Low temperature behaviour of 3-D spin glasses in a magnetic field

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Submitted on 1 Jan 1990

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Low temperature behaviour of 3-D spin glasses in a

magnetic field

Sergio Caracciolo, Giorgio Parisi, Stefano Patarnello, Nicolas Sourlas

To cite this version:

Low

_temperature

behaviour of 3-D

_spin

_glasses

in

a

magnetic

field

Sergio

Caracciolo

_{(a), Giorgio}

Parisi

_(b),

Stefano Patarnello

_(c)

and Nicolas Sourlas Laboratoire de

_{Physique Théorique}

de l’Ecole Normale

_{Supérieure (*),}

24 rue _Lhomond, 75231

Paris Cedex 05, France

(Received

on March 14, 1990,

accepted

on

_May

_15,

₁₉₉₀₎

Résumé. 2014 Nous

_présentons

les résultats de deux ensembles de simulations

_numériques

de verres

de

spins d’Ising

tridimensionnels en

_présence

d’un

_{champ magnétique.}

Dans le

premier

ensemble nous calculons, entre autres

_quantités,

la distribution de

_probabilité

du recouvrement des

configurations

des

_{spins P (q ),}

la distribution de

_probabilité

du recouvrement des

configurations

des liens

_{Pe(qe) et}

la

_{susceptibilité}

verre de

spin

~SG pour différents volumes et

_{températures.}

Les

résultats sont en bon accord avec la théorie du

_champ

moyen :

P (q )

et

_Pe(qe)

sont non triviaux et non _{automoyennants} et _{~SG augmente} avec la taille linéaire L du

_système

comme ~SG ~ L03C9. Nous

estimons 03C9 = 1,8 ± _0,25 à T = 0,83. Dans le deuxième ensemble de simulations nous introdui-sons un

_{petit couplage}

03B5 entre deux

_copies

du

_système

et leur recouvrement

_Q(03B5)

est calculé en

fonction d’03B5. La pente de

Q(03B5)

à

_l’origine

augmente avec L, en accord avec les autres

simulations. Nos données ne sont _pas

_compatibles

avec les alternatives à la théorie du

champ

moyen

proposées

à ce

_jour.

Abstract. 2014 We _present the results of two sets of numerical simulations of 3-d

Ising spin-glasses

in the presence of an uniform

magnetic

field. In _{the first set,} among other

quantities,

we _compute

the

_{spin-spin overlap probability}

distribution

_{P (q ),}

the link-link

_{overlap probability}

distribution

Pe(qe)

and the

_{spin-glass susceptibility}

_~SG for different volumes and _{temperatures.} The results

are in

good

agreement with mean-field behaviour :

P(q)

and

_Pe(qe)

are non trivial and non

self-averaging

and _~SG shows an increase with the linear size L of the _system as

_{~SG ~ L03C9.}

Our estimate is 03C9 = 1,8 ± 0.25 at T = 0.83. In the second set of simulations a small

coupling

03B5 is

introduced between two

_copies

of the _system and the copy

overlap Q (03B5)

is

_computed

as a function of 03B5.

_Q(03B5)

becomes _steeper around 03B5 ~ 0 as L increases, in _agreement with the

_previous

set of simulations. Our data seem

_{anyhow incompatible}

with the alternatives to mean-field

theory

proposed

so far.

Classification

Physics

Abstracts 75.50 - 75.10

Introduction.

Spin glasses

are well understood in the infinite range

limit,

where the mean field

_theory,

based on the

_replica

or on the

_cavity

_approach

_[1],

should

_give

the exact results.

Quite

likely

the

same behaviour is found in short range models in the limit of infinite dimensions. In this case,

also at nonzero

magnetic

field,

there are two

_{phases :}

the usual

_paramagnetic

high-temperature

phase,

and a low

_temperature,

_{glassy phase,}

characterized

_by

a

_large

number

(strictly speaking infinite)

of pure

equilibrium

states. The

_dynamics

of the

_system

is

_quite

complex.

For intermediate times the

_system

remains in a

_given

_pure state and the time correlation functions

_decay

_{asymptotically (within}

this time

_range)

with a _power

_{law ;}

at much

greater

times the

_system

_hops

_{between different pure}

_equilibrium

states. The

_hopping

time

diverges

with the size of the

_system,

and this feature makes numerical simulations of

equilibrium properties quite

awkward.

All these states have the same

thermodynamical properties,

in

particular they

have the same free _energy

_density,

but

_they

differ in their local

magnetizations ;

a

_quantity

useful to

compare different states is the

overlap

parameter

q"10

defined as follows :

where

_{m "}

is the local

_{magnetization}

in the sate « at the site i and N is the total number of sites. The

_{self-overlap q"a}

turns out to be

_independent

of the state a and is

_usually

called the Edward-Anderson order

_parameter

qEA. Another useful

quantity

is the link-link

overlap,

defined as :

Here U

_{uj> e}

is the value of the correlation between

_spins

at sites i

_{and j}

_averaged

in the pure state a and

_J..

is the

_coupling

between these

_spins.

It is not

_completely

understood whether the mean field

_picture

remains correct also in a

short range model at a

given

finite dimension. While this is

possible

for

large

dimensions,

there must be a lower critical dimension below which this

_picture

fails. In

_particular

it is

believed that in two dimensions there is no

_{glassy phase.}

The behaviour in an extemal

magnetic

field is even more

_complex

to foresee : as in finite dimension the amount of

frustration is

_finite,

there exists a critical field

_h,

such that for h

_{:--. h c}

no

_phase

transition occurs. Is

_{hc =F- 0 ?}

It is very difficult to understand the situation in three dimensions.

_Very

_long

computer

simulations

_[3,

_4]

at zero external

_magnetic

field,

hint at the existence of a

phase

transition,

from the

_high

temperature

phase

with zero _qEA, to a low

_temperature

phase

with non zero

qEA ; numerical simulations of

relatively

few instances of the interactions at low

_enough

temperatures

(when

the correlations

_length

is

_larger

than the size of the

_lattice)

show a

mean-field like behaviour both for the statics

_[5]

and for the

_{dynamics [6].}

But it is not _easy to

extrapolate

to the very

large

volume

behaviour ;

moreover the

_experimental

data are

consistent with the occurrence of a

_phase

transition

although

the

possibility

of a

_gradual

freezing

is difficult to exclude.

Most

_{unfortunately}

there is even less information on the behaviour of the

_system

at nonzero

magnetic

field : it would be

_extremely

_interesting

to know if the

_{glassy phase predicted by}

the

mean field

_theory

survives or not under these circumstances. Some

_{qualitative scaling}

destroy

such a

_glassy

_phase

_and

therefore no

_phase

transition occurs in this case. The main

assumption

on which this

simplified

model rests, is that the excitations of the

_system

at low

temperature

are due to the

_flipping

of

_independent,

localized

_{regions (« droplets ») ;}

due to

frustration,

the interface _energy of such an excitation would _{grow very}

weakly

with the size of

the

_droplet,

whilst the

_gain

in _energy for any finite

magnetic

field to reverse all

_spins

in a

droplet

in a

_given

direction would

_always

_{be favorable. Therefore any}

_ordering

other than the uniform one would be unstable

_(this

is the so-called

_{droplet model).}

So far this

_picture

has not

received a direct confirmation

_(e.g.,

via numerical

_simulation).

An

_{important quantity}

to

study

in this context is the

_probability

distribution

_{Pe (qe)}

of the link-link

_overlaps

qe introduced in

(Eq. (2)).

When a domain of volume v and surface s is

_{flipped, q changes by}

an amount

_proportional

to _v,

_{while qe}

_{changes by}

an amount

_proportional

to s. Thus

_by

comparing qe and q

it is

_possible

to

_disentangle

surface from volume effects.

In _{this paper} we have

investigated

whether some of the

predictions

of the mean field

_theory

for the low

_temperature,

_{glassy phase}

are confirmed in three dimensions at nonzero

_magnetic

field.

_Indeed,

our numerical simulations seem to confirm many of the features proper to mean

field

_theory.

We have measured the

_{spin glass}

_{susceptibility XSG}

and we found that it exhibits

a critical behaviour at low

_temperature.

_{This may be}

_expressed

via an

_{approximate scaling}

law of the form _{Y SG -}

_{L ’,}

where L is the linear size of the

_system

and ù) is a

_positive

_exponent.

The

_probability

distribution

_{P (q )}

seems to be non trivial and non

selfaveraging

in the infinite

volume limit.

_{Pe (q,)}

exhibits a similar behaviour.

In section 2 we introduce the model Hamiltonian and

_provide

some technical details of the numerical simulations. In section 3 we

_present

our numerical results. These results are

discussed in section 4. A brief

_description

of the results of the

_present

simulations has

_already

appeared

elsewhere

_[9].

Simulation framework.

The

_question

of the existence of _{many pure} states and their

_overlaps

may be studied

considering

two identical

_replicas

of the same

_system,

whose Hamiltonian is defined on the

sites of a cubic lattice with

_{periodic boundary}

conditions :

We indicate the two

_replicas

of the

_system

_using

the two sets of

_{spins a}

and T. The notation

(j)

stands for the sum over

_pairs

of nearest

_neighbour

sites. The first two summations describe the standard Hamiltonian for a

_spin-glass

in a uniform

_magnetic

field

_(for

two

identical

_replicas).

The

_coupling

_constants

are

_{statistically independent}

random

variables,

which take with

_{equal probability}

the values ± 1. There is also

_occasionally

a _very small

coupling

e between the two

_replicas

for reasons that will become clear later. Our

_analysis

is based on a

_large

set of Monte-Carlo simulations of the

_system

described

_by

the Hamiltonian

(3).

We used the

Metropolis et

al.

_algorithm.

In all our simulations the

_magnetic

field was

kept

fixed at the value h = 0.2. We did two kinds of numerical

experiments.

A first _set

(which

in the

_following

will be referred to as set

_A)

concerns

_{uncoupled copies (8}

= 0 in formula

_(3)).

Here measurements were

_performed

at three different

_{volumes, V}

=

L 3

L =

6, 10,

14. To

achieve statistical

_equilibrium,

we started with random initial conditions at

_temperature

T = 4

and then

_slowly

cooled the

_system

_{by increasing}

the inverse

_temperature

_/3

=

1 / T

by

successive

_steps

_{of AB =}

0.05. Measurements were

_performed

at

_{/3 = 0.65,}

_{0.75, 0.85,}

_1,

1.2. For each volume a total of

_Nj

different realizations of the

_couplings

_.l¡j

were

_randomly

configurations

of the

_spins)

were simulated. For L = 6

Nj

was set to

_32,

whereas for L = 10

and L =

14,

Nj =

16.

The definitions of

_{P (q)}

and

_{Pe (qe )}

_given

in

_{equations (1)}

and

₍₂₎

are not _{very useful for}

numerical purposes because

_{they require}

the

_knowledge

of the local

_{magnetizations}

_{for every} state a. It is well known

_[1]

however that there exists an alternative

definition,

thermodynami-cally equivalent

to the

_previous

one, where this

knowledge

is not

_{required. According}

to this

definition it is

_enough

to measure the

_probability

distribution of the

_overlaps

of individual

configurations

of the

_{copies provided}

that these

_{configurations obey}

the Boltzmann distribution. So the

_{Pj(q) probability}

distribution has been

_{computed numerically}

as the

probability

distribution of the

_{configuration overlap}

which for the

_copies

a

and 8

is

Similarly

we measure

_{P e, J(qe),}

the

_probability

distribution of the link-link

_overlaps

qe :

Remember that in our

simulations .l;} =

1. We did these measurements for several

_pairs

of such

_copies

for _every one of the

_{Nj Hamiltonians.}

We denote

_{by P (q )}

and

_{Pe (q )}

_{the average}

over the J’s. We also

_computed

the

_{spin glass susceptibility :}

where ( > j

means the average over the J’s.

In the second set of simulations

(we

will denote this as set

_B)

we considered

_coupled

replicas,

i.e. the E term in

(Eq. (3))

different from zero. The average

overlap

among two

replicas

is now a function of the

_{coupling strength}

Q

=

Q

(E).

The reason for

_introducing

this

coupling

between

_copies

is the

_following.

Consider the case of a

_ferromagnet.

At the

_phase

transition there is a

_{spontaneous symmetry}

_breaking

with the appearance of a

_spontaneous

magnetization.

It is

_possible

to

_study

this

_symmetry

_{breaking by measuring}

_{the response of the}

system

to an extemal

_symmetry

breaking

field. One can

_study

the

_{magnetization}

m (h ),

the order

_parameter

of this

_problem,

as a function of an extemal

_magnetic

field h

_(the

variable

_conjugate

to the order

_parameter).

Below the critical

temperature,

m (h)

becomes

steeper

and

_steeper

around h - 0 as the volume increases and it

develops

a

_{discontinuity}

at

h = 0 in the infinite volume limit. The

magnitude

of this

discontinuity

is twice the

spontaneous

magnetization.

In the

_spin-glass

case the order

_parameter

is the

_overlap

q. The variable

_conjugate

to the order

_parameter

is the

_coupling

e between two

_copies.

If there is a

phase

transition at a

_temperature

_T,

we

_expect

that for T --

_{T,,, Q (E)}

will show a similar

behaviour around E =

0,

i.e. it will become _a

_steeper

function of e as the volume increases

and

_develops

a

discontinuity

if the transition in E is first order.

Q (e) provides

additional

information which may be

independent

from that

_{supplied by P (q) [10].}

Remark that for

£

_{= 0,}

..t

The thermalization schedule for the set B of

_experiments

was similar to the

_previous

one. Simulations were

_performed

for V =

43,

63,

83

and

103.

The e term is chosen small

compared

this

_purpose

we chose e such that

EQ (e) - E « hm (h) - h 2.

The values of e used in these

simulations

_ranged

from s = - 0.0025 _{to e} = 0.01. In this B set of _runs for each e we had a

single pair

of

_copies

for all the

_Nj

= 128 different Hamiltonians we simulated.

It is not _easy to ensure thermal

_equilibrium

for

_{spin-glasses.}

It is well known that very

long

relaxation times characterize these

_systems,

and indeed no thermalization check is safe from

criticism. If there exist several

_valleys

one would

_like,

for non

_{self-averaging quantities,}

to

disentangle

thermalization within a

_valley

from

_valley

to

_valley

transitions. To check thermalization we did the

_following.

First we checked that the

_{self-averaging quantities}

(namely

the

_{energy E}

and the

_{magnetization m)}

did not show any drift

during

the measurements. This is illustrated in

_figures

1 and 2.

_Secondly

in the case of the B runs

_(we

recall that the thermalization schedule was the same for the set A and the set

_{B run),}

we

picked,

at

_random,

a few

_pairs

of

_copies

and looked at them

_{separately. Figure}

3 shows the

recording

of the

_overlap

of one

_{single pair}

versus simulation time and

figure

4 and

figure

5

show

_respectively

its

_probability

distribution

_during

the first and the last

quarter

of the

measuring

time. What is characteristic in

_figure

3 is that the

_systems

move

_by

sudden

_{jumps (it}

is

_tempting

to

_interpret

those as

_valley

to

_{valley transitions)}

and the

_overlap

comes back to the

original

value at much later times. This

_strongly

suggests,

in our

_opinion,

that the

_overlaps

we measure

_correspond

to

_equilibrium.

We consider this as an evidence that the simulation times are of the order of the

_typical

times

_{to jump}

from one

_valley

to another. It is

_interesting

also to

notice in

_figures

4 and 5 that the

_probability

distribution of the

_overlap

_{is very broad}

_(we

would like to _say that several « states » are

_visited),

and that this distribution is

essentially

the same for the first and the last

_quarter

of the

_measuring

time. This is another

_{strong argument}

for

_equilibrium.

For small sizes we see

occasionaly

that q -+ - q, i.e. the

system

completely

reverses

_itself,

before

_coming

back to the

_{original position.}

For several

43

systems

there are

only

two allowed values of the

_{overlap q}

and - q, which means that there is

only

one state,

while for

_larger

_systems

we find

several different values of q. This is in

agreement

with

Fig.

1. _{Internal energy,}

_averaged

over the different _systems, as a function of Monte-Carlo time, at

Fig.

2. -

Magnetization, averaged

over the different _systems, as a function of Monte-Carlo time, at /3 = 1. The time scale is in

arbitrary

units.

Fig.

3.

_{- Spin-spin overlap}

of one

_particular

copy, as a function of Monte-Carlo time. The time scale is

Fig.

4. -

Histogram

of the

_{spin-spin overlap}

of one

_particular

_copy,

_during

the first quarter of the

measuring

time.

Fig.

5. -

Histogram

of the

spin-spin overlap

of the same _copy as in

_figure

4,

during

the last quarter of the

_measuring

time.

previous

simulations

_[5]

which also show an increase of the number of states with volume. Note that the relative

overlaps

of these states are

_{significantly}

different.

Finally

we measured the time average of the local

_{magnetizations}

of individual

_copies

IL t =

(Ilt)

_{Lut ( T)}

where T is the Monte-Carlo « time » and we

_required

that

T = _1, t t

where

_{Na is}

the number of

_{copies ;}

i.e. we measure the average

overlap

in two

_completely

independent

ways. The first

by

time

averaging

a

_single

_copy and the second

_{by using}

the

short,

the first determination is too

_large

because a _copy does not have time to thermalize and every

spin

remains

strongly

correlated with its initial

position,

while the other is too

_small,

as

the different

_copies

start from different random

_{configurations}

and would be almost

orthogonal

to each other.

Equation (8)

is a

_highly

non trivial numerical constraint.

Indeed,

by

looking

to

_{P (q) (see Figs. 8),}

one can estimate qEA, the maximum value

of q

for which

P (q) is

non zero and which

_corresponds

to thermalization within a

_{single valley.}

On the other

hand

_{dq qP (q) -}

0.43,

very different from qEA. As

equation (8)

is well verified

by

all our

data,

we conclude that what we are

_measuring

are the

_{equilibrium properties}

of our

_systems.

This thermalization test _{is very} similar to that of Bhatt and

_Young

_[4].

Results.

We start

_{by presenting}

the results of the set A of runs, i.e. the case where there is no

_coupling

between

_copies.

Before

_discussing

in detail the low

_temperature

measurements, it is

instructive to show the

_exchange

energy

(i.e.

(Jij

a;

oj»

and the

magnetization

as a function

of

_temperature.

This is done in

_figures

6 and 7

_{respectively.}

We see that

_starting

at a certain

temperature

which is

_{quite high,}

the

_{magnetization}

_stops

_growing

with

_/3

while the

_exchange

energy continues to increase. This behaviour of the

magnetization (which

is reminescent of mean-field

_theory),

has

already

been observed

_{experimentally [11].}

Fig.

6.

_{- Average magnetization}

as a function of the inverse _temperature /3, for L = 14.

Fig.

7.

_{- Average exchange}

energy as a function of the inverse _{temperature 6,} for L = 14.

Figures

8a and 8b show the

_(averaged

over the

_{J’s) P (q )}

_{at 8}

= 1.2

and P

= 0.65

respectively,

for L =

6,

10 and 14.

For 8

= 0.65 the behaviour of

P (q ) depends strongly

on

the size of the

_system.

For L =

6 P (q )

is non

trivial,

while it is like a 8-function for

L = 14. This indicates that at this

_temperature

the correlation

length

is in between those two

Fig.

8a. -

Average P (q )

as a function

_{of q at 8}

= 1.2. The dashed line is for L = 6, the dot-and-dash

line for L = 10 and the continuous line for L = 14.

Fig.

8b. -

Average

P

_{(q )}

as a function

of q

at ₈ = 0.65. The dashed line is for L = 6, the dot-and-dash

sizes,

in

_agreement

with

_previous

simulations

[3]. For f3 =

1.2 the

_{P (q)’s}

for the three sizes

are non trivial and

qualitatively

similar to each other.

Figure

9 _{shows the average link-link}

overlap

Pe (qe )

for L = 10 and for

8 =

1. We _see that not

only

the

configuration overlap

but

also the link-link

overlap

has a non trivial distribution. As we will see

later,

the two

_overlaps

are in fact

_strongly

correlated.

Fig.

9.

_{- Average}

_{Pe (qe )}

as a function

_{of qe}

at p = 1.0 and L = 10.

Another

_important

observation is the absence of

_{self-averaging}

of

_{P (q ) (the}

same is true

for

_Pe(qe),

see

_later),

as in mean-field

_{theory [12, 13].}

This can be seen

by looking

at different

PJ(q)’s (three

of them are shown in

_{figure 10)}

or can be measured more

quantitatively by

At

_/3

= 1 _we found that S =

0.81,

0.82,

0.86 for L =

6,

10 and 14

respectively : increasing

the

volume does not decrease the absence of

_{self-averaging}

in P

_(q).

One of the most

_striking

results we observed was the

apparent

correlation between

spin-spin

and link-link

_overlaps.

In order to

_clarify

this

_important

_point,

we did detailed

measurements of the conditional

_{probability P (qe ;}

_{q ),}

i.e. the

_probability

for the link-link

overlap

to take the

_{value qe}

when the

_{spin-spin overlap equals}

_q. We found that this

probability

is _very

_peaked

and

_{self-averaging.}

To a _very

good approximation

it is a delta function

where

_{a, b}

and c are constants

_depending

on the

_temperature.

The values of these constants

are a = 0.317 _±

0.003,

b = 0.27 ± 0.01 and c = 0.41 _± 0.02

for 13

= 0.65 and a = 0.44 ±

Fig.

10. -

Pj(q)

for three different realizations of the

couplings

at T = 1 and L = 14.

data,

we show P

_{(qe ; q ) averaged}

over the different

_coupling

realizations in

_figure

11. As we

will see

later,

in mean-field

_{theory qe}

is a

_quadratic

function of _q.

When

_{measuring P (q )}

we can

_{simultaneously}

extract the

_{spin-glass susceptibility}

The results are

_given

in

_figure

12 where we

_give

the

_{log-log plot}

of _XSG as a function of the

linear size L of the

_systems.

We

_plot

the results for three values of the inverse

_temperature,

/3

= 0.65

(the

lower

points)

/3 =

1

and 8 =

1.2

(the higher points).

For /3 =

1 and

/3

= 1.2 the

points

seem to be

_aligned

on a

_straight

_line,

i.e. our results are

_compatible

with the

_assumption

_{of scaling}

for X sG, while this seems not to be the case

_{for /3}

= 0.65. The errors

shown in the

_figure

_represent

_only

the J to J fluctuations.

_Assuming

naive

_scaling

at

/3

=

1.0,

we

_get

_{XSG - L’}

with = 1.55 ± 0.25 while _at

_/3

= 1.2 we

_get

= 1.8 ± 0.25.

We measured also the

_probability

distribution

_{Pi (e)}

_{of the average} energy for a link. More

precisely

for _every realization of the Hamiltonian and for every

_{link (ij )}

of the lattice we

measured the _average of the

_exchange

_energy

where ( )

is the time average, the sum is over the different

_copies

of the same Hamiltonian and

_Nc

is the number of

_{copies. Pî(e)}

is the

_probability

distribution

_{of eij}

over the lattice

links. We found that

_Pp(e)

is self

_averaging.

Our results are shown in

figure

13. If the

dynamics

is driven

_by

the successive

_flipping

of the same domains on the lattice we

_expect

to

find a

_{secondary peak}

in this distribution around e = 0.

Clearly

no such

_peak

is found in our simulations. This

_again

is an indication for a more

_{complicated dynamics}

than the one

Fig.

11. P

_{(qe ;}

_{q ) (see}

text for its

_{definition) averaged}

over the différent realizations of the

couplings

at B - 1.0 and L = 10. The

magnitude

of

_{P (qe ; q )}

is

_proportional

to the size of the

points.

Fig.

12.

_{- Log-log plot}

of the

_{spin-glass susceptibility}

versus the size of the _system. The upper

_points

Fig.

13. -

Pl (e)

(see

text for its

_{definition) averaged}

over the different realizations of the

_couplings

for

L = 10. The continuous line is for Q = 1.0, the dot-dashed line for 13 = 0.85 and the dashed line for

/3=0.65.

Let us now discuss

ultrametricity [12].

For every realization of the J’s we

_peaked

three

copies (in

several different

_ways)

and we calculated the

_{corresponding three-configuration}

overlaps

ql::--q2:--q3. When

0. 5 , q 1 10. 6,

we measured the

_probability

distribution

!S (,6 q )

of

_{b q}

= q2 - q 3 as a function of the volume. This method

_{of looking}

for

_{ultrametricity}

is the same as in Bhatt and

Young

[14].

The results are shown in

_figure

14. There is a definite

trend for

_{’ ( 8 q )}

to become

_steeper

as the volume increases. This is

_compatible

_{with the space} of states

_becoming

ultrametric in the infinite volume

limit,

but one should be careful before

jumping

to

conclusions,

as we will discuss later.

In the set B of runs we measured the average

_{configuration overlap (2(e)}

as a function of

the small

_coupling

e

between

two

_copies.

The result is shown in

_figure

15 for the

_following

values of the linear size of the

_{system :}

L =

4,

6,

8 and 10.

(2 (e )

becomes _a

_steeper

function of

E as the volume increases in

agreement

with the results on _{X sG} of the runs with

_uncoupled

Fig.

14. -

J(8q)

for L = 6

(dashed line),

L = 10

(dotted line)

and L = 14

(continuous line)

_at

,s = 1.2.

Fig.

15. -

Discussion.

In this

_chapter

we would like to discuss the

_{following points.}

a)

The

correspondence

between

_{configuration overlaps}

and link-link

_overlaps.

- One of our most

_interesting

results is the one to one

correspondence

between

configuration overlaps q

and link-link

_overlaps

_qe. This has

_already

been observed in connection with the

_study

of the robustness of

_{ultrametricity}

both

_{numerically [15]}

and in the framework of the

Sherrington-Kirkpatrick (SK)

model

_{[15, 16].}

In the latter case the

_{correspondence}

can be made very

precise.

As is well

known,

when all the

_spins

are

_{coupled together, only}

the first two moments

of the J distribution are relevant in the

_{thermodynamic}

limit. So we can consider

_only

the case

of the.l;j = :t

1 distribution. For two

_{given configurations}

of the

_spins

_U and _ri their link-link

overlap

is qe =

(IIN 2)

o-

cry

Tu

rj

where the sum over i

_{and j}

runs over all the sites of the

ij

system.

The

_{right-hand expression}

is

nothing

but the square of the

spin-spin

ovérlap.

So in the SK model we have _qe =

q 2

for

every

spin configuration.

One can also show on

_general

grounds

the

_{following inequality :}

qe --

e 21

where e is the average energy of a link. This result

is shown in the

_appendix.

It seems to be a contradiction between this

inequality

and the SK

model result _qe

_{= q 2}

_{for q}

= 0. In

fact,

because of the choice of

normalization, e

= 0 in

the

thermodynamic

limit of the SK model and there is no contradiction. For two identical

configurations qe

= q = 1. This relation

provides

a check to our numerical results.

Is _{there any relevance of} the

_previous

results for a

_{droplet-like picture ?}

When two

configurations

differ

_by

the reversal of a

_compact

domain of

_{spins (a droplet),}

_{q = 1 - 2 v / V}

while qe

= 1 - 2

s /3

V where v is the number of

_spins

inside the

_{domain, s}

the number of

links at the interface between the domain and the rest of the

_system

_(the

« surface » of the

droplet)

and V is the total volume of the

_system.

In order

_{for q and qe}

to be

_{substantially}

different from one, both the volume and the surface of the domains must have fractal dimension

_equal

to three. For this reason this result

_together

with the

_strong

correlation

between q and qe

are difficult to

_get

in the framework of a

_{simple droplet}

model.

b) Scaling andfinite

volume

_effects.

- There are

_clearly

several

_signs

of finite volume effects

in our simulations. The most obvious is in the volume

_dependence

of

_{P (q).}

There is a tail in

P ( q )

for

_{negative q’s}

and this tail

_strongly

decreases with the volume of the

_system.

There is also a volume

_dependence

of qEA, the maximum value of q. One would like to make a

decomposition

P (q, h ) = P. (q, h ) + AP (q, h, L )

where

_{P. (q, h )}

is the infinite volume result and a

_scaling

ansatz for the finite volume correction

_{OP ( q,}

h,

L ).

The

_positivity

of

P ( q )

for _any volume and the lack of data for many different sizes and different values of the

magnetic

field did not allow us to do so.

The same finite volume effects were

_present

in the B set of numerical

experiments.

We would like to make a

_scaling

ansatz _{for the average}

_{overlap a (e )}

similar to the one valid in the

ferromagnetic

case. In that case it is well known that the

_{magnetization}

as a function of an

external

_magnetic

field scales as

_{m (h, L )}

=

f (hL ’), w

= 3. A similar _ansatz in _{our case}

would be

_{a{e, h,}

_{L )}

=

f (EL’,

hL 8).

As _we do not have data for different values of

h,

_we

have,

in a first

_{approximation, ignored}

the

_possibility

of

_dependence

on

hL,6.

In

_figure

16 we

show the same data as in

figure

15

plotted against

the scaled variable EL’. We took cv = 1.55 as determined

_by

the volume

_dependence

of the

_spin-glass

_{susceptibility}

at e = 0. The data are less

scattered,

however there is still some volume

_{dependence ;}

indeed

such an ansatz

_implies

that for

E = 0,

a{ e) is L independent.

But we do find a volume

dependence

of

₍₂₍₀₎

which means that for the volumes we

_study

there are still corrections to

Fig.

16. - The same as in

figure

15 but with e rescaled

(see text).

difficulty

of

_writing

down a

_{simple scaling}

law forbids us to reach a definite conclusion on this

important point.

Of course one should also consider the more radical

_possibility

that

spin-glasses

have a more

_complicated

behaviour than the

_systems

without disorder we are familiar

with

_(a

multifractal

_behaviour,

for

_example)

and that naive finite size

_scaling

is never valid.

c)

Critical exponent

for

the

_{spin-glass susceptibility.}

- As the discussion in the

previous

paragraph

suggests,

the best critical

_exponent

we can measure is a kind of effective

_exponent.

On the other hand the

_{log-log plot}

of the

_{spin-glass susceptibility}

versus the

size,

shown in

figure

12,

seems to

_give

a

_straight

line thus

_suggesting

a

_{simple scaling}

behaviour. We would

like to be able to estimate the

_systematic

error on W due to corrections to naive

_scaling

but

lacking

data for

_larger

sizes makes it

_extremely

difficult. Just to

_get

an idea of their eventual

importance,

we made the

_assumption

that the

_{negative q}

tail of

_{P (q )}

was due to the non

leading

finite volume effects

_only.

We then

_computed

_{X sG}

_{again by taking}

the moments of

P (q )

for

_{positive q only (i.e.}

we

_{imposed by}

hand

_{P (q )}

= 0

_{for q -- 0).}

The

_{resulting log-log}

plot

is shown in

_figure

17.

_Again

the

_points

seem to

_align

on a

_straight

_line,

but we

_get

a

quite

different value for the

_exponent

w = 2.23 ± 0.1 _at

f3

= 1. This confirms the

sensitivity

of w _on

the

_systematic

_{uncertainties. The data for the energy}

_{overlap give}

a

_quite

similar value.

We remind the reader that in mean-field

_{theory cv}

= 3. As

X sG is

simply

related to the

second cumulant of

_{P (q),}

one can view it as a measure of the width of

P (q ),

w = 3

meaning

that the width is volume

independent.

We find smaller value of w

(but

we saw

that the

_systematic

uncertainties are

_large)

which hints to a

shrinking

of

P (q )

as the volume

increases. It would be very

interesting

to simulate

_larger

_systems

to control the finite volume

systematic

errors. If the result that w >- 0 is

_confirmed,

the function

_û(c)

is

_singular

at e = O. If 0 : {ù : 3 the function

_0(c)

_may or _may not be continuous at e = 0. In the latter

case the

replica

_symmetry

is

broken,

while in the former one we face a situation similar to that of the Kosterlitz-Thouless

_phase

in the two-dimensional

xy-model

where the correlation

Fig.

17.

_{- Log-log plot}

of the

_{spin-glass susceptibility}

when

_{negative q’s}

are

_{ignored (see text)}

versus the size of the _system. The upper

points correspond

to 6 - 1 and the lower

points

to /3 = 0.65.

d) Ultrametricity.

-

Ultrametricity

is _very hard to test

_numerically.

Our results for the volume

_dependence

of

_{’ ( 8 q )}

seem to be a

_convincing

evidence for

ultrametricity,

but we would like to mention two reasons for caution. One is the existence of the

triangular

inequalities.

The other is the

_change

of the

_{shape of P (q )}

as we increase the volume. Because

of the

_shrinking

of

_{the q}

0 tail of P

_{(q )}

as the volume

increases, 6 q

_{= q2 - q 3 is allowed} to

take

_larger

values for smaller volumes. This

_{« P (q )}

effect »,

combined with the

_triangular

inequalities,

could in

_{principle explain}

some of the volume

_dependence

of

_{’ ( 8 q ).}

Conclusions.

We

_presented

the results of extensive simulations of three-dimensional

_{Ising spin-glasses}

in the presence of an extemal

_magnetic

field. All our results are

_compatible

with a mean-field

like behaviour. Both the

_{configuration overlap q}

and the link-link

_{overlap qe}

have non trivial

and non

_{self-averaging}

_probability

distributions. There is numerical evidence for

ultrametrici-ty.

Our results are

_compatible

with the existence of a

_phase

transition in the presence of an

extemal

_magnetic

field. We measured the critical

_exponent

of the

_{spin-glass susceptibility}

and

between qe

and _q,

_{namely qe}

= a

+ bq 2 + cq 4

and that the coefficients

_{a, b}

and c are

self-averaging.

There are limitations to the above results. Because of

_strong

finite volume

effects,

we cannot

_produce

a

_{simple scaling picture}

for the transition.

The behaviour of the finite volume

_systems

we simulated is in

_good

_agreement

with the

predictions

of mean-field

_theory

with broken

replica

symmetry.

As

_already

stated we cannot

discriminate between a continuous or a _{discontinuous transition}

_{in e,}

_although

_the

_properties

we observed makes the latter more

_likely.

Our data seem

_{anyhow incompatible}

with the

alternatives to mean-field

_{theory proposed}

so far. We feel that a reasonable theoretical

approach

to finite dimensional

_spin-glass

should use as a zeroth

approximation

the

_theory

with broken

_replica

_symmetry,

because it

_captures

the behaviour of finite-volume

_systems.

Indeed some _very recent theoretical

_{approaches [17, 18] point}

in this direction.

Acknowledgments.

The numerical simulations were

_performed

on the IBM

_3090/VF

_computers

of CNUSC at

Montpellier,

of CNUCE in Pisa and of the ECSEC IBM center in Rome.

We thank D.

_Fisher,

M. Mézard and M. A. Virasoro for several useful discussions and G. Radicati di Brozolo for

help

in the

_optimization

of the _{programme. N.} S.

acknowledges

the

warm

_hospitality

of the ECSEC center where this work started. Also S. P.

acknowledges

the

hospitality

of the Laboratoire de

_{Physique Théorique}

de l’ENS.

Appendix.

In this

_appendix

we _prove the

_inequality

_{q, -- e 2where qe}

is the link-link

_overlap

and e is the average energy per link. We have

where a and T are the

spins

of the two

_{configurations.}

The sums

_{over (ij)}

and

(kf)

run over all the links of the lattice and

_Ne

is the number of links. So

where

Remark that

_{(M2)( ij)}

_{(ki) =}

_M(

ij) ( ki) , i.e. M is a

projector.

Then lI = 1 - M is the

projector

to the

_orthogonal

_{space. It is easy then} to show that the

_eigenvalues

of II are all

equal

to one

_except

for one of them which is

_equal

to zero. In other words the matrix II is

positive

semi-definite.

_Averaging

over

_{long running}

times and over different

_copies

of the

system,

ensures that

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