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ergodicity breaking occurs in the spin-glass phase, defined as the phase. Where the. Edwards-Anderson parameter is non-zero. We see however no reason.
J.

Phys.

France

I

(1992)

2

1705-1713

1992,

SEPTEMBER

1705

PAGE

Classification

Physics

Abstracts

05.40

75.40

64.70

Communication

Short

breaking

ergodicity

Weak J. P.

Service

systems

disordered

in

Bouchaud

Physique

de

(Received

May

25

We

is

in

final form

phenomenological

a

the

lifetimes

probability

law

of the

distribution.

J9June

model

Cedex,

91191Gif-sur-Yvette

for

France

J992)

the

dynamics

A

distributed in

functional over

disordered

of

metastable states are many We show that aging occurs

simple hypothesis leads to a new with spin-glass experiments agreement

infmite.

remarkable

spite of fifteen Dynamical

accepted

that

CEA-Saclay,

Condensd,

l'Etat

J992,

postulate

power lifetime

average which is in

de

present

We

Abstract.

systems. broad, a

aging

and

this

fornl

nearly

five

(complex) according to

model for

when

the

relaxation

the

decades

in

time.

theory of equilibrium spin-glasses is not yet settled likely to be the dominant aspect in experiments. One of the striking aspects of the dynamics of spin-glasses in their low most temperature phase is the aging phenomenonpeculiar and awkward feature from the rather a thermodynamics point of view : the relaxation of a depends on its history. More system precisely, if a system is field-cooled below its spin-glass magnetization the temperature, relaxation depends on the Waiting time t~ between and the switch off of the the quench magnetic field [4, 5, 6]. Similar effects are observed on the viscoelastic properties of polymer melts [7], magnetic properties of HTC [8] and superconductors recently on the more relaxation after a heat pulse in Charge Density Wave systems [9]. Analytical fits of the magnetization function relaxation of time have been proposed. In [6], it is proposed that as a initial «stationary» the relaxation is a power-law With a small (negative) part of the For times longer than the Waiting time, relaxation is Well fitted by a « stretched » exponent. exponential decay, provided that an effective time is introduced [6]. In [4, 5], however, a of the stretched exponential «real» time found for relatively times. short Many was phenomenological theories have been devised for the stretched exponential decay to account lo-14], and for the slow part and aging [6, 15-18]. We however feel that the basic mechanism underlying aging has not been fully appreciated (see however [17]) although it is, in our opinion, one of the constitutive properties of spin-glasses. The aim of this note is to suggest aging is related ergodicity breaking that which peculiar and has, in these to systems, a perhaps unexpected meaning. We find on according to our simple models thatsome definition, see below Weak » ergodicity breaking in the spin-glass phase, defined occurs « In

[1, 2, 3].

as

the

Why taken

phase the as

two an

Where should

years of effects

the be

operational

dispute, are,

the

however,

Edwards-Anderson linked

in

definition

general, of

the

parameter is suggest that

and

spin-glass

non-zero.

the

transition.

We

see

appearance

however of

aging

no

reason

could

be

JOURNAL

1706

reproduce both We magnetization decay form The analytical We find in particular

t

and

t~

«

that

t

the

to

stringent the

mo exp i-

=

regimes

two

describing theory. As We

our

»

the

states

«

is

between

is

x) (t/t~)~ ~~i

y/(I

which

and

time

of

»

In

two

states.

the

probability

the shall

energy We thus

certain

a

states

«

the

all

previously

those

proposed.

0

w

x

w

(tit~)-Y =

strongly

are

connected, since the relaxation. This part of the

initial see,

these

expect

trapping see

find

is

that

rough, these

above

exponent

feature

should

configurations,

Which

minima

traps

act

?

simple picture

«

Which

»

shall

We

surrounded

are

states

as

extremely

is

system

local

a

both

[6].

of

data

disordered

these

With

agreement

is

be

get hold

r.

words,

to

time

metastable

to

landscape

also

disconnected

are

other

remarkable

our

corresponding

following (Fig. I,

the

any that

the

barriers.

distribution

the

landscape fo below

».

during

system

What

Since

energy

from

theory is in of [4. 5], and the very long landscape of a finite energy

minima

local

many

high

rather

fast

the

time experiments Widely accepted that the

rough, With call loosely of

of

of

parts

decay

law

parameters

test

short

«

It is

by

that

Note

t~.

»

related

(t»t~)

time

:

m(t) for

long

and the time (t«t~) short hypotheses. quite reasonable different of this decay is however the

power

a

N° 9

I

under

m(t) for

PHYSIQUE

DE

the

we

times

[11]) ;

fo

:

is

A

exists

a

for

percolation

«

the

energy

»

energy level

required to « hop » energy benveen the traps is very

minimal

the

that

assume

system

~(r) there

dynamics

the

between

metastable

two

states

is

negligible.

r

( a

r

b)

AGING

io

r

Fig. I. fo which drawing When zero

the

a) is

is

Schematic

one

view

minimal

the

dimensional

extemal

magnetization

field

is cut,

states.

of

energy : in

the

energy needed to

landscape

go from mountains of

: one

holes

are

drilled

metastable

reality height » f~ also landscape acquires a slope non-zero

the

state

exist which

below to

the

reference

another.

Note

between

drives

different the

system

energy that this states.

b)

towards

N° 9

ERGODICITY

WEAK

the picture, energy the spin glass phase

this

In

f.

fo

In

model,

BREAKING

of

distribution

the

barrier

very

of

both

low

f's

~f )

P where

fact

is

x

Which

proliferate

that

is

N

is

Random

exponential

is

NIT

exp

number

equal Energy

Model

[1, 23,

III

:

REM,

(1)

I, fo is the

0 and

between the

x

T/T~

=

landscape is temperature independent. This energy x(T) is non-trivial [I]. Its shape is Well approximated

Where

breaking [5]. scheme Assuming that AE= P ( f ) df : ~ (T) dr

fo- f,

that

and

roexp(AE/T),

r

1707

depth of the trap, (REM) and of the SK

the

to

SYSTEMS

fo)lT

(f

x

In

constant.

a

thus

DISORDERED

IN

reference

Which is

free

the

model,

and

AE

the

=

dependent

temperature

a

levels

AGING

AND

by

the

level

to

the

in

the

SK

case

one-step

«

replica

»

using

finds,

above

connected

the

not

one

=

is

(I)

and

=

~(y)

large

for

[w is

r.

models-

where

TF/«

x

the the

is

F

be

may

decays

for

that

real

slowly

more

for

spin-glasses the large negative f, (f )

P

(

with

0.

~

this

In

~ (r)

case,

reads

8 is

where

a

The

o

IT)

will

lead

probability example

as

fi(- f

+'~

)~

distribution

~~

rir~j-

jog

air =

logarithmic,

to

ideas

the

of

Fisher

low

of

other

many

20-22],

with

free

states

energy

law

power

a

(3)

when

xwl

('

+

(4)