**HAL Id: jpa-00232116**

**https://hal.archives-ouvertes.fr/jpa-00232116**

### Submitted on 1 Jan 1982

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**What is the static Edwards-Anderson order parameter**

**in the mean-field theory of spin glasses ?**

### H.-J. Sommers

**To cite this version:**

### H.-J. Sommers.

### What is the static Edwards-Anderson order parameter in the mean-field

### the-ory of spin glasses ?.

### Journal de Physique Lettres, Edp sciences, 1982, 43 (21), pp.719-725.

### �10.1051/jphyslet:019820043021071900�. �jpa-00232116�

L-719

### What

### is

### the static

### Edwards-Anderson

### order

_{parameter}

### in the

### mean-field

_{theory }

### of

_{spin }

_{glasses ?}

H.-J. Sommers

Institut

_{Laue-Langevin, }

156X, 38042 Grenoble Cedex, France
(Reçu le _{29 juin }1982, accepte le 17 _{septembre 1982)}

Résumé. 2014 Nous présentons une preuve, par la méthode des

### répliques,

de la### proposition

suivante : la valeur moyenne### quadratique

du_{spin }local dans le modèle de verre de

_{spin }de

_{Sherrington }

et
### Kirkpatrick

est_{égale }au minimum q(0) des

_{paramètres }d’ordre q(x) de Parisi. Ceci correspond à

### l’interprétation dynamique

de_{Sompolinsky }

concernant _{q(x). }On montre que q(0) obéit à une loi

de

_{puissance q(0) ~ }

b2/3 pour de faibles champs ### appliqués b

et pour des domaines de températures 0 T_{Tc. }De

_{plus }on calcule l’aimantation en

_{champ }faible à toutes

_{températures. }

Les calculs
sont basés sur une relation exacte pour la ### susceptibilité

locale renormalisée,### qui

remplace l’hypothèsede Parisi et Toulouse.

Abstract. 2014 We

present a

_{replica }

derivation for the claim that the ### averaged

_{square of the local }

### spin

### expectation

value in the_{long-range spin }

_{glass }model is

_{equal }to the minimum,

### q(0)

of Parisi’s orderparameters

### q(x).

This_{corresponds }to the

_{dynamic interpretation }

of _{q(x) }due to

_{Sompolinsky. }

We
show that _{q(0) }

_{obeys }

a power law ### q(0) ~

b2/3 for low external fields b and all temperatures 0 T_{Tc.}Furthermore we calculate the low field

### magnetization

for all_{temperatures. }The calculations are

based on an exact relation for the renormalized local

_{susceptibility }

which _{replaces }the Parisi-Toulouse

### hypothesis.

### LE

### JOURNAL

### DE

_{PHYSIQUE - LETTRES}

J. _{Physique }-
LETTRES 43 ( 1982) L-719 - L-725
Classification
Physics Abstracts
75.40
1 er NOVEMBRE _{1982, }]

Edwards and Anderson

_{[1] suggested originally }

the time _{persistent }

_{part }of the local

_{spin }

cor-relation as the

### spin glass

order_{parameter}

We consider for

_{simplicity }

an _{Ising }

model _{(Si = }

_{± }

_{1). }

The bar denotes the _{configuration }

_{average}

with _{respect }to the random

_{exchange }

interactions and the brackets denote the thermal or _{}

dyna-mical _{average. In }the mean-field

_{approximation }

made _{by }

Edwards and Anderson and
### Sher-rington

and_{Kirkpatrick }

_{[2] }

_{q }

was _{replaced by }

the static Edwards-Anderson order parameter
Edwards and Anderson

_{implicitly }

assumed the _{equivalence }

of both definitions since for _{very}

### long

times the_{dynamical }

correlations should _{decay. q }

is the order _{parameter }which can be calculated

_{by }

Monte-Carlo _{computer simulations }

_{[3]. }

_{Unfortunately }

the definition _{(1) }

seems to be
L-720 JOURNAL DE _{PHYSIQUE - }LETTRES

ill-defined for

_{long-range spin }

_{glasses. }

_{Sompolinsky }

_{[4] suggested }

a continuum of order _{parameters}

where _{LX }are relaxation times

_{arranged }

in a _{decreasing }

order _{(0 }

x ### 1)

which all_{go }to

_{infinity}

in the

_{thermodynamic }

limit but _{q(x) }

remains a monotonous function between 0 and 1. _{}

### Sompo-linsky

introduced a second order_{parameter }

_{d (x) }

where _{~( 1 - }

_{~(1) }

+ _{J(x)) }

is the local _{}

### suscepti-bility

on a time_{scalerx. }

He ### derived,

with the_{help }

of certain _{assumptions, }

a selfconsistent _{theory}

which turned out to be _{equivalent }

to the _{theory }

of Parisi _{[5] }

with _{J’(x) = - xq’(x) }

_{subject }

to
- d

### ’(x)jq’(x) being

monotonic. However in the Parisi_{theory }

the definition of x is _{slightly }

different
so that _{q(x) }

is constant outside a certain interval where it increases _{monotonicly. }

The definition
of the _{boundary }

values is invariant. Thus we have two well defined limits ### q(O)

and_{~(1). }

It was
### suggested

### [6, 7]

that_{~(1) }

is _{equal }

to q since this accounts well for the results of _{computer }simu-lations

_{[8]. }

_{However, }

it has been _{pointed }

out _{by }

_{Young }

and _{Kirkpatrick }

_{([9], }

hereafter referred
to as

_{YK) }

that this _{interpretation }

is _{wrong }since

_{they }

determined _{numerically }

an _{inequality}

for the internal _{energy per }

_{spin, }

u,
which is violated

### by

the_{Parisi-Sompolinsky }

solution _{if q = ~(1), }

since LJ _{’(x) }

0.
We are

_{considering }

an _{Ising }

model of N ### interacting

### spins

in external### fields bi

with theHamil-tonian

where the

_{long-range }

interactions have the distribution
### Inequality

### (4)

can be_{proved }

_{rigorously following }

_{Bray }

and Moore _{[10]}

The second

_{equality }

follows _{by partial integration, }

the _{inequality }

follows _{decomposing Si Sj >}

2

into

_{Xij }

_{+ }

_{Si ~ }

_{S~ ~ }

and _{using }

the _{positivity }

of the matrix _{Xij }and

_{of ( ~ (( }

_{~ )~ 2013 ~ ~ ~)) . }

.
### B’

/### According

to the_{theory }

of _{Sompolinsky ~(1) }

is the _{quantity }

that is calculated in _{computer}

simulations ### and q = q(0).

The last claim has never been written down but it is the naturalconse-quence of

### Sompolinsky’s theory.

We### present

below a_{replica }

derivation of this statement which
also reflects _{why }

the _{replica }

derivation of the local _{susceptibility }

does not _{give }

the result which
is obtained for finite N _{by varying only }

the one local field at the site considered. The _{replica}

derivation _{gives }

the renormalized local _{susceptibility }

discussed below.
YK claimed

_{that q }

cannot be _{simply expressed by }

_{q(x). }

However _{they calculated q using }

a
restricted ensemble of ### spins having

a_{positive projection }

onto the _{magnetization, }

since _{then q}

as function of the external field b _{extrapolates }

_{linearly }

to a finite value for & -~ 0. _{However,}

there exist _{many }

_{possibilities }

of such restricted ensembles which all _{give }

different results for
b --> 0 _{(a }

few have been _{presented }

_{by }

_{YK). }

_{They }

seem to _{correspond }

to the different relaxation
times of _{Sompolinsky. }

The linear _{extrapolation }

for & -~ 0 _{may }

_{correspond }

to a certain _{q(x).}

One calculation of YK without restriction is well _{compatible }

with a _{power }law

### dependence of q

for & -~ 0. It shows that the linear

_{extrapolation }

calculated with the restricted ensemble mentioned
above starts at a field which is an order of

_{magnitude }

_{greater }than the field where the

_{magnetization}

### rapidly

goes to zero. We will show below that_{q(O) }

has a _{power }law

### dependence

for all _{temperatures }T > 0 below the critical

_{temperature }

_{T~. }

This has been shown _{by }

Parisi _{[6]}

for T ~ _{T~. }

Besides it follows that _{q(O) = }

0 for b = 0 which coincides with the result of
Mor-genstem and Binder in two and three dimensions

_{[11]. }

Thus the _{spin glass phase }

is critical for
all _{temperatures }

0 T _{7~. }

The above critical index is universal and _{temperature }

### independent.

### Sompolinsky

and_{Zippelius [12] }

obtained a _{temperature }

### dependent

critical index for the low### frequency

behaviour of the_{dynamical spin }

correlation. ### However,

the connection of the_{dynamical}

### theory

with the full_{equilibrium theory }

is not _{quite }

clear because of the ill-defined limits for
t ~ oo

### (Eq. (1)).

The above critical index### (Eq. (8))

is based on an exact relation for the renormalized local_{susceptibility }

which is valid for all _{temperatures }

and external fields in the _{spin glass phase}

and has been derived _{by }

the author from the _{Sompolinsky equations. }

It _{replaces }

the
Parisi-Toulouse _{hypothesis [13] }

and the _{Bray-Moore }

criterion for a soft mode _{[14]. }

Furthermore we
calculate the

_{magnetization }

and _{susceptibility }

for low fields and _{explain }

the difference between
local _{susceptibility, }

renormalized local _{susceptibility }

and total _{susceptibility.}

We start with a

_{replica }

derivation ### of q = q(O).

The_{replica }

method consists of _{averaging}

Tr W" with a

### positive integer n and W

=### exp( - #H)

and### trying

to obtain### log

Tr W from thecontinuation ~ -~ 0. We may also

### directly

calculate derivatives of the free energy### by

the### replica

method. A crucial

_{point }

is that Tr W" can be written as an _{integral }

over ### variables

_{~,}

0 a

### /3 n,

which can be evaluated for N -~ oo### by

saddle### point integration.

Let us dividethe n

_{replicas { a } }

into two _{groups }

_{{ }

_{CXI }

_{}, { }

_{a2 } }

_{with n 1, n2 }

_{replicas}

Assume now that the local external field

### bil~

in the first_{group }is different from the field

### b~2~

inthe second _{group. }Then

The index a or _{a 1, }_{a2 }indicates which

### replica

the### spin S a belongs

to._{S~ ~ 1, Sl ~ 2 }

mean the
### expectation

values in the fields_{b~l~, i }

_{respectively. }

The traces should be normalized so that
Tr 1 = 1. W" in

### equation (10)

_{means }

### actually M~’. W 2

with different fields in the two_{groups.}

We have

_{explicitly }

used the fact that _{b~ 1 ~ }

is _{independent }

of _{a 1 }and

_{bi (2) }

is _{independent }

of _{(X2. }Thus

Tr Wn

_{Sf1 Sf2 }

is a constant.
L-722 JOURNAL DE

_{PHYSIQUE - }

LETTRES
We are interested in the limit N -+ oo. The r.h.s. of

### equation

### ( 11 )

can be evaluated_{by }

the method
of

### steepest

descents if we_{interchange }

the limits n -+ 0 and N -+ oo. This is the standard _{way }to

### proceed.

The result becomes_{proportional }

to the normalization factor Tr Wn _{which goes }to 1

for n --~ 0. We use the fact

_{that ! ~~ ~ 2 }

is _{independent }

of i. _{Then it is easy }to see

### (e.g.

### by

the methodof Ref

_{[7]) }

that the result is
where

_{q~~a2 }

means the value at the _{stationary point. }

This statement is not trivial in the case of
### replica

### symmetry

### breaking,

since_{equation (12) }

is _{only }

valid _{if qa ~ a~ }

is constant _{for a }

_{belonging}

to the first _{group }and

_{(t2 to }

the second group. _{qa 1 ~~ }

is not maximized as claimed _{by }

Thou-less et al.

_{[7]. }

For a _{system }

of two _{replicas }

_{equation (12) }

coincides with the order _{parameter}

definition of Blandin et al.

_{[15, 16]. }

The two fields _{bi 1 ~ }

and _{b~z~ }

have _{only }

been introduced for the
### partition

of the_{replicas }

into two _{groups. }Further

_{replica }

_{symmetry }

_{breaking }

is _{produced }

_{by}

### interchanging

the limits n -+ 0 and N -~ oo.Now we look for a

### replica

_{symmetry }

### breaking

solution which contains a_{partition }

such that _{(12)}

is

_{independent }

_{of (t1 }

and _{a2. }Furthermore

_{qa~a= }

should be _{independent }

of the choice of the _{partition}

if several

_{partitions }

with the desired _{property }

exist. The solution should _{go }to Parisi’s solution for

_{~ }

-+ ### b~~~

and indeed Parisi’s solution has the### properties

which are needed. The_{symmetry}

### breaking

scheme of Parisi consists in the_{following. }

The zero _{step }

is _{qa~ - }

_{qo }for C( =1=

_{~3. }

In a first
### step

the n_{replicas }

are divided into ### nlm,

_{groups }

### of ml

### replicas

with a_{new q, }

_{within }

each _{group}as indicated in

_{figure }

1. In a next _{step }

the same ### procedure

is### applied

to_{the q }

_{matrices }

and so on.
We see from

### figure

1 that the### only

### possibility

for### dividing

the### replicas

into two groups with thedesired

_{property }

is that _{nlm, }

and _{n2/ml }

are ### integers.

Thus we_{get }

for the Parisi solution in the
limit n -+ 0

In our

_{approach }

the _{partition }

_{(nl, n2) }

is _{given }

first and then the limit N ~--~ oo is ### taken,

which### produces

the stable_{replica }

_{symmetry }

_{breaking }

scheme. Therefore the _{partition }

is not _{completely}

### arbitrary

if we start from Parisi’s solution.We can use the same method to find the averages of

### higher

powers of### Si > :

### Fig.

1. - Schematic form of the matrixqa~ after the first iteration step of Parisi. One

_{possible partition}

which must be constant with

_{respect }

to _{a 1, }

_{(X2’ }..., oe~.

### Again

the### only

### possibility

is that the### decom-position

of the_{replicas }

into m groups must be commensurate with _{the first iteration step of}Parisi. It

_{immediately }

follows
where

_{mo(y) is }

obtained from all the next iteration _{steps }

in the same _{way }as in Parisi’s work

### [17].

Thus

where

_{4J(y, }

_{x) }

_{obeys }

the Parisi differential _{equation}

with the

_{boundary }

condition
### Equation

### (15)

is similar to the_{Sherrington-Kirkpatrick }

case. The ### only

difference is that### mo(Y) =1= tgh

y.### mo( y)

is renormalized due to the interaction### according

to### equation (17)

and isthus

_{additionally }

a functional of ### q’(x).

We now derive the low field

### properties

### using

a relation which has been derived### by

the authorfrom the

_{Sompolinsky }

_{equations}

This

_{equation }

is valid for all external fields and it _{represents }

the correct formulation of the
Parisi-. Toulouse

### hypothesis

### [ 13] :

the### averaged

square of the renormalized local### susceptibility

is constant.It corrects also the

_{Bray-Moore }

criterion for a soft mode ### [14].

_{However, }

one can show that an
### analogous

### equation

exists for all x, for x = 1 it takes the form of### Bray

and Moore.### Equation (19)

follows from the

_{selfconsistency }

_{equation }

for _{q(x) }

differentiated with _{respect }

to _{x.for }

x -+ 0
### (in

the_{Sompolinsky formulation). }

A more detailed derivation will be ### published

elsewhere. Inaddition we have

### equation

### (13)

### Expanding

both_{equations }

with _{respect }to

### Pb

=_{fib }

_{+ }

### /~Jz q(0)

_{we }

_{get :}

and

### ~(0)

turns out to be of order_{(Pb)2/3. }

We _{multiply equation (21) }

with _{(pb)~ }

subtract both
equa-tions and take into account _{only }

terms _{up }to the order

_{(Pb)2. }

_{Performing }

the _{simple }

Gaussian
L-724 JOURNAL DE _{PHYSIQUE - }LETTRES

In the last

_{equation }

we _{may }

### replace

### mo3~(o)

_{by }

the value for ### b --+ 0,

i.e._{neglecting }

the
correc-tions of

_{q’(x) }

due to the external field. Thus we have _{proved }

the _{power }

_{law, }

_{equation }

_{(8), }

for all
0 T

_{Tc. }

The coefficient is determined ### by

the Parisi solution for b = 0. For T -~_{T~ }

we
have
which coincides with the Parisi result

_{[6].}

Let us now

_{expand }

the _{magnetization }

for b -~ 0
From

_{equation (21) }

we find _{mo(o) }

_{up }to the desired order.

_{Introducing }

this into _{equation }

_{(25)}

we obtain
### Up

to this order we findThis is the usual relation between the

_{magnetization }

and the _{susceptibility }

for low fields.
### However,

because of the_{singular }

### behaviour,

### equation (23),

this is not the total_{susceptibility}

Parisi

### [6]

did not_{distinguish }

between the two _{susceptibilities. mo }

is the renormalized local
### susceptibility,

it is the derivative of the_{magnetization }

with _{respect }to the external field for fixed order

_{parameters }

_{q(O), q’(x). }

Thus it does not contain the critical ### dependence

on the### magnetic

field. For fixed order

_{parameters }

it can be _{expanded }

in _{powers }of the external field. On the other

### hand,

for the total_{susceptibility }

the _{dependence }

of the _{order parameters }on the external field

is needed. The unrenormalized local

_{susceptibility obeys }

a ### simple

Curie law for b = 0 since### q(O)

=### 0,

however this is_{not }

_{a }

### macroscopic thermodynamic

### quantity

in the### spin glass phase.

From the

_{replica }

calculation it is _{easily }

seen that
The above

_{replica }

derivation shows that the _{right }

hand side is _{only equal }

to the
unrenor-malized local

_{susceptibility, ~ Si )/b~bi }

= 1 - _{( Si }

_{)2 }

if _{~ }

is _{constant, i.e. without }

_{replica}

References

### [1]

EDWARDS, S. F. and ANDERSON, P. W,, J._{Phys. }

F 5 (1975) 965.
### [2]

SHERRINGTON, D. and KIRKPATRICK, S.,### Phys.

Rev. Lett. 35 (1975) 1792.### [3]

KIRKPATRICK, S. and SHERRINGTON, D.,### Phys.

Rev. B 17_{(1978) }

4384.
### [4]

SOMPOLINSKY, H.,### Phys.

Rev. Lett. 47_{(1981) }935.

### [5]

DE DOMINICIS, C., GABAY, M. and DUPLANTIER, B., J._{Phys, }

A 15 _{(1982) }L47.

### [6]

PARISI, G., J._{Phys. }A 13

_{(1980) }1887.

### [7]

THOULESS, D. J., DE ALMEIDA, J. R. L. and KOSTERLITZ, J. M., J._{Phys. C }

13 ### (1980)

3771.### [8]

PARISI,### G.,

Philos._{Mag. B }41 (1980) 677.

### [9]

YOUNG, A._{P., }

_{KIRKPATRICK, S., Phys. }Rev. B 25

### (1982)

440.### [10]

BRAY, A. J., MOORE, M. A., J._{Phys. C }

13 ### (1980)

419.[11] MORGENSTERN, I. and _{BINDER, }_{K., }

_{Phys. }

Rev. Lett. 43 _{(1979) }1615,

### Phys.

Rev. B 22 (1980) 288, Z._{Phys.}

B 39 (1980) 227.