On computer simulations for spin glasses to test mean field predictions

HAL Id: jpa-00246357

https://hal.archives-ouvertes.fr/jpa-00246357

Submitted on 1 Jan 1991

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

On computer simulations for spin glasses to test mean field predictions

Sergio Caracciolo, Giorgio Parisi, Stefano Patarnello, Nicolas Sourlas

To cite this version:

Sergio Caracciolo, Giorgio Parisi, Stefano Patarnello, Nicolas Sourlas. On computer simulations for

spin glasses to test mean field predictions. Journal de Physique I, EDP Sciences, 1991, 1 (5), pp.627-

628. �10.1051/jp1:1991158�. �jpa-00246357�

L

Phys.

I 1

(1991)

627 -628 MAi 1991, PAGE 627

Classification

PhysidsAbsnncts

75.50 75.10

On computer simulations for spin glasses to test

_mean

field

predictions

Sergio

Caracciolo

(I ), Giorgio

Parhi

(2),

^{Stefano Patamello}

_(~)

and Nicolas Sourlas (4>*

(1)

^{scuola Normale}

superiare

and INFN Sezione di

Pisa,

Piazza dei

Cavalieri,

Pisa

56110,

Italia

(2) Dipartimento

di Fisica dell'Universith di Roma II, and INFN sezione di Tor

Vergata,

Ma E.

Camevale,

Roma

00173,

Italia

(~)

^IBM

ECSEC,

Ma del

Giorgione 159,

Roma

00147,

Italia

(4)

Institute for Advanced

study, Princeton,

NJ

08540,

U.s.A.

(Received11

Mm.ch 1991,

accepted14

March

1991)

Huse et al.

[I]

commented _on the results of _our recent numerical simulations of three dimen- sional

spin glasses

_[2]. It _seems to us that

they

have not

properly

understood the

spirit

of _our

paper

and this is at the

origin

of many ^of their remarks. Our aim was to test

(in

the

region

accessible to

present general _purpose computers)

the

predictions

of mean field

theory. Although

we are

eventually

interested in the infinite volume

limit,

we believe that information _on the behaviour of finite she

systems

may ^be very ^useful in

testing

theories. The main result of _our

paper

was to find that

up

^{to lattices of size 14~ the results of the numerical} simulations _were

compatible

^with the

predictions

of mean field

theory (incidentally

we noted that the same data could not be

explained using

the

existing dropled models). Any

one of _our tests

(with

the

exception

of the

magnetization

data,

for which Fisher and Huse

postulate

it ban accident of the J ₌ +I

model)

^could be

subject

to criticism because of

systematic

or statistical uncertainties and this is discussed at

length

in our

paper.

^{However if all the tests} are taken

together,

^{the evidence becomes much}

stronger.

^{It is} very

unlikely,

we

believe,

that the

agreement

we find in all these very different tests is due to _a

conspiracy

of the data.

Here _we discuss _some of the technical

points

raised in the _comment.

(* ^{Pennanent address:} Laboratoire de

Physique Th60rique

de l'Ecole Normale

supdrieure,

24 rue

Lhomond,

75231 Paris Cedex

05,

France.

628 JOURNAL DE PHYSIQUE I N°5

a)

Lomtion of the transition

temperature

Fisher and Huse argue that the

temperature

we used was too

high

and

they

came to this con-

clusion

using

mean field

theory

to

get

a

quantitative prediction

of the

phase diagram

in the H-T

plane.

This

approach

involves uncontrolled

approximations

and it

usually gives

bad results: for instance the critical

temperature

at H

=

0,

D ₌ 3

computed

with the _mean field

approximation

is wrong

by nearly

a factor 2. A direct

computation

of the

phase boundary (of

the

type

we have

done) unfortunately

cannot be avoided. We also notice that for finite size

systems

^{the transition} is rounded and the correlation

length

becomes

larger

than the size of the

system

at a

temperature slightly larger

than the critical

temperature

^{in the}

bulk;

in such _a situation the

system

^behaves

qualitatively

^similar to a

system slightly

below the transition

temperature.

b)

Non

triviality

of

Pi q)

We agree ^that we do not

prove

^that

P(q)

is _non

trivafl,

and this _was

widely

commented in _our

paper.

^{We stressed that} we could be in _a

phase

different from the

high temperature phase

in which the function

P(q)

is

trivial, replica _syrnmetry

is broken in the sense of reference [3] ^{and the}

susceptibflity

is

infinite;

there is _no trace of this

proposal

left in their comment.

c)

Non

self-averaging

of

P(q)

We

agree

that S may remain different from _zero also if

P(q)

is

asymptotically ^trivial;

^indeed we have not used the data _on S in _our papers ^to argue

against

this

possibility,

but _we have

presented

them as additional informations.

It _seems to us that there is _no shortcut with

respect

to numerical simulations to obtain infor- mation _on the low

temperature phase.

We all agree ^that a new much faster simulation

algorithm

would be welcomed.

References

[1] HusE D.A. and FiSHER

D-s-,

On the behadour of

ising spin glasses

in _a uniform

magnetic fiefs,

_pre-

ceding

_paper, J

Phys.

I 1

(1991).

[2] CARAcctoLo

s.,

PARisi G., PATARNELLO s. and souRLAs

N., Earophys.

Leii. ii

(1990) 783;

J

Phys.

France Sl

(1990)

1877.

[3] MtzARD

M.,

PARisi G. and VIRASORO

M.A~,L Phys.

Frvnce 50

(1989)

3317.