Numerical studies of the spin-flip dynamics in the SK-model

HAL Id: jpa-00246394

https://hal.archives-ouvertes.fr/jpa-00246394

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.

Numerical studies of the spin-flip dynamics in the SK-model

G. Kohring, M. Schreckenberg

To cite this version:

G. Kohring, M. Schreckenberg. Numerical studies of the spin-flip dynamics in the SK-model. Journal

de Physique I, EDP Sciences, 1991, 1 (8), pp.1087-1091. �10.1051/jp1:1991192�. �jpa-00246394�

f Phys. lfrance 1

(1991)

1087-1t9l Aotrr1991, PAGE 1087

Classification Physics ^Abstracts

05.50 M.60C 75.10N

Shortcommmication

Numerical studies of the spin-flip dynamics in the SK-model

G.A~

Kohring

and M.

Schreckenberg

Institut for Theoretische

Physik,Universitat

zu KbIn,

Z0lpicherstr.

77, D-5tK© KfiIn 41, Gcrnlany

(Receveid15Mqy

1991,

accepted 2iMqy1991)

Abstract. The

spin-flip dynamics

of the sK-model is studied

numerically

with systems ^of up ^to 10~

spins.

For

sequential dynamics

at T

= o we find _{a remanent}

magnetization

of o. 085 + o.oos which is about half the value found earlier for much smaller systems. Furthermore _{we have calculated the} percentage ^of spins which flip n times

during

the descent into _a metastable state.

Remarkably,

the number of

spins

which _never

flip (n

₌

o)

is

asymptotically equal

to the number of

spins

which

flip only

once

(n

₌ 1). Therefore the dominant contribution to the remanent

magnetization

comes bom

the

spins

which flip twice

(n

=

2).

We have found the convergence time, with the system sizes used here, to

diverge

^faster than _a

simple

_power law.

The relaxational behavior of

spin glasses

is

mainly

influenced

by

a

large (exponential)

number of metastable _states which detain the

system

^from

getting

into

equilibrium (for

reviews _see

[1, 2]).

Therefore the

long-time

^{limit of}

macroscopic quantities

^like the

magnetization depend

on the way the

system

was

prepared initially. If,

for

example,

a

magnetic

field is

applied

and then switched

oflj

a nonzero

overlap

between the final and the initial state can be

observed,

known as the remanent

magnetization.

It is also an

interesting quantity

because it serkes _{as an} order

parameter

^{in the}

asymmetric

SK-model [3,

4].

The difficulties for

analytical approaches

to the

problem lay

in the fact

tha~

in

principle,

the whole set of

dynamical equations

has to be solved u1 order _to take the influence of the initial conditions

on the final state into account. In the case of the SK-model no exact results are known

fnd

one is forced to attack the

problem by _computer

simulations of systems with the

largest possible

sizes. Earlier

investigations

with system ^sizes up to 1024

spins

led to a remanent

magnetization

of 0.14 + 0.01 at zero

temperature

[5]. ^Here we

report

^{the results of} simulations of

systems

^{with sizes} up to 105

spins

which show that the value of the infinite

system

^{is about} half this value

(qualitatively

this was

anticipated by

P

Spitzner [fl).

Another

interesting question

about the SK-model _comes ~om

analytical

studies of the sequen- tial

spin glass dynamics

on

Cayley

trees:

during

the descent into a metastable state no

spin flips

more than twice [7, 8]. ^Since our simulations _were also

perforated sequentially

we

analyzed

the

percentage

^of

spins

which will

flip

_n times before the

system

^is

trapped

into _a metastable _state.

Those

spins

which

flip

an even

(odd)

number of times

give

a

positive (negative)

contn~ution _to

1088 JOURNALDEPHYSIQUEI N°8

the remanent

magnetization. Remarkably

we find that the

percentage

^of

spin

which _never

flip,

n = 0, ^{and which}

flip

_{once, n =} I, b

equal

ⁱⁿ the infinite system.

In the numerical

simulations,

_{we use}

Ising couplings

between the

spins,

because

only

the first

two moments of the

coupling

distribution should contn~ute _to the

properties

of the SK-model

in the

thernlodynamic

limit. Thb allows _us to make _use of efficient

multi-spin coding techniques

[9,

10].

^These

techniques

work very well up ^to system ^{sizes which exhaust the main} memory of the

computer. (On

the HLRZ'S

Cray-YMP/832

this _occurs for

system

^sizes at about 3.5 _x 10~

spins,

and for the

University

of

Cologne's

NEC-SX3/11 at about 2.4 _x

10~,) Bu~

in order to reach the

large system

^sizes necessary ^{for the}

present work~

we have had to resort to a scheme

whereby

the

couplings

are recalculated at each iteration

step.

^{Under the constraint of}

symmetric couplings,

this is

posswle by generating

_a

completely

^random

matrix, J(;, using

a

lbusworth-Kirkpatrick-Stoll shift-register

random number

generator

^with

lags

of 17 and 11, and then

defining

the

symmetric

couplin# _{by: J;;}

_«

_J];

₊

Jj;. Again

we make _use of the fact that

only

the first _{two moments} of the

coupling

distribution _are

important. ~The complete

^Fbrtran program ^{b available from the}

authors.)

In

figure

I we show results for the remanent

magnetization,

_mm as a function of the inverse

system

size. The data

presented

here _was

averaged

over

approximately

^10~

starting

states at small system

sizes,

and

only

10 states for the vely

largest

_system.

Except

for the last data

point,

the _error bars

would be smaller than the

symbols

themselves.

o

o o o 8

£

a

a

a a°

~ Da

an lo~ ^lo~no

~-i

N°8 SPIN-FLIP DYNAMICS IN THE sK-MODEL 1109

The best fit in terms of the smallest _sum of the

squared

deviations is

given by

a

logarithmic

correction to a

simple

_power:

miN)

_{= moo} +

$

+

~~$l~~ ¹¹⁾

where

hi

and

A2

are of order

unity

and _{a m} 0.27.

Assuming equation (I)

to be

valid,

_we found

the remanent

magnetization

to be 0.085 + 0.005. Thb value b

quite

similar to the value of 0.06

which was found in the

Hopfield-model

for _« > «c [10].

We have also checked the value of the remanent

magnetization

for

parallel dynamics

and _we

again

find that _a

simple

combination of powers ^{does not fit the data} very well.

Again using

the above fit fornlula _we find _mm

= 0.18 + 0.01. Thb result b _not inconsbtent Wth the results in _one dimension which show that the _remanent

magnethation

for

parallel dynamics

is twice that for

sequential dynamics _[11].

In

trying

to understand the result of the remanent

magnetization

we also have measured, as is shown in

figure

2, the percentage ^of

spins

which

flip exactly

_n times before the system ^becomes stable. The most

interesting _aspect

of the data is the numerical

equivalence,

for

large

_systems, be-

tween the

spins

which _never

flip (n

₌

0)

and the

spins

which

flip

once

(n

₌

I).

Therefore the main contribution to the remanent

magnethation

comes from the

spins

^which

flip

^twice n = 2, TMs

o

~~ Da ^°

4 ,o

VY11

^~~

.° ° . ° ° °

0l ,3 ilw

~Q

~m

~

. AA AA a4 4a an aa _a

* 4

on oo oo oo oo oo

lo~ lo~ lo~ lo~

~-l

Fig.

~ -Average percentage ^of

spins

which flip n times

during

the descent into _a metastable state.

(D)

n = 0,

(.)

_{n =} 1,

(6)

_{n =} 2 and

(o)

_{n =} 3.

Again

the errors not shown _are of the same size _as the

symbols.

1W& JOURNALDEPHYSIQUEI N°8

percentage

^{should then}

give

_{an upper} bound _{to mm}

and,

since the

percentage

^of

spins

which

flip

n times b observed to decrease

monotonicly

^Wth

increasulg

_n, we can obtain _a lower bound _on mm

using

the

percentage

^of

spins

which

flip

three times

(n

₌

3). (A

note of caution here: _we have observed that the

percentage

^of n = I

flip spins

reaches a maximum at about N

~ 5 _x 10~,

we can not rule _out such behavior for

larger

_n

flips

^{when N} >

10~.)

In the last

figure

_we show _a

plot

of the convergence ^times as a function of the system ^{sizes. From} the data _we conclude that the convergence ^time grows ^{faster than} a

simple

_power law _over the range ^of

system

^{sizes simulated here. Thb b} a much faster

growth

than

logarithmic growth

found in the

Hopfield-model

^with « < ac [10].

Finally,

the

present

simulations

suggest,

^that system ^sizes up ^to 105

spins

are not

sufficiently large

to

unambiguously

determine the

asymptotic properties

of the SK-model.

a a a a a a

6 a

~ a

V ~~

a

a

a a

io~ io~ lo'

N

Fig- ^3. Average convergence time _{as a} function of the system size.

Acknowledgements.

The authors are indebted _to M. Neschen for many

stimulating

discussions. TMs work _was per- formed within the research program ^{of the}

Sonderforschungsbereich

Ml KoIn-Aachen-Jolich

supported by

the Deutsche

ForschungsgemeinschafL

GAK thanks the

BMFT(0326657D)

for fi- nancial

support

and we thank the

University

of

Cologne

for time _on the NEC-SX3/11 and the HLRZ Jolich for time on the

Cray-YMP/832.

N°8 SPIN.FLIP DYNAMICS IN THE SK-MODEL 1t91

References

iii

CHOWDURY D.,

Spin glasses

and other ~ustrated systems

(Singapore:

World scientific, 1986).

[2] MtzAm M., PAR1Si G. and ViRAscRo M.A~,

spin

glass theory and beyond (sin gapore: World scien- tific, 1987).

[3] ^{SPrI2NER E and KINzEL} W, Z Phys. B 77

(1989)

511.

[4] PFENNINO ~, RJmER H. and SCHRECKENBERO M.,f Phys. ^lfrance 1

(1991)

323.

[5j KmzEL W,Phys. Rev B 33

(1986)

5166.

[q

SPn2NER P,

(private communication).

ii

SCHRECKENBERO M.,J Phys. ^{A: Math Gen.}

(1991)

in prow.

[8] SCHRECKENBERO M. and IIIEOER H., Z Pij~s. B

(1991)

in press.

[9] ouvEiRA PM.C.,

Computing

Boolean statistical Models

(singapore:

World Scientific, 1990).

for _a shorter review, see: STAUFFER D., f Phys. A: Math Gem 24

(1991)

909.

[10] ^KoHR1No G.A~,J Stat. Phys. 59

(1990)

1077;J Phys. A: Math Gem 23

(1990)

2237.

ii ii

GARDNER E., DERRIDA B. and MornsHAw P, f

Phyi.

France All

(1987)

741.

[12] ^{MACKENZIE N.D. and YouNo} A.P, Phys. Rev Leti 49

(1982)

3ol; J Phys. C: Sofid State Phys. 16

(1983)

5321;

KINzELBACH H. and HORNER H., Z Phys. B

,

in press.