Zero temperature dynamics of a spin glass chain

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Zero temperature dynamics of a spin glass chain

P. Krapivsky

To cite this version:

P. Krapivsky. Zero temperature dynamics of a spin glass chain. Journal de Physique I, EDP Sciences,

1991, 1 (7), pp.1013-1018. �10.1051/jp1:1991185�. �jpa-00246381�

Classification

Physics

Abstracts

05.20

Zero temperature dynamics of

_a

spin glass chain

P. L.

Krapivsky

Central

Aerohydrodynamics

Institute,

Academy

of Sciences of the USSR, 140160 Zhukovsky-3, Moscow

region,

U-S-S-R-

(Received

25

February

1991,

accepted

21 March

1991)

Abstract. We consider an

Ising spin glass

chain at zero temperature.

Introducing

a time-

dependence

in the model

by

_means of Glauber

dynamics,

_we have calculated one-spin and two-

spin

correlation functions

exactly.

For

example,

_we have found that if _one starts at time

t ₌ 0 with all the

spins aligned,

the

magnetization

evolves as

[I

_{+ exp}

(- t)

_{+ exp}

(-

2

t)]/3.

Our results _are valid for _a

spin glass

chain with _an

arbitrary symmetric

and continuous distribution of

nearest-neighbour

interactions.

1. Introductiton.

In _recent years considerable interest has

developed

in the

study

^of zero-temperature

dynamics

of

spin glasses

and of neural networks

[1-5]. They

exhibit

qualitatively

the _same features

(many

metastable states, remanence

effects, etc.)

_as

spin glasses

at low temperatures.

However, they

_are much

simpler

to

study

from _a theoretical

point

of view because the effect of thermal noise is eliminated.

They

_may also have _a

practical advantage

if _one wants to build

pattern recognition

devices.

The _reason that

zero-temperature dynamics

is non-trivial and

interesting

is the existence of

a

huge

number of metastable states

(for

_a

review,

_{see, e-g-,}

[6]).

Metastable states _are

spin configurations

which _are

separated by

_energy barriers from other

configurations.

These

metastable states _are

responsible

for _remanence effects

[7],

_very slow relaxations and

sensitivity

to initial conditions.

At the moment, _one knows how to calculate the number of metastable states and the sizes of their basins of attraction for _a one-dimensional

spin glass

at zero temperature

[8-10].

The

particular dynamics

used in references

[8-10]

_was

single spin-flip dynamics.

Some of these results have also been

generalized

to more

complicated dynamics [I I].

All these studies

give important

results

conceming

the remanent

magnetization,

the

residual entropy, etc. at time t

= cc. In this paper, we will

develop

_an

approach

to kinetic behaviour of _a one-dimensional

spin glass

chain with Glauber

dynamics.

We will consider _an

Ising spin glass

chain with Hamiltonian

H=-£J~,~~jS~S~~j, S~=±I. (I)

1014 JOURNAL DE

PHYSIQUE

I M 7

Our results _are

independent

of the details of the distribution

p(/~)

^{of the}

exchange

interactions

/~.

^The

only

condition for the result to hold is that the distribution _p is

symmetric,

p

/~ )

_{= p}

(- /~),

and does not contain any delta function. So _our results will be valid e,g, for

a Gaussian distribution

p(/~)

=

(2wJ~)~~'~ exp(- /(/2J~)

and for _a flat distribution

p

(/j

_" 0 ₂

/j /J)

2.

Dynamics.

A linear

Ising spin glass

chain with Hamiltonian

(I)

has _no natural intrinsic

dynamics.

One way of

introducing

_a time

dependence

in this model is

by

means of Glauber

dynamics [13].

We

postulate

a master

equation

for the

probability P(S, t)

of

finding

_a

particular

realization of the chain with

spin configuration

S

=

(S~)

at time t,

P

(S~

=

£

_Wk

(S~)

P

(S~) Wk(s)

P

(s)j (2)

In this

equation ll~(S)

denotes the rate for the system to

jump

from _a

configuration

S _{to a}

configuration ^S~

obtained from S

by flipping spin

k. The transition _rates

W~(S)

for

going

from S _to

S~ obey

the detailed

balancing

condition

w~(sk) Peq(sk)

=

w~(S~ Peq(s~ (3)

so that in the limit t

~ cc the

equilibrium

distribution

P~(S~ ~exp(- H[Sl/l~J

will be attained.

However,

there _are then still _{many ways} in which the

W~(S)

_can be chosen such that detailed balance condition is satisfied. We shall follow Glauber and choose

~~~

_~~

~~~~~ ~~~

_~

~~ ~~~

_~~~ ~~~

where

1~

_~

~k,k+i

+

~k-i,k

1~

_~

~k,k+1~ ~k-i,k

~~ ~~

2 ^~~ T ^~ 2 ^~~ T ^~

~~ ~~~~

~

~_~

~

~

~ ~~~~

~

~

T

~

_~

~ ~~~~

Note that

by

the definition

(4)

time is measured in units of

elementary spin flip

time. For _a

ferromagnetic Ising

chain _e~

= &~ = tanh ~ ~

and

consequently

these rates _are identical

2 T

to those of reference

[13].

From the master

equation

_we shall derive evolution

equations

for the

spin

correlation functions

iii..

i~

)

m

~j

S~~.

$ P(S~. (6)

s

~

Taking

the time derivative of

equation (6)

and

using

the master

equation (2)

and relations

(4)

one finds

~ ⁿ

i <ii in I

= n

<ii in

+

Z

_i~ip

I.. _ip

+ i

I

₊

_bi~ I.. _tp

i

Ii (7)

p= i

From this

equation

we see that _an

advantage

of the choice

(4)

for the transition rates is that the evolution

equations (7)

for

n-spin

correlation functions

(6)

contain

only n-spin

correlation

functions. This makes the

problem

solvable at _zero temperature as we shall show below.

3.

One-spin

correlatiton functions.

At T

=

0 the

dynamics

becomes

extremely simple. Equation (5) implies

~k "

°, ~k

~ S~II

(~k-i,k)

~t

(~k-i,k'

_~

'~k,k+1' (8~)

~k

_~

°,

_{~k ~} S~ll

(~k,k+i)

~t

'~k,k+1(

_~

'~k-i,k' (8b)

It is therefore convenient to divide all bonds

along

the chain into three

types, namely

_strong

bonds,

medium bonds and weak bonds. We say that

J~~~j

is a strong ^{bond if}

J~,_~

~ j ^~ max

( J~

j,~

[, J~

~ j,~_~ ~

).

So _a bond is strong ^if

iti

two

neighbouring

bonds _are weaker.

Similarly

_a bond is weak if its two

neighbouring

bonds _are stronger.

Lastly

_a bond is a

medium bond if _one of its

neighbours

is stronger and the other

neighbour

is weaker. This classification is violated when

[J~

_{j ~ =}

[J~

_~

~ j

[,

^but our

assumption

that the distribution

p(/~)

does not contain _any delta

~function ~shows

_{that such}

a violation _occurs with _zero

probability.

Further _we shall call parts ^{of the chain between} two consecutive weak bonds

by

clusters. So the chain is broken into clusters of medium and strong bonds delimited

by

consecutive weak

bonds. A

typical

cluster is

depicted

in

figure

I. It contains the

only

_one

strong

^{bond and} an

arbitrary

number of medium bonds.

Zero temperature

dynamics

of

spins

in _an

arbitrary

cluster is determined

by

the

spins

of this cluster

only.

So the chain is broken _{onto a} system ^of

noninteracting

clusters.

Let _us consider the kinetic behaviour of

one-spin

correlation function in the cluster

depicted

in

figure

I.

Combining equations (7)

and

(8) gives

( _(k)

=

(k)

₊

e~(k

₊ I

)

at I _w k _mm

(9a)

~

(k)

=

(k)

+

&~(k

I

)

at _m + I _w k

w L.

(9b)

Here _we have used the notation of

figure I,

L is the

length

of the cluster and

(m,

_{m +} I the

only

_strong bond in the cluster.

For

simplicity,

_we

stipulate

that at t = 0 the chain is

fully magnetized,

I.e.

(k)

₌

^I

for all k.

Other initial conditions _can also be dealt with- in the

following

_we shall denote _s

= e~ =

&~~

i and

c~ =

jkj

_exp

(t) (10)

Solving (9) yields

c~ = exp

(St)

c~_j = I ₊ e~_j

s(c~- I)

Cm-2 ~ l ₊ ~m-2 ⁺ ~m-2

~m-I(Cm

l

~S~)

^~~~~~

Cm-3 "1+ _{~m-3 + ~m-3 ~m-2}

t121+

_{~m-3 ~m-2 ~m-I}

_S(C~

I _~S~

l12'),

etc, at km _m, and similar results

cm + i = exP

(St)

Cm+2 ~ l ₊

~m+25(Cm+1~ l) (lib)

c~~~ =1+

&~~~

t +

&~~~ &~~~(c~~j

I

-St),etc.

atkmm+1.

1016 JOURNAL DE

PHYSIQUE

I M 7

(Jk~i<:(

j

_m ml

L

Fig.

I. A

typical

cluster with the strong bond (m, _{m +} ).

These correlation functions become

extremely simple

after _{an average over} the random

couplings. Equations (8) imply

that in such _an average

e~'s

ⁱⁿ

(I la)

and

&~'s

ⁱⁿ

(I16)

take the values _± I with

equal probability

and

independently. Performing

the averages of the solution

(I I)

_one finds _:

<k)

=

(~~P~~ _[1

^{~~ ~~}

^~#'~'

^'~+

⁽¹²⁾

+

exp(-

2

t)]/2

at k

= m and k

= m +

From the

knowledge

of the kinetic behaviour of

one-spin

correlation functions in _a cluster of

length

_{L, one can} deduce the relaxation of the

magnetization

m(t)

= exp

(- t) ~ £ (L

2

) X~

+

[I

_{+ exp}

(-

2

t)] £ X~. (13)

2

~2

Here

X~

is the

density

of clusters of

length L,

which _was

computed by

Derrida and Gardner

[10]

X

=

2~(L

I

(L

+ 2

)/(L

₊ 3

) (14)

A direct calculation of the _{sums on} the

fight-hand

side of

equation (13)

then

yields

w w

~j ^{(L 2) X~}

=

~j ^X~

=

1/3.

Thus _we obtain

L=2 L=2

m(t)

=

ii

+

exp(- t)

_{+ exp}

(-

2

t)i13. (15)

Observe that remanent

magnetization

at time t

= cc

equals

to

1/3.

TMs result _was first found

by

Femandez and Medina

[12]

and then rederived and

generalized

in references

[8-11].

4.

Two-spin

correlatiton functitons.

We _{now turn to} the calculation of

two-spin

correlation functions at _zero

temperature.

^For a

pair

^of

spins

from distinct clusters _one has the obvious result

liJ)

=

Ill lj) (16)

For _a

pair

of

spins

inside _a cluster

equations (7)

and

(8)

_can be rewritten _as

~

c~~ ⁼ e~ c~

~ j _~ + ej c~_{~ ~} j

(I

_w I

~

j

_{w m}

) (17a)

dt

~

c~~ ⁼ &~ c~ j _~ + &~ c~~ j

(m

₊ I _w I _~

j

w L

(17b)

dt

~

c~j ⁼ e~ c~ _~ i j ⁺ &~ c~_~ j

(I

_w I mm, m + I _w

j

_w L

) (17c)

dt

with

c~~ ⁼

(ij) exp(2 t). (18)

Hereinafter _{we assume} that I

ij.

Other correlators _can be reconstructed from the relations

c~~ ⁼ c~~

and _c~~

=

exp(2 t).

We

begin by calculating

the

special two-spin

correlation functions in which _one

spin belongs

to the strong ^bond

(m,

_{m +} I

).

From

(17),

_a closed system ^of

equations

is obtained for the

rows

j

= m and

j

= m + I. It is convenient to present these

equations

in _a vector form _:

with

A~ =

~~+

A~* ₌

~~

(20)

Cim Cim+1

A~

we determine

directly

from

(17) A~

=

Uexp(2 t)

at _s

=

(21a)

A~

=

U ₊

V[exp(2 t) I]

at s =

(21b)

where _we introduced the shorthands

U=

(()

_and V=

(~j).

Solving (19)

with the initial conditions

(21) yields

A~ =

Uexp(t) F~(t)

at _{s =}

(22a)

~41 " U exp

(-

t F~

(t

₊

_Ve~

_e

~ j G

~

t)

at _{s =}

(22b)

where

F~(t)=

I +e~t+. +e~..

e~_~t~~~~~/(m-I-I)!+e~.

_{e~_j}

_~j

~

t~/nj

n=m-1

and

G~(t)

= 2 exp

(t) ~o ~j ^t~+

^'

+~P/(k

_{+ +} 2_p

One _can also calculate

two-spin

correlation functions in the columns I _=m and

I

= m + I.

They

_are described

by

formulas such _as

(22a)

and

(22b).

One _can further calculate all other

two-spin

correlation functions

recurrently.

For

example,

in the

triangle

I wi _<

j

_mm, we have

already

found the _{upper row}

j

_{= m, see}

(21)

and

(22). Using (17a)

_we then find the correlators in the row

j

₌ m

I,

etc.

1018 JOURNAL DE

PHYSIQUE

I M 7

After

performing

the disorder average of these

unwieldy

correlators _we arrive at the

extremely

_compact final result

(ij)

=

Ii ) (j)

at

(ij)

#

(m

_{m +}

(23)

with

one-spin

correlation functions

given by (12),

and

(m

_{m +}

)

= exp

(-

2

t) (24)

Thus _our results

imply

the absence of correlations between

spins

of distinct clusters

(see (16))

and also the absence of correlations between

spins

of the _same cluster after

performing

the disorder _average

(see (23))

with the

only exception

of

spins

connected

by

a strong ^bond.

Acknowledgment.

I wish to thank N. Slavnov for _warm

hospitality

in Institute of

Computer

Mathematics and Informational

Technologies.

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