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Dynamical Phase Transitions in 3-Dimensional Spin
Glasses
Bernard Derrida, G Weisbuch
To cite this version:
Bernard Derrida, G Weisbuch. Dynamical Phase Transitions in 3-Dimensional Spin Glasses. EPL -
Europhysics Letters, European Physical Society/EDP Sciences/Società Italiana di Fisica/IOP Pub-
lishing, 1987, 4 (6), pp.657-662. �10.1209/0295-5075/4/6/004�. �hal-03285584�
EUROPHYSICS LETTERS
Europhys. Lett.,
4
(6), pp. 657-662 (1987)
15 September 1987
Dynamical Phase Transitions in 3-Dimensional Spin Glasses.
B.
DERRIDA
(*)
and
G.
WEISBUCH
(**)
(*)
SPT
CEN-Saclay
-
91191
Gif-sur-Yvette, France
(**)
GPS ENS
-
24 rue Lhomnd,
75251
Paris
Cedex
5,
France
(received 27 April 1987; accepted in final form
11
June 1987)
PACS.
05.40.
-
Fluctuation phenomena, random processes, and Brownian motion.
Abstract.
-We
study the time evolution
of
the distance between two configurations submitted
to the same thermal noise for the
3d
k
J
Ising spin glass. We observe three temperature
regimes: a high-temperature regime where the distances vanishes in the long-time limit.
An
intermediate-temperature
regime where the distance has a nomm limit independent
of
the
initial distance.
A
low-temperature regime where the distance in the long time limit seems to
depend upon the initial distance. For the sake
of
comparison, we have repeated our simulations
for the ferromagnetic case.
Most of the physical properties of spin glasses which can be observed
in
the real world are
dynamical effects due to slow relaxation processes (see ref.
[l,
21 for a review). After a long
debate about the existence or the nonexistence of a spin glass phase for 3-dimensional spin
glasses, the most extensive numerical simulations [3] which have been done
so
far indicate
that there is a spin glass phase for temperatures
T
<
T*g=
1.2
J
for the
f
J
model, but that
slow relaxation effects appear in a large range
of
temperatures
TSg
<
T
<
TF
2:
4.5
J,
where
the data (spin autocorrelation function) can be fitted by stretched exponentials. Theoretical
arguments[4] based on reasons similar to those which lead to Griffiths singularities[5]
predict that below the critical temperature
TF
of
the ferromagnet, nonexponential decays
should be observed.
In this letter, we present numerical data on the evolution of the distance
(D(t))
between
two configurations which are submitted to the same thermal noise, for the
3d
f
J
Ising spin
glass on a cubic lattice. When measuring
(D(t))
after
a certain time (generally 500 iteration
steps), we observe
3
regimes:
a
high-temperature regime
T
>
TI
(with
TI
=
4.1
J)
where,
(D(t))
vanishes inde-
pendently of the initial value
D(0);
an
intermediate regime
T2
<
T
<
TI
(with
T2
=
1.8
J)
where,
(D(t))
is nonzero and does
not depend on the initial value
D(0);
a low-temperature regime
T
<
T2,
where
(D(t))
depends on
D(0).
658
EUROPHYSICS
LETTERS
The simulations having been made for relatively short times
(t
G
500),
small systems
(NG864),
and few samples
(MS800),
our
determination of
TI
and
T2
although not very
accurate could be consistent with
TI
=
TF
and
T2
=
TSg.
The distance
D(t),
in the limit
t
+
00,
then appears as a useful order parameter for spin
glasses, since
it
gives a clear signature of the intermediate phase
T2
c
T< TI.
Methods.
-
Our
numerical simulations are done for
a
system of
L3
Ising spins on a cubic
lattice of linear dimension
L
with periodic boundary conditions. The nearest-neighbour
interactions
Jij
are randomly chosen
(1)
1
1
2
2
p
(Jij)
=-S(Jij
-
J>
+
-6(Jij
+
J)
.
The interactions
Jz~.
are quenched and symmetric
(Jij
=
JjJ.
the local fields
hi(t>
are computed according
to
A
spin configuration
{Si(t)}
evolves according to the following rule:
at
each time step
t,
all
and the spins are then updated according to
Si(t
+
1)
=
+
1
with probability
1
Si
(t
+
1)
=
-
1
with probability
where
T
is the temperature of the system. The dynamics
are
parallel dynamics. However,
if
we choose the linear size
L
to be even, the system
is
decomposed into two independent
sublattices which ignore each other and, therefore, stand as two different samples. One can
easily check that dynamics
(3)
lead to the right thermal equilibrium for each sublattice in the
long-time limit
(i.e.
the correlation functions between the spins of each sublattice averaged
over time are the same as if they were computed at thermal equilibrium on the
full
lattice).
We consider two different initial configurations
{Si(0)}
and
{SXO)}
at
time
t
=
0,
and we
let them evolve according to exactly the same rules: the
JG
used to compute the fields
hi(t)
and
hXt)
are the same and the random numbers used in
(3)
to decide whether
Si
and
Sl
are
+
or
-
1
are the same (in particular
if
hi(t)
=
h:(t),
then
Si(t
+
1)
=
S:(t
+
1)).
A
similar method
was used recently to study the spreading of the damage caused by one spin flip
[61.
We then
measure the distance
D(t)
between the two configurations
as
the number of spins which are
different
(i.e.
such that
Si(t)
=
-
SXt))
The two sublattices are independent since at each time step they just exchange each other
and for each cube of linear dimension
L,
we consider that we have two samples
of
size
N
=
L312
spins.
In order to average
D(t)
over disorder and thermal fluctuations, we repeat the
simulations and generate
M
samples by constructing
MI2
cubes.
If
two configurations
B.
DERRIDA
et
al.:
DYNAMICAL PHASE TRANSITIONS IN %DIMENSIONAL SPIN GLASSES
659
become identical at time
t,
they remain identical at any later time. When we generate
M
samples the
first
quantity we can measure
is
Ml(t),
the number of samples such that
{&(t)}
and
{Sl(t)}
are
still different at time
t:
one then defines a survival probability
P(t)
by
(For
any finite system at finite temperature
P(t)
goes to
0
as the time goes to infinity.
However, in the simulations described below, there is a large range of time when
P(t)
remains almost constant. This behaviour is similar to the behaviour of the magnetization in
finite
systems.)
We then measure the average distance
(D(t))
over those
Ml(t)
samples which have
survived and, therefore,
where
D,(t)
is
the distance measured at time
t
for the s-th sample.
were used:
is
then
1.
In order to study how
(D(t))
depends upon
D(O),
three different sets of initial conditions
A)
Configuration
{Si(0)}
is
random and configuration
{Sl(O)}
=
{
-
&CO)}
for all
i.
D(0)
B)
Configurations
{Si(0)}
and
{S:(O)}
are random and independent.
D(0)
is
then
1/2.
C)
Configuration
{Si(O>}
is
random and configuration
{S:(O)}
is identical to
{Si(O)}
except
for
one spin on each sublattice.
D(0)
is then
UN.
The simulations have been performed for cubes of linear dimension
L
=
8
(each sublattice
having
256
spins) and
L
=
12
(each sublattice having 864 spins). The results are averaged
over
800
samples
for
N
=
256 and
200
samples for
N
=
864. The numerical effort is thus
roughly the same for the two sizes.
Spin
glass
results.
-
Figure
1
shows the survival probability
P(t)
as a function of
temperature T for the three sets of initial conditions
A),
B)
and
C),
after
500
time steps. Two
regimes can be observed. Above Tl=4.5J,
P(t)
is
0,
whatever
D(0).
Below
T1,
we
see
in
cases
A)
and
B)
a sharp increase
of
P(t)
up to
1.
Two
different initial configurations never
become identical. Even more surprisingly in case
C),
we see that two initial configurations
which differ by
a
single spin have a probability of the order of 60 percent to remain different.
The results
do
not seem to depend upon the size of the system, at least when we compare
them for the cases
L
=
8
and
L
=
12.
The
results
(not represented here) after
100
steps are
very
similar
except
for
the transition region
T
=
T1.
Improving the quality
of
fig.
1
is not
easy since the error bar decreases like
M-lI2,
but does not decrease with the system size
N.
Distances
(D(t))
are plotted in
fig.
2.
They exhibit three different regimes.
For
T>Tl,
(D(t))
vanishes for all three cases
A),
B),
and
C).
In the range T2
<
T
<
TI (with
T2
=
1.8
J),
(D(t))
does not depend upon the
set
of
initial
conditions
A),
B) or
C)
or
upon the system size; by comparing
fig.
2a)
and
b),
we see that
(D(t))
has not evolved between times
t
=
100
and
t
=
500
and, therefore, seems to have
already reached its long-time
limit.
660
0.8-
h
U
a
0.6-
0.4-
0.2
-
EUROPHYSICS
LETTERS
0
4
0.0.0~
.O
0.0
O.
0 0.
0
P
4
00
4
0
I
w
0
1
2
3
4
T(J)
Fig. 1.
-
3d
k
J
spin glass case. Survival probability,
P(t),
that two initially different configurations
remain different after
t
times steps (here
t
=
500),
as a function
of
temperature
7'.
The smoothness of
the curves gives an idea of error bars which are not figured
(0.07
for
L=
12 and
0.035
for
L=8).
(White signs correspond to cubes of linear size
L
=
8 and black signs to cubes of linear size
L
=
12.
Triangles are for initially opposed configurations, case
A),
squares for random configurations, case
B),
and diamonds for configurations differing initially by only 1 spin, case
C).
The triangles are masked by
the squares when they coincide).
Lastly, in the range
T< T2,
(D(t))
does depend upon
D(0).
We see, however, that
(D(t))
does not change with the system size
L
and has very little change with time except in
case
C)
where the difference between the two configurations takes a longer time to spread
for the largest system
L
=
12.
The existence
of
large range
of
temperatures where
(D(t))
seems to have reached an
equilibrium value independent of system size and initial conditions is in fact the main result
of
this paper.
T2
clearly depends upon iteration time and system size. One expects
T2
to
Fig. 2.
-
Spin glass case. Distance
(D(t))
as a function of
2'.
(The signs on the plots have the same
meaning as in fig.
1.)
t
=
100
for
a)
and 500
for
b).
In the temperature range between 2
J
and
4
J,
the
data for the three sets of initial conditions coincide.