Energy barriers in SK spin-glass model

HAL Id: jpa-00211063

https://hal.archives-ouvertes.fr/jpa-00211063

Submitted on 1 Jan 1989

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished 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.

Energy barriers in SK spin-glass model

D. Vertechi, M.A. Virasoro

To cite this version:

D. Vertechi, M.A. Virasoro. Energy barriers in SK spin-glass model. Journal de Physique, 1989, 50

(17), pp.2325-2332. �10.1051/jphys:0198900500170232500�. �jpa-00211063�

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.

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}

0,34 ± 0,08.

Abstract.

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. ^For

low 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}

0.34 ± 0.08.

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 http://dx.doi.org/10.1051/jphys:0198900500170232500

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.

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 evaluation of 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 function F. 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 original 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

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.

2330

where ai = ^± 1, .) J stands for the average ^over the bond distribution and det Jp 1 is ^the

normalization of the delta functions ; the evaluation of (5.4) ^{is now} in progress.

At any rate, ^as far as barriers are concerned, ^the interesting ^{case is} given by ^finite

P when Nu _{oo ;} in fact for large P the number of negative eigenvalues of the Hessian is

approximately P /2 [16] while barriers are related to saddle points ^{with few} (0 (1 ) ) negative eigenvalues. ^{So the} energy and the magnetic fields of barriers in the discrete system ^differ only ^{for small terms} 0 (1/ Ù) from the values attained in the corresponding ^saddle points

of the continuous case.

In our simulations we tried to make the correspondence between the saddle points ^{and the}

barrier configurations found with the algorithm of section 2, forcing the variables satisfying equation (4.1) ^to the values ± 1. This operation often leads to the correct identification but sometimes it causes some magnetic ^field appearing ⁱⁿ (3.4) ^a change ^of sign, ^{so that the} proper

correspondence may involve ^a much more complicated rearrangement ^{of the} configuration.

Saddle points ^are locally optimal solutions of the problem ^{in the} continuous case but when the system is forced to the discrete space ^{one cannot} exclude the appearance of different, ^lower lying, ^barrier configurations.

6. Morse theory.

When dealing ^with smooth surfaces it can be useful to invoke the results of Morse theory [17]

concerning ^the topological analysis of surfaces. For instance, given ^certain boundary conditions, ^{one can define a} topological ^{invariant for a} smooth function f given by :

that is by ^{the sum} upon stationary points ^of f ^{contained in a} domain, ^of (-1 )m,

m being the number of negative eigenvalues ^of the Hessian calculated in these points.

We used this result as an efficiency ^test for the numerical search of saddle-point configurations. ^{Given two} minima of energy E (Eq. (2.2)) ^we considered the restriction of the

hypercube [_ l@ 1]N ^obtained freezing ^those spins which have the same value in the minima.

We have been considering ^{2 to} 9-dimensional subspaces (barriers ^{in such} subspaces ^could provide ^that hierarchy of constraints which Palmer et al. [8] suggest ^{as a mechanism for slow}

dynamics) ; up ^to 6 dimensions we were able to perform ^an exhaustive search for stationary points ^{as described in the} previous section ; with these points ^we always matched the correct invariant (equal ^{to + 1 in our} case). ^In bigger subspaces, ^{where an} exhaustive search is

computationally ^too heavy, ^{we tried to find the} stationary points of function F (Eq. (3.1)) ^{at a}

finite value of {3 through repeated minimizations of function G (Eq. (3.5)). Unfortunately ^the

latter approach ^{led to the correct result} only ^{in small} subspaces ^{where the} previous ^{one is still} practicable ; ^{in one case} with 9 variables we missed the correct invariant even after 200 minimization trials, when each of the six stationary points that could be found had been retrieved at least three times.

7. Numerical results.

Using ^the algorithm ^{described in section} 2, ^{we studied} the relation between the Hamming

distance dab and the energy barrier Eab separating ^two metastable states a, b. ^{We considered} respectively 1 000, 500, ²⁰⁰ samples ^for systems ^{of size N}

48, 64, ^96. During ^the simulations

we found that, ^on the average, barrier heights sensitively depends ^on the energy of the

metastable states (the ^lower these states, the higher ^the barrier). ^{To have} comparable results,

we considered, ^{at various} N, ^minima lying, ^on the average, in the ^same energy density range,

taking ^{into account the} 5, 8, ^{20 lowest} lying metastable states for N

48, 64, ^96.

Measuring the barrier from the higher ^{of the} corresponding ^{minima, we} performed ^{a white}

average of barriers ^{at a} fixed value of da6, obtaining ^the plot ⁱⁿ figure ^1.

Fig. ^1.

Mean value of energy barriers ^versus normalized distance d/N ^{for N}

⁴⁸ (crosses),

N

64 (diamonds), ^N

⁹⁶ (squares).

The height of barriers appears ^to be, ^on the average, ^a non-decreasing function of the

distance, suggesting ^a possible one-to-one correspondence between the hierarchical tree related to energy barriers and the one of the distances. Such relation would explain ^the

ultrametric property ^{of the states in} phase space emerging ^{from the} analytic solution of the model.

To make this correspondence ^{exact in the} thermodynamic limit, ^one needs the relative fluctuations in the barrier height go ^to zero, ^{at a} fixed distance, when N --+ 00.

Our data, ^{shown in table} I, ^{do not exhibit} such convergence but ^more data and greater ^sizes would be necessary ^to make this test significant.

Table I.

Relative fluctuation in energy barriers for ^{some values} of normalized distance.

2332

It is worth noting ^{that we} expect ^a significant dependence of the barrier on the distance only

when the corresponding metastable states do not differ much in energy. If ^one minimum is much higher ^{than the other, we} expect ^{a barrier near to} the upper state, almost independent

from the position ^{of the lower one.} Numerical simulations show that the vast majority ^{of the}

minima lies on the slopes ^of big valleys ^{so that}

local barriers

could significantly ^contribute

to the mean value. We checked the relevance of these barriers limiting ^the computation ^{of the}

mean value to minima with a difference _{in energy} getting smaller and smaller. From this test

we had that discrepancies ^are important only ^for normalized distance d/N ^0.15.

The plot ⁱⁿ figure ^{1 exhibit a} plateau ^for d/N 2: ^{0.5 ;} this behaviour is due to the presence of the lowest time-reversal barrier (between ^{a state} and its time-reversed state) [11], ^{which is}

common to many couples of local minima.

With our numerical data we could estimate a scaling form for barrier heights. Since it is believed that the scaling of time-reversal barriers is not the same of the others [5], ^{we limited}

the comparison ^to the interval 0.25 d /N ^0.40. Assuming ^barrier heights growing ^{with the}

number of spin ^{N as} N ", ^{we had 0.26 a 0.42.}

Acknowledgments.

We acknowledge ^B. Derrida, ^M. Mézard, N. Sourlas for useful discussions ; ^{we are} especially grateful ^to G. Parisi for many stimulating suggestions.

One of us (MAV) acknowledges ^a fellowship by the J. Simon Guggenheim ^Memorial

Fundation.

References

[1] SHERRINGTON D. and KIRKPATRICK, Phys. Rev. Lett. 32 (1975) ^1792.

[2] ^{PARISI G.,} Phys. Rev. Lett. 43 (1979) 1754 ; ^J. Phys. A ¹³ (1980) L117, 1101, ^1887.

[3] ^PARISI G., Phys. Rev. Lett. 50 (1983) ^1946.

[4] ^{MÉZARD M., PARISI} G., ^SOURLAS N., TOULOUSE G. and VIRASORO M. A., Phys. ^{Rev. Lett. 52} (1984) 1156 ; ^J. Phys. ^{France 45} (1984) ^843.

[5] MACKENZIE N. D. and YOUNG A. P., Phys. Rev. Lett. 49 (1982) ^301.

[6] SOMPOLINSKY H., Phys. Rev. Lett. 47 (1981) ^935.

[7] ^PALADIN G., MÉZARD M. and DE DOMINICIS C., J. Phys. Lett. France 46 (1985) ^L985.

[8] ^{PALMER R. G., STEIN D. L.,} ABRAHMS E. and ANDERSON P. W., Phys. Rev. Lett. 53 (1984) ^958.

[9] CHAMBERLIN R. V., Phys. ^{Rev. B 30} (1984) ^5393.

[10] ^MÉZARD M., PARISI G. and VIRASORO M. A., Spin ^Glass Theory ^and Beyond (World Scientific, Singapore) ^1988.

[11] ^{NEMOTO K., J.} Phys. ^{A 21} (1988) ^L287.

[12] THOULESS D. J., ANDERSON P. W. and PALMER R. G., ^Philos. Mag. ³⁵ (1977) ^593.

[13] ^RAMMAL R., TOULOUSE G. and VIRASORO M. A., ^{Rev. Mod.} Phys. ⁵⁸ (1986) ^765.

[14] ^PARISI G., private communication.

[15] PALMER R. G. and POND C. M., J. Phys. ^{F 9} (1979) ^1451.

[16] EDWARDS S. F. and JONES R. C., ^J. Phys. ^{A 9} (1976) ^1595.

[17] BOTT R. and MATHER J., ^Battelle Rencontres, Eds. C. M. De Witt and J. Wheeler (New York,

Amsterdam, ^{W. A.} Benjamin) 1967, p. 460.