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

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

J. Vannimenus, G. Toulouse, G. Parisi

To cite this version:

J. Vannimenus, G. Toulouse, G. Parisi. Study of a simple hypothesis for the mean-field theory of spin-glasses. Journal de Physique, 1981, 42 (4), pp.565-571. �10.1051/jphys:01981004204056500�.

�jpa-00209043�

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

J. Vannimenus, ^{G. Toulouse}

Laboratoire de Physique de l’Ecole Normale Supérieure, 24, ^rue Lhomond, 75231 Paris 05, ^France

.

and G. Parisi

INFN, ^Frascati (Roma), Italy

(Reçu ^{le 30 octobre} 1980, accepté le 8 décembre 1980)

Résumé. 2014 Nous présentons ^{une étude détaillée des} conséquences ^d’une hypothèse simple pour le modèle de

Sherrington ^et Kirkpatrick

^{à 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 simula- tions 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.

A detailed study ^is presented of the consequences of a simple hypothesis ^{for the} Sherrington-Kirk- patrick model, namely ^{that in the} spin-glass phase ^the entropy ^is independent ^{of the} applied magnetic ^{field. This} hypothesis ^{leads to} predictions ^{in excellent} agreement ^with the results of available Monte-Carlo simulations, ^for

the ground-state energy, the entropy ^{in zero} field, the energy and spontaneous magnetization ^{as a function of the}

interaction mean value Jo.

The relevance of mean-field theory ^{to real} spin-glass materials is discussed, underlying ^{the extant areas of} agreement and suggesting ^new experiments.

Classification

Physics ^Abstracts

05.50

75.50K

1. Introduction.

This _paper is the third in a

series [1, 2] examining the consequences of a simple Hypothesis ^{for the} spin-glass phase, ^{of the infinité} ranged Sherrington-Kirkpatrick ^{(S-K) model} [3, 4].

Interest in this model started with the physical ^idea

that its solution is of mean-field type and that it should provide ^{the best} definition of what mean-

field means for a spin-glass.

We are therefore led to address two issues :

i) ^{How close to the true solution of} the S-K model does this hypothesis lead ? This involves a comparison

between the existing Monte-Carlo data and ^our pre- dictions.

ii) ^{What comes out of the} comparison ^with experi-

mental measurements on real (three-dimensional) spin-glass materials ?

2. The S-K model and the projection Hypothesis.

The Hamiltonian of the S-K model in a uniform

magnetic ^{field is}

for N Ising spins Si. ^{The bond} interactions Jij ^are

taken as independent ^random variables, ^{with a Gaus-}

sian random law of mean value Jo/N ^and of variance

J/k ^(the normalizations ensure a sensible thermo-

dynamic 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} replica- symmetry breaking.

The projection (or PaT) hypothesis consists in

assuming that within the spin-glass phase (or replica- symmetry ^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 http://dx.doi.org/10.1051/jphys:01981004204056500

566

where q is the Edwards-Anderson (E-A) ^order para- meter.

Expression (2) implies that the free _{energy F} has

an additive character :

3. Fits with exiating Monte-Carlo data for the S-K model.

Early Monte-Carlo 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 replica-symmetry 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 expan- sions at low temperatures ( T ~ 0) ^{and around the}

transition temperature (T - 1) ^{can be obtained and}

have been derived [1]. ^{Between these limits, the calcula-}

tion can be done numerically. The results for the entropy S( T) ^are displayed ⁱⁿ figure ^1. They ^fall slightly

below the Monte-Carlo 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. ^1.

Entropy ^{of the} S-K ^model computed ^{via the} projection hypothesis (H

0). ^The triangles represent the Monte-Carlo data of Kirkpatrick ^and Sherrington, ^{obtained on} 500-spin 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 zero-temperature

MS-K (H) ^{derived from the} original ^S-K theory ^without replica-symmetry breaking is also drawn for compari-

son. The initial slope corresponds ^{to a} susceptibility X(O) ^{= 1 in the} spin-glass phase, ⁱⁿ agreement ^with

Fig. ^2.

Zero-temperature magnetization ’predicted by ^the pro-

jection hypothesis (full curve), compared ^{to the} magnetization computed ^without replica-symmetry breaking (dot-dashed curve).

Parisi’s scheme and with the most recent computer simulations [5, 6] (Fig. 2).

e Ground-state _energy, Jo ^{= T = H = 0.}

The ground-state ^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 ground-state energy without replica-symmetry breaking ^{Es-’ = -} (2/n)1/2 :

The sign of (M 0 S - K _ M ) corresponds ^{to the} expecta- tion that the true ground-state energy is higher ^than E S-K (Fig. 2). ^{One hnds} with the PaT hypothesis :

hence

whereas Parisi [5] ^obtains Eo = - 0.763 3 ± ^0.000 1,

a remarkably ^close âgreement. [In ^fact, (ÊMIDT)

is _very small on the critical line and the result for Eo

is altered by ^{less than 10-4 if for} instance the maximum value of MS-K at given ^{H is used in} equation (7),

instead of M(H) given by ^the projection hypothesis.]

e Ground-state _energy as a function of J(» ^{H = 0.}

It was shown in reference [2] ^{how a} simple ^thermo- dynamic argument ^{allows to} derive the general ^case Jo * 0 from the case Jo ^{= 0. For the} ground-state

energy Eo(Jo), this leads to the prédiction :

Fig. ^3.

Ground-state energy per spin Eo ^{as a} function of Jo.

The triangles ^{with error bars} represent the Monte-Carlo results

on 500-spin samples [Ref. 4].

The complete ^{results are} plotted ⁱⁿ figure 3, together

with the Monte-Carlo data of Kirkpatrick ^{and Sher-} rington [4] (500-spin samples). ^The agreement ^is considerably better than with the S-K theory ^without replica-symmetry breaking ; ^it would still be improved (for Jo 1) by taking ^{into account the size effects in}

the Monte-Carlo data which are expected ^{to become}

more pronounced ^{in the} spin-glass phase.

Spontaneous magnetization ^{as a} function of JO.

A non-zero ground-state 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}

Monte-Carlo data [4], obtained with a field H ⁼ 0.01.

The agreement ^is considerably better than with the

non-replica-symmetry 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.

Fig. ^4.

Zero-temperature 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} Monte-Carlo data in a

held H

0.01 [Ref. 4].

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} spin-glass ^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, corres-

ponding ^{to the} exactly ^soluble spherical ^model [7].

Comparing ^these two extreme limits, ^we recognize

that the susceptibility ^is predicted ^{to be constant in}

the spin-glass 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 mean-field theory ^{for a real} (three-dimensional)

material. By ^its définition, mean-field 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 so-called lower critical dimension)

and for instance the cusp in the specihc ^heat predicted

for the S-K model is not observed in real materials : We should therefore trust better the mean-field theory deep ^{in the low} température spin-glass phase ^{than in}

the critical region. Besides, ^the fluctuations are expect- ed to be weaker in systems with long-range ^inter-

actions : this favours the standard spin-glass materials,

such as CuMn and similar alloys, ^{for a} comparison

with mean-field 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}

568

been found repeatedly ^on CuMn alloys ^{of various}

concentrations [8-12], ^{when the} measurements are

made with cooling under fixed applied ^field (field- cooling). ^{In these} conditions, ^the magnetization ^is

found to be at equilibrium ^i.e. time-independent,

whereas it is not when the field is applied ^{at low tem-} peratures ^{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 field-cooling ^is reversible).

Another _very important prediction ^{of the mean-}

field theory ^{is the existence of a critical line} H c( T),

below which replica-symmetry breaking ^takes place.

Note that this prediction ^goes against ^an established

folklore, claiming ^{that the} spin-glass transition disap-

pears 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 deter- mination which are best illustrated by the sketch made in figure ^{5. In finite} field, mean-field 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 spin-glass 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 con-

sequence, the définition of Hc(T) by the end in the spin- glass plateau ^for M(T) ^{is not} very precise [12].

An alternative experimental definition of the insta-

bility ^{line is} through ^{the onset of} irreversibility ^effects [14], ^and it will be interesting ^to compare both deter- minations.

A third domain of comparison ^{between mean-field} theory and experiments ^concerns the anomalous power laws predicted by ^{the PaT} hypothesis. ^{Thus it is} predicted ^{that the} equilibrium magnetization ^M

under small field H behaves like :

Detailed comparison ^with experiments ^{would be} very valuable and is under _way [12].

Finally, it would be helpful ^{to have more} experi-

mental data on the existence and properties ^{of the}

Fig. ^5.

^{Sketch of} mean-field theory predictions, showing quali- tatively ^the position of the transition line Hc(T) ^{and the} fuzzy ^cross-

over line Hp(T).

mixed spin-glass-ferromagnetic 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. Replica-symmetry breaking ^and projection pro-

perties.

^{It has been} shown, in section 3, ^{that the} projection hypothesis (2), ^{and its} consequence (3),

lead to predictions for various thermodynamic quan- tities which are very close to the truth. There is also

some evidence that the other projection hypothesis (4),

for the E-A 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} replica-symmetry breaking ^ana- lysis, ^{and how} they ^{could be} intimately ^{related one to}

the other.

5.1 REPLICA-SYMMETRY BREAKING. - For this _pur- pose, ^we shall briefly ^{review some features} of the Parisi

replica-symmetry breaking ^scheme [5]. ^{In this} scheme,

the order parameter ^{at a} given temperature ^{is a func-} tion q(x), ^{0 x} 1, ^{which is} expected ^{to be conti-}

nuous and monotonously non-decreasing ; its extremal value is understood as being the E-A order parameter :

The formulae which relate the zero field suscepti- bility x and the energy t ^{to the} (E-A) ^order parameter q

in the oaramaenetic nhase.

have simple generalizations, ^{in this} replica-symmetry

breaking ^{scheme :}

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 posi-

tion of x is a function of temperature ^{in the} spin-glass 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.

Consider ^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 :

A simple prediction ^{can be} derived from this formula.

At low temperatures, provided q(T) ^{behaves as}

q - 1

a T 2... ^the _position ^{of the break} _point ^x

in q(x) ^{will tend to} 1/2 :

As we shall see, this prediction ^{of the} scaling assump- tion 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 func-}

tion 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

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 assump- tion.

5.3 COMPARISON WITH THE VARIATIONAL CALCULA- TION OF q(x).

The 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)

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}

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}

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 ineffi- cient 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 func- tion q(x, T) satisfies the scaling assumption ^{within the}

numerical precision. The function q(x) ^{has been} para- metrized as

and we have looked for the values of _xn which maxi- mize 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} para- meters. As a by-product ^{of the} calculation, ^the comput- ed value of the susceptibility is x ^{= 1.00011} (for

T ⁼ 0.2 and k ⁼ 20) ⁱⁿ good agreement ^{with the}

suggestion that x - ^{1 in the} spin-glass 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

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 E-A}

order parameter q, according ^to (22).

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 implica-

tions. Carrying (22) ^into (19) ^and deriving ^with respect

to T, ^one gets :

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 scal-

ing assumption, ^a relation between a thermodynamic quantity, ^the specihc ^heat, and the E-A order _para- meter. 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 equa- tion (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 assump- tion (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 :}

and using ^relation (21) ^one gets ^{the additive} property of the free energy (5), from which (2) ^and (3) ^{are obtain-}

ed. Since relation (28) ^{is not} exactly verified, it is clear that the set of equation (29) ^{has to be} only approxima- tely ^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.

It was pointed

out in reference [1] ^{that the} position ^{of the maximum}

with respect ^{to H of the} entropy SS - K(T, H) computed

without replica-symmetry 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} computatio-

nal results, using ^both équations independently.

These results are displayed ⁱⁿ figure ^{8, where the}

locus of the maximum of SS-K ^{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 replica-symmetry breaking. [Indeed, according

to hypothesis (2) ^the spin-glass 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 replica-symmetry breaking ^be

related to the impossibility ^{for a} disordered system such as a spin-glass ^{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 replica-symmetry breaking Hc(T). ^The entropy SS_K computed ^without replica-

symmetry 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 discus- sions on the present work and its relevance to the

experimental results, ^{and we thank S.} Kirkpatrick

for kindly providing ^the original figures representing

his Monte-Carlo results.

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