What is the static Edwards-Anderson order parameter in the mean-field theory of spin glasses ?

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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’hypothèse

de 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 order

parameters

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 the

Hamil-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 natural

conse-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 the

continuation ~ -~ 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 divide

the 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~

in

the 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 method

of 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 the

desired

_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 matrix

qa~ 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 is

thus

_additionally

a functional of

q’(x).

We now derive the low field

properties

using

a relation which has been derived

by

the author

from 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. In

addition 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 find

This 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.

[12]

SOMPOLINSKY, H. and ZIPPELTUS, A.,

Phys.

Rev. Lett. 47

₍₁₉₈₁₎

359.

[13]

PARISI, G. and TOULOUSE, G., J.

_{Physique-Lett.}

41

₍₁₉₈₀₎

L361.

[14]

BRAY, A. J. and MOORE, M. A., J.

_Phys.

C 12

₍₁₉₇₉₎

L441.

[15]

BLANDIN, A., J.

_{Physique Colloq.}

35

₍₁₉₇₈₎

C6-1499.

[16]

BLANDIN, A., GABAY, M. and GAREL, T., J.

_{Phys. C}

13

₍₁₉₈₀₎

403.

[17]

PARISI, G., J.

_Phys.

A 13 (1980) L115.