Stretched exponential decay in an asymmetric $\mathsf{+ / -}~J$ spin chain

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Stretched exponential decay in an asymmetric +/ − J spin chain

H. Rieger

To cite this version:

H. Rieger. Stretched exponential decay in an asymmetric +/ − J spin chain. Journal de Physique I,

EDP Sciences, 1991, 1 (1), pp.13-18. �10.1051/jp1:1991113�. �jpa-00246301�

Classification

PhysicsAbstracts

05.50

Shom Communication

Stretched exponential decay in

_an

asymmetric _{+ /} J spin

chain

H.

Rieger(*)

Institut fur Theoretische

Physik,

Universit3t zu Knin, 5000 Knin 41, FR.G.

(Received

24 October 199f

accepted

29 October1990

)

Abstract. The

dynamics

of_a one-dimensional

spin glass

with al)owance of asymmetric interactions among

neighbouring spins

_is considered. The

algebraic decay

of the remanent

magnetisation

of the

infinite +J

spin

chain at zero temperature ^is no

longer

present ^for a

nonvanishing probability

of

asymmetric

^bonds. Thus the

asymmetric

^+J

spin

chain shows an asymmetry ^induced

phase

transition at _zero temperature. ^The

analytical investigation

of the model as well as computer simulations

strongly

suggest ^that the

decay

is stretched

exponential

when the

probability

of

antisymmetric bond-pairs

is not zero.

The

investigation

of disordered

spin _systems

with

asymmetric

bonds has

gained

_some interest in recent years ^motivated

by

their relevance for neural network models

[I].

^Within mean field

theory,

that means for the

asymmetric

SK model

[2],

it was

conjectured

that

asymmetry destroys

the

spin glass

behaviour. This _can be proven ^{for linearized models}

[2, 3],

^{but remains} controversial for

Ising spins _[3-7].

in this

paper

we consider _a one-dimensional version of these

spin systems

and we shall _see

that, although easily formulated,

the model has _some

fascinating properties

and is

comparable

in its

complexity

for instance with the random

binary alloy [8, 9].

The model consists of N

Ising spins

_{«I =} +I

arranged

in a chain

(with cyclic boundary

condi-

tionS)

and

interacting

with nearest

neighbours

via the local fields

h,

₌

Ji,;-ia;-i

+

J;,I+ia;+I (I)

The interaction

Strengths J;,I+i

can be +J or J with

probability 1/2 and,

in contrast to

magnetic

models

[lo, 11], they

need not to be

Symmetric,

I-e- it _can be

J;,,+i # _J,+i~~

With

probability

p ^E

lo, Ii

it is

Ji,I+i

₌

Ji+i,I

and with

probability

I

pit

is

J;,;+i

₌

-J,+i,I

The

dynamics

of the

System

is the conventional

Glauber-dynamics [10]

: the

Spins (interacting

with _a

heat-bath)

are

up-dated

in _a random

Sequential

_manner and the transition

probabilities

_are

w

(a, a()

=

ii

+

a(

tanh

flh~]

,

(2)

(*)

^Address

aflerJanuaiy1,

1991 Institute for

physical

science and

techno)ogy, University

of

Mary)and, College Park,

MD

20742,

U.S.A.

JOURNAL DE PHYSIQUE I T I, M I, JANViER 1991 2

14 JOURNAL DE PHYSIQUE I N°1

where

fl

is the inverse

temperature.

^{In the} case of

vanishing temperature

we have

simply

the

following

rule _: the new value of _a; is +I if

h;

=

2J,

-1 if

h,

= -2J and ₊ I _or I with

probability

la if

h,

₌ 0.

By writing

down the Master

equation

for the

probability

dhtribution P

(a, t)

of

spin

con-

figurations

_{a, one}

gets

^the

following

linear differential

equation

for the

magnetisation

_ni,

it)

=

(«;)(t)

₌

j~~ a;P (a, t)

of the

spins

with ₇

=

tgh 2flJ

and the matrix r

given by

0

PI

0 0

9p

PI

0

9]

0

0

9j

0

~"

14)

0

9(_~

9(

0

9jj_~

⁰

where

9f

=

sign J,+i,<

and

fl/_~

=

sign J,,;-I

Note that

only

^{the random} variables

9/

and

9/

_are correlated with

(9/9/

=

2p

1. The

temperature dependence

enters

only trivially

via 7 =

tgh 2flJ

< 1. As one can

easily

see the absolute values of the

eigenvalues

of r _are smaller

than 2 and therefore _m

(t decays always exponentially

for

nonvanishing temperature.

Hence _we consider

only

the _case of _zero

temperature (7

₌

1)

^from now on.

The matrix r is

triangular apart

^{from the elements}

TIN

and

rNi emerging through

the

cyclic boundary

conditions. Furthermore it is not

symmetric

but h similar to a

complex symmetric

matrix

r,

where

f,,,+i

=

I,+i,<

=

@.

_{Hence for}

a

symmetric bond-pair (J,,,+i

₌

J,+i,<)

one has

I,,,+i

= +I and for _an

antisymmetric bond-pair

one has

I;,;+i

= i = I.

In the

purely symmetric

case

(p

= I the matrix r _can

easily

be

diagonalized.

If

9)

₉

(

= +1

the

periodic boundary

condition

produces

no frustration and the

eigenvalues

of r _are

given by in

= 2 _cos

(27rn IN

for _n

=

I,

...,

N,

the

eigenvalue

1

= 2

corresponds

to the

time-independent

solution of

(3).

The

density

of

eigenvalues

of r

diverges

therefore in the limit N

- cc at 1

= 2

algebraically,vith

_an

_exponent la, leading

to an

algebraic decay

of the remanent

magnetisation

N

Jfp=j(t)

₌

~j (a,(t)a,(0))

_c~

ill

_{for t >1.}

₍₅₎

<=1

This is well known from the

zero-temperature

^{behaviour of the} one-dimensional

ferromagnet [10].

We

begin

the

investigation

of the _p

#

I _case with the

following

observation _: let Re r be the real

part

^{of the} transformed matrix Re

I

_{Then the}

olf~diagonals

of Re

I

are

composed

of chains of

length

lk

consisting only

of +I and chains of

lengths ik consisting only

of

0, corresponding

to

connected

parts

^{of the}

spin

chain that have

only symmetric, respectively antisymmetric, bond-pairs (note

that

j~~ (lk

⁺

ik)

=

N).

Thus Re

I

_is

everywhere

zero

apart

^from

(lk

+

I)

x

(lk

+

I)

block~

Hence,

according

to

the

theorem

of

Bendixson which can befound

in

_tandard

of

linear algebra),

the _greatest real _part of

the

eigenvalues

of r is smaller than 2

where

lmax

is

the

length

of

the_longest

part

of

the^hainconsisting of nly

symmetric bond^irs.

With other^{ords, for}

an

^nfinitechain

with not

morethan lmax

symmetric

^nd~pairs

in succession,

we have

:

Mi~~~ <

where _c

= I + cos 7r

/

_(lm~x +

1)

> 0 and C is _a constanL Furthermore

equation (3)

has _no time

independent

solution

except

^{m=0 In the limit N} - cc the maximum

length

lmax

diverges

and

(6)

becomes trivial because _{c -} 0. Now _one needs informations about the

density

of

eigenvalues

of r in the

vicinity

of1 ₌ 2.

Let _us first have _a look _on the

corresponding

diluted +J

spin chain,

where all

pairs

of anti~

symmetric

bonds _are removed _so that the

system

^{consists of} isolated chains of

length

lk ^with

only symmetric

bonds. The

probability

of _a

spin

_to

belong

to a chain of

length

I is

(I p)2p~~~

^and

therefore the remanent

magnetisation

of the diluted

spin

chain can be calculated

by averaging

the

decay

of the _remanent

magnetisations

of finite chains over all chain

lengths

with the above

weights.

This

yields

a stretched

exponential decay _[12]

Afd;i(t)

c~ exp

(t IT)

^~/~ for _t »

1, (7)

where _T is _a characteristic time

depending

_on the dilution _p. The

decay

at

long

times is determined

by

the

extremely long

chains

weighted

with their

probability.

^{The effect of}

intrtlducing again

the

antisymmetric

^bond

pairs

at both ends of _a finite chain is to

produce

a disturbance

propagating

within time in the direction of the center of the chain. Note that

always

one of the

spins neigh~

bouring

an

antisymmetric

bond

pairs

has _zero field and therefore

flips

^with

probability la.

Thus

we

expect

^the ma

gnetisation

to

decay

_even faster within these finite chains in the presence of _an-

tisymmetric

bond

pairs.

^In what follows

present

^the

arguments,

^which

support

^{the claim that the}

asymptotic

behaviour of the remanent

magnetisation

of the

asymmetric spin

chain is stretched

exponential

as in the diluted version of the chain.

Consider

firstly

the

periodic

^chain with _an

antisymmetric

bond

pair following

k

symmetric

bond

pairs.

^The

Laplace

transform

ji(z

_{of the remanent}

magnetisation

_can be

represented

with the

help

^of a continued fraction.

Analyzing

the

singularites

of this continued fraction _one

gets

^{k I}

poles

at zn = -I ₊ cos _7rn

/k in

=

I,...,

k I and

2(k I) algebraic

cuts

arranged

in such _a way ^that the

asymptotic

behaviour is

essentially

determined

by

the

poles.

Hence the remanent

magnetisation decays

to zero in the _same way as it

decays

to its time

independent

value in the diluted chain

(note

that in the

pert(>dically

diluted chain there is in addition _to the above

pole

one at _{z =} 0

corresponding

to the

nonvanishing time-independent

solution of

Eq. (3)).

16 JOURNALDEPHYSIQUEI N°1

Returning

to the

aperiodic,

random model _{one can}

try

to

get

more

insight

into the

asymptotic decay by investigating

the

spectrum(I

of the matrix r Since it is

only complex~symmetric

the

eigenvalues

are

generally complex.

We have calculated

numerically

the

spectrum

^{of various real-} isations of the matrix r with sizes up ^{to N} = 500.

Plotting

these

eigenvalues

ⁱⁿ the

complex plane yields pictures resembling

Julia~sets and the

support

^{of the}

spectral density

seems to be fractal.

This should be

expected

with

regard

to similar

systems (e.g.

the random

binary alloy

chains

[8, 9]).

^{The situation is} even more

complicated,

because _now the

spectrum

^is

complex. Already

at this

stage

one could

reject

the

hope

for _a

complete analytical

^solution of this

problem,

but let _us first have _a closer look to

analytical

tools that _are available.

I.

Dyson's

formalism

[13].

Since the random variables

9f

_are uncorrelated for different site indices I _{one can} derive _a func~

tional

equation

for the

generating

function

fl(I)

=

f d~l log (I I)

_p

(I)

of the

spectral

den-

sity p(I). Using

a more refined formalism

developed

in

[14]

one

gets

^the

following equation

for the

quantity D(u,

^I that contains the informations about

p(I

via

ail)

₌

D(cc, I)

D(u, I)

₌

pD ~-l ^~, l)

+

ii p)D ~l

⁺

^~, l)

+

log ~-l ^{D(cc, 1). (8)}

" 't 't

This

equation

is similar to that

occuring

in the context of the

disordered, binary

harmonic chain and _can

only

be solved for the

special

cases p = 0 _{and p}

=

I,

and

gives

us a hint to the

complexity

of the

problem [8].

2. lkansfer matrix method

[9, is, 16].

The Ansatz _m

it

= a e~~ for

equation (3) yields

_a recursion relation for a,+i Hence _{one can} write

(a;+i a;)~

₌

f (a,, a,-1)~,

where

2j

is _a 2 _x 2 transfer matrix.

Choosing

fixed

boundary

conditions instead of

cyclic,

which should not make any ^{difference in the limit N} - cc, ^one is confronted with the task of

investigating

a

product

of

noncommuting

random matrices that _must transform _one

given

vector into itself. This _can

only

be achieved for

special

^values of

I,

which are the

eigenvalues.

In

analogy

to the random

binary alloy

we were able to prove ^the existence of

special frequencies

within the

spectrum,

^{where the}

spectral density

must vanhh. At these fre~

quencies

_one should find Lifshitz

singularities [15]

^{of the}

spectral density, especially

at 1

= 0 and this leads to the stretched

exponential

behaviour

(7).

3. Method of moments

[17].

This method

gives

us the

strongest

^{evidence for} our claim. One considers the _moments of the

spectral density

ii")

₌

/ _{d~l p(I)I"}

# lim

(

lt r"

(9)

N-co ₌

(~ ^At this

point

one should be _aware of the fact that the matrix r is not Hermi tian and thus the

algebraic multiplicity

of _an

eigenvalue

can be

higher

than the

geometric multiplicity.

If _we exclude these

degenerate

cases from the ensemble of

matrices,

_{one can}

diagonalize

r

by

an

orthogonal

tranformation.

and _uses the fact that lt r"

=

j~~~,

_{~~~ j~,}

__,

~j

rp~p~rp~p~...rp~p~,

^I-e- a sum over all closed walks of

length

_~

weighted by

the moments of the random variables

9f.

Since the matrix r is

triangular

the walk h one~dimensional with nearest

neighbour,steps only.

Now _{one can}

try

to

compare

^{these moments with those of the}

corresponding

^diluted

matrix,

where all

antisymmetric

bond

pairs

are set to zero. This

procedure changes only

the

weights

of the dif§erent walks and one can derive _an

estimate, by

which the moments of the

spectral density

behave like those of a

diluted +J

spin chain, yielding

a stretched

exponential decay

of the remanent ma

gnetisation.

4. Numerical simulations.

Finally

we made

preliminary

simulations of the

zero~temperature dynamics

of the

asymmetric

+ J

spin

^chain for

system

^sizes up ^to

10~.

We found _some evidences for _a

non-algebraic decay

of the remanent

magnetisation

_{for p}

# I,

and _a stretched

exponential (7)

with _{v m} 0.3 and _{T m} I fits rather well. To make more

quantitative

statements about for instance the characteristic time _T in

dependence

of _{p, we} do not have

good enough

statistics. lb

improve

the numerical results within

acceptable computer time,

we intend to use multi

spin coding techniques [18].

Conclusion.

By investigating

the

asymmetric

+J

spin

chain with different

analytical

methods _as well _{as nu~}

merical calculations and simulations _we found

strong

indications for _an

asymptotic

stretched expo~

nential

decay

of the remanent

magnetisation.

Within this

paper only

results _were

given,

a detailed version will be

published

elsewhere

[18]. Choosing

a continuous distribution of the

couplings

in~

stead of the discrete +J distribution considered here

destroys

this

zero~temperature

^behaviour

and one

gets

a

nonvanishing

remanent

magnetisation _[19]

_even if

asymmetry

^is

present [20].

^This h due to the fact that

always

_one of the two

incoming

bonds is

stronger

^{and thus the field is} never

zero. The

generalisation

the _one dimensional model to

higher

dimensions would be

interesting

especially

vith

regard

to the above mentioned

controversy concerning

the

spin glass

transition in the

asymmetric

SK model and will be discussed in the future

[18].

Acknowledgements.

The author is

indepted

to B. Derrida and M.

Schreckenberg

for many

stimulating

discussions.

Valuable hints

by

^{J. Duarte}

concerning

the simulations _are also

acknowledged.

This work _was

performed

within the research program ^{of the}

Sonderforschungsbereich

341 K6ln~Aachen-Jiilich

supported by

^the Deutsche

Forschungsgemeinschaft.

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