Memory effects in the dynamic response of a random two-spin Ising system

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Memory effects in the dynamic response of a random two-spin Ising system

M. Nifle, H. Hilhorst

To cite this version:

M. Nifle, H. Hilhorst. Memory effects in the dynamic response of a random two-spin Ising system.

Journal de Physique I, EDP Sciences, 1991, 1 (1), pp.63-77. �10.1051/jp1:1991115�. �jpa-00246304�

Classification

Physics

Abstracts

02.50 75.10N

Memory effects in the dynamic _response of

_a

random two-spin Ising _system

M. Nifle and H. J. Hilhorst

Laboratoire de

Physique Thdorique

et Hautes Energies

(*),

Bitiment 211, Universitd de Paris- Sud, 91405

Orsay,

France

(Received10 July

1990,

accepted14 September 1990)

Rksvmk. Motivd par des effiets de m6moire observables dans les _verres de

spin

l'on btudie _un

systdme

moddle extrdmement

simplifid.

Il _se compose de deux

spins d'Ising

I

dynamique

de Glauber, dont la fonction de corrdlation I

l'dquilibre

varie

rapidement

et a16atoirement _en fonction du

champ

extdrieur. Comrne dans les _verres de

spin,

_une

r6ponse dynamique

_non lindaire

apparait ddji

dans le

rdgime

lindaire des

propribtds statiques.

On calcule (I) les

susceptibilitds

altematives lindaire et _non lindaire _en

champ

zdro et

(it)

la

susceptibilitd

altemative lindaire _en fonction du _faux de variation d'un

champ primaire

I variation lente. Le

probldme mathdmatique

consiste _{en une}

dquation

diffdrentielle

stochastique

_avec des corrdlations

jemporelles

de

longue portde.

Pour _un

champ

oscillant

d'arrplitude Ho

suffisamment

grande

(mais

toujours

dans le

rdgime statiquement lindaire)

_ces corrdlations conduisent I des termes

correctifs _non

analytiques Hi log Ho

dans la

susceptibilitd dynamique.

Abstract, Motivated

by magnetic

_memory effects observable in

spin glasses

_we

study

_an

extremely simplified

model system. ^{It consists of} two

Ising spins

with Glauber

dynamics,

whose equilibrium correlation is _a

rapidly

and

randomly changing

function of the extemal field. As in spin glasses, _a nonlinear

dynamic

_{response appears even} in the

regime

of linear static

properties.

We calculate (I) the linear and nonlinear _ac

susceptibility

in _zero field and (it) the linear _ac

susceptibility

_{as a} function of the rate of

change

of _a

slowly varying

background field.

Mathematically

the

problem

is to deal with _a stochastic differential

equation

with

long-ranged

correlations in time. For an

oscillating

field of

sufficiently large amplitude Ho (but

^still in the

statically

linear

regime)

these correlations lead to

nonanalytic

correction terms

Hi

^'

log Ho

in the

dynamic susceptibility.

1. Introduction.

We

study

the

dynamical

_response of _a two-level

system

with energy levels ± J. The

system

^has the

particularity

that J

depends randomly,

with a very short autocorrelation interval r~ _«

l,

_on the value of _an extemal

parameter

^{H. As such this}

system

is _one of the

simplest examples

of what _one

might

call random

equilibrium

models. These _are models whose

equilibrium

state is

hypersensitive (and

in the

thermodynamic

limit

infinitely sensitive)

to the value of _{one or more} external

parameters, Many spin glass

models

belong

to this class. The

(*) Laboratoire associb _au C-N-R-S-

hypersensitivity

of the correlation functions of _a

spin glass

both to

temperature

and to

magnetic

field has been established in _mean field

theory by

Parisi

[I],

within the

droplet

model for finite-dimensional

spin glasses by Bray

and Moore

[2]

^{and Fisher and} Huse

[3, 4],

and in _a Gaussian

approximation

around _mean field

theory by

Kondor

[5].

Instead of

using

the term

hypersensitivity, Bray

and Moore [2]

speak

of the chaotic nature of the

equilibrium

state; we shall

prefer

to say that the

equilibrium depends randomnly

_on the extemal

parameters.

The randomness of the

equilibrium

state has strong consequences for the

dynamics.

In

spin glasses

it is

responsible

for the

extremely

^slow

magnetic

relaxation. More in

particular

it is essential for the

explanation [6, 8]

of _a rich

variety

of nonlinear

aging phenomena

in small

magnetic

fields

[9-11].

The heuristic

theory

of reference

[6]

deals with

arbitrary

time-

dependent

fields and temperatures, ^but

explicit

results _are limited to _cases where the field

and/or

the

temperature undergo

_{one or a} ^few discrete

jumps.

In reference

[8]

^also

exclusively jumps

_are considered. A one-dimensional

Ising spin glass

with

randomly temperature

dependent

correlations _was defined

by Koper

and Hilhorst

[12]

and its

dynamical

behavior

investigated

for the

special

_case of _a

linearly varying temperature [13]. Mathematically speaking

this

special

case avoids the

complication

of randomness that is self-correlated in time.

It is

clearly desirable,

both for

practical

and for theoretical _reasons, to

investigate

what

happens

in

arbitrarily time-dependent fields,

and _more in

particular

in

periodic

_ones. We therefore formulate _a

simple

two-level model

using spin glass language,

and

study

the effects

on it of _an

arbitrary time-dependent magnetic

field

H(t).

The

dynamics

of the model is

govemed by

_a Glauber

type [14]

master

equation.

Two types ^{of fields}

play

_a role _a finite field

H(t)

of

typical

value

(H(t) Ho

and _a

probing

field

h(t)

=

ho

^e"~ of «infinitesimal»

amplitude,

_as used in

experiments

which determine the _ac

susceptibility.

A

parameter

^{il sets the time scale} on which

H(t) varies,

the unit of time

being

of the order of the

flip

time of _an individual

spin.

If

H(t)

oscillates with

frequency il,

then the

coupling J(H(t))

is autocorrelated _over

arbitrarily long

time intervals.

These autocorrelations _are the main

problem

addressed in this work.

In sections 2 and 3 _we define the model and _state its main

properties.

In section

4,

_we formulate its

dynamics.

Its _ac

susceptibility

is calculated in the limits il _« I

(Sect. 5)

and

rHo»

I

(Sect. 6).

In section 7 _we calculate the _ac

susceptibility

in _a

slowly varying background

field. Section 8 contains _a summary and

conclusions,

and may be read

independently.

2. A random

equilibrium

model of two

Ising spins.

We consider two

Ising spins,

_sj and s~, that _may take the values ₊ I and I. The inverse

temperature

p

w I

/kB

T is _a fixed

parameter.

^When

placed

in _a

magnetic

field

H,

the

spins

are

coupled by

_an effective interaction

J(H),

_so that the Hamiltonian is

3C(sj, s~)

=

J(H)

_{sj s~}

H(sj

₊

s~) (2.1)

This effective Hamiltonian should be

thought

of _as

arising

from _a true

microscopic

_one via _a renormalization

procedure (see

_e.g.

Bray

and Moore

[15])

that _we shall not detail. As _a consequence of the randomness and the frustration at the

microscopic level, J(H)

is _a random function of H.

Finding

the exact statistics of

J(H)

is _a difficult

problem

that

belongs

to

equilibrium spin glass theory.

Here _{we are} interested in the

dynamical

behavior that follows when the statistical

properties

of

J(H)

_are

given.

Motivated

by

mathematical convenience _we express these in _terms of the variable

z(H)

w

tanh

pJ(H).

We

postulate

(I) z(0)

= ± zo _m + tanh

pJo

with

probabilities

2 , 2

(it) z(H) changes sign ~jumps

between _±

zo)

with

probability

r dH in each infinitesimal 2

field interval dH.

All

remaining

statistical

properties

_can be derived from

(I)

and

(it),

_e.g,

@f

= 0

(2.2)

z(Hi)z(H~) =z(e~~"~" _(2.3)

where the overbar indicates _{an average over} all random functions

z(H).

The

spins

_{si and s2 may}

represent

the net

magnetic

moments of two

neighboring regions

in _a

spin glass.

If these _are of linear size

L,

then

Jo

and r will increase without limits

[2]

when L

gets larger.

We therefore

expect,

and _use e.g, in section

6,

that I

IF

is small with

respect

to the

typical

value

Ho

of the

magnetic

fields that _we shall consider _:

I _«

rHo. ^(2.4)

Furthermore,

in

experiments exhibiting aging [9-11]

the

magnetic

_energy of _a

spin

is

always

small

compared

to its thermal energy :

pHo

«1

(2.5)

and _{we use} this relation

throughout

this work.

3.

Equifibriuul.

The

equilibrium properties

of this

two-spin

_{system are} trivial.

Letting (... )

~

denote _a thermal average with respect to 3C _we have

(sj

₌

(s~)

=

sinh 2

p

^H

/ (cosh

2

p

H _{+ e} ^~

P~(~°) (3.1)

(sj s~)

₌

(cosh

2

pH

_e ^~

P~(~~) / (cosh

2

pH

₊ e~ ^~fl~~~~)

(3.2)

Obviously

these

equilibrium

_averages

depend,

via

J(H), randomly

_on the field H, For later

reference _we record here the

purely ferromagnetic

and

antiferromagnetic

_zero field

susceptibilities,

obtainable from

(3.I)

for

J(H)

=

Jo

and

J(H)

=

Jo, respectively,

viz.

XF,AF " 2

fl /(I

₊

e~~~~)

=

fl (1

±

20) (~.~)

4.

Dynandcs.

4.I TIME _EVOLUTION EQUATIONS. Let

P(sj,

_{s~ ;}

t)

be the

probability

to find the

spins

at

time t in the

configurations (sj, s~),

We

postulate

for P the master

equation dP(si,s~; t)

~ ⁼

wi(sit-si)P(-si,s~;i)+ w~(s~l-s~)P(si,-s~;i)-

t

Wi(-

_si

si)

₊

W~(-

_s~

s~)) P(si,

_s~

t) (4.1)

Here

ll§(- al «)

is the _{rate at} which

spin

I

flips

from _« to _-«

(with

I

=1,2;

« =

±1).

We shall make the Glauber

[14]

choice

Jl~

(-

_{si sj}

=

[I

_z

(H)

_si

_s~]

I s, tanh

pH] (I

= 1,

2) (4.2)

which fixes the unit of time _on the scale of the

spin flip

time.

The

expression (4.2)

satisfies detailed

balancing

_: if H and

p

_are

fixed, P(sj,

_s~

t)

will

approach

the canonical

equilibrium

distribution. We note,

however,

that in

(4.2)

the field H may be _an

arbitrary

function of time.

For

magnetic

fields

(H(t)

_s

Ho

we may, in view of

(2.5), simplify (4.2) by writing

tanh

pH(t)

m

pH(t) (4.3)

It is not

possible, however,

to linearize the random function

z(H)

in

H,

_so that the master

equation (4.I)

remains nonlinear in the field. This

nonlinearity

is at the root of the nonlinear response, and is

why

this model

represents

a

spin glass

albeit in

rudimentary

form.

From the master

equation (4.I) together

with

(4.2)

one

easily

derives

equations

for the

time-dependent

moments

'~~~~

"

~~l)1

^~

~~2)i~

'

~~'~~)

g(t)

w

(sj s~)~, ^(4.4b)

where

(... )

is _an average with respect to P

(sj,

_s~ t

).

One

finds,

_upon

linearizing

as in

(4.3),

~()~~

=

2

PH(t) 211 z(t)I m(t)

2

PH(t) z(t)

_g

(t) (4.5a)

d[(t)

=

4

z(t)

_{4 g}

(1). (4.5b)

Here and henceforth

z(t)

is shorthand for

z(H(t))

when _no confusion _can arise.

Equation (4.5)

is a set of _two

coupled

linear stochastic differential

equations.

Its

difficulty,

as

compared

to the usual type ^of

equations

in this class

(see

_e.g. Ref.

[16]),

is that the stochastic function

z(t)

_may be self-correlated for

arbitrarily large

time differences.

The

time-dependent magnetization m(t)

_can be solved from

equations (4.5).

For

arbitrary

initial conditions at _t

= co we find

1 2(1 t') ₊ 2 di" z(t") i'

m(t)

=

2

p

dt'e

"

l 4

dt"'e~~(~'~

~"~

z(t') z(t"') H(t'). (4.6)

-w -w

This formula is the

starting point

for the

analyses

of section 5. It is _a

fully explicit

solution for the

magnetization

for _any

given

realization of the random function

z(H(t)).

The

problem

left is to

perform

the average on all random functions _z in order to obtain the

quantity

of final

interest, if.

This

problem

is considered in section 6.

4.2 ALTERNATIVE TIME EVOLUTION EQUATIONS FOR MONOTONE

H(t).

When

H(t)

is _a

monotone function of

time,

there is _a standard method

[16]

to derive _an altemative _set of time evolution

equations.

To obtain these _one makes _use of the bimodal _nature of the random

function

z(H).

For monotone

H(t),

the

quantity

_z

jumps

between zo and _zo at a rate

~r[fi(t)),

_without

any correlation with past

jumps.

There

exists, therefore,

_a master 2

equation

for the

joint probability P(sj,s~;

_{a ;}

t)

of

finding,

at time t, the system ⁱⁿ

(si, s~)

while

z(t)

has the value azo

(for

_a

= ±

I).

It reads dP

(sj,

_{s~; a} t

)

=

wi(sit-si)P(-sj,s~;

«

i)+ w~(s~l-s~)P(si,-s~;

«

;i)-

di

(wj(-

_si

lsi)

₊

_w~(-

_s~

ls~)j P(sj,

_{s~ «}

i)-

-(rlHl _iP(si,s~;« ;t)-P(si,s~;-« ;t)1. (4.7)

Here the W terms

trivially generalize

those of

equation (4.4)

and the r

)fi)

terms _are due to the

jumps

of

z(t).

Let

m"(t)

and

g~ (t),

with _a

= ±, be averages

analogous

to those of

equations (4.4),

but with

respect

to

P(si,

_{s~ ; a ; t}

).

From

(4.7)

_one

finds,

after

linearizing

_as in

(4,3),

~~()~~

^{= «zo 4}

^{g« (t)} jr _{jH(i) iga(i)}

g- ^a

(i)1 (4.g)

which

yields,

^for

arbitrary

initial conditions at t ₌ co,

ii

g* ( t)

= ± 2 zo dt' _e~^~~~ ~'~ ~' ~~~~ ~~~'~

(4.9)

m

showing

that at all times

@

=

g~ (t)

_{+ g~}

(t)

= °

(4.10)

For m*

(t)

_we

find, using

that

g+ (t)

= g~

(t),

d

jm+(t)

^~

~° ~

~~~~~~~~ ~~~~~~~~ (m+(t)j

~

~

'~

(~)

_r

)fi(t)

_{I + zo +} r

)fi(t)

'~

(~)

+

flH(t)[1- 2zog+ (t)]

(~ ^(4.ll)

The

equations (4.9)

and

(4.I I)

_{are sure,} that

is,

nonrandom.

They

constitute _an alternative

description

of the time

evolution,

valid for monotone

H(t).

The

problem

left is to find their solution. The relation

if

=

m+

(t)

₊ m

(t) (4.12)

then

yields

the

quantity

of final interest. This

problem

is considered in section 5.

4.3 THE Ac SUSCEPTIBILITY IN THE NONRANDOM cAsE. For later reference _we record here

the

expression

for the _zero field _ac

susceptibility XJ(il

of the _same

two-spin

model but with constant

(H-independent) coupling

J in _a field

H(t)

=

Ho ^e'~~

It is

easily

found to be

i _-z- in

xj(11)

=

P (1 z2)

^~

(4.13)

(1-z)~+-n~

where _z

« tanh

pJ.

For il

=

0 and J

= ±

Jo

this reduces to the static

susceptibility (3.3).

S. The

magnetization

in _a low

frequency

field.

S. THE Low-FREQUENCY ^LIMIT il _« I. Let _as before

Ho

^{be the}

typical

value of the time-

dependent magnetic

field

H(t),

and let il the

typical

time scale _on which it varies. we shall find

m(t)

in the low

frequency

limit

il«1, (5.I)

I-e- when _a

large

number of

spin flips

takes

place

before

H(t)

varies

appreciably.

This condition still allows from _an

arbitrary

value of the ratio

ilrHo.

Two limit

regimes

_may be

distinguished.

(I)

For

ilrHo

«

I,

there _are many

spin flips

between two

jumps

of the

coupling

constant

J,

and the

system

^is most of the time close to the

equilibrium

determined

by

the instantaneous values

J(H(t))

and

H(t).

(ii)

For

ilrHo» I,

_on the contrary, ^there are many

jumps

of the

coupling

constant between two

spin flips,

and the system ^{is in} a

stationary

state in which the

spins feel,

to lowest

approximation, only

the average

coupling ^f

between them since

f= 0, they

_are like free

spins.

The considerations of the _next subsection _are for

arbitrary ilrHo.

5.2 CALCULATION _OF

m(t)

FOR fl _« I. The

starting point

for the small-R

expansion

of

m(t)

is

provided by equations (4.9)

and

(4.ll).

In

(4.9)

we

expand

H(i')

=

H(i) (t t') H(1)

₊

(5.2)

and _use the fact that

~~~

il ~

Ho (5.3)

dt~

to

neglect

all terms

beyound

the first derivative.

The result of the

t'integral

then is that

j

^zo

g~ (t)

m ,

(5.4)

+

~

^r

£i(t)

up ^to corrections of relative order il. This

expression

_{can now} be inserted in

(4.

I

I),

which _can in _tum be solved for small il. This is done in

appendix

A.

The result is that

@

=

X'H(t)

₊ X"

H(t) (5.5)

For r _« I

(see later),

the

quantities

X' and RX" in this

expression

become identical to the real and

imaginary

_part,

respectively,

of the usual

dynamic susceptibility. However,

for

general r,

the

quantities

X' and X" _are

given by

(I ^-z/+ r)fi) (1+ ^r)fi)

X'

=

p

^{~ ~}

(5.6a)

(1+~r)fl)) (I-z/+ _~r)fi))

4 2

(1-z/+ ^r)£i) ₍₁₊ ^r)fi) _)~+z(j

X"

=

p

^{~ ~}

~

(5.6b)

~

(l

+

r)fi) (I ^-z(+ r)fi)

4 2

again

_up to corrections of relative order il, The

important

feature of this result is that X' and X"

depend

_on the time

dependent

field via r

)h).

As the

amplitude Ho increases,

_a

dynamical nonlinearity

sets in when r

)h)

becomes of the order of

unity,

I-e- when

How (5,7)

5.3 APPLICATION TO A PERIODIC LOW FREQUENCY FIELD. It is of interest to

apply

the

general

result

(5.6)

to the

special

_case of the

oscillating

field

H(t)

=

Ho ^e'~~ (5.8)

It is understood that the associated

coupling

constant is

J(Hocos ilt).

The field

(5.8)

is monotone

only

in time intervals of half _a

period,

and therefore _{errors are} incurred where these intervals

join.

Since _a

spin flip

erases the _memory of

previous

values of

J,

these _errors

are of _a relative order

equal

to the

spin flip

time divided

by

the

half-period,

that

is,

of relative order il. Since this is also the _error in

(5.6),

_{we may use}

(5.8)

in

(5.6). (In

Sect.

6,

_we shall find that this is true

only

for il -0 at fixed

rHo;

if _one first takes the limit of

large rHo,

then the contribution from the extrema is

actually

il

log rHo

times the contribution from the monotone

intervals).

The result is that for the

oscillating

field

(5.8)

the

dynamic susceptibility

is

given by

k(il

_;

t)

= X'+ iilX"

(5.9)

with X' and X" _as in

(5.6),

but with r

)h)

=

rHo

sin

ilt(.

Hence the response ^to a

harmonic

signal

is nonlinear in the

wnplitude

and nonharmonic. The two _cases

(I)

and

(it)

mentioned above _now

correspond

to two limits.

(I)

For r

= 0

(no disorder) equation (5.9) reduces,

_as it

should,

to the

time-independent expression

i i ₊

z]

e(n)

=

p

i in

~

(5.lo)

2 zo

which is the low

frequency

^limit of the

susceptibility (4.13)

at constant

J, averaged

_over

J= _±

Jo.

(ii)

For r

- co one finds

I(n)

=

p ii

ⁱⁿ

,

(5.ii)

2

which is the low

frequency

limit of

(4.13)

at the average

coupling ^f=

0. Hence the limit r - co is _a free

spin

limit.

Equation (5.6)

describes the full _crossover between these _two limits.

Finally

it is worth

noticing

that in the limit T- _co, I.e. zo -

0,

^the r

)h) dependence

of X' and X" in

(5.6) disappears

_: this is

also,

_as of _course it should

be,

_a limit of free

spins.

6. The

magnetization

in _a nonmonotone field _:

expansion

for

rl§

»1.

6, I EXPANSION IN POWERS OF z. When

H(t)

is not of _a

simple sign

in the time interval of

interest,

correlations between

magnetic

field values at different times _come in. An

approach

based _on the

equations

of section 4.2 may then _no

longer

be

possible,

and _we have _to take instead the _more

general

result

(4.6)

_{as a}

starting point.

The

general problem

of

averaging

this

expression

_on all random functions is too

hard,

and therefore _we must look

again

for _an

expansion

method. We

expand (4.6)

in powers of

z and get, upon

averaging,

ii

+f

= 2

p -

dt' _e~^~~~ ~'~ l ₊ 2

dti dt~ z(ti) z(t~)

w

i>

1>

4

~

dt"

e~~(~'~

~°

@fit)

₊

H(t'). (6.I)

'm

The first term in

(6.I) yields

a contribution

corresponding

to free

spins.

The dots indicate terms of fourth and

higher

order in _z. we shall look for conditions that will allow to

neglect

these. Since the average of _a

product

of two z's is small when the correlation interval

I/r

is

small,

we

expect (6.I)

to lead to a

large-r expansion. Although

the purpose of this section is to

study

time intervals in which

fi(t)

_{is not of}

a fixed

sign,

_we shall first show how the

expansion

works when

£i(t)

_{# 0.}

6.2

LARGE-rHo

EXPANSION FOR MONOTONE

H(t).

We first consider the contribution _to

(6.I)

from the second term in the square brackets.

Calling

this term

I~

and

using (2.3)

_{we can}

write it _as

I~

=

2

z/ _{dti dt~}

_exp r

(t~

ii

£i(ti

₊

_(t~

11)~

ji(ti)

₊

_(6.2)

~ _i~

2

The n-th term in the

expansion

in

(6.2)

is of order

rHo

il

~(t~

₁₁)~. ^{In section 5 the second} and

higher

derivatives could be

neglected

because of the condition il « I. One has _to argue

differently

_now. The first term in

(6.2)

shows that the time differences that contribute

effectively

to the

integral

_{on t~ are} restricted to t2 ii w I

/r fi(tj

l

In rHo.

We shall take

nrHo

»

(6.3a)

but without

imposing

_any condition _on il. The n-th term in

(6.2)

then is

effectively

of order

rHo

il

~(ilrHo)~

^~

=

l

/(rHo)~

^' Hence the second and

higher

derivatives _are

negligible

if in addition to

(6.3)

the condition

rHo

_»

(6.3b)

is satisfied.

Upon replacing

the bounds of the t~

integral by

± co we find

Ii

=

4

z/ ~

dtj /

(I

^{+ O}

( _(6.4a)

1' l~

H(tj)

0

By

an

analogous

calculation _we find for the third _term in square brackets in

(6.1)

4

z/

_{i i}

I~

=

I

+ O + O

(6.4b)

r

fl( t)

^J2

I~Ho I~Ho

One _can convince oneself that _an average

involving

_a

product

of 2 _n factors _z will

contribute,

in

leading order,

_a factor

I/(rilHo)~.

The result of this subsection therefore is that for

ilrHo

» I and

rHo

» I

i

Ii ^di~

⁴

^zj

@

=

2

p

dt' e~~(~ _~'~ l ₊ 4

z/ _H(t') (6.5)

m i' r

£i(tj

_r

£i(t')

This shows that the _average

magnetization

is related _to the extemal field

by

_a nontrivial memory kernel.

Once the result

(6.5) established,

_{one may} consider it in the limit il _« I. After _one

Taylor expands H(t'),

I

/fi(ti),

_and I

/£i(t')

around the time t, the

t'integration

_can be carried out.

One finds

(at

least _to

leading

order in the small

quantities

il and

I/rHo)

a result

equal

to

(5.5)

and the

large-rHo

limit of

(5.6).

Hence the limits of this and the

previous

section commute.

6.3

LARGE-rHo

EXPANSION FOR NONMONOTONE

H(t).

We _{are now}

ready

to _see how the

expansion (6.I)

works when

£i(t)

is not of _a

simple sign

in the time interval of interest. Our considerations will be restricted to the

oscillating

field

H(t)

=

H~ e>Dt, (6.6)

but _can be

applied

without

great difficulty

to _more

general

nonmonotone fields. With

(6.6), equation (6.I)

_can be cast into the form

+

=

k(11

_;

t) Ho e'~~, (6.7)

where

I(il t)

=

~

f

+ X

i(il

_;

t)

₊

x~(il

t

)

₊

(6.8)

Here the first term is the free

spin susceptibility,

and _xi and _{X2 are} corrections due to the second and third _term in

(6.I), respectively. Explicitly,

ii

^{i i}

Xi

(il

t

)

= 4

p

dt' e~ l~ +'~~~~ _~'~

dtj dt~ z(ti z(t~) (6.9a)

m

i' ~

i i~

X2(il

_;

t)

=

8 p

-

dt'e~ ^(~+'~~(~ ~'~ dt"

e~~(~'~

~"~

z(t' z(t") (6.9b)

m

-

m

Both

expressions

_are evaluated in

appendix B,

in the limit

rHo»1, n2rHo»1. _(6.io)

The calculation makes _use of the fact that the time

integrals

in

(6.9),

in the limit

(6.10), acquire

their main contribution _near the extrema of

H(t). Explicitly,

the result of

appendix

B

is,

for

j

= 1,

2,

Xj(il

_;

t)

=

16

pz/(2

+ in

f~~

_e~ ~~+~~~(~~°~"/ ~~

((il) (il~ _{rHo)~ log rHo} (6.I1) with, using

the abbreviation _y

=

e~~"/ ~,

((il)

=

~ Y~~

~

j

₌ 1, ^2.

(6.12)

(1+ y)(i

_y

J)

These functions behave _as

((0)

= ,

((il

m

il

/4 grj

as il

- co

(6.13)

The result

(6. II)

shows that the

large-rHo expansion

is

nonanalytic.

Several comments are in

place.

First,

_upon

comparing (6.7), (6.8),

and

(6. II)

to the

corresponding

result for the monotone case,

equation (6.5),

_{one sees}

that,

not

counting

the free

spin

term, ^the contribution _to the

susceptibility

from the extrema

outweighs

the contribution from the monotone intervals

by

a

factor il

log rHo.

^{Hence in the} nonmonotone case the limits il _« I and

rHo

_» I _no

longer

commute.

Obviously,

in

general

nonmonotone field

H(t)

each extremum _t~~~ will

give

_a

correction term to the free

spin

result which is

proportional

to

)ji(t~~~) )~

'

log rHo.

Secondly,

the

periodic

field

Hoe~~~

has _an infinite sequence of

maxima,

all at H

=

Ho (and

^of

minima,

all at H

=

Ho).

The correlations between the

J(H)

values _near all

these maxima

(minima)

lead to infinite series of terms that _can be summed and

give

the

factors

lj(il)

and

(6.ll).

The ratio of these series is

~e~~"/ ~,

_{so that} when the field oscillates

rapidly

_on the scale of _a

spin flip

time

(il

_» I

),

the correlations may extend _over

arbitrarily

_many maxima

(minima).

Thirdly,

the result

(6. II)

is nonharmonic due _to the

exponential

factor with _t mod

gr

In

appearing

in it. This type ^of

nonharmonicity

is

distinctly different, however,

from the response at a

frequency

3

il,

discussed

by Tanaguchi

et al.

[17],

which _can be observed _near the

freezing

temperature ^{and is} a consequence of the

nonlinearity

of the static

susceptibility.

7. Ac

susceptibility

in a nonconstant

background

field.

In this section _we

investigate

how the

susceptibility

of the present random

equilibrium

model is affected

by

_one of the external parameters not

being stationary.

The

analogous question

was considered

by Koper

and Hilhorst

[13]

for _an

Ising

chain with

randomly

temperature

dependent interactions,

in the presence of _a constant rate of

temperature change, t.

_It

was found there that the _ac

susceptibility

increases _{as a} function of

)f).

In our model the

magnetic

field is the

interesting variable,

and _{so we}

study

^its ac

susceptibility

in _a nonconstant

magnetic background

field

H(t).

It is measured via a

rapidly oscillating

_« infinitesimal _»

probing

field

ho

e~~~, so that the total field is

H~~~(i)

=

H(i)

+

ho

e<wi

(7.1)

It is understood that

il _«

w

(7.2)

where il is

again

the time scale _on which

H(t)

varies. For convenience _we restrict ourselves to il _« I. The

amplitude _ho

is

supposed

_so small that the

jump

rate r

)fi(t)

of J is

only

2

negligibly

affected

by

it.

The

starting point

is

again equation (4.I I),

but _now with

H(t) replaced by H~~~(t)

in the

inhomogeneous

term ; the

expression (4.9)

for

g+ (t)

remains unmodified. The solution of

equation (4. II)

so

modified, averaged

over the

randomness,

will be called

@,

and is of the form

m~~~(t)

₌

+

₊

I(w H) _{ho e'~~ (7.3)}

with

@) given by (5.5)

and

(5.6). Going again through

the calculation that led to

(5.6),

but

now without

linearizing

the

probing

field in time

(see appendix A),

_one obtains

I-z/+~r)H) I+~iw+~r)fl)

y(w fl)

=

p

^~

~

~ ~

(7.4)

1+(r)H) (I+~iw) ^-z(+~ (l+~iw) ^r)fl)

2 2 2

One _may check that for small _w this reduces to

y(w

_;

£i)

=

X'+ iwX" with X' and X"

given by (5.6).

For

large

r

)k)

the

expression (7.4)

_can be

expanded

_as

y(w;H)=

_x

I+-w~

2

2+~w~

2 x

°_ iw

1

~°

~~

+ O _

(7.5)

r)H)

_I

+-w~ r)H)

_{I +}

-w~ (r)H))~

4 4

This shows that both Re _x and Im _{X are}

increasing

functions of the rate of

magnetic

field

change )k).

_As in the _case of reference

[13],

the

interpretation

is that _a

slowly varying

external parameter

(temperature

_or

magnetic field) _prevents

the system ^{to build} up

equilibrium correlations,

and that

spins

out of

equilibrium

with their environment

respond

more

easily

to the

probing

field than

spins

in

equilibrium.

8. Conclusion.

We have studied _an

extremely simplified

model of _a

magnetic spin glass,

in which all

frustration and randomness is

represented by

a

single

^stochastic

coupling

constant

J(H).

Oscillations of the

magnetic

field H may therefore _cause,

potentially, arbitrarily long

correlations in time.

Mathematically

the time evolution

equations

of this model

system

can be cast into two different forms.

Firstly they

_can be reduced to the set of two linear

inhomogeneous

stochastic

differential

equations, equations (4.5).

These contain the

driving magnetic

field

H(t),

and the stochastic coefficient

z(t)mz(H(t))

=

tanh

pJ(H(t)),

which is autocorrelated for field

differences less that

~l/r.

Hence

z(t)

has _an instantaneous autocorrelation time

I

IF _{)£i(t) ).} Equations (4.5)

_can be solved for _any realization

z(H(t)),

but the

remaining problem

is to average the final

expression. Secondly,

the

problem

_can be cast in the form of two

coupled ordinary

differential

equations, equation (4.ll).

This

equation

contains _a time

dependent

matrix and the

problem

is to solve it.

Letting

il ^' be the

typical

time scale _on which

H(t) varies,

^and

Ho

its characteristic

value,

we have determined the average

magnetization +f

for _a

given H(t)

in the

following

_cases.

(I)

In section

5,

_we consider the limit il _« I. In this limit

long

correlations in time

exist,

but

a solution is

possible

since their effect is

effectively suppressed beyond

time differences

exceeding

the relaxation time of the free

(undriven)

_system, which is l. This limit includes

the _cases

ilrHo«

I and

ilrHo» I,

which _are

interpreted

in section 5.I in terms of

stationary

states close to and far from

equilibrium, respectively.

The full _crossover between

the two cases is described.

(it)

In section

6,

we consider the limit

rHo

»

I, r)fi(t) ilrHo

_» I. In this limit the instantaneous correlation time is short. It is also the limit in which the two

spins

_are

effectively nearly uncoupled,

so that the

expression

for

@

takes the form of the free

spin

result

plus

corrections. In the

region

of the

(il, rHo) plane

where both

(I)

and

(it)

are satisfied the results of the _two calculations coincide.

(iii)

The calculation

(it)

_{may no}

longer

be

valid, however,

_near time where

h(t)

=

0,

such

as in the _extrema of _an

oscillating driving

field

Ho e'~~

In section 6.3 _we consider this _case.

Under the conditions

rHo

» I and il^~

rHo

» I it is found that the _extrema

give

_a correction

to the free

spin

behavior which dominates the

remaining

corrections

by

_a factor

il ^'

log rHo.

(iv) Finally,

_we have considered in section 7 the response

@)

_{to a}

_{rapidly varying}

infinitesimal field

ho e'~~ superimposed

_{on a} slow

background

field

H(t).

It is found that the

susceptibility I(w £i),

in the limit of

large r,

is _an

increasing

function of the rate of

change )H)

^{of the}

magnetic field, just

as it _was found _to be

increasing

^with the _rate of temperature

change

in

previous

work.

Although

such behavior has been shown here

only

for _an

extremely simplified

model system, one should expect ^the same effect to _occur

qualitatively

in real

spin glasses,

and

possibly

in _more

general

disordered

systems.

Appendix

A.

Derivation of

(S,ti~.

we whish to find the solution

m~(t)

of

equation (4.ll)

in _a small-n

expansion,

where D~ ^' is the time scale _on which

H(t)

varies.

The idea behinds the

expansion

is that in order to find

m~(t),

_we need _to consider

preceding

instants of

time, t', only

such that t t'w

I,

where I is the

spin flip time,

after which the memory is

effectively

erased. Hence for il

WI,

it is sufficient to consider all

quantities

that _{vary on a} scale

il~~

_either

as

effectively

constant, _or, at most, as

linearly varying

with time.

In order to find _m ^~

(t)

_we consider

equations (4.

II

)

and

(5.4)

with t

replaced everywhere by

t'. Then _we

expand H(t')

_as ⁱⁿ

(5.2),

and write

similarly h(t')

=

fl(t)

_{+ After}

one

suppresses all second and

higher

derivatives of

H(t),

_one obtains

)

_t

l'~~(~'~)

⁼

^-2A(1) l'~~(~'~)

⁺

m~

(t')

_m-

(t')

+

P lH(t) (t t') H(t)I _Ii

_{2 zo}

_{g+ (t)I} (Al)

where

A(t)

is the matrix in

equation (2.ll).

The

equation (Al)

becomes

diagonal

after _a

rotation in the

plane

over an

angle p(t)

such that

tan p

(t)

= x~

(x~

₊

z/)~'~ zo] (A2)

x _m r

k(t) (A3)

The matrix

A(t)

has the

eigenvalues

The

equation

_can then be

integrated componentwise,

which

gives

for

@)

= m~

(t)

+ m_

(t)

the result

@

=

P Ii

_{2 zo}

g+ (t)I z (i

_{+ «} sin 2 _w

(t))

[Ap i(i) _H(t) p2(t)1i(t)j. ^(A5)

Tedious but

straightforward algebra

then leads _to

(5.6).

Appendix

B

Derivation of

(6,ll).

The

expressions (6.9)

_{are our}

starting point.

From

(2.3)

and

(6.5)

_we have

Z(tl) Z(t2)

"

Z/C

^{~~°~ ~" ~~' ~" ~~~}

(Bl)

This

expression

is invariant under the substitution

(ti, t~)

_-

(tj

+ ni gr

In,

_{t~ + n~ gr}

In ),

where nj and n~ are

integers

whose _sum is _even. It follows in

particular

that the

xi

(il

_;

t)

and

x~(il

_;

t) depend

on t

only

via t mod _gr

In.

we consider first the

expression (6.9b).

After _a transformation of the variables of

integration

it takes the form

m m

X2(il t)

=

8

p o dt'e~~~'~~

~'

~

dt"

e~~~" ~") (82)

We

put

now

t ₌ ngr

In

₊

r ,

n an

integer,

t e

[0,

_gr

In (83)

In order to take

advantage

of the invariance

properties

of

(Bl),

we consider

separately

the

following integration intervals,

t'e

[0,

_gr

/2

il ₊

r] (case

_p

=

0)

t<ei(2p-1)«/2n+r,(2p+1)«/2n+r) p=1,2;. (84)

For _a

given

t' _we put t'

= pgr

In

+ _r + r' r'e

[-

_gr

/2 il,

_gr

/2 il)

t"

= pgr

In

₊

r +

I" (85)

and consider

separately I"

e

[r',

_gr

In (case

_{q =}

0)

I"

_e

[(2

_q I

)gr In, (2

_q + I

)

_gr

In

_q

=

1, 2,.. (86)

The result is that

(82)

_can be written _as

« «

X2(il

_;

t)

=

8

p

e~ ~~+'~~ _~

£ (-

l_~P

e~~P"/

~

£ ^e~~~"/

^~ _J~~

(87)

p=o q=o

with,

for p, q =

1, 2,

,

«/2D «/D

j j d~, ~~,,

~(2-iD)r'-4r"

~ ~, _~

~,,) (~~)

pq-

-«/2D -«/D

If p =

0 the lower limit of the r'

integral

is

replaced by

_r, and if q =

0 the lower limit of the r"

integral

is

replaced by

r'.

We substitute _now

(Bl)

in

(88)

and notice that for

large

r the main contribution _comes from the

neighborhood

of r'

= r = 0. In order to make _an

asymptotic expansion

_we

Taylor

expand

the cosinus around _zero and pass ^to the _new variables of

integration

x=r<n/j, _y=r"n /j. ₍₈₉₎

Expanding

for

large

r _we find

-(«fi «fi

_'

jx2_~2j

J

=

z/(il

^~

_rHo)~ ~+(«fi

dx

~-«fi dy

_e ^~ _x

~l~~~l~fil~°nip. _(Bio)

If the conditions

(6.10)

_are

fulfilled,

the correction terms _are

negligible

and _one finds

J

m 4

z/(il

^~

_{rHo)~ log rHo (Bl I)}

By extending

the

analysis

_one finds that all

(~

^behave

asymptotically

as

J,

except ^for an extra

factor when _q

=

0. Substitution of

(Bll)

in

(87) yields (6.ll)

for

j

= 2.

2

Next _we consider the

expression (6.9a)

rewritten _as

m i> i~

Xi(il t)

=

4

p o

dt'e~ ^(~+'~~

o dtj o dt~ z(t ii) =(t t~). (B12)

Equation (Bl)

and the

preceding analysis

enable _us to conclude the

following.

The innermost double

integral

in

(B12)

will

receive,

in the

asymptotic

limit

(6.10),

_a contribution J from each

point (t-

_ii,

t-t~)

within the domain of

integration

and of the form

(nj «In,

n~ gr

In )

^with _nj,n~

integers

and _{nj + n~ even.} The number of such

points depends

_on

t' and _on

r = t mod _ar

In.

It

equals

2

p~

for t'e

((2

_p I

)

_gr

In

_{+ r, 2 pgr}

In

+ r

p~+ _~p +1)~

for t'e

(2

_pgr

In

_{+ r,}

(2

_p +

I) «In

₊

r) (B13)

where p =

0,

1,

2, Upon considering

the

t'integral

in each ofthese intervals

separately

_one

finds,

after

straightforward algebra,

the result

(6.ll)

for

j

=

1.

References

[1] ^PARISI G.,

Physica

124A (1984) 523.

[2] ^{BRAY A. J. and MOORE M.} A.,

Phys.

Rev. Len. 58

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