Local Mean Field Dynamics of Ising Spin Glasses

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Local Mean Field Dynamics of Ising Spin Glasses

Tran Hung, Mai Li, Marek Cieplak

To cite this version:

Tran Hung, Mai Li, Marek Cieplak. Local Mean Field Dynamics of Ising Spin Glasses. Journal de

Physique I, EDP Sciences, 1995, 5 (1), pp.71-83. �10.1051/jp1:1995115�. �jpa-00247045�

Classification

Physics

Abstracts

75.10N 75.10H

Local Mean Field Dynamics of Ising Spin Glasses

Tran

Quang Hung(*),

Mai Suan

Li(**)

and Marek

Cieplak

Institute of

Physics,

Polish

Academy

of

Sciences,

02-668

Warsaw,

Poland

(Received

6 June 1994, revised 6 October,

accepted

10

October1994)

Abstract. The Glauber

dynamics

of three-dimensional

Ising spin glasses

_are studied _numer- ically _m the local _mean field approximation. The _aging eifect is observed in both field cooled and

zero field cooled regimes but the remanent

magnetization

_never reaches true

equilibrium.

The dynamic

susceptiblity

behaves like _in experiments. A double

peak

structure in the real part ^of the

susceptibility plotted

_{as a} function of temperature _{may appear} for

non-symmetric

distribu-

tions of the

exchauge couplings.

This, however, does not indicate reentrancy. The temperature

dependence

of the

dynamic specific

heat is opposite to that found

theoretically

_in small clusters.

l. Introduction

Intensive research _on

Ising

_spin

glasses (SG)

has led to _a basic

understanding

of their

equi-

librium

properties Ill. Theory

of the

dynamic behavior, however,

is less

developed.

Thus

it _seems worthwhile _to

explore predictions

of

simple approximations

which deal with the

dy-

namics. An

approximation

which _we focus _on here is the local _mean field

approach.

This method has

proved

to be _an

adequate

tool for

studying equilibrium properties

of short _range three dimensional SG'S and random field

systems [2, 3].

^{This invites} one to

apply

it _now to

dynamics

_as defined

by

the Glauber model _[4] and descnbed in _more details in Section 2. We

study

the three dimensional Edwards-Anderson

Ising

SG and consider the

following subjects:

the

phenomenon

of _agmg

(in

Sect.

3),

behavior of the

dynamic susceptibility (in

Sect.

3),

the

phenomenon

of reentrancy

(in

Sect.

4),

and

finally,

behavior of the

dynamic specific

heat

(m

Sect.

5).

The technical

advantage

of _our

approach

_over Monte Carlo _is

that,

in

principle,

it allows for much

larger system

sizes and time scales which may

help

to establish _true trends. Ex- act results for the

dynamic susceptibility là-?]

^and

specific

heat [8] ^can be obtained

only

for small clusters and the behavior of these

quantities

_can be related to the structure of local energy

(*)

On leave from Hanoi Technical University.

(**)

On leave from Thai Nguyen Techuical Institute.

©

Les Editions de

Physique

1995

minima. The studies of clusters _are

illuminating

but the

problem

with clusters is that

they display

_no finite critical

temperature

so the cntical behavior cannot be

investigated.

Dur basic results _can be summarized _as follows:

a)

The

aging

effect

[9-13]

^is

present

^{within the} local _mean field

approach.

What it _means is

that the remanant

magnetization depends

_on how

long

_a

magnetic

^field has been

applied

before

switching

it off.

However,

the

magnetization

does not

decay

to _zero

monotonically. Instead,

it reaches _a

non-equilibrium plateau reflecting shortcommgs

of the method.

b)

Dur results _on the

dynamic susceptibility

in SG'S _{are m a}

qualitative _agreement

with the experiments.

c)

For

non-symmetric

distributions of the

exchange couplings

_as studied in the _context of the

SG-ferromagnet

transitions the real

part

^of trie

dynamic susceptibility displays

_a two-

peak

structure _as a function of

temperature,

T. We

speculate

that in expenments ^{this could}

be taken _{as a} manifestation of the

reentrancy phenomenon _[14].

We

show, however,

that this

interpretation

is not correct.

d)

The

frequency dependence

of the

dynaiuic specific

heat is shown to be similar to that found for the

six-spm

cluster [8]. ^The

temperature dependence, however,

is

opposite:

the

maxima in the real and

imagmary

_{parts move} towards

higher temperatures

on

decreasing

the

frequency.

The

dynamic specific

heat has been measured in

glasses [15,

_{16] but} not

yet

^{in SG'S.}

It is

hoped

that this paper will

trigger

interest in such

experiments.

2. Mortel and Metl~od

We consider the Hamiltonian

~ "

~ _J>jS,Sj

_H

~S~, ₍₁₎

<q> ~

where S~ =

+1,

<

ii

> denotes summation _over nearest

neighbors,

the

couphngs fg

_are

Gaussian distributed with trie _mean

Jo

and trie

dispersion

J. Trie

magnetic

field considered bene is time

dependent

and

H =

Ho

₊

Hi

cosuJt.

(2)

According

to Glauber [4],

dynamics

of trie

Ising system

_may be

govemed by

the

followmg equation

(1+

_To ^~ _<

S~(t)

> = < tanh

flhi

> +

(1-

_<

Si

tanh

flh~ >) tanh(flH)

,

(3)

where

hi

=

~j _{fg Sj}

is the

exchange

field

acting

_{on spm}

Si.

^In what follows _we will

adopt

the

J

local mean field

approximation

_m which _<

tanhflh~

> is

replaced by tanh(fl

_{< h~}

>).

This

brings equation (3)

into the form

(i+To))m~(t)

=

tanin(fl~fjmj)

J

+

tanin(flH) Il

_m~

tanh(fl ~j _fg

mg )]

,

(4)

J

where

titi " < S~ >

(5)

In the static

limit, equation (4)

reduces to the standard local _mean field

equation _[2],

1-e- to the TAP

equation

without the

Onsager

reaction _term

II?I.

It should be noted that inclusion of the reaction term in

systems

^{with short} _range

couplings

would lead to

unphysical

results

(the

field cooled state, ^for

example,

_may

acquire negative

_entropy at low

T) _{[2, 18].}

To solve

equation (4)

we will _use the fourth-order

Runge-Kutta

method with the

step

_size control

[19].

3.

Aging

EoEect

There have been several

phenomenological _attempts

to descnbe the

physical

mechanism of agmg

[20-25]. Recently,

it bas been shown that _a mean-field

dynamical

model

(a spherical

SG with

multi-spin interaction)

_can exhibit

aging

effect in trie

thermodynamic

limit

[26].

^Monte Carlo

study

of the 3D Edwards-Anderson mortel

[27, 28] gives quantitative agreement

with

experiments.

In this section _we deal with the standard SG

(Jo

#

o,

J

# o)

and ask what would be the

predictions

of the local _mean field

approximation.

We first consider field cooled

(FC)

and then

zero filed cooled

(ZFC) regimes.

FC REGIME. In _a

typical experiment _[9, loi

in which the FC

aging

^effect is observed _one first cools the system ^{down below the}

freezing temperature, Tg,

^and

keeps

the

magnetic

field

constant. After _a certain

waiting time, tw,

the field is switched off and the thermc-remanent

magnetization, Mrm (t),

_is measured _{as a} function of t. The

asymptotic

^time

decay

of this quan-

tity,

well below

Tg,

is found to follow _an

algebraic

law. This is

found,

for

instance,

in the short range SG

Feo ômno ôTi03 (II,

_12] and in _an

amorphous

metallic SG

(Fe~Niji-~~)76P16B6Al~

[13].

We consider several ways of

cooling.

In the first way, the

system

^is

cooled, using

the static local _mean field

equation,

from T

=

7J/kB (T

>

Tg

m

5J/kB)

down to T

=

2.5J/kB (the temperature step

^is

o.o5J/kB)

and

Ho

# 0.5J. The field is then

kept

fixed for _a

waiting

time tw and trie

dynamic equations

are used. The field is

subsequently

switched off. Trie remanent

magnetization Mrm(t)

is calculated _{as a} function of time t

(from

the _moment when the field is switched

off).

We find that if the

coohng

is done _m the static way the agmg effect is absent:

Mrm(t)

reaches the _same

plateau

at

long

time scales for different

waiting times,

1-e- there _{is no}

dependence

_on tw.

In the second _{way, we} cool the

system

_in the field from above to below

Tg

very

qmckly.

In this _{case a}

spin configuration

alter trie

coohng

is

essentially

random

(spins

take values of +1

randomly)

with _{a zero}

magnetization.

This

configuration

is used _{as a}

starting configuration

to solve

equation (4).

A similar

approach

has been used in trie Monte Carlo simulations of

Rieger [28].

We _can show that within the local _mean field

approximation

trie

aging

effect is absent in

systems

^{with L} < 4 but for

larger

L's it is _more and more

pronounced.

The time

dependence

of

Mrm(t)

obtained for _vanous

waiting

^times is shown _m

Figures

1 and 2 for L

= 10 and L

= là

systems respectively. Clearly,

the

aging

effect is

present

but

Mrm(t)

reaches _a

plateau

instead of

going

to _zero

monotonously.

We have considered times up ^to

105To

and _{saw no}

changes

_m

Mrm(t).

We _now consider the third _way of

coohng

when the

coohng

rate is finite. One starts from

high temperatures

^{and then decreases T}

linearly

with t,

1e.,

T(t)

=

Tst ~~ ~~~t

tcooi ,

where T~t and

Tend

_are

starting

and final

temperatures respectively.

tc~~j ^is a

coohng

time. At T =

2.5J/kB

the field is

kept

for _some

waiting

time and then it is swiched off.

Figure

3 shows

0.4 0.4

3D SG, L=la 3D SG, L=15

,',

0.3 RANDOM CONFIGURATION 0.3 ' RANDOM CONFIGURATION

'.,

',

'

',

a÷ a÷

Î

_f~ ~=100 _f~ ^' ~=100

0.2 ~=10 0.2

(

~"10

ZÎ ^~"~

ZÎ ',

^~~~

',

"

o-i ai

o-o O.O

-1 0 2 3 4 5 -1 0 2 3 4 5

log

t

log

t

Fig.

l

Fig.

2

Fig.

l. The time

dependence

of

Mrm(t)

for the L

= 10 system

(Ho

= 0.5J, T

=

2.5J/kB)

for diiferent

waiting

times _as indicated. The initial

spin configuration

is random

(tca~i

_-

0)

but it is identical for each of the waiting times.

Mrm(t)

reaches _a plateau at long time scales.

Fig.

2. Same _as in

Figure

1 but for L ₌ là.

results for the L ₌ 10 system and tc~~j = To. The _agmg effect is present but it

gets

^weaker

compared

to the _case of the very

quick cooling.

ZFC REGIME. The

aging

elfect _may be aise observed in the ZFC

expenments:

the

sample

is

rapidly

cooled from above to below

Tg

m zero field and

then,

after _a time

tw,

_a small field

is

applied. Then,

the increase in

M(t

+

tw)

_is measured _{as a} function of t.

Consider the second way of

coohng:

the

system

is cooled down very

quickly

to T

=

2.5J/kB

but without the field. After

tw,

the field H

= 0.5J is switched _on. The agmg effect in the ZFC

regime

of the L

= la system is shown in

Figure

4.

Again M(t)

has _a

plateau

at

large

t. The

aging

effect for other

cooling

_ways is shown to be similar to that for the FC

regime.

We have considered various values of

Ho

The results _are

qualitatively

the _{same as} those

presented

in

Figures

1-4. Thus within the local _mean field

approximation

_{one con} observe the

aging

effect but this

approximation

is not sufficient to monitor any further evolution towards

equilibrium.

The fluctuations _are smoothed out and the

system

stays frozen in _a local minimum.

These results _are not

changed

when _an

improved

_mean field method is

adopted,

_as shown in the

Appendix.

It should be noted that _more

comphcated

_{agmg scenanos}

involving

"tem-

perature jumps"

in the _manner of

expenments

^{of reference} _[29] have been also studied

by

the Monte Carlo simulations

[30].

4. AC

Susceptibility

of tl~e Standard SG

The

experimental

literature on AC

susceptibility

of SG is qmte substantial

(see,

_e-g-,

Iii ).

The basic

finding

^is that when _one fixes _uJ and

plots x'

_{as a} function of T then _one

gets

a curve

with _{a maximum.} For

large

uJ's the _maximum is rather broad. On

decreasing

_uJ the

position,

04 0.3

3D SG, L=10

3D SG, L"10

°.~ tcoai~Î

o-z

àÎ

_{~=100 'J}

0 2 _~"10

à

~~~ ~-

~=5

0.1 ~"10

~=io0 0. i

ZFC

regime,

H=o.5J

O.O O.O

-1 0 2 3 4 5 -1 0 2 3 4 5

~°~ log

t

Fig.

3

Fig.

4

Fig.

3. The aging eifect for the L

= 10 system and icaai _{" TO}

(Ho

_" 0.5J, T

=

2.5J/kB).

The system _is cooled down from T

=

8J/kB

to T

=

2.5J/kB.

The temperature _varies with t

linearly.

Fig.

4. The agiug eifect in ZFC

regime

at T

=

2.5J/kB (H

= 0.5J, L

=

10).

The results

correpond

to the _very

quick cooling

_regime

(icaai

_-

0).

o-z o-z

improved

LMF

~=ioo

z~ z~ ^LMF

~

~-zoo ~

S o i ~UIOO S

o i

~j

~=20

~j

à _à

improved LMF

O.O o_o

0 2 3 4 5 _{o 2} 3 4 5

Îog

t

iog

t

Fig.

5

Fig.

6

Fig. 5. The time

dependence

of

Mrrr(t)

obtained in the

improved

local

mean field

approximation

for the L

= 10 system

(Ho

₌ 0.5J, T

=

2.5J/kB)

for diflerent waiting times. The solid, dotted and dashed _curves

correspond

to tw = 20To,100To and 200To

respectively.

The initial spin

configuration

^is

random.

Mrrr(t)

reaches _a

plateau

at

long

times.

Fig.

6. The time

dependence

of

Mrrr(t)

for tw ₌ 100To, T ₌

2.5J/kB,

and Ho ₌ 0.5J. The solid and dashed _curves

correspond

to trie

improved

local _mean field and local _mean field

approximations respectively.

The asterisk marks the value of tp

(tp

cs 155To and 485To _in the local _mean field and improved local

mean field

approximations respectively).

Tw,

of tue maximum _moves towards lower

temperatures

^alrd it

sharpens

_up. In tue DC limit tue maximum

acquires

a

cusp-like

_appearance and Tw -

Tg.

^Tue

absorptive susceptibility x",

_on tue other

hand,

bas _a small but distinct

anomaly

around

Tw.

This

anomaly

looks like _a skewed

peak

or a kink. In order to calculate tue

dynamic susceptibility

for _a state _as close to tue

equilibrium

_as

possible,

_we

adopt

tue

following procedure.

Tue system ^is cooled down

statically

to _a

temperature

^under consideration and then _a small AC field

(Ho

₌

0)

^is

switched _on, and tue Glauber

dynamics

_are

implemented.

Thus for _{a given} T tue

starting spin configuration

^{is obtained from tue} static solution of tue local _mean field

equations.

Tue AC field is switched _on at t

= 0. We start to monitor tue

magnetization

after _some time to, ^chosen

so that ail transient

exponentials

_can be considered

extinct,

and _we fit it to

M(t)

=

Hi IX' sin(uJt)

_X"

cos(uJt)j (6)

In _{our case one} needs to about

3To,

where To is a

period _(To

"

27r/w),

and tue _average is done

typically

_over 7

periods

which bas been found to be _more than suflicient. We take

Hi

to be

equal

to 0.01J: smaller values of

Hi

leave tue result almost

unchanged.

Figure

7 shows tue

temperature dependence

of tue

susceptibility

for selected values of _w

and for

Jo

_" 0. Tue results _are

averaged

_over 10

samples.

Tue maxima of

x'

and

x"

move

toward lower

temperatures shghtly

_{as w} is decreased

indicating

that tue effective

paramagnet-

SG transition

temperature Tg(w)

^{decreases with}

decreasing

w. It should be noted that tue

corresponding

^{maximum of tue static}

susceptibility x(w

=

0)

is located _at T _m 3.8J. Within tue local _mean field

approximation

this

position

does not coincide with tue

paramagnet-SG

transition

temperature Tg

m 5J which _may be determined

by

other criteria: from tue _maxi-

mum of tue nonhnear

susceptibility

_or from tue condition that tue Edwards-Anderson order

parameter

^vanishes _[31]

(Monte

Carlo calculations

give

_an estimate of

Tg

m J

iii ).

Tue results

presented

in

Figure

7 _are very similar to those obtained for the standard SG

Aui-~Mn~

with

x = o.0298

[32]

^{and for}

Euo.2Sro.85

_[33] and also to what

happens

in tue cluster

systems except

that in small clusters tue

w - 0 limit

corresponds

to a curve with _{no cusp.}

In

Figure

8

X'

and _{X" are}

plotted

_versus

logw

^for T

=

2.5J/kB

and

3.5J/kB.

At low

frequencies X' depends

_on

log

_{w very}

weakly.

As _w goes ^to zero tue

absorption

part

x"

vanishes.

This part of tue

susceptibility

_appears to

satisfy

_an

approximate

relation

X"(uJ)

m

-bôX'(uJ)/ôlnuJ ₍₇₎

with b

= 1.23 + 0.05. Tue

right-hand

of

equation (II)

_is also shown _in

Figure

8.

Lundgren

et ai. [34] derived _a b of

7r/2

which dilfers from _our result. Tue value of

7r/2 depends

_on tue

assumption

of

log-uniform

relaxation times which does not seem to be accurate _in tue local

mean field

approximation.

Figure

9 shows tue Cole-Cole

plot

for

X'

and

x"

^for two selected values of T below

Tg.

^These

curves bave _a

shape

of _a semicircle.

5.

'Reentrancy'

One of tue

interesting problems

of tue SG

physics

_{is a}

question

^{of tue}

reentrancy:

is there

a transition from tue

ferromagnetic phase

to tue SG

phase

_on

lowering temperature.

So

far,

theoretical

studies

of static

properties

of

short-range Ising

SG'S

give

_no evidence ioT the _reen-

trancy phenomenon [14].

One may

suggest

^{that the} reentrancy may be

dynamical

in nature,

i e.

resulting

from

considering

finite observation times. In order to check

this,

_we bave stud- ied tue

temperature dependence

of

x(w)

in

systems

with

asymmetrically

distributed

exchange

couplings

for _various

frequencies.

0A ,-,

°'~

,'p~,

'~

~~°

0.3

T=2.5J/kB

~ o oi

T=3.5J/kB

0.3 ~ ^---,

~ ^. 0.05 _{o_z}

~ Ù-1

o-z o-i

o-o o-1

0-10 ,~'

' '

o io ~fh

z

~ ~,' (

g~,1

~ _'

g _~ ,'

~~ ii _{0 05} ~

'r _''

~*~ ô

~~ ô '

~~

~ , ~

~

~~" &~~

~

~/~l

_j

^Î

0.00 _" ^~

o.05 '* * 1

/*

_é * Â

~

~ ~ ~

Il

~

~ ~'~~~4

~3 ~2 ~l 0 2

"

logio

lù

°'°°

~ z 4 6 8 la

kBT/J _~ig

₈

Fig.

7

Fig.

7. Temperature

dependence

of trie _ac

susceptibility

of trie standard _spin

glass. Open triangles,

closed

triangles

and stars

correspond

to _wTo

= o-1, 0.05 and 0.01

respectively.

Trie _maximum of

x'

and x"

slightly

_moves towards lower temperatures. ^{Trie dashed} curve

corresponds

to trie static _case of _{w =} 0. Results _are

averaged

_over 10

samples.

Fig.

8. Plot of

x'

and

x"

_~ersus

loguJ

for T

=

2.5J/kB (dashed litre)

and T

=

3.5J/kB (solid fine).

Closed

triangles corresponds

to

-box'(w)/ôInw

for T ₌

3.5J/kB,

where b

= 1.23 + 0.05. Within trie

error bars b is trie _same for both temperatures.

o.15

.

T=2.5J/kB

.

T=3.5J/kB

o.io

o.05

o.oo

Ù-o o-1 o-Z 0.3 0.4

X'

Fig.

9. Trie Cole-Cole

plot,

_le-, x" _w.

x'

for T

=

2.5J/kB (closed triangles)

and T ₌

3.5J/kB

(closed circles).

1.i

Jo=o.9J Jo=i.iJ

0.5

LJTO"o.05 LJTO ^"o.i

O.g

0.4 0.7

0.5 0.3

o.7

~

o-fi

~

o.5

0.3

o-o o-1

6 7 6 7 8 9 io

k~T/J k~T/J

Fig.

10

Fig.

Il

Fig.

10. Temperature

dependence

of

x'

and x" for Jo

" 0.9J and _wTo

" 0.05 for 3

samples (L

=

10).

For each

sample x'

bas _two

peaks

which

probably correspond

to trie

paramagnet-ferromagnet

and trie reentrant transitions.

Fig.

ii. Trie _{same as in}

Figure

8 but for Jo _# I.lJ and _wTo

# Ù-1.

It _appears that the

theory

of

short-range Ising

SG'S _{gives no} evidence for the

reentrancy

in the static

regime [14].

^The

question

_we ask in this section is whether

reentrancy phenomenon

is

possible

_on short time scales. To _answer this

question

_one bas to calculate tue AC

susceptibility

for various values of tue _mean,

Jo,

of tue

couplings.

Within tue local _mean field

approximation

tue

ferromagnetically

ordered

phase

exists at low T's for

Jo

> 0.6J

[31]. Figure

10 shows tue T

dependence

of

X'

and

x" (Jo

=

0.9J,

_{wTo "}

0.05)

for 3 dilferent

samples

^with L

= 10. A

similar

plot

for

Jo

" I.IJ and _wTo

= o-1 is shown _in

Figure

Il. For each of these

samples

we observe two

peaks

in

x'

_{as a} function of T. Tue

temperatures corresponding

to tue two

peaks

^will be denoted

by TÙ/~

and

Tù/~

_{where tue latter is lower. We calculated}

TÙÎ~

and

TÎÎ/x

in each of tue 10

samples

and took their _average. Tue results _are shown in

Figure

12.

The dilference between

Tù/x

_and

_Tùàx

_{is found to decrease}

on

lowering

_w and the two

peaks

comcide at wTo < 0.008. A

possible interpretation

of the _two maxima _structure could be that

TÙ/x corresponds

to transition between

paramagnet

^and

ferromagnet,

and

Tflx

_{to tue}

transition between

ferromagnet

and SG. It is

possible

that this _is how tue

experiments

bave been

interpreted

when tue reentrancy was involved.

However,

it appears that such

interpretation

would be

misleading

for several _reasons.

First,

tue

Jo-dependence

of

TÎÎ/x

_{is not at ail like}

predicted by

tue _mean field theories

(infinite

range

systems)

of

reentrancy iii TÎÎÎ~

increases

with

Jo

instead of

decreasing. Second,

tue two

peaks

_appear also _in tue

equilibrium

SG

regime.

ù~i~

8 o.i p~

o.05 o.coi

à

~

É

~

FM

SG

0

O.O O.5 1-O

Jo/J

Fig.

12. The T Jo

phase diagram.

Trie solid fine

corresponds

to trie

equilibrium

_case

(trie

_curve

is obtained from maximum of trie

susceptibility). PM,

FM and SG denote trie paramagnet,

spin glass

and

ferromagnetic phases respectively.

Trie

dotted,

dashed and

long-dashed

fines correspond to TÙÎX and

Tflx

_{obtained for}

wTo = 0.1, 0.05 and 0.01

respectively.

Trie diflerence between two temperatures

is found to decrease _on

lowering

_w. For _wT < 0.008 TÙÎX and

Tflx

_coincide.

Thus _origins of any ^true

reentrancy

^{in short} _range

systems

^{remain misterious} or are a result of mistaken

interpretation

of

experimental

data. In

addition,

tue

puysics

of tue

two-peak

structure remains to be elucidated. Most

likely

it relates to some cluster excitations.

6.

Dynamic Specific

Heat

Tue

specific-ueat spectroscopy

^{bas been} demonstrated to be _a useful tool for

studying

relaxation processes in

supercooled liquids [15, 16].

^In a

typical experiment [15]

one immerses _a ueater into tue

liquid

and

apphes

_a sinusoidal current at

frequency

_w. Tue _power

dissipated

contains

a DC

component

and _an AC

component

at

frequency

_w,

ranging

between 0.2 Hz and 6 kHz.

Tue AC

component,

^{results in}

puase-suifted temperature

oscillations and defines tue

complex specific

ueat. Tue DC component, on tue otuer

uand,

is not

relevant, provided

the thermal

response is measured

locally.

Birge

and

Nagel

_[15]

pointed

out tuat in unfrustrated materials and standard

hquids

tue real part ^{of tue}

specific

ueat

c'(w)

does _not

practically depend

_{on w} and tue

imaginary part

is almost _zero. In tue _case of

SG'S, uowever,

and

presumably

of otuer

systems

^witu many modes of

configurational relaxation,

tue

dynamic specific

ueat may be used _as anotuer

probe

of SG

dynamics [8].

^{Indeed tue}

dynamic specific

ueat

couples

to tue time evolution of _even-spin correlations wuereas tue

dynamic susceptibility

_{is given} in terms of

odd-spin

correlations.

By working

witu _a

six-spin

^cluster

Cieplak

and Szamel demonstrated _[8] tuat tue

temperature

and

frequency dependence

of tue

dynamic specific

ueat is similar to tuat of tue

dynamic

susceptibility

except tuat tue former _is not alfected

by

_processes

coupled

to tue very

longest

relaxation time in tue

system.

Wuen _one fixes _w and

plots

c' and c" _{as a} function of T tuen _one

gets

a curve witu _a maximum. On

decreasing

_w tue

position

^{of tue} maximum moves towards

jower

temperatures

^{and it}

suarpens

_up. For _a fixed

T,

_{one can} show tuat in tue _{w -} 0

limit,

c' reacues _a

plateau

wuereas c" vanisues [8].

Tue

question

_we ask _{now is} wuat is tue beuavior of tue

dynamic specific

ueat of tue

system

undergoing

tue transition to tue SG

phase

at T

#

0? In order to _answer tuis _one suould find tue time evolution of tue internai energy because tuis is tue

quantity

wuicu defines tue

dynamic specific

ueat. In tue local _mean field

approacu,

tue _energy is determined not from

two-spin

correlations but from tue local

magnetizations.

Tuis could be _{a reason}

wuy

_our results will be found dilferent from tuose obtained for clusters. Tue otuer _reason could be tue existence of _a finite critical temperature.

In tue

experiments

on

liquids,

_an

oscillatory

ueat _causes

temperature

oscillations. For _a tueoretical

study

of SG'S it is _more convenient to define tue _same

dynamic specific

ueat

by inverting

tue situation. We thus propose tuat tue

temperature

^{of tue}

spin

system ^varies

periodically

witu time:

T(t)

= T ₊

ôTsin(wt)

,

(8)

wuere ôT is assumed to be small relative to T.

Tue

perturbation

in T results _{in a}

oscillatory

evolution of tue internai energy. Tue

oscillatory part

^{of the} average

Hamiltonian,

_< H >, ^will be denoted

by

à _< H _>. When transients die out _one can define c' and c" _as follows

à _< H _>

IN

₌ ôT

[c'(w) sin(wt) c"(w) cos(wt) (9)

Tue time evolution of à _< H _> bas to be tuen determined and _{we use}

equation (4)

witu H

= 0

for tuis _purpose.

Tue initial

configuration

for tue time evolution bas been obtained in _an

equilibrium

_way for eacu T. We monitor à _< H _> and fit it to

(9)

after _some time

to,

cuosen _so tuat ail transient

exponentials

_can be considered extinct. In _{our case one} needs ta of about

5To,

wuere

Ta

is _a

period

of the

temperature

oscillation. Dur calculations have been carried eut for the 3D

system

of size L ₌ 10. A suflicient

averaging

of

c'(w)

and

c"(w)

_is

accomphshed

_over 10

periods.

The

amplitude

ôT is taken to be

equal

to

0.01J/kB (the

results _remain almost the _same for smaller

ôT).

Figure

13 shows the

temperature dependence

of the

dynamic specific

heat for selected values of _w. The maxima of c' _{and c"}

move toward

higher temperatures

_{as w} is decreased. It should be noted that _an

opposite tendency

has been observed bath

experimentally

and within the

dynamic

local _mean field

approximation

for the

dynamic magnetic susceptibility.

It is aise

opposite

to what has been observed for the

6-spin

cluster. The

longer

the

period

of

oscillations,

the doser c' resembles the

equilibrium specific

heat. The

plots

of c" _versus T are similar to those of c'

exept

^that the _w

= 0 limit

corresponds

to c"

= 0. Thus the smaller _w, the smaller

is the _maximum.

We _now tutu to the discussion of the

frequency dependence

of the

dynamic specific

heat.

Figures

14 and 15 show c' and c" _versus

logiow, respectively.

These _curves resemble those for the

dynamic susceptibility.

At low

frequencies

^c' becomes saturated. The dilference between

our results and the results obtained for the

toy

model _{[8] is} observed for c". _In the _case of the

six-spin cluster,

c" may have two maxima at low

temperatures

which reflect the elfect of two

separated long

relaxation modes. In _{our case one} has

only

_one maximum This may be

explained by

tue fact tuat _our

system

bas many

long

relaxation times and tue elfect of

individual modes is smootued eut. From

Figures

14 and 15 it is clear tuat _any maximum

in

c" _is bound to be located _{at a}

frequency

wituin _a

region

of increase of c'.

Figure

16 shows tue Cale-Cale

plots

_on wuicu c" _is drawn _{as a} function of c'. For ail

temperatures

^tue

plot

bas _a circular

suape.

In tue _case of tue _six-spin cluster tue _{curve cari} consist of two

overlapping

semicircles [8]. ^{Dur results for tue} Cale-Cale

plots

_are similar to tuose for tue

dynamic susceptibility (see Fig. 9).

It would be

interesting

to confront tue results _on

dynamic specific

ueat witu

experiments.

ù-fi ù.3

,-,~

^L~i~ L~i~

', ₀

-- D.I

' _-- D,1

~-~- 0.25

ù.4 _--- 0.25 _Ù-Z 0.5

, -- 0.5

r i,

o-z o-i

ù-ù _ù-ù

ù 2 4 6 8 ù 2 4 6 8

kBT/J kBT/J

a) b)

Fig.

13. Trie real and

imaginary

parts ^{of trie}

dynamic specific

heat for trie 3D system

(L =10)

_as

a function of T. Trie dasued fine

corresponds

to the

equilibrium specific

heat obtained in trie local

mean field

approximation.

Trie stars, _open and closed

triangles

correspond to _wTo

" Ù-1, 0.25 and Ù-à,

respectively.

ù.3

~

~

~~~

0.4

ù-Z

~ l 75

C1,

~ 2.5

o-z

o-i

ù-ù ù-ù

-3 -2 -1 ù 1 2 -3 -2 -1 ù 1 2

logio

~ù

logio

lù

Fig.

14

Fig.

15

Fig.

14. Trie real part ^{of trie}

dynamic specific

heat _w.

Îogio

_w for T

=

2.5J/kB,1.75J/kB

and

I.oJlkB

_as indicated. At low frequencies c' reaches _a

plateau.

Fig. 15. Same _{as in}

Figure

14 but for the imaginary part ^of the

dynamic specific

heat. It goes ^to

zero in the

w - 0 limit.

Acknowledgments

We

acknowledge partial support

^{from the Polish} agency KBN

(grant

number 2P302 127

07).

ù.3

2.5J/kB

ù-Z C

1.75J/kB

o-i

I,ùJlkB

o-ù

ù-ù o-Z ù.4 ù-fi

C~

Fig.

16. The Cole-Cole

plot,

_i-e-, c" _vs, c' for selected T ilrdicated lrext to the

curves. For ail

temperatures ^{considered these} curves have _a semicircular

shape.

Appendix

Improved

Local Mean Field

Approximation

Here _we discuss the aging elfect wituin the

improved

local _mean field

approximation _[35].

Tuis

approacu

is

equivalent

to _an

approximate

evaluation of _<

tanu(h,)

_> witu _a factorized statistical

weigut P(S)

₌

fl~ (l+

_{< S~} >

S,)

_or to

expanding tanu(h,)

_{as a sum over} ail

possible products

of

Sj (up

to tue fiftu order _in D ₌

3), taking

into account tuat

S)

= 1.

Adopting

tue first _{way we cari write}

equation (3)

in tue form

(i +To()mi(t)

=

11 fli(i

₊

SJmJ)/21tanh(fI~JIJSJ)

ls, j _{j j}

+

(1-

_mi

£ fl[(1+ _Sjmj)/2] tanu(fl£ _{J,jsj)) tanu(flH) (10)}

ls, j j j

Using

tue last

equation

_we bave considered tue aging elfect in tue FC _regime.

Figure

5 shows tue time

dependence

of

Mrm(t)

for selected

waiting

times at T

=

2.5J/kB (the cooling

down to this

temperature

is assumed to be toc

quick

_so that _{one can use} the random

spin configu-

ration _{as a}

starting configuration). Clearly,

similar to tue results obtained in tue local _mean field

approximation, Mrm(t)

reacues _a

plateau

at

long

times. Tue time scale

tp required

to

reacu _a

plateau

in tue

improved

local _mean field

approximation

_is,

uowever, larger

tuan tue

corresponding

time scale obtained in tue

simpler approximation.

Tuis _can be _seen in

Figure

6 wuere

Mrr~z(t)

calculated

by

tue two metuods _is suown for tw =

l0oTo,

T

=

2.5J/kB

and L = 10. In tuis _case tue local _mean field

approximation

_gives

tp

m

155To

wuereas tue

improved

local _mean field leads to

tp

m

485To.

Thus tue time scales _are enuanced but tue

approacu

to a true

equilibrium

is still absent.

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Young

A. P., Rm. Med.

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Classification

Physics

Abstracts

02.00 05.00 64.00

The B.A.M. Storage Capacity

H.

Engliscu(1),

V.

Mastropietro(~)

and B.

Tirozzi(3)

(~) Universitàt

Leipzig,

Inst. für

Informatik,

D-04109

Leipzig, Germany

(~) Dipartimento di Fisica, Università di

Pisa,

56100 Pisa, Italia

(~)

Dipartimento

di

Matematica,

Universita' di Roma "La

Sapien2a",

00185 Italia

(Received

22

February

1994, revised 23

September 1994, accepted

4 October

1994)

Abstract. Trie Bidirectional Associative

Memory (B.A.M.)

is

a neural network which _can

store and associate

pairs

of data _in trie form of two patterns using an ilrput network of M

neurons and

an outputlretwork with Nlreurolrs.

Despite

its ilrterest there

are no theoretical

investigations

about this model. We obtailr trie

equatiolrs

of state _{in a}

rigorous

_{way usilrg} olrly trie

assumption

that trie Edwards-Alrderson parameters ^associated to trie _two networks

are

self-averaging:

this

important

property

corresponds

to trie

replica

_symmetry

hypothesis

_in trie

replica

calculations. A _comparison between trie methods used in trie hterature _is made and trie connection of _our derivation with Peretto's method is shown. Trie storage

capacity

of the

B-A-M- is

computed

whelr N

= M alrd _a boulrd

on it is denved when N

# M,

in contrast with trie

strongly

diluted _case in which trie critical capacity _is ^{unbounded for}

N/M

_- 0 or - ce.

l. Introduction

Recently

_some models bave been

proposed

in order to _recover remarkable

properties

of tue uuman

brain,

_in

particular

tue

capacity

of

remembering sometuing starting

^from _an

incomplete

or

approximate

information. The

Hopfield

model is for instance _one of such models. In this model the state of the brain

corresponds

to _a set of variables

a,(t)

= +1 which describe the

state of tue1-tu _neuron, E

1, N,

at the time t. Some

biological considerations,

for which we

refer to

(10), _suggest

tuat tue

a~(t) obey

tue

following dynamics:

ai(t +1)

₌

sign(£ _{J, jaj(t))}

J

wuere

f,j

=

(1IN) £(_~ gfg(

and tue gH are p

patterns,

_/J E

1,

_p ^and

(gH)~

e

gf

_e +1. Each of these patterns is a piece of information to store into tue

system.

Since _{we want} to make

a mortel

storing

_any kind of information _{we assume} that the

gf

_are

independent, identically distributed,

random variables with

probability

distribution

P(g$

=

1)

=

P(gf

=

-1)

=

1/2.

We say that the

system

recognizes some

approximate

information

if,

choosen

ai(0)

₌

e,g)

wuere _e, is _a random

variable

such tuat

Pie,

= 1) = 1- _q, tuer

a,(t)

_-t-ce

g). By numencal

Q

Les Editions de

Physique

1995