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)