves like : Fig. 5. Sketch of meanfield theory predictions, showing qualitatively the position of the transition line Hc(T) and the fuzzy crossover line Hp(T). mixed spinglassferromagnetic phase [2]. However, this is typically a domain where new features might emerge for Heisenberg spins (due to the possibility of canting for the spins), so that further progress should be made by both theory and experiments on this problem. 5. Replicasymmetry breaking and projection properties. has been shown, in section 3, that the projection hypothesis (2), and its consequence (3), lead to predictions for various thermodynamic quantities which are very close to the truth. There is also some evidence that the other projection hypothesis (4), for the EA order parameter, is rather good. At this stage it is interesting to consider how these two projection properties could appear (exactly or approximately) in a replicasymmetry breaking analysis, and how they could be intimately related one to the other. 5.1 REPLICASYMMETRY BREAKING. For this purpose, we shall briefly review some features of the Parisi replicasymmetry breaking scheme [5]. In this scheme, the order parameter at a given temperature is a function q(x), 0 x 1, which is expected to be continuous and monotonously nondecreasing ; its extremal value is understood as being the EA order parameter : The formulae which relate the zero field susceptibility x and the energy t to the (EA) order parameter q in the oaramaenetic nhase. Detailed comparison with experiments would be very valuable and is under way [12]., Finally, it would be helpful to have more experimental data on the existence and properties of the have simple generalizations, in this replicasymmetry breaking scheme :

6 a Consider The 569 Since relations (17) and (19) hold in the presence of a finite field H, it should be specified that the total energy is so that the free energy F(T, H ) can be written Now, the variational calculations [5] using staircase approximations for the function q(x) suggest that, in zero field, this function is monotonously increasing for 0 x x and constant for x x 1. The position of x is a function of temperature in the spinglass phase, with x( T > 1) > 0 so that, naturally, the x dependence of q(x) disappears in the paramagnetic phase. 5.2 A SCALING ASSUMPTION. now the consequences of the following scaling assumption for the function q(x, T). Suppose that Carrying (22) into (18), using the hint that x(t) 1 and taking the derivative with respect to temperature, one obtains : Fig. 6. The scaling function q f (x/t ), computed with the use of the projection hypothesis via equation (24). obtained may be used as a test of the scaling assumption. 5.3 COMPARISON WITH THE VARIATIONAL CALCULA TION OF q(x). crucial test of the validity of equation (22) is provided by the comparison of its predictions with the results of a direct variational calculation of q(x, T), but it is necessary to recall first some aspects of the variational method. In the approach of reference [5] the function q(x) can be found by maximizing an effective free energy which functionally depends on q(x) A simple prediction can be derived from this formula. At low temperatures, provided q(t) behaves as q 1 T 2... the position of the break point x in q(x) will tend to 1/2 : Rather accurate values for the free energy càn be obtained using simple trial functions where q(x) is a piecewise constant function. It is more difficult from a numerical point of view to find good results for the function q(x). Indeed, it we write As we shall see, this prediction of the scaling assumption is well supported by the variational calculations. It is clear from equation (23) that the function f(x/t) can be found from the knowledge of the function q(t). Indeed the two equations and give the function f (u) implicitly. If we use as input the function q(t) given by equation (4), we find the result shown in figure 6 for the function f(u). Since q(t) as given by the projection hypothesis is certainly an accurate approximation, the function f(u) so has zero as accumulation point for its eigenvalues. This phenomenon is very interesting and has many important physical consequences. It also implies that there are directions in the functional space for which F[q] changes by only a very small amount : In other words F is very flat near its maximum (it has been argued in reference [15] that F has really a maximum only if q(x) is monotonous). A small numerical error

7 It 570 in computing F may strongly affect the form of qo(xà moreover the standard computer programs to minimize a function of n variables are rather inefficient if the region near the maximum is too flat. In spite of these difficulties, we have performed some numerical work along this approach, and the results are consistent with equation (22), i.e. the function q(x, T) satisfies the scaling assumption within the numerical precision. The function q(x) has been parametrized as in q(x) is consistently observed in both sets of results, and the small discrepancy at the lower temperature is due to the choice for the variational function (p which becomes less accurate when q(x) 1. Equations (22) and (23) are thus satisfied with good accuracy. Unfortunately it has not yet been possible to prove the validity of equation (22) starting directly from the analytic formulae of référence [5]. 5.4 FURTHER IMPLICATIONS AND COMMENTS. The scaling assumption has further interesting implications. Carrying (22) into (19) and deriving with respect to T, one gets :. and we have looked for the values of xn which maximize the functional FM, for différent values of K. To avoid the minimization of a function of too many parameters, the {xn} have been represented by a smooth function x ~(q) depending on a few parameters. As a byproduct of the calculation, the computed value of the susceptibility is x (for T 0.2 and k 20) in good agreement with the suggestion that x 1 in the spinglass phase. The final results are shown in figure 7. The light staircase curve represents the actual function used in the variational calculation (for T o.l and K 11), while the smooth curves represent the functions 9(q) for T 0.1 and T 0.2 (K 11). These curves are in remarkable agreement with the function q(x) computed from the function f shown in figure 6 (for clarity only a few points have been drawn on the figure). In particular the existence of an inflexion point where C is the spécifie heat. Elimination of dq/d T between (27) and (23) yields hnally the relation Thus we have obtained, as a consequence of the scaling assumption, a relation between a thermodynamic quantity, the specihc heat, and the EA order parameter. However, if we calculate separately C and q from the two hypotheses (2) and (4), projecting from the instability line, we hnd that relation (28) is not strictly verified. The discrepancy is most pronounced at low temperatures, but it is to be noted that equation (28) is a very stringent test there : The projection is made from very high fields and a minute variation of the entropy with the magnetic field may then add up to a sizeable error. In the presence of a finite field, the scaling assumption (22) can be generalized in the following way : The existence of a plateau, for small values of x, in q(x) is suggested by the variational calculations [5]. Now, bringing (29) into (19), it is easily seen that 03B5(T, H) can be written as : Fig. 7. Comparison between the predictions of the scaling assumption (22) and the direct variational calculation of q(x, T). The full lines represent the smooth interpolating functions used in the variational approach (see text) and the results of the scaling assumption are denoted by circles (for T 0.1) and by crosses (T 0.2). The dashed curve corresponds to the locus x(q) of the break point above which q(x, T) is constant and equal to the EA order parameter q, according to (22). and using relation (21) one gets the additive property of the free energy (5), from which (2) and (3) are obtained. Since relation (28) is not exactly verified, it is clear that the set of equation (29) has to be only approximately verified. Obviously, these various remarks and relations do not yet add up to a fully consistent theory. Perhaps they give a fair account of the general qualitative pattem. 6. On the cause of the transition. was pointed out in reference [1] that the position of the maximum

8 571 with respect to H of the entropy SS K(T, H) computed without replicasymmetry breaking, lies close to the instability line. The study of the precise location of this maximum needs some care and the Maxwell relation : provides a useful consistency check on the computational results, using both équations independently. These results are displayed in figure 8, where the locus of the maximum of SSK is seen to lie always extremely close to but below the instability line : Thus the transition seems to occur in order to freeze the entropy and to prevent it from decreasing when the field is lowered, as would be the case in the absence of replicasymmetry breaking. [Indeed, according to hypothesis (2) the spinglass phase would appear as a truly «frozen» phase, in the sense of the Nernst principle, since it would be impossible to cool it via an adiabatic demagnetization procedure.] This behaviour is remarkable because there is a priori no general thermodynamic reason why as/ah or OMIDT could not be positive. Consider for instance an antiferromagnet in a small uniform held parallel to the staggered magnetization : The longitudinal susceptibility increases with temperature and ôm/ôt is positive in low fields., Could the origin of replicasymmetry breaking be related to the impossibility for a disordered system such as a spinglass to have a positive am/8t? The proximity of the two curves on figure 8, though perhaps a coincidence, lends support to this idea. Fig. 8. The full line is the critical line for replicasymmetry breaking Hc(T). The entropy SS_K computed without replicasymmetry breaking shows a maximum with respect to H just below Hc(T). Full circles denote points where ASIDH 0 and crosses points where ôm/ôt 0. Acknowledgments. We are extremely grateful to P. Monod and H. Bouchiat for many fruitful discussions on the present work and its relevance to the experimental results, and we thank S. Kirkpatrick for kindly providing the original figures representing his MonteCarlo results. References [1] PARISI, G., TOULOUSE, G., J. Physique Lett. 41 (1980) L361. [2] TOULOUSE, G., J. Physique Lett. 41 (1980) L447. [3] SHERRINGTON, D., KIRKPATRICK, S., Phys. Rev. Lett. 32 (1975) [4] KIRKPATRICK, S., SHERRINGTON, D., Phys. Rev. B 17 (1978) [5] PARISI, G., Phys. Rev. Lett. 23 (1979) 1754; J. Phys. A 13 (1980) 1101; J. Phys. A 13 (1980) 1887 ; Philos. Mag. B 41 (1980) 677. [6J BANTILAN, F. T., PALMER, R. G., J. Phys. F (Metal Physics) 11 (1981) 261. [7] KOSTERLITZ, J. M., THOULESS, D. J., JONES, R. C., Phys. Rev. Lett. 36 (1976) [8] HIRSCHKOFF, E. C., SYMKO, O. G., WHEATLEY, J. C., J. Low Temp. Phys. 5 (1971) 155. [9] THOLENCE, J. L., TOURNIER, R., J. Physique Colloq. 35 (1974) C4229. [10] GUY, C. N., J. Phys. F (Metal Physics) 7 (1977) [11] SHOICHI NAGATA, KEESOM, P. H., HARRISON, H. R., Phys. Rev. B 19 (1979) [12] MONOD, P., BOUCHIAT, H., private communication. [13] KNITTER, R. W., KOUVEL, J. S., J. Magn. Magn. Mat. 21 (1980) L316. [14] CHAMBERLIN, R. V., HARDIMAN, M., ORBACH, R., presented at the Int. Conf. on the Physics of Transition Metals, Leeds (U. K.), Aug [15] THOULESS, D. J., DE ALMEIDA, J. R. L., KOSTERLITZ, J. M., J. Phys. C (Solid State Physics) 13 (1980) 3271.