Rigorous bounds for 2D disordered ising models A. Georges, D. Hansel, P. Le Doussal, J.M. Maillard, J.P. Bouchaud To cite this version: A. Georges, D. Hansel, P. Le Doussal, J.M. Maillard, J.P. Bouchaud. Rigorous bounds for 2D disordered ising models. Journal de Physique, 1986, 47 (6), pp.947-953. <10.1051/jphys:01986004706094700>. <jpa-00210291> HAL Id: jpa-00210291 https://hal.archives-ouvertes.fr/jpa-00210291 Submitted on 1 Jan 1986 HAL is a multi-disciplinary 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. J. Physique 47 (1986) 947-953 JUIN 1986, 947 Classification Physics Abstracts 75.40 05.50 - Rigorous A. bounds for 2D disordered Georges (+), D. Hansel (++), Ising P. Le Doussal models (+), J. M. Maillard (*) and J. P. Bouchaud (**) (+) Laboratoire de Physique Théorique de l’ENS, LP du CNRS, 24, rue Lhomond, 75231 Paris Cedex 05, France (+ +) Centre de Physique Théorique de l’Ecole Polytechnique, GR du CNRS no 48, route de Saclay, 91128 Palaiseau Cedex, France de Physique Théorique et Hautes Energies, LA au CNRS et Université P. et M. Curie, 4, place Jussieu, tour 16, 1er (**) Laboratoire de Spectroscopie Hertzienne de l’ENS, LA au CNRS, 24, rue Lhomond, 75231 Paris Cedex 05, France (*) Laboratoire (Reçu le 3 dgcembre 1985, accepté le 18 fgvrier étage, 75230 Paris Cedex 05, France 1986) Résumé. Nous développons une méthode améliorant de façon systématique les bornes recuites (pour l’énergie libre et l’etat fondamental) et l’appliquons aux modèles d’Ising désordonnés. Cette méthode prend en compte les effets géométriques de frustration et conduit, à la suite d’un procédé d’optimisation, à des bornes dont le comportement critique diffère de celui du modèle pur. Les lignes critiques déduites de cette étude s’avèrent être des expressions algébriques simples en accord remarquable avec les résultats exacts déjà connus. 2014 free energy and the ground state An optimization procedure leads to new bounds for which the critical behaviour differs from the pure model behaviour. Geometrical effects, such as frustration, can also be included in this method. We obtain simple expressions for the critical lines which agree remarkably well with previously known exact results. A method systematically improving the annealed bounds (for the Abstract. energy) is presented and illustrated by the example of disordered Ising models. 2014 1. Introduction. Because of the complexity of the disordered systems it is natural to develop methods which use as much as possible the known results available for pure models, with, of course, a special emphasis on the exact results. Apart from very few exceptions, the analytical approaches for two (or more than two) finite dimensional models are either rigorous inequalities or approximation schemes whose validity is difficult to judge. It is the aim of this paper to present a method in which those two aspects are combined. This method applies to all disordered systems once an exact solution is available for the corresponding pure model (even on a restricted region of the parameter space). Here, we apply it to two 2D disordered Ising models : the dilute bond model and the spin glass model. A relevant quantity for these problems is the quenched average In Z &#x3E; of the logarithm of the partition function. The well-known annealed bound for this quantity follows from the convexity of the logarithm This is certainly also a free energy in the high temperature region. However, since it amounts to replacing the high temperature variables tanh (PJj) by their mean values, one cannot expect a critical behaviour different from the pure model one. Furthermore such an approximation scheme does not take into account some essential physical effects (such as frustration [1]) which become relevant at low tempe- simple annealed &#x3E;&#x3E; bound good approximation of the true « ratures. Our method is a systematic improvement of the annealed scheme leading to rigorous bounds which have the two following properties : i) a « bootstrap type » optimization process leads to a modification of the critical behaviour which amounts to a Fisher renormalization [2] of the expo- nents ; ii) geometrical effects are Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01986004706094700 explicitly taken into 948 (such as frustration on a plaquette in the spin glass case). account The optimized bound is then given by (5) with 2. Method. Let us consider an elementary cell a of the lattice (which can be a single bond, a plaquette or any bigger pattern) and denote by W0153 its associated Boltzmann where t tanh pJij and c cosh fljij and we have weight. For a fixed configuration of bonds, the par- suppressed the indices i, j when not explicitly needed. tition function reads : F I (T) is a function which can be deduced from the Onsager-Houtappel [3] solution of the triangular Ising model = The usual annealed approximation (1) amounts to choosing a single bond as elementary cell a and to replace W,, by its mean values Wa &#x3E;. Let us introduce an arbitrary positive function f ({ Jij(a) }) of all the bonds which lie in the elementary cell, being the same for all the cells. One can write : weight for each cell has thus been replaced by f Wa. Now one can obtain an upper bound on In Z &#x3E; by exactly averaging the first term in the sum and using the annealed procedure for the new weight f Wa : In the particular values : = case where P(Jij) takes only two The Boltzmann where Na is the number of elementary cells a. This upper bound, that we denote by In Z f, depends on the particular choice for the function f. The best of all these bounds is given by : where the minimum is taken over all positive functions. In this bound the original disordered model has been replaced by a pure one, with an effective Boltzmann weight fWIZ). This method can be used in practice whenever some exact piece of information can be found for this new pure model. In this respect the choice of function f can be restricted in order to keep, or make, the model solvable. To be explicit we now illustrate these general ideas by the example of the random bond Ising model on the triangular lattice, with a probability distribution P(Jij) of the couplings. We introduce three different types of rigorous bounds, corresponding to different choices for a and f. 2.1 TYPE-I BouND. In this case one chooses each bond Jij to be the elementary cell a, which is the simplest possible choice. W then reads : - one can always choose the function f Minimization with respect to fixing A by : f simply to be : amounts to the optimized bound (7) being In Z f taken for this value A* of A. We thus see that we make use of the exact solution of the pure model, both to determine the optimum value A* of A and to compute the bound itself, using A*. The equation of the critical variety for this bound is simply given by : where t is the critical value of the pure model 2 - vi 3 (tc = for the triangular lattice). A straightforward calculatioin shows that this « bootstrap » procedure modifies the original critical behaviour of the pure model (logarithmic singularity) into a cusplike behaviour. More generally, for a pure model with a non-zero exponent a, the thermal exponent of the optimized bound is : 949 This is precisely a Fisher-type renormalization in the presence of an external constraint. This last result is reminiscent of the approximations of the spin glass problem developed by Syozi [4]. In this approach, one treats on the same footing the thermodynamical and disordered averages, by introducing a grand partition function 3 which depends on the temperature and on some chemical potential M. J is then The mean concentration p of bonds Jij determined as : For this type-II bound, we do not use any optimization scheme but make a particular choice for f, which is naturally suggested by equation (15), namely f = ø -1. The remarkable point is that averaging the new Boltzmann weight (!W &#x3E; still leads to an exactly solvable model for the triangular case. This leads to : = particular cases of binary probability distribution (9), where the Junction f reduces to a single parameter A, equation (11) is somewhat similar to (14) (A playing the role of p). However, let us point 6ut the following differences with Syozi-type effective medium approaches : i) for continuous probability distribution, instead Ti variables a very simple expression in terms of the [5] (all the il ) are equal) : The critical variety where F2 has We notice that in the minimization with respect to a single variable (chemical potential), we minimize in a whole functional space ; ii) our method is a controlled approximation scheme in the sense that we obtain rigorous bounds on the free energy. of a 2.2 TYPE-II BOUND. - We now choose the hatched = tanh pJi is : are ignored). 2.3 TrnE-III BouNDs. Type-III approximation generalizes both types I and II : in the case of spin glasses, it takes account of the frustration of each elementary cell in an optimal way. The arbitrary function f(Tl, ’t2, i3) now depends on the three couplings of an elementary triangle. In Z f is still an exactly anisotropic solvable model : - as : where with ti (16) and the critical behaviour for this type-II bound is, of course, the same as the pure one. Clearly, this bound is not only simple to obtain, but is also well suited to take frustration effects inside each elementary cell into account. Indeed the dominant high temperature contribution of each elementary plaquette In 0 &#x3E; is averaged exactly (however, correlations effects between different frustrated pla- triangle depicted in figure 1 as the elementary cell a. quettes The Boltzmann weight associated with each cell a can be written for (see Fig. 1). where F2 is the anisotropic version of (17) (see [5]) : Minimization with respect to f can be achieved by a functional derivation of In Z f. This leads to the following simple (and symmetric) function, at the minimum : Fig. 1. 2013 cell of the coupling constants and the elementary anisotropic triangular Ising model. The three 950 where y, 6 are given by the relation : and the critical variety is now : high temperature limit fo goes to 1 and we the type-II bound. The type-III bound is however always better than both type-I and type-II, and has the same critical exponent a’ as type-1. In the recover 3. Results. In this section we detail the applications of our 3 types of bounds for dilute Ising models and for Ising ± J spin glass. 3.1 DILUTE ISING MODEL. In this case we concentrate on the phase diagram for the binary distribution (9b). Critical curves for type-I and III approximations are given in figure 2 for the triangular lattice. The two approximations give almost indistinguishable curves, except at very low temperatures (where, as we shall see, type-III only is in agreement with the known exact Pc). Moreover, type-III is extremely close to real space RG calculations [6] or effective medium- type approximations [7]. Exact results available for this critical curve are 2. The paramagnetic-ferromagnetic (P/F) critical for the triangular dilute Ising model in the (p, T) plane, for type I and type III approximations. An enlargement of the low temperature region is also represented. Fig. - curve the following [8] : i) in the low temperature limit, Pc is the threshold of the percolation problem of the triangular lattice, given by : namely The asymptotic behaviour near p Pc can be obtained exactly from some duality arguments [8] and, in the case of the triangular lattice, is given by : = ii) tc = Fig. 2bis. The type I the square lattice. on - P/F curve for the dilute Ising model information. Equation (23) reads : with _ t In the pure limit p - 1 the critical value is 2 and the slope is given by [8] : J3 T = t/1 + t2 and a = 2 t + t2. Straightforward The observed agreement with numerical results calculation shows that this equation is in complete agreement with pc, tc and the slope dpEquations (27)leads fully justifies the comparison of the analytical expressions for the type-III critical curve with this exact dT P=l to an asymptotic beha- 951 viour at low temperatures singular [10]) and imposes constraints on the phase diagram. The type-III critical line is given by is not given by : structure of the The constant is different from the exact one but is very close to it (less than 2 %). For such an approximation scheme, agreement in the pure limit is not surprising. On the other hand, it is remarkable to notice that equation (27), which was deduced from the exact solution of a free fermion model, completely coincides in the low temperature limit with the equation (24a) which is a consequence of a star triangle relation on the percolation model (or equivalently q-Potts model in the limit q -+ 1). The type-I bound can be extended in a straightforward way to the case of the square lattice. (We display the corresponding critical curve in Fig. 2bis). This and curve dtTp- P= I 1 also gives = 8 - 6 the exactly known p. =t 2 slope, and is in remarkable agreement with RG calculations [9]. 3.2 + J SPIN GLASSES. - We consider binary distribution of equation (9a). with b I and III = t/(1 + t + t2) and = tl(1 - t + t2). Types certainly closer to the true ferromagnetic/paramagnetic (F/P) critical curve than the simple annealed and the type-II lines. In the pure limit, types I and III agree with the exact values tc 2 are = Type-I and very close to each other and differ only in the low temperature regions where the reentrance feature of type-I (already noticed by many authors [4]) is lost when frustration effects are taken into account. Finally, as discussed in [12], a precise location of Nishimori’s line should shed some light on the value of PC. Indeed Nishimori has shown the existence of the following topological constraint for the F/P transition line. Let us consider the intersection point (p*, t *) of Nishimori’s line and the true F/P line : one has, for the true Pc at T 0, curves are = now the 3. 2.1 Critical varieties. The critical curves for the triangular Ising model are plotted in figure 3. They correspond to the simple annealed (1), type I, II, III bounds. We have also plotted Nishimori’s line where the exact internal energy is known (and - Fortunately enough, the intersection of Nishimori’s line and both types I and III are extremely close so that one can safely estimate the lower bound p* to be p* -- 0.8. On the square lattice the same analysis for type-I bound leads to the lower bound p* -- 0.87 on p,, in good agreement with numerical simulations which give p,, - 0.9 [13, 14]. 3.2.2 Upper and lower bounds of the quenched free , Our approximation scheme gives a set of energy. better and better lower bounds for the quenched free energy of the triangular random Ising model. We now choose to explicitly display the results in two regions of particular interest in the phase diagram : Nishimori’s line and p 1/2. Restricted to Nishimori’s line at tanh pJ 2 p - 1, there exist two upper bounds that provide a test of the accuracy of the method. Figure 4 gives the different upper and lower bounds for the free energy of the model restricted to Nishimori’s line. The two upper bounds are related to the knowledge of the exact internal energy and to the exact calculation of In l/Z), respectively. The lower bounds correspond to the simple annealed and type I, II, III bounds. The true quenched free energy lies in the hatched region. One remarks again that types I and III give rather close results. Another region of special interest is p 1/2. We display, in figure 5, the lower bounds on the free energy as a function of the temperature for p 1/2. For low temperatures the behaviour of type II is - = = The P/F critical curves for the triangular ± J Fig. 3. Ising spin glass in the (p, T) plane, for the simple annealed SA, type (I), types (II), (III) approximations. Nishimori’s line N is also represented and the corresponding value for p* is given. - -III = = 952 Bounds for the free energy of the triangular Ising Nishimori’s line as function of p. Lower bounds : simple annealed SA, types (I), (II), (III). Upper bounds : internal energy U (dashed dotted line), and In I/Z &#x3E; (dashed line). The true free energy lies in the hatched region. Fig. 4. model - on Fig. 6. -Bounds for the ground state energy of the triangular Ising model : the lower bound of Toulouse and Vannimenus (dotted line), the simple annealed SA and type-11 lower bounds. Type-III is very close to type (II) and is not represented. We also give the ferromagnetic upper bound E*. Figure 6 shows bound (30) for the sample annealed and type-II cases, together with (31). We have also indicated the obvious ferromagnetic upper bound : One remarks that type-II bound has two different regimes : i) for low frustration (p &#x3E; 0.8) the maximum 0 and (30) value of the free energy is reached at T coincides exactly with (31); ii) for higher frustration (0.2 p 0.8) the temperature for which the maximum is reached in (30) changes abruptly so that type-II bound splits off from (31) when (31) becomes worse than the simple = 5. Lower bounds on the free energy as a function of the temperature T for p 1/2 (Triangular Ising Model) : simple annealed SA, type II. We also give the bound obtained by considering a bigger pattern made of three hatched Fig. - = triangles. greatly improved as compared with the « simple annealed » approximation. 3. 2. 3 Lower bounds on the ground state energy. As remarked by Vannimenus and Toulouse [14], a lower bound FA on the free energy provides, at the same time, a lower bound on the ground state energy : annealed bound. Here again the entropy of this bound remains positive forp not too far from 0 or 1. This is in contrast with the simple annealed approximation and in this region this could provide an approximation of the ground state entropy. - Furthermore, a simple counting argument for the on the triangular lattice [15] : frustration of the plaquettes gives another lower bound 4. Comments and speculations. It has been shown that the usual annealed bound can be efficiently improved by using a simple systematic optimization scheme and by including geometrical effects (such as frustration effects). For the dilute Ising model, for instance, simple algebraic expressions are obtained for the critical curves which are in remarkable agreement with both known exact results and available simulations. The leading idea was to make the maximum use of exact information available on pure models in 953 order to get rigorous bounds for the corresponding disordered model. Here we have dealt with the situation where the pure case is an integrable model. This idea can be easily generalized : let us suppose for example that some exact information is available only on a submanifold of the parameter space (partition function on the critical variety and disorder variety of the two-dimensional pure Potts model, ( 1/Z&#x3E; on Nishimori’s line for Ising spin glass and Potts glass [12]). It is then possible, using the function f, introduced above by (3), to bring the effective parameters back on this submanifold and thus to obtain an exact bound on the free energy. As an illustration of these ideas, one can make use of the upper bound recently derived [12] for the free energy restricted to Nishimori’s line. One can extend this bound in the whole parameter space, to get for the binary distribution (9a) : Moreover these ideas could also apply when only numerical data are available (Monte Carlo simulations, ...). Another field of application for such an idea is the estimation of the Lyapounov exponent of a product of random matrices [16]. Acknowledgments. We thank G. Toulouse for careful reading of the manuscript. We also acknowledge fruitful discussions with M. Gabay, T. Garel and R. Rammal. References [1] [2] [3] [4] TOULOUSE, G., Comm. Phys. 2 (1977) 115. FISHER, M. E., Phys. Rev. 176 (1968) 257. HOUTAPPEL, R. M. F., Physica 16 (1950) 425. KASAY, Y. and SYOZI, I., Prog. Theor. Phys. 50 (1973) 1182. [5] GEORGES, A., HANSEL, D., LARD, J. Lett. [6] [7] [8] M., LE DOUSSAL, P. and MAIL- to appear in J. Phys. A Math. Gen. 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[15] TOULOUSE, G. and VANNIMENUS, J., Phys. Rep. 67 N° 1 (1980) 47-54. [16] BOUCHAUD, J. P., GEORGES, A., HANSEL, D., LE DousSAL, P. and MAILLARD, J. M., to appear in J. Phys. A Lett.