Damage Spreading in ±J Asymmetric Ising Spin Glasses

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Damage Spreading in ±J Asymmetric Ising Spin Glasses

R.M.C. de Almeida, L. Bernadi, I. Campbell

To cite this version:

R.M.C. de Almeida, L. Bernadi, I. Campbell. Damage Spreading in ±J Asymmetric Ising Spin Glasses.

Journal de Physique I, EDP Sciences, 1995, 5 (3), pp.355-364. �10.1051/jp1:1995106�. �jpa-00247060�

J. Pllys. I éYaJ1ce 5

(1995)

355-364 _MARCH 1995, PAGE 355

Classification Pllysics Abstracts

75.10N 75.40M

Damage Spreading in +J Asymmetric Ising Spin Glasses

R-M-C- de Almeida

(~),

L. Bemardi

(~)

^{and I.A.}

Campbell (~)

(~) Instituto de Fisica, Universidade Federal do Rio Grande do Sui, C.P. 15051, 91500 Porto Alegre, RS, Brazil

(~) Physique des Solides, Université Paris-Sud, 91405 Orsay, France

(Received

28 April 1994, revised 2 November 1994, accepted 28

November1994)

Abstract, Spin glasses in dimensions 2, 3, 4 and _mean field _are studied numerically _as

a function of the degree of asymmetry of the interactions between _spins. Damage spreading

data _are used ta _assess the eoEects _on the spin glass ordering of introducing _asymmetry in the

càuplings.

In ail dimensions the critical temperature for damage spreading _is rather insensitive ta the asymmetry, ^but the spin glass ordering is strongly suppressed by _any limite degree of

asymmetry. We discuss the physical significance of the damage spreading parameter.

l. Introduction

The diverse collective behaviors shown

by

diiferent

spin-glass-like

_systerns have raised _rnany

interesting

theoretical and

practical questions.

The diiferences between these rnodels _are due to diiferent assurnptions on the variables S~ or on the interactions between

thern, Jij.

In

particular,

neural networks have attracted attention to

asymmetric

interaction matrices where the

couphngs

between two spins cari act

diiferently

in each direction (1.e.

Jj~

is trot set

equal

to

J~j

).

We will refer to

symmetric spin glasses

whenever ail J~j =

Jj~ (SISG)

and

fully asymmetric

spin

glasses

^when

Jj~

are chosen

independently

of J~j

(AISG).

This

problem

represents ^also an

interesting generalization

of spin

glass

models

iii. "Asymmetric" couplings

_{are a way} of

defining

an

equilibriurn

ensemble via

dynarnic

rules instead of uia _a Harniltonian and the

"temperature"

T is _a pararneter of the

dynarnic

rules. We _cari note that neural networks have

asymrnetric couplings.

There _are few

analytic

theories of

asyrnrnetric

_versions of

spin glass

rnodels: there _is, for

exarnple,

the work

by

Crisanti and

Sornpolinsky

_{[2] or}

by

Hertz et ai. [3]. ^These calculations deal with

spherical

versions of the

asyrnrnetric

SK rnodel [4] and both find that for limite tem-

peratures ^and any limite amount of asyrnrnetry ^{there is} no

divergence

_in the relaxation titres.

However, in reference [2], a spin

glass

transition at T

= 0 is found but it is attributed to an

artifact of the

spherical

model.

Also,

_a

perturbative expansion

m the asymmetry parameter

yields

_a

non-vanishing

autc-correlation function at zero temperature

[5,6].

^Numerical simula- tions have also been

performed:

Crisanti and

Sompolinsky

_[7] found

that,

for

Ising

systems ⁱⁿ the _mean field limit and Gaussian distributed

couplings J~j,

^{the E-A order} parameter

decays

to

JOURNAL DE PHYSIQUEL -T 5,N°3, MARCH 1995 4

@ Les Editions de Physique ¹⁹⁹⁵

zero for ail

degrees

of asymmetry and limite temperature

T,

whila at T

=

0,

_as the asymmetry parameter decreases from its maximum to limite

values,

the number of

single spin flip

stable states increases from _a few to

exponentially large

values.

However,

the basins of attraction of these stable fixed points are

extremely

small _so the life time of the

paramagnetic phase

is

exponentially long:

a

phase

transition would be detected

only

when the observation times _are

larger

thon the

typical

relaxation times. On the other

hard, Spitzner

and Kinzel [8] simulated the _same system ^{and found that} at T

= 0 and for

non-vanishing

values of the asymmetry _pa-

rameter the remarient

magnetization

becomes

langer

than _zero,

indicating

_a

change

of behavior with

decreasing

asymmetry.

Schreckenberg

and

Rieger

_[9,

loi

have make

analytical

studies of the

dynamics

in the ID

spin glass induding

asymmetry.

Information _cari be obtained _on this

problem through

numerical determinations of

quantities

such _as the autocorrelation function

q(t)

whose

non-vanishing

values at

long

times indicate

spin- glass phase,

^{and the}

darnage spreading

distance

D(t) il1-15].

Derrida and his coworkers

il 2,14,16,17]

have shown that there is _a

damage spreading

transi-

tion at _a non-zero temperature in _an infinite _range

fully asymmetrized spin glass,

and have esti-

mated the critical temperature for

damage spreading

_in the 2d AISG. In _recent work

Pfenning

et ai. [18] ^simulated an infinite _range

spin glass

with +J

couplings through

T

= 0

sequential updating

and found _a

spin glass ordering

when the asymmetry is

sufliciently

low.

In this _{paper we} present autocorrelation function and

damage spreading

data for +J ISG in dimensions 2,

3,

4 and _mean field for variable asymmetry and limite temperatures. We will

lay particular emphasis

_on the correction between the

damage spreading

parameter

D(T)

and the

spin glass ordering.

2. Simulations

We

performed

simulations for

spin glasses

with +J interactions, for dimensions 2, 3, 4 and

mean field. For limite dimensions the

spins

_{were on}

hypercubic

lattices with _near

neighbor

interactions, ^and

periodic boundary

conditions.

Samples

of

250~, 20~,

^12~ and 800

spins

_were

used.

For _a fraction 2a of the

couplings, _Jj~

and J~j were chosen

randomly

and

independently;

for the other

(1- 2a) couphngs,

_{J~j was} chosen

randomly

and

Jji

was set

equal

to

J~j.

^The asymmetry parameter thus _ruas from _a

= 0 for the SISG to a = o-S for the AISG. We used the heat bath

dynarnics,

+1 with

probability il

_{+ exp}

(-

^~

~j _{Jij Sj}

_)] ~~

~~

i

a~(t

₊ 1) ₌

(1)

-l with

probability il

+ exp

(+

^~

~j _J~jSj

_)]~~

~~

J

Updating

_was

performed

for _sites in random order in the _mean field _case, and _{on two} sub- lattices

altemately

for limite dimensions. For the

damage spreading,

two diiferent initial _con-

figurations A(0)

and

B(0)

_were

chosen,

and the sites in both lattices _were

updated using

at each step ^the same random number

[12,13]. Following

de

Arcangelis

et ai. [13] we first _an- nealed at the

measuring

temperature T to establish the initial

configurations A(0)

and

B(0)

in thermal

equilibrium,

and then

proceed

with the

damage spreading

autocorrelation function measurements.

N°3 ASYMMETRIC SPIN GLASSES ₃₅₇

As

usual,

the autocorrelation function is defined

by

N

q(t)

₌

p ~ai(0)ai(t) ₍₂₎

and the

damage spreading

distance between the

configurations A(t)

and

B(t) by D(t)

=

j _{~ rat(t) ut (t)i~ (3)}

D(t)

is the fraction of

diiferent'spms

_or

_Hamming

distance between the _two

configurations.

We _cari note that if A and B _are identical D

= 0; and if ail

ut

=

-af

then D

= 1; ^{and if}

there _{is no} correlation between

spin

orientations in A and B then D

=

1/2.

Following

Derrida and Weisbuch [12] we considered three choices of initial conditions:

a) A(0)

and

B(0)

_are chosen

independently;

b) B(0)

is the mirror

image

^of

A(0)

and

c) B(0)

diifers from

A(0)

for _one

single

inverted spin.

We will denote the

long

time

limiting

value of

D(t)

at temperature T _as

D(T).

We have

suggested ils,

20] ^{that the diiferent}

regimes

observed for

D(T) correspond

to dif- ferent

morphologies

of the available

phase

_space.

Thus,

if

D(T)

= 0 the

phase

_space is

"geo- metrically" simple,

if

D(T)

> 0 it is

complex

_or

"labyrinthine".

As T tends to Tg ^the

phase

space takes _on

increasing complexity

_so that _as T _- Tg ^from

above, D(T)

-

1/2.

It is

important

to note however that this

picture

is trot

really

appropnate to the AISG _case.

By choosing

a fixed set of sites _as

being occupied

in the above

hypercube discussion,

_{we are}

treating

the system as if each

configuration

had _a well defined _energy. This _is trot the _case for the

AISG,

which _cannot be described

by equilibrium

statistical

mechanics,

and _we should be

careful _m the _use of this

analogy.

3. Simulation Results

3.1. THE FULLY ASYMMETRIC CASE. Here _we discuss the

limiting

_{case a =} Ù-S, in the

different dimensions.

Figure

^{shows the}

long

time limit of the

Harnming

distance

D(T)

_{as a}

function of temperature for d

= 2, 3 and 4 and for _mean field. These values do trot

depend

on the initial conditions

(choices

of

A(0)

and

B(0)),

and the time scales for

reaching

the

equilibrium

_are rather short _even at low temperatures. We _con note that in each dimension there is _a well defined

damage

temperature or

dynamic

transition temperature TD such that

D(T)

tends to _zero for T _> TD and to a limite value for T _< TD. Trie values of TD

given

in Table I _are

only slightly

lower than those for trie SISG in each dimension

[16, 20].

^If we

Table 1. Va1tles

of

the

damage

critical temperattlre TD

for

the

fully symmetric

ISG

(Refs.

f16, 20j)

^{and the}

asymmetrical

AISG

(the

Éd restllt is in agreement ^with

Rel. fl$j).

d SSG ASG

2 1.70 1.6

3 3.92 3.7

4 6.05 5.9

D

^~~

ÎÎ

s

~ a=0.0

~ a=0.5

3

o-1 ~~. _{~ ~}

~

i

~ ~ ~ ~

T

^~

~0

f 2 3 4

d

Fig. l Fig. 2

Fig. l. Hamming distance D _{as a} function of temperature T with _{a =} ù-à for the 2d, 3d, 4d and

mean field mortels.

Fig. 2. Plot of the temperature TD at which the Hamming distance D goes ^ta zero for the fully symmetric

(a

= 0) and fully asymmetric

(a

= ù-à) _spm glasses.

D(T=0)

06

o 5

04

0 3

oz

OI

o o

o oo o z5 o 50 o 75 où

Î Id

Fig. 3. The T ₌ 0 value of the Hammmg distance D _{as a} function of the _inverse of the dimension,

1Id

for _a

= ù-à-

plot TD(d), Figure 2,

_{we cari} estimate _a lower critical dimension at which

TD(d)

_goes to zero:

dl

_m 1.3 _as

against dl

_m 1.2 [20]. ^{We bave checked that for d} = 1

D(T)

_goes to _zero for ail values of T.

For the _mean field _case the value ofTD is

high

but is diflicult to estimate because of random

N°3 ASYMMETRIC SPIN GLASSES ₃₅₉

fluctuation eifects. At lower temperatures, in contrast to the SISG _case, there is _no

major slowing

down of the

relaxation,

_{so we cari} establish accurate values of

D(T)

in

relatively

short

time _ruas. Trie low temperature ^limits

D(0) depend

_on d and reach o-S

precisely

for _mean

field. This _mean field result agrees with the data of

Spitzner

and Kinzel [8] ^{and of}

Pfenning

et ai. [18].

By plotting Dd(0)

_{as a} function

of1Id, Figure

3, we cari agoni estimate the lower critical dimension

dl

_m 1.3.

3.2. THE PARTIALLY ASYMMETRIC CASE. It is clear that asymmetry in the

coupling

matrix accelerates relaxation and tends to suppress the

glass

transition.

For 2d SISG there is _a

general

agreement that the

spin glass

transition _occurs at Tg = 0; with asymmetry _we should expect a

fortiori

_no frozen

spin glass phase. Extrapolation

of

damage

distance values

D(T)

for the SISG _{as a} function of temperature

[20],

or

D(0)

_{as a} function asymmetry a to _zero asymmetry, _give

D(0)

= o-S for the 2d

SISG,

_1-e-,

D(Tg)

₌ o-S- From

Figure

⁴ it is dear that for _any limite _a,

D(0)

is less than

1/2. Also,

the _average correlation time _r defined _as

r =

/°° ^q(i)

^di

⁽⁴⁾

is

presented

in

Figure

5 _{as a} function of temperature for diiferent

asymmetries: although

r increases for low temperatures as a

decreases,

there _appears to be _no

divergence

of the

autocorrelation function relaxation time _as T _- 0. Both observations indicate that there is

no

spin glass

^order in 2d for _any limite _a. It is hard to obtain

D(T)

values for the 2d SISG below T ₌ 1 because of

long

relaxation time

problems. By measuring samples

with small but

non-zero values of _a and

extrapolating

to a = 0, we cari get round this technical obstacle.

The 3d SISG presents _{a spin}

glass

transition at Tg = 1.175 [19]. ^The introduction of _a fraction of

asymmetric

bonds _cari be

expected

to lower this temperature or to suppress this transition.

Figure

6 shows the

plot

of

D(T)

_{as a} function of the asymmetry and temperature.

D(T=0)

o 55

Me _an Fie id

o 50

0 45

0 40 3d

0 35

° ~°

2d

0 25

00 01 02 03 04 05

~l

Fig. 4. Plot of the limit value

D(T

₌ 0) for 2d, 3d and _mean field mortels _{as a} fuuctiou of trie asymmetry pararneter _a.

1000 ^{" " "}

.

w

«

~

io ~

o

o

~ o

~

a .

. . . .

.

à 1

T

Fig.

with = ù-1, _0.2, 0.3, 0A, and ù-à

from top to bottom.

D(T)

o.5

0 ₄

03

02

* _a= ID

o a=15

~ ~

' a= 30

a= 50

00

00 T

ig.

ature ^{T trie} 3d _spinglass, with variable_asymmetry.

Trie of q(t) at _low emperatures is very rapid

the SISG

limit

a

= 0 is as shown by _the

average orrelation time data in Figure _7.

D(0) dearly tends _to

the ^alueo-S

near

_the SISG

limit. As _we suggested ils], a ^cessary

condition

for

the

elaxation time for q _{to diverge} at temperature T _is

D(T) =

1/2. From

data of

Figures 6 and 7,

the

3d

SG

with partial ^at

T

₌

0

has D(0)

a laxation

time

_hich does

trot diverge for a >

0.10.

This

means the _spin lass ordenng

is

N°3 ASYMMETRIC SPIN GLASSES ₃₆₁

ioeo W _,

«

a ~

a

a

& .

h

io a

o

o o

° o

, , ,

' '

a .

1 0 _0.2

0.4 0.6

iooo

«

a

a io

~ a

~

o o ~

o ^{a a}

' a o

, a ^a _, o

. . . ,

1

0 0.2 0.4

0.6

0.8

1 T

with symmetry a = -1, .2, .3, 0A,

igure

we efine as trie _{where trie} relaxation time _the mean _field AISG has

no Tg. It that trie ondition D = 1/2 is not

simply related

to

references ^8,18], evidence is _given for spin glass ordering at

zero

emperature below a cntical

oncentration which ould be less than a = 0.10 for

trie

N ₌ 800 sample size

used.

We

find limite average ^orrelation times r,

and

trie

limite relaxation times for q(t),

at

T =

0.2

for symmetric oncentrations

a

down to and induding _a =

compatible with refs.

[2-4,8,18]

but does

trot ^solve trie roblem of hether

there is

a

_regime

with limite Tg for _limite (small) values _of a.

4.

We

have

performed

extensive

^umerical simulations of _dimensions 2d,

3d

and infinite range

Ising spin

ystems with random

+J

_couplings of _ariable

fuit asymmetry

The introduction

of

asymmetry only slightly owers the _dynamic

transition"

emperature TD

at

_which the damage spreading arameter

D

_goes ^to

zero

in comparison _to

the

models.

The asymmetry has a much

more

dramaticeifect

on

the spin _glass

for the 2d

model, which

has ^a

transition at _Tg =

0

in the

fully symmetric

appears

case, we have that _there is no _spin glass

transition

at

Tg

=

smaller values of

a,

the _{question of} the xistence

of

_a ^pin _glass^ransition

^at

limite a

remains

open. The infinite _{range model}

has been stated

[8,18] to

have

spin lass ordering at T =

a < 0.15. We bave found

that trie utocorrelation function time _does trot diverge for a _{> o-1}

N°3 ASYMMETRIC SPIN GLASSES ₃₆₃

at T ₌ 0.2. Trie

damage spreading

_parameter

D(T)

_on trie other hand tends to o-S _as T tends to zero for the whole range of _a between 0,5 and 0.1.

The results

give

^further information _on the

phenomenology

of the

darnage spreading

_param-

eter. In each system studied _{as a} function of the parameters a and

T,

we have monitored the

long

^time limit value

D(T).

For 2d and

3d, extrapolating

to _a

= 0 confirms the

conjecture ils]

that

D(T)

tends _to

1/2

_as T tends _to Tg ^{in the}

symmetric

ISG. For D <

1/2,

the _average correlation time does not

diverge,

_so the system has not attained the ordered

spin glass

state.

In the infinite _range

model, D(0)

tends to o-S for ail _{a even}

though

the _average correlation time

diverges

at T

= 0

only

for small _a.

Nevertheless, extrapolation

of the data taken _at finite

a to a = 0 confirms that for the

symmetric

_mean field ISG

D(T)

tends to

1/2

at T _m 0.7

[20],

rather than at the

Sherrington-Kirkpatrick

model value Tg = 1. This may be _a limite size

eifect;

for

practical

_reasons it would be _very diflicult _to make measurements _on

samples

much

larger

than the N

= 800

samples

_we have worked with.

Finally

the decrease in relaxation times caused

by

the introduction of asymmetry in the

couplings

_may

provide

_{a way} to obtain

spin glass

transition temperatures with

higher

accuracy

by extrapolating

the _a

= 0 value from asymmetric _spin

glasses

simulation data in 3d _or 4d

samples,

for

example.

Work in this direction is _now in progress.

Acknowledgements

The calculations _were

performed

thanks to _a computer ^time grant

provided by

the Institut du

Developpement

et des Ressources _en

Informatique Scientifique.

^{This research bas bene-}

fitted of _a

partial

financial support ^from the French-Brazilian CAPES-COFECUB _program.

R-M-C-A- thanks the Laboratoire de

Physique

des

Solides, Orsay,

for trie kind

hospitality

and

acknowledges

financial support from Brazilian

agencies

Conselho Nacional de Desenvolvimento Cientifico _e

Tecnolôgico (CNPq)

and Finaciadora de Estudos _e

Projetos (FINEP).

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