Energy barriers in SK spin-glass model

Energy barriers in SK spin-glass model
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  2325 Energy barriers in SK spin-glass model D. Vertechi and M.  A. Virasoro Dipartimento di Fisica dell’Università di Roma « La Sapienza », Piazzale  Aldo Moro 2, 00185, Roma, Italy INFN, Sezione di Roma, Italy (Reçu le 6 avril 1989, accepté le 17 mai 1989) Résumé. 2014 Nous étudions les hauteurs de barrière séparant des états métastables pour le modèle de verre de spin avec symétrie d’Ising et portée infinie. Une configuration de barrière correspond à un col de {mi, i = 1, ..., N, - 1 ~ mi ~ 1} de la surface d’énergie qui interpole de façon régulière l’énergie dans l’hypercube. Pour des barrières d’énergie faibles, qui sont importantes pour la dynamique, le nombre de directions descendantes au col est fini dans la limite N ~ ~. Nous trouvons que ces directions sont contenues dans un sous-espace linéaire de l’hypercube [20141,1]N engendré par les directions pour lesquelles 03A3j Jij mj = 0. Nous avons fait des simulations numeriques par deux algorithmes différents. Les résultats sont cohérents. Si nous supposons que les hauteurs de barrière croissent avec la taille N du système comme N03B1, nous trouvons 03B1 = 0,34 ± 0,08.  Abstract. 2014 The height of barriers separating metastable states is studied for the infinite range Ising spin-glass model.  A barrier configuration corresponds to a saddle point {mi, i = 1, ..., N, 2014 1 ~ mi ~ 1} of an energy surface that smoothly interpolates the energy in the hypercube. Forlow energy barriers, which are relevant for the dynamics, the number of independent descending directions from the saddle point is finite when N ~ ~. It is found that these descending directions are contained in the linear subspace of the hypercube [20141,1]N generated by directions for which 03A3j Jij mj = 0. Numerical estimates were performed by two distinct algorithms which lead to consistent results.  Assuming barrier heights growing with system size N as N03B1 we find 03B1 = 0.34 ± 0.08. J. Phys. France 50 (1989) 2325-2332 1cr SEPTEMBRE 1989, Classification Physics  Abstracts 05.20 - 75.50L 1. Introduction. One of the characteristic features of spin glasses is the existence of many states of minimum free energy almost degenerate, separated by very high free energy barriers and unrelated by a symmetry one to another.  An appropriate infinite-ranged Ising spin-glass model was proposed by Sherrington and Kirkpatrick (SK model [1]). It is believed that the SK model can be solved by means of replica method for which Parisi’s replica symmetry breaking ansatz [2] produces a stable mean field solution. Many of the thermal equilibrium properties of the model have been studied using this approach.   Article published online by EDP Sciences and available at  2326 Replica symmetry breaking is physically related to the breaking of ergodicity [3] ; furthermore one of the consequences of Parisi’s ansatz is a particular hierarchical organization of the states in phase space called ultrametric [4]. The dynamics of the SK model is greatly complicated by the breaking of ergodicity [5] ; in the infinite volume limit the time evolution is confined to one of the available pure states on any finite time-scale. Many efforts have been devoted to the comprehension of the dynamics on infinite time-scales, so that transitions between various states are allowed [6]. In practice the precise understanding of this regime requires a detailed knowledge of properties of spin glasses at finite (though large) size, such as the height of the free energy barriers between the states, which is not available at the moment. The complicated structure of the free energy surface seems to be responsible for anomalously slow dynamics [7, 8] and could be relevant to understand the dependence of relaxations on particular waiting-times [9]. Spin-glass models have many aspects in common with combinatorial optimization problems [10] ; the investigation of free energy barriers in SK model may favour a detailed comprehension of the structure of some of these problems, possibly explaining the efficiency of « simulated annealing » algorithms. Energy barriers introduce in a natural way a hierarchical organization of the energy minima [7] which could be related to the ultrametric organization of states in phase space.  A  numerical evaluation of energy barriers between metastable states in SK model at zero temperature has been reported in a recent paper by Nemoto [11] ; the analysis is carried out considering all the metastable states in a given sample and averaging over samples. However the data in reference [11] do not show a significant scaling form for barrier heights since only rather small systems are considered (up to 24 spins). In the present work we extend the analysis of energy barriers to systems of greater size (up to 96 spins), considering only low energy metastable states. We use two distinct algorithms to evaluate barrier heights which lead to consistent results. We find that in a barrier configuration the descending directions of the energy surface are contained in the linear subspace of the hypercube [-1,1 ]N generated by directions with a small (0(1/ IN» magnetic field. From the simulations we derive that the average energy barrier between metastable states is a non-decreasing function of the Hamming distance ; the connection of this result with the ultrametric property of states is examined.  Assuming barrier heights increasing with system size N as N ", we estimate a = 0.34 ± 0.08 for our numerical data. 2. Energy barriers for oi = ± 1. We consider zero temperature solutions of TAP equations [12] to identify metastable states. Setting the external field equal to zero TAP equations and energy E are written as : where o-j = ± 1 and {Jij} are independent random Gaussian variables with zero mean and variance 1 /N, N being the number of spins. We limit our investigation to the low energy solutions in each system, choosing respectively the 5, 8, 20 lowest lying metastable states for systems with N = 48, 64, 96. The algorithm used to find these solutions consists in performing deterministic descents starting from many randomly generated configurations.  2327 Given these states we try to determine the energy barrier Eab separating the states a, b, defined by where a path is a sequence of configurations connected by one-spin-flip passages. The minimization is taken over all possible paths connecting the states. Our main approach to this complicated combinatorial optimization problem consists in using a deterministic descent algorithm to carry out the minimization.  As in Nemoto [11] one introduces the one-spin-flip operator Pi, defined by Pi ai = - ai.  A path connecting the states a, b can then be described as a product of Pi satisfying thé relation : where n is the length of the path. Starting from a randomly generated minimum length path, our iterative improvement algorithm is based upon two elementary moves ; the first one consists in commuting the operators which lead to the highest configuration in the path if this operation reduces the barrier. The second one consists in adding to the path two new identical operators in the region of maximum height, choosing the pair which gives the maximum reduction of the barrier. We have considered more complicated « deformations » of the path but, as a preliminary analysis of the results did not show significant improvements while the increase in computer-time was considerable, we decided to drop them. It is obvious that barriers defined by (2.3) satisfy the inequality so that barriers are described by a hierarchical tree indexed by {Eab} . So, after the evaluationof Eab for each pair of metastable states considered, we constructed the minimal spanning tree [13] to obtain {Eab} . 3. Barriers in a smoothed energy surface.  A different approach to the problem of energy barriers consists in looking for the saddle- points of a smoothed energy surface given by : which corresponds to TAP free energy [12] without the reaction term. We point out that the mi’s are not the actual finite temperature magnetization, neither the F is the correct free energy but just an auxiliary function.  A stationary point for function F satisfies the equations : with In the limit 13 -+ oo, with hi finite equation (3.2) becomes :  2328 as in (2.1). So the minima of the F coincide with zero temperature solutions of TAP equations when /3 ---> oo while for small /3 the surface becomes smooth enough to allow a numerical search for saddle point configurations.  At a fixed value for /3 (typically f3 = 14,8,,) we numerically minimized the function whose absolute minima with G = 0 corresponds to stationary points of the functionF. Given a stationary point we established through the diagonalization of the Hessian whether we had a minimum or a saddle point, obtaining both, up to saddle points with three negative eigenvalues. 4. Characterization of barrier configurations. To recover the connection with the srcinal spin system we looked at f3 --+ 00 limit of saddle point configurations of function F (Eq. (3.1)).  As a first step we notice that while equation (3.2) is related to stationary points, equation (3.4) only describes the minima of the F, suggesting that to have saddle points one needs hi --+ 0 in some site when f3 --+ oo (numerical simulations were found in agreement with this scheme). In fact the stationarity condition is satisfied, in f3 --+ oo limit, if The generic element of the Hessian matrix of function F is : where 8ij is the Kronecker delta. The structure of the Hessian is rather simple when 6 ---> oo ; in fact variables satisfying equation (3.4) lead to diagonal terms positive and divergent while variables satisfying equation (4.1) lead to vanishing diagonal terms. Since the contribution of non-diagonal terms to the eigenvalues is finite, the search for negative eigenvalues can be restricted to the linear subspace generated by variables satisfying (4.1). The submatrix of the Hessian related to these variables has zero trace so that there must be eigenvalues of both positive and negative sign (having all the eigenvalues equal to zero is an event of zero probability). So the union of the eigenspaces related to the negative eigenvalues o f the Hessian is a linear subspace strictly contained in the space of variables with hi = 0. It is an interesting remark [14] that similar results can be obtained directly in the discrete system ; in fact, using definition (2.3), a barrier configuration must be higher than the configuration which preceeds or follows it in the path and differ from them just in one site ; calling these sites 1 and 2 this property many be written as : with  2329 The stability of the path with respect to commutation of inversion operators gives : Equations (4.3) and (4.5) lead to : and N is even one has : For Gaussian distributed couplings equation (4.6) gives : while, in general, one has hi = 0 (1 ) for low energy configurations [15]. 5. Exhaustive search for stationary points. It is worth noting that the considerations of the previous section allow, in principle, a complete enumeration of the stationary points of function F in 8 ---> 00 limit for any finite system. In fact assuming condition (4.1) satisfied in P sites (sites 1,..., P for simplicity) and equation (3.4) valid in the other sites one has : with so that variables satisfying (4.1) must solve the linear system (5.1). Then, given a spin configuration in N - P sites, one must solve (5.1) ; if the solution satisfies 1 mi 1 1 for i = 1, ..., P and each of the ri in the other sites has the same sign of the corresponding spin one has found a stationary point. Performing these steps for all possible spin configuration of a given choice of sites and for all possible choices of sites gives all the stationary points at a fixed P. Repeating this procedure for P = 0,..., N one completely determines all the stationary points of the function for a given set of Jij’s. Following the same line of reasoning as before one may obtain an analytic computation of the number of stationary points with a given a = P /N and f = F/N when N --+ ao (1) ; one must consider : (1) We gratefully acknowledge B. Derrida for crucial observations about this point.
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