Thermo-cycling experiments with the three-dimensional Ising spin glass model

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Thermo-cycling experiments with the three-dimensional Ising spin glass model

Heiko Rieger

To cite this version:

Heiko Rieger. Thermo-cycling experiments with the three-dimensional Ising spin glass model. Journal

de Physique I, EDP Sciences, 1994, 4 (6), pp.883-892. �10.1051/jp1:1994229�. �jpa-00246952�

J. Phys. Ilfance 4

(1994)

883-892 JUNE1994, PAGE 883

Classification

Physics Abstracts

75.10N 75.50L 75.40G

Thermo-cycling experiments with the three-dimensional Ising spin glass mortel

Heiko

Rieger(*)

Institut für Theoretische Physik, Universitàt

zu KôIn, 50937 KôIn, Germany

(Received

3 February 1994, accepted 15

February1994)

Abstract. A charactenstic feature of the non-equilibrium dynamics of real spin glasses at low temperatures _are strong aging eflects. These phenomena _can be mampulated by changing the externat parameters in various _{ways: a}

thermo-cycling

expenment consists for _instance of _a short heat puise during the waiting time, by which the relaxation might be strongly aflected. Results of numencal expenments of this kind, performed _via Monte Carlo simulations of the three- dimensional Ising spin glass model, _are presented. Trie theoretical implications _are discussed and the scenario found is compared with trie experimental situation.

1. Introduction.

The

investigation

of trie _so called

aging phenomenon

in

spin glasses

has been _a

major

focus of research

activity

since the seminal work of

Lundgren

et al.

iii

of1983. It lias been realized

by

trie

experimentalists

_[2] that

magnetic

properties of

spin glasses depend strongly

_on the time

they

_spent below the

freezing

temperature

T~.

^{These eflects} are a consequence of trie

extremely

slow

dynamics

of spin

glasses

at low temperatures,

regardless

of the existence of _an

equilibrium phase

transition

(for

_an overview of their

equilibrium properties

_see

[3]).

Following

trie arguments of the

droplet theory

_{[4] one}

imagines

the

dynamical

_process at low temperatures to be

governed by growth

of

domains,

which will reach _some

typical

size after _a certain

waiting

time. This

lengthltime

scale becomes manifest in _{a crossover} observable _e-g- in trie thermc-remanent

magnetization decay (see

aise [5]). ^{On trie other}

hand, inspired by

Parisi's solution of trie mean-field model of

spin glasses

_[fil, it bas been

proposed

that trie many metastable _states

existing

in trie

rougir free-energy landscape

of

spin glasses might

be

organized

in _a hierarchical _way

iii. Dynamics

then

explores

states with

decreasing free-energy,

and the escape from these

valleys

becomes

harder,

which _means that it stays _a

longer

time within these

valleys.

In this way trie

depth

of the

valley

reached

dunng

the

waiting

time becomes manifest

again

_{as a crossover} in observable

quantities (see

aise

[8j).

(*) e-mail: [email protected]ù-koeIn.de.

Thermc-cyding experiments

consist of _two temperature

changes during

trie time in which trie material is

aged

in trie spin

glass phase

_[9j: either _a short heat

puise

is

applied

to the spin

glass during

trie

waiting

time after which trie relaxation of _e.g. trie thermo-remanent

magnetization

is

measured,

_{or a} short

negative

temperature

cycle

is

performed,

which is trie _{same as a} heat

puise

but with _a

negative

temperature ^shift

during

trie

puise.

It lias been

pointed

out

il Ii

that this kind of experiments _cari discriminate between trie

droplet picture

and trie hierarchical picture.

Within trie hierarchical

picture

_a heat

puise experiment

and _a

negative

temperature _cy-

ding experiment

should bave _an asymmetric outcome: The hierarchical

organization

of trie metastable states

depends

_on temperature in such _{a way} that trie

free-energy-valleys split

into

several,

steeper

valleys

with

decreasing

temperature. Here _an escape (1.e. ^{relaxation of} trie system ^{into other} parts ^of

configuration space)

becomes _{even more}

diliicult, resulting

in _a

partial freezing

of trie

aging

_process

during

_a

negative

temperature

cycle. Upon heating

several

valleys

melt

together,

which facilitates trie relaxation. After

completion

of the

cycle

trie system is

again

in _{a narrow}

valley

with _a

higher free-energy,

similar to trie

beginning

of the experiment and thus

restarting

trie

aging

_process.

Experiments reported

in [7], ^where a metallic

spin glass Ag(Mn)

lias been

used,

and in [9, II,

loi,

where _an

insulating spin glass Cdcri.7Ino.354

lias

been

used,

_can be

nicely interpreted

within this

picture.

On trie other hand within trie

droplet theory

_a heat

puise experiment

and _a

negative

tem- perature

cycling experiment

should bave

qualitatively

_{a symmetric outcome:} in bath _cases trie domains that bave _{grown so} far should be reduced in size

(or

_even

destroyed) by

trie tempera-

ture

cycle,

and thus reinitialize trie

aging

_process. This is _a consequence of trie fact that

spatial

correlations _among

equilibrium

_spin

glass

states at diflerent temperatures below Tg are short

ranged

with _a

typical length

scale

decreasing rapidly

with

increasing

temperature ^diflerence [4, 5, 14].

Experiments

in

Cu(Mn) spin glasses

_[12] and

Cu(Mn) spin glass

^films _[13] bave

reported

_a scenario that

speak

in favor of this

interpretation.

At first

sight

it _seems to be odd that trie _same

experiment

done with similar materials lead to very diflerent conclusions. Since _m both _cases also metallic

spin glasses

bave been

used,

like for instance

Ag(Mn)

in

[loi

and

Cu(Mn)

in [13], it

might

be hard _to find trie _reasons for trie

discrepancies

^within trie materials that bave been used. What is _more

probable

_is that

a different kind of

analysis

bas lead to diflerent interpretations: ^{In trie first} case [7,

9-llj, supporting

trie hierarchical

picture,

trie _curves for trie

decay

of the quantities ^of interest itself bave been used to

analyze

trie effect of _a heat

puise.

In contrast to this in trie second _case

[12, 13], supporting

trie

droplet

model _or trie idea of _an

overlap length,

trie derivative of trie

curves with respect to trie

logarithm

of time and its connection to trie distribution of relaxation

times lias been trie basis of trie argument.

Besides these _two

phenomenological approaches

cited above _{some success} lias been made very

recently

in trie

analytical

and numerical

investigation

of microscopic mean-field models of

aging

in

spin glasses [15-17]

or

related, simplified

models

[18,

19]

(note

also trie adiabatic

cooling approach presented

in

[20]).

However, _up to _now trie

study

of microscopic models of

aging phenomena

in two- _or three-dimensional model of _spm

glasses

had to be

performed exdusively

_via numerical simulations [21,

22].

^Here not

only

_a

qualitative

agreement _among

simulation results and

simple aging-experiment (without

temperature

cycle)

could be estab-

lished,

but also numerical values for various exponents

determining

trie functional form of the

decay

of trie thermo-remanent

magnetization

bave been shown in [22] ^{to be trie} sonne in certain

spin glasses (as

_e-g-

IFe15N185)75PioBoA13

and

Feo.5Mno.5Ti03)

and trie three-dimensional Edwards-Anderson model with heat-bath

dynamics.

One

might

^{ask whether} temperature

cyding

_m Monte-Carlo simulations will also show such

a remarkable concurrence with trie

experimental

scenario. In this paper we present ^trie results

N°6 THE THREE-DIMENSIONAL ISING SPIN GLASS MODEL 885

of such simulations. In trie _next section _we introduce the

model, give

_some ^numerical details and define trie

quantities

that bave been calculated. In sections 3 and 4 _we present the

results,

which _are discussed in section 5.

2. Numerical

procedure.

Trie system ^under consideration is trie _{saine as} in [22], ^trie three-dimensional

Ising spin glass

model with nearest

neighbor

interactions and _a discrete

I+J)

bond distribution. Its Hamilto- nian is

7i =

~j _J~a~aj

_h

£

a~

,

Il

(iJ) i

where trie

spins

a~ = +1 occupy trie sites of _a L _x L _x L

simple

cubic lattice with

periodic boundary

conditions and trie random nearest

neighbor

interactions

Jjj

take _on the values +1

or -1 with

probability 1/2.

Trie

quantity

h is _an externat

magnetic

field. Trie

dynamics

is trie

so called

Metropolis algorithm,

defined

by

trie

probability

for

single

spin

flips w(a~

_-

-a~)

=

min(1, exp(-AE/T))

,

(2)

where AE is trie _energy difference between trie old _state and trie _new state in which trie

spin

at site1 is

flipped.

Trie most eliicient

implementation

of this

algorithm

with

sequential up-date

on a

Cray-YMP

lias been used

(see

_{[23] for}

details).

To mimic the temperature

cyding

experiments [9-13] ^trie

following procedure

lias been ap-

plied:

First trie system is initialized in _a random state at a temperature ^T < T~ m 1.2 (T~

is trie

freezing

_or

spin glass

transition temperature ^{of trie} model

(1),

_see

[24], however,

note also

[25]),

which

corresponds

to _a fast

quench

from trie

paramagnetic phase

into trie spin

glass phase.

Then trie system ^is

kept

at trie temperature ^{T for} a first

waiting

time twi,

during

which trie initial

aging

^takes

place.

Note that time is measured in Monte-Carlo _sweeps

through

trie lattice. Then trie temperature is

changed

to

Tp,

^which is

larger

than T in _a heat

puise

_experi-

ment and smaller than T in _a

negative

temperature

cycle experiment.

Trie system is

kept

at trie temperature Tp ^for a time

tp. Finally

trie

cycle

is

completed by changing

trie temperature

back _agam to trie initial temperature

T,

where trie system ^is

aged again

for _a time tw2. ^After this time

(note

that _up to this point ta~e = twi

+tp

+ tw2 Monte-Carlo _sweeps bave been

done).

We

performed

two different kinds of experiments: ^In trie first trie whole simulation is per- formed in _zero

field,

and _we stored trie

spin configuration

at time toge ^{and measured its}

overlap

with trie spin

configurations

of trie system t Monte-Carlo steps later:

Clt>rage)

_"

£ _1°ilt

₊

rage)°iltage))

13)

1

Here (. means a thermal _average

Ii.e.

_{an average over} different realizations of trie ther-

mal noise, but trie _same initial

configuration)

and trie bar _{means an} average over dilferent realizations of trie bond-disorder.

In trie second

experiment

_we

keep

trie system ^within an externat field h

during

trie whole temperature

cyding procedure

and switch it off after trie trie

procedure

bas been

completed

(1.e. ^after

toge).

^{From that} moment _on trie thermo-remanent

magnetization

~ÎTRM_{(~, ~age)}

fi

^{(Ut(~ + ~age))} ,

~

~ (~)

i

is measured. This

field-cooling

experiment is

exactly

what is done with real

spin glasses

[9-13].

The linear system ^{size of the}

samples

is L ₌ 32 (1.e. ~- 3 _x 10~

spins),

and _we

averaged

over 256 different realizations of the disorder. There _{are nu} limite size effects observable within the time scale of la° Monte-Carlo steps, which _means that the

typical

correlation

length (or

linear domain size within the

language

of the

droplet picture)

is still smaller than half of the linear system size after _t

=

10~. _{We believe that our results do not}

depend significantly

_on the choice of the

dynamics (2).

Let _us

adopt

trie

point

of view that

spin glasses

_are critical for ail temperatures ^below the

spin glass

transition temperature Tg

(note

that the correlation

length

in trie frozen

phase

is infinite for ail

temperatures).

In this _{case we} would expect that _any

microscopic dynamics

without

order-parameter

conservation

(model

A in trie classification of

Hohenberg

and

Halpenn

_[27] will

give

^the _saine ^{universal results for} ail temperatures below Tg as

long

_as the

spins

_are of

Ising

type and the interactions _are short

ranged (so,

for

instance,

also in trie _case of the

short-ranged Ising spin glass Feo.5Mno.5Ti03 [26]).

^As soon as one

considers e-g-

Heisenberg spins

_or RKKY-interactions trie quantitative behavior

might change, although

_we bave _{nor reason} to believe that the

qualitative

picture ^{of trie} results

presented

here

changes significantly.

3. The correlation function

C(t, t~ge).

The autocorrelation function

C(t,

_toge) defined in

equation (3)

measures the

overlap

of

spin configurations

at time _{t +} toge ^{with that achieved alter}

aging

the system ^for _a ^time twi at

temperature

T, exerting

a heat

puise

of duration

tp

^with temperature

Tp

^and

finally aging

trie system

again

for _a time tw2 at temperature ^{T. In}

figure

1 _we choose T

=

o-1,

twi =

lo~,

tp

=

lo~,

_tw2

= 10~

(a), lo~(b)

and various heat

puise

temperatures

Tp.

^In

figure

1 _one observes that the short heat

puise

diminishes trie correlations and

C(t,

_t~~e) varies

smoothly

between

trie two curves obtained

by

_{Tp =} T

(no

heat

puise)

and

Tp

= oo. The latter _curve is identical

to that obtained

by simple aging

with

waiting

time tw2 since Tp = oo

destroys

ail correlations grown

during

trie first

waiting

time twi ^{Thus the heat}

puise

tends _to reinitialize

aging, however,

not

completely

_as

long

_{as Tp} is _Dot

high enough.

This is in agreement ^{with the}

experiments [9-11]

^and

[12, 13],

^{but the} temeratures at which

aging

is

fully

reinitialized is much smaller _m

trie

experiments (see especially

[9,

loi

thon in _our results.

In

figure

2 trie _saule parameters as above _are used _up to the duration of trie

puise,

which is

now

tp

=

lo~. _Note that _Dow trie heat

puise

is

only

_one decade shorter than trie first

waiting

time twi and _one observes differences _to

figure

1: For Tp = I.o and 1.3 the correlations _are

larger

at

long

times t than those without heat

puise (one

_cari observe _a similar effect in trie

experimentally

obtained data in

Fig.

8 of Ref.

iii).

One

possible interpretation

is that _on

one side trie

longer

heat

puise destroys

_some of the correlations

ongmating

^from trie first

aging (note

that for small _t ail _curves with Tp > T lie below Tp =

T),

but drives trie system ^into

energetically

_more favorable states

(deeper valleys)

like in simulated

annealing

_[29]. Thus it is harder for the system to relax from the

vicinity

of the _state reached after t~ge, ^{which enhances} the correlations at

long

times.

This picture is

supported by figure 3,

where T ₌

o-1, Tp

₌ I.o and the _sum of first

waiting

time and duration of the heat

puise

_is

kept

constant: twi +

tp

₌ 1000. The

longer

the heat

puise

the

larger

trie correlations

C(t,

_t~ge) at

large

times t. For comparison we have inserted _a

plot

of the function

C(t,

tw ₌ lob _obtained

by simple aging

with _a much

longer waiting

time tw ~10~ _» t~ge.

This effect is

completely

absent if _one

performs

_a

negative

temperature

cyding

experiment with trie _same data for twi,

tp

^and tw2. Trie result for T

= 0.9 and Tp = o-1

(note

that _now

Tp

<

T)

is

depicted

in

figure

4: For

increasing

duration of trie "cola"

puise

the correlation

function

C(t, t~ge)

is

dearly

diminished. It _seems that _aging, which is relevant for the dynam-

N°6 THE THREE-DIMENSIONAL ISING SPIN GLASS MODEL 887

T=o.7,

t~i

₌ ^io'

,

t~

₌ ^io~

,

t~z ₌ io~

10°

/5 w

« n

« a

-~ na

£

^°°%

çJ a n

~

oT~ ^T °q~~

-T~ io 3. _i 6, 9 2 2 25

n T~ «

°

lÔ~

la' lo~ lo~ la' lo~ lo~

T=0.7,

t~i =10', tp ^=10~, t~~

^=10~

io°

/5

j

~

£

~J

oTp T

-T~ 0 3, 6, 9, 2 2, Z à

UT ce

~yi

lo~ lo~ lo~ lo~ lo~ lo~

1

Fig. l. The spin autocorrelation function

C(t,

toge) defined in (3) in dependence of t

(number

of

MC-steps)

for _vanous heat puise temperatures Tp. The other temperature cycling parameters _are

T ₌ o.7, twi _"

lo~,

_tp

= lo~ and tw2 _" lo~ _in the _upper figure and tw2 "10~ _{in the lower}

figure.

From top ta bottom _{it is} Tp ₌ T

(o),

Tp ₌ 1-o, 1.3, 1.G, 1.9, 2.2, 2.5

(fuit fines)

and Tp _{= oo}

ID).

The

errer bars _are sigmficantly smaller than the symbols for the _curves plotted with points.

ics at trie final temperature

T,

is frozen

during

trie

negative

temperature

cycle,

trie system cannot reach

valleys

_as

deep

_as those it would

explore during simple aging

at temperature ^T

(corresponding

to trie

tp

₌ ^o

curve). However,

this

freezing

_is

only partial,

_since

compared

with trie

simple aging

_curve

C(t,

_tw

=

tw2)

the correlations _are still

higher.

^Let _us condude this section with this observation of _a dear asymmetry between heat

pulse experiments

shown in

figures

1-3 and

negative

temperature

cyding

experiments shown in

figure

4.

4. The

magnetization M(t, _t~~e).

The correlation function that bas been

investigated

in trie last section is hard to measure

in

experiments

with real spm

glasses.

Nevertheless it bas _a

physical

meaning and it

yields

T=0.7, t~i =10'

,

tp

^=10~

,

t~~

^=10~

io°

D °Q~

~~OE

D Q~

Î

~~%

~ D

« D

-4 D~

+i ~%

~/ D

Cl Q~

OTp T °q~~

-T~ =10,13 16 20

UT ^°

~

lo~~

lo~ lo~ lo~ 10~ 10~ lo~

t

T=0.7,

t~i

^=10'

,

tp

^=10~

t~~

^=10~

io°

Q Q

/s °°q~~

~ _Q

°4 D

+fl

°?q~

Q

ii Cl

= 13

z _{= ~}

t

ig.

temperatures

from ta

bottom at t _+~ 10~) Tp ₌ T (o), Tp = I.Ù, 1.3,

1.fi, ^2.0

(fuit fines) and

Tp = cc in),

trie same

as

theorem, see _[21, 22] )

and

^ditional

information in

In

this

section we erform trie procedure already mentioned in

section

2,

exactly trie xperimental

situation

^escribed

in [9, 12,

13]: _trie temperature ycle is clone

a

weak

external

field, by hich

a

gnetization

is

_induced. After

trie

of trie

trie field is _switched off ^and _trie

decay

_of trie

(thermo)-remanent agnetization

(4)

is easured.

We show in trie ollowingresults for

rather

^trongagnetic

fields (h =

^{.5), for}

reason that trie data

are

less cattered, since trie

signal

magnetization)

is We

per-

formed also imulations for _h =

0.2

and h

= 0.1,

hich give ualitatively the

differences

originating in trie

fact ^that h ~ _0.5

is certainly outside the

linear

_{response regime}

are

not

servable on these time ^ales.

In figure 5 _we depicted trie results

aging temperature ^as T

=

o-1and twi = 10~,

trie

duration of trie pulse was

N°6 THE THREE-DIMENSIONAL ISING SPIN GLASS MODEL 889

HeaJ.Puise Experimenl T=0 7,T _p=10 t_wi+1_?='°°°.1_w2=I00 NegaJiveTempetature Cycle T~09 T_p=07 1_wi+J _p=1000 t_w2=100

D ~~

D

~ D

~~9'~

~~"~~~~~~° ~~éb"~~Î~~~

~%l' ^~ ~ z

D ",

~~c_

~~Î' ^{~~ Q "'} '.

""

~é

Q ', ~,

~"

D

~~ ', ~Q

°

'D ', O

~

' '.~ ~+ ", ^~ô~

t P-o _o

In

"'l'Î-Q

)

~

~

",,

_".»

À t_j-10-- ^{à .° _p= ~}

~

Î~~ÎÎÎ ^{' ". "} t~P~i ~~

~

~

',, _~,~

ÎÎ

' P"500

ÎÎ*ÎÎÎ ^{' ', jp"900 ~ <,}

C(t Ioe5) C

', Cil W2( D ° 1,

b ~q~ ^',

à ^',

' D

~ô. _~

io ioo iooo ioooo iooooo io ioo iooo ioooo iooooo

t

Fig. 3 Fig. 4

Fig. 3.

C(t,

_toge) in dependence of t with T

= 0.7, Tp ₌ 1.0 and tw2 " 100. The _sum twi +tp = 1000

is constant, from bottom ta top it is tp = 0

(o),

_tp

= 10, 100, 300, 500 and 900

(full fines).

The top

curve

in)

is just for comparison it is the function

C(t,

tw ₌ 10~) obtained from simple _aging with

waiting time tw ₌ 10~ _» toge.

Fig. 4.

C(t,

rage) from _a negative temperature cycling expenment with T

= 0.9, Tp ₌ 0.7

(< T!)

and tw2 ₌ 100. As in figure 3 the _sum twi + tp = 1000 is constant, from top ta bottom it is tp = 0

(o),

tp ₌ 10, 100, 500 and 900

(fuit fines).

The bottom

curve (D) is just for comparison ^it is the

function

C(t,

tw ₌ 10~) ^{obtained from} simple aging with waiting time tw ₌ 10~

= tw2.

trie final

waiting

time is tw2 ₌ 10~. _For comparison trie remanent

magnetization

obtained from

simple aging

at temperature ^T = o-1and

waiting

time tw

= 10 is shown. One observes that trie heat

pulse

diminishes trie

magnetization

for _times smaller thon 0.1 twi> like it does with trie correlations _m

figure

1. Trie

aging

_process is agam

only partially remitialized,

trie temperature of trie heat

pulse

bas to be _very

high

to

nullify

trie

magnetization

obtained

during

trie first

waiting

time tw. This is _a consequence of trie

high magnetic field,

which makes trie system "stiffer" with respect to a heat

pulse.

For smaller

magnetic

fields trie temperatures

needed _to

completely

re-initialize trie

aging

_{process are}

significantly smaller,

which should not be confused with trie observation that for

higher

fields

aging

effects _are less

pronounced [28].

Furthermore _one observes that for t > 0.2 twi trie

heat-pulse

is able to enhance trie _mag- netization, an effect that becomes _more pronounced trie

longer

trie heat

pulse

is. This effect

can be

interpreted

in trie _same way as in trie

preceding

section about trie correlation function

C(t, _t~~e):

The heat pulse diminishes trie initial

magnetization M(0, toge) slightly

^and

destroys

some of trie

magnetic

correlations grown

during

twi

Simultaneously

it drives trie system into

energetically

_more favorable states, ^which bave _a

non-vanishing magnetization

and _{are sur-} rounded

by higher free-energy

barriers. After trie

completion

of trie temperature

cycle

trie initial

magnetization

_is

smaller,

but it takes

longer

to relax from this

magnetized

state and to

approach

zero

magnetization.

Again,

if this picture ^is correct in _{essence, one}

might

_{expect a} different outcome _{in a}

negative

temperature

cycle

experiment. ^In

figure

6 _we show such _an

experiment

with T

= 0.9 and

cycle

TRM, Heal Puise H=05, T=0 7,t_wI =10A4,1_P=ioe2, t_w2=1VI TRM NegaJ,ve Temp -Cycle H=0 5, T~0 9,T_p=06 1_wt =10A3 J_~2=ioe2

T_p-T o ~

01 T_p=10

~

flp~ ^D

,~

(

^{_p=i 3} ~-P ~~~~

"° T$=Î tjl

=10A4

', ~o T_p=22 ~~j§- ^,_0Aj

'. ', ~.?~ ^T_p=25 ^{1-P- _P= o}

', fl

~ T_p=30

~ ~ ~

"_ ', ", ^{(J,J_w=10( Q}

f

','

',~ ^'Q

~i ^{006 '.} '.,

~ ~~~ ^o

',,~ Ù, ^° _~

~, ^{° '. ,, ~}

~ a '.

0 04 a

~ ~ ~~

D~~~

-,

O

'. ^o

~

~

o~ ".,_ °°o~~

0 02 °DDc~ _{~ ~~} ^o

a

~

o

Daa~~~

~ D

o o

io ioo iooo ioooo iooooo io ioo iooo ioooo iooooo

t t

Fig. 5 Fig. 6

Fig. 5. The thermo-remarient magnetization

MTRM(t,

toge) ^defined in (4) _in dependence of t

(num-

ber of MC-steps) for various heat puise temperatures Tp. The other temperature cycling parameters

are T

= 0.7, twi _"

lo~,

_tp

=

lo~ _{and tw2}

" lo~. _{From top ta bottom}

(at

_t

=

loo)

it is Tp = T

(o),

Tp = I.o, 1.3, 1.G, 1.9, 2.2, 2.5

(fuit fines)

and Tp = cc

in).

The

errer bars _are significantly smaller

than the symbols for the _curves plotted with points.

Fig. 6.

MTRM(t,toge)

from

a negative temperature

cycling

expenment with T

= o.9, Tp ₌ 0.6

(< T!),

twi = 10~ _{and tw2}

#

10~. _From bottom _ta top

(at

t ₌ 1000) ^{it is} tp ₌ 0

in),

tp ₌ 10~, 10~, 10~ and 10~. _{The top}

curve (o) _is just for comparison it is the function

MTRM(t,

tw ₌ 10~) obtained

from simple _agmg with waiting time tw ₌ 10~.

temperature Tp = o-fi

(note

that Tp ^{< T}

now).

Trie initial

waiting

time is twi ₌

lo~,

trie final

waiting

time is tw2 _# lo~. _For

increasmg

heat

pulse length tp

^trie remanent

magnetization

is either

unchanged

or

slightly

increased for ail times t. Thus trie negative temperature

cycle

does not

destroy

_any of trie

magnetic

correlations that bave build up

during

trie initial

aging

process. Trie

magnetized

domains continue _{to grow}

during

trie

cycle (however,

at _a much

smaller

rate),

for

comparison

trie

magnetization

_curve obtained for

simple aging

at T

= o.9

with _a

waiting

time tw =

lob _is

shown,

which shows _a still

larger magnetization

than that of

negative

temperature

cyding

with

tp

₌ ^105.

We condude that TRM-measurements in temperature

cyding

experiments manifests

again

an asymmetry ^{between heat}

pulse

and negative temperature

cycle

experiments and therefore

yield

trie _same picture as that obtained from trie calculation of trie autocorrelation function

C(t,

_t~~e) described in trie last section.

5. Discussion.

By calculating

trie autocorrelation function

C(t,

_t~~e) and trie thermc-remarient

magnetization MTRM Ii,

_{t~~e) we} tried to

explore

trie effect of temperature

cyding

_on trie

aging

_process ^within

trie three-dimensional

Ising spin glass

model. We demonstrated that

by

_a heat

pulse,

which is short

compared

to trie initial

waiting time,

the

aging

process is

partially

re-initialized. On

N°6 THE THREE-DIMENSIONAL ISING SPIN GLASS MODEL 891

the other

side,

_a

negative

temperature

cycle experiment partially

freezes the system into trie

(domain)-state

reached

during

the initial

aging

_process. Some

experiments

_on real spin

glasses

show _a much dearer outcome [9,

iii, which, nevertheless, might

be

interpreted

to _concur with

our observation. And

finally

_as

pointed

out in

il Ii

this asymmetry would

pledge

in favor of the hierarchical

picture

mentioned in trie introduction and

against

the

droplet picture.

However,

_our results _{are on a}

qualitative

level and the

dynamical

_processes involved _are still

microscopic

_{on a}

logarithmic

time scale. Thus it

might

be hard to

verify

_one

phenomenological, macroscopic theory

and

falsify

another _on the

ground

^of our numerical

data, although they

have been obtained with the most eflicient

existing algorithm implemented

_{on one} of the fastest computers ^{available. Trie} parameter space for this kind of

experiments

is

essentially

six-dimensional

(T, Tp,

twi,

tp,

tw2 and

h),

therefore _a

systematic investigation,

as was done for

simple aging experiments

in trie _same model

[22],

seems to be forbidden. Hence _we had _to confine ourselves to demonstrate what kind of scenario for temperature

cyding

experiments is obtained for trie

model,

time scales and

quantities

under considerations and _can

only

offer _a

possibly speculative interpretation.

Furthermore _we would like to

point

out that _very strong crossover

phenomena

_are observable within _our results _{as soon as} trie duration of trie heat

pulse

^{of trie} final

waiting

time become

comparable

to trie initial

aging

time. We

interpreted

them within _a

picture

of _a relaxation in

a

rough free-energy landscape,

which

again

seems to be most

appropriate

for trie results _we obtained. This

picture

is rather flexible and is able to

explain

_a lot of features in _a frustrated system real _or

theoretical,

and

regardless

of the existence of _a

phase

transition. In _a

theory

that is based _on trie assumption of _a relaxation within _a

complicated free-energy landscape

no

quantitative prediction

about the

growth

of

spatial

correlations

during

the

aging

_process is made. This feature is _on trie other side trie basic

ingredience

^{of trie}

droplet

model [4].

Although

both theories _{seem to} make

contradicting predictions je-g-

the symmetry or asymmetry ^{of heat}

pulse

and

negative temperature-cycling experiments)

_our

impression

is that

they

have _more in

common thon

usually

^admitted.

We think that it

might

be very useful _to try to find _a

synthesis

of both

models,

Dot

only

in order to be able to describe trie

growth

of

spatial

correlations and their destruction

by

_a heat

pulse

and their

freezing during

_a

negative

temperature

cycle.

^Domain

growth

has _not been

investigated by

direct measurements up ^to now

(for experiments

that

investigate

this _matter

indirectly

_{see [13, 30,}

31]). However,

in numerical simulations _one bas _an immediate _access to trie

quantities

^of interest and work _on this

subject

is _m progress [32]. ^{It is} our

impression,

obtained from trie results

presented

in this _paper and in other

publications [21, 22],

that the simulation of finite-dimensional

Ising spin glass

models _can make relevant predictions for real

spin glasses,

too, and will _prove to be _{a very} useful tool in

testing

and

improving

phenomenological

theories for them.

Acknowledgements.

I _am indebted _ta M.

Schreckenberg

and E. Vincent for

critically reading

trie

manuscript

^and

I _am

grateful

to E.

Vincent,

J.

Hamman,

J. P. Bouchaud and M. Mézard for _a

stimulating

discussion. I would like to thank the HLRZ at the research _center in Jühch for the generous

allocation of computing time

(approximately

250 CPU

hours)

_on the

Cray

YMP. This work

was

performed

within the SFB 341 KôIn-Aachen-Jülich.

References

iii

Lundgren L., Svedlindh R., Nordblad P. and Beckman O., Phys. Rev. Lett. 51

(1983)

911.

[2] Lundgren L., Nordblad P., Svedlindh R. and Beckman O., J. Appt. Phys. 57

(1985)

3371;

Hoogerbeets R., Luc Wei-Li and Orbach R., Phys. Rev. Lett. 55

(1985)

III.

[3] Binder K. and Young A.P., Rev. Med. Phys. 58

(1986)

801.

[4] ^Fisher D.S. and Huse D.A., Phys. Rev. B 38

(1988)

373; Phys. Rev. B 38

(1988)

386.

[5] Koper G-J-M- and Hilhorst H-J-, J. Phys. France 49

(1988)

429.

[6] ^Parisi G., Pliys. Rev. Lett. 43

(1979)

1574; Phys. Rev. Lett. 50

(1983)

1946.

[7] ^Lederman M., Orbach R., Hammann J-M-, Ocio M. and Vincent E., Phys. Rev. B 44

(1991)

7403.

[8] ^Bouchaud J-P-, J. Phys. 1Ifa~lce 2

(1992)

1705;

Bouchaud J-P-, Vincent E. and Hammann J., J. Phys. I France 4

(1994)

139.

[9] Refregier P., Vincent E., Hammann J. and Ocio M., J. Phys. France 46

(1987)

1533.

[10] ^Vincent E., Hammann J. and Ocio M., chapter 7

(pp. 207-236)

of Recent Progress in Random

Magnets, D. H. Ryan editor Ed.

(World

Scient., Singapore,

1992).

iii]

Lefloch F., Hammann J., Ocio M. and Vincent E., Europhys. Lett. 18

(1992)

647.

[12] Granberg P., Lundgren L. and Nordblad P., J. Mag~letism a~ld Mag~letic Materials 92

(1990)

228.

[13] ^Mattson J., Djurberg C., Nordblad P., Haines L., Stubi R. and Cowen J-A-, Phys. Rev. B 47

(1993)

14626.

[14j Bray A.J. and Moore M.A., Pliys. Rev. Lett. 58

(1987)

57.

[15] ^Crisanti A., Horner H. and Sommers H-J-, Z. Pliys. B 92

(1993)

257.

[16] Cugliandolo L. and Kurchan J., Pliys. Rev. Lett. 71

(1993)

173; preprint

coud-mat/9311016.

il?]

Cughandolo L., Kurchan J. and Ritort F., Pliys. Rev. B 49

(1994)

6331.

[18] ^{Pansi G. and Marinari} E., J. Phys. A 26

(1993)

Ll149.

[19] ^{Franz S. and Mézard} M., Europliys. Lett. 26

(1994)

209.

[20] ^Iofle L.B., Phys. Rev. B 38

(1988)

5181;

Freixa-Pascual M. and Horner H., Z. Phys. B 80 (1990) 95.

[21] ^Andersson J-O-, Mattson J. and Svedlindh P., Pliys. Rev. B 46

(1992)

8297.

[22] Rieger H., J. Phys. A 26

(1993)

L615; and ta be published.

[23] ^{Ito N. and Kanada} Y., Supercomputer 25

(1988)

31.

Heuer H-O-, Camp. Phys. Comm. 59

(1990)

387;

Rieger H., J. Stat. Phys. 70

(1993)

lo63.

[24] ^{Bhatt R-N- and} Young A.P., Phys. Rev. Lett 54

(1985)

924;

Ogielski A.T. and Morgenstern I., Phys. Rev. Lett 54

(1985)

928.

[25] ^Marinari E., Pansi G. and Ritort F., preprint coud-mat

/9310041.

[26] ^Gunnarson K., Svedlindh P-j Nordblad P., Lundgren L., Aruga H. and Ito A., Phys. Rev. Lett.

61

(1988)

754.

[27] Hohenberg P-C- and Halperin B-I-, Rev. Med. Pliys. 49

(1977)

435.

[28] ^This can aise be _{seen in} trie expenments: ^Vincent E., private communication.

[29] Kirkpatrick S., Gelatt C.D. and Vecchi M.P., Science 220

(1983)

671.

[30] Granberg P., Sandlund L., Nordblad P., Svedlindh P. and Lundgren L., Pliys. Rev. B 38

(1988)

7097.

[31] ^Schins A., Arts A.F.M. and de Wijn H-W-, Pliys. Rev. Lett. 70

(1993)

2340;

Schins A., Dons E-M-, Arts A.F.M., de Wijn H-W-, Vincent E., Leylekian L. and Hammann J., Phys. Rev. B 48

(1993)

16524.

[32] Rieger H., Steckemetz B. and Schreckenberg M., _in preparation.