Dynamics in a spin glass model with large order parameter fluctuations

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Dynamics in a spin glass model with large order parameter fluctuations

A. Jagannathan

To cite this version:

A. Jagannathan. Dynamics in a spin glass model with large order parameter fluctuations. Journal de

Physique I, EDP Sciences, 1991, 1 (5), pp.659-667. �10.1051/jp1:1991160�. �jpa-00246360�

(2)

Classification

Physics

Abstracts

75.10N 75.40G 75.50L

Dynamics in

_a

spin glass model with large order parameter

fluctuations

A.

Jagannathan

Centre de

Physique Th60rique (*),

l3cole

Polytechnique,

91128 Palaiseau Cedex, France

(Received

^I October 1990, revised J7 December 1990,

accepted

30

January 1991)

Abs«act. A

dynamical

extension of the _mean

spherical

model for

spin glasses

is

presented.

Features of

nonselfaveraging

and

large

fluctuations present ⁱⁿ the

original

model _are _seen to result in the existence of _a spectrum ^of ^time ^scales ^of relaxation,

leading

to

non-exponential spin

correlations.

Spherical

models _may

arise, generally speaking,

in situations where the

system

under consideration _possesses certain local constraints that _are difficult to treat

exactly. Then,

_a

possible approximation

consists in

replacing

the set of local constraints

by

_a

single, global,

constraint. The _name

spherical

arises from the first _use of this

approximation

in

treating

the

Ising ferromagnet [I]

when Berlin and Kac solved the model

by replacing

the local

Ising spin

variables

by

continuous variables that satisfied _a

single global spherical

normalization. In other contexts, one has the slave boson

technique

in models for

strongly

correlated electron

systems,

^where ^the ^fixed ^local

(onsite) particle

number constraint is

replaced by

a

global

number conservation of

particles

over the entire lattice

[2].

More

recently,

_among models for

Heisenberg antiferromagnets utilizing

the

Schwinger

boson

representation,

the

spherical approximation

has been used to

approximate

the local boson

occupation

number constraint

[3].

^In all these _cases, it is assumed that the local variables will not be too much affected

by

the

admittedly

extreme

simplification

made

by

the

spherical approximation,

and that fluctuations remain in _some _sense small. This

assumption

_was _proven correct in _some of the _cases above

by

an

explicit

calculation of fluctuations. The interest of this paper lies in the _case where this

assumption

is _no

longer

true, and this _occurs in the _use of the

spherical

model for random systems.

The

present study

considers _a

possible dynamics

for _a

spin glass

system that is

subject

to the weak version of the

spherical length

constraint. The unusual fluctuation

properties

in the

model,

described

below,

are

expected

to

give

rise to unconventional

dynamical properties

in

spite

of the

simplicity

of the model.

Compared

with results obtained for

statics,

the results for its

dynamical properties

_are

necessarily

_on less firm

ground,

_as

they

_are model

dependent,

and

one can

certainly

conceive of _a number of

quite

different

dynamical

extensions of the

original

(*)

CNRS

Laboratory

UPR 14.

(3)

660 JOURNAL DE PHYSIQUE I N 5

model. Here _we

present

^what we

hope

to be _a

plausible, physically

^relevant but _not

rigorously justified, dynamics

for the _mean

spherical spin glass

^model. The motivation is

finally,

of

course, to

gain

_some

understanding

of the time

dependent properties

of random systems, which _are

typically nonselfaveraging,

have many

coexisting

time scales of

behavior,

and _are still

poorly

understood.

The

spherical

model of the infinite range

spin glass

_possesses _a

phase

transition

analogous

to that of Bose-Einstein condensation in _a gas of

noninteracting

Bose

particles.

The

spherical ferromagnet

in dimensions greater ^than two does _so _as

well,

about which _more is said later. In the _case of the bosons it has been shown that the choice of

ensemble,

canonical

(constraining

the

particle

number to be _a fixed

value)

_or

grand

canonical

(where

the constraint is

applied

to the average number of

particles)

does not affect the

thermodynamic

_averages but

gives

rise instead _to _very different fluctuation

properties

in the condensed

phase [4].

^A ^similar situation arises in the _case of the

spin

model. The

analogue

of

relaxing

the

particle

number constraint

is,

in the _case of the

spin model,

the relaxation of the strict

spherical spin length

constraint in favour of the _mean

spherical

constraint

(where

the constraint is satisfied

only

_on the

average).

The latter _case

corresponds

to

taking

the

grand

canonical ensemble for the Bose gas, where the average number of

particles

is

fixed, using

_a chemical

potential,

but fluctuations around this number _occur. In

general,

_one

expects

that these fluctuations _are

small, typically finite,

in the

thermodynamic limit,

_so that total number of

particles

is _a well defined

thermodynamic

observable. However this is not the _case in the ideal Bose gas, and its

spin analogue,

the

spherical

model. This

property

^of ^the

spin glass

is not

altogether

_a trivial

result,

and there are some subtleties involved _as will become evident further

below,

when _we discuss the effects of external fields and spontaneous symmetry

breaking

in the model.

In the _mean

spherical spin glass

^model

it

_has _been _shown _that _there _is

a nontrivial distribution of

possible

^values of the order

parameter,

^which ^is ^the

expectation

value of _a certain

appropriately

defined

spin

variable

[5].

In contrast to the

corresponding Ising

_or S~

models,

or indeed the strict

spherical model,

where the order parameter

density

is well defined its fluctuations

being typically

of order I

IN

where N is the system size in the _case of the _mean

spherical model,

the

possible

values _run _over _a wide range. Associated with each

possible

value of the order parameter ^and the

corresponding

value of the

system

^free energy, is _a

probability density.

The fact that this

probability density

has _a finite

width,

_even in the limit N

going

to

infinity,

leads to

nonselfaveraging

behavior and to the

possibility

of

defining

_an order parameter function such _as the function for the

spin glass

order

parameter Q

in

[5].

In contrast, ^when the

length

constraint is

strictly

enforced

(as

in [6] ^where ^the

spherical

model for the

spin glass

was first

considered, by

Kosterlitz et

al.)

fluctuations _are of order I

IN compared

with the average, and the order

parameter

^does ^have a well defined value in the

thermodynamic

limit.

Finally,

to address the

question

of nonrandom

models,

the

property

^of

large

fluctuations is shared

by

the uniform

ferromagnet,

which also

undergoes

a form of Bose

condensation,

_as noted in

[7]. Notably however,

in this _case the

application

of _an infinitesimal extemal field has the

property

^of

sharpening

the distribution of

magnetization

to _a

spike

of

vanishing

width in the

thermodynamic

limit. Hence for the

ferromagnet,

_one

_expects

that the order parameter is

a well defined

quantity, except

ⁱⁿ

strictly

_zero extemal field. In the

spin glass

_an infinitesimal external field does not have this effect and fluctuations remain

important.

It should be noted that the fluctuations in the _mean

spherical

model that lead to the definition of _an order parameter ^function ^have a different

interpretation

from those described

by

the Parisi solution for the

Ising spin glass [8, 9].

In the _case of the

spherical model,

the fluctuations _are

entirely

due to the

length

fluctuations

permitted by

the relaxation of the

spherical

constraint. If _one _were to hold the overall

spin length fixed,

_as _we remarked

before,

(4)

these fluctuations would be

insignificant. However,

the fact that there is _an

underlying disorder,

embodied in the random interactions between

spins

is

crucially important

for the existence of

nonselfaveraging

in the

spherical spin glass.

This is the _reason for the difference

in behavior from the

ferromagnet

of Berlin and Kac

[I],

which

(when

considered in the

presence of _an infinitesimal external

field,

which _can later be set to

zero)

does have the

selfaveraging

property even in the low

temperature phase.

Thus the _mean

spherical

model for the

spin glass

is of

particular

interest. It admits the definition of _an order

parameter

^function that resembles that of the

Ising spin glass,

but unlike the latter _owes its existence

only indirectly

to the disorder. One may

obviously object

that the

analysis

here may be far from

applicable

to any real system,

dealing

_as ^it does with _a

highly

artificial

(although _arguments

_may be made in its

favor) length

constraint _on the

spins, and,

in

addition,

considers the _mean field _or infinite _range limit

(although that,

_as

well,

_may be

argued

not to be unreasonable in real

systems). Nevertheless,

there may be valuable

insights

to be

gained given

the

interesting

structure described in this introduction. One great

advantage

of the model is that the fundamental distribution

underlying

the

physics

_can be

readily calculated,

and thus it

provides

_an

alternative, exactly soluble,

model of _a

system

^with

large

fluctuations and

nonselfaveraging

behavior.

And,

_as _we will

show, interesting

time

dependences

in several different

regimes,

such _as the stretched

exponential decay

at short times emerge in _a natural way.

we

begin

with _an outline of the

model,

and then present ^the ^main ^results obtained.

Model for relaxational

dynandcs

in the

Spherical spin glass.

The Hamiltonian of the

Spherical Spin glass

model is H=

-1/2 £ (jS, ),

_every

pair

(I, j

of spins

being

assumed to interact with _some fixed random

coupling

,,j

(~.

The

spherical length

constraint is that the continuous

spin

variables

($) satisfy

£ S)

=

N. In this infinite _range

model,

the

couplings

_are all taken from _a

single probability

distribution

P[(~]=l/fi~ exp(-N((/2J~).

The different realizations of the

N _x N matrix of

couplings J, corresponding

to different

samples,

then form _a Gaussian random matrix ensemble whose

eigenvalue spectrum

is known in the limit that N

- co

[10].

Written in

diagonal form,

this Hamiltonian is H=

-1/2 £AS(

where the are the

eigenvalues

of the matrix J. The _new

spin

modes

(S~)

_are related to the site variables

if by

_an

orthogonal

transformation

$

₌

£a;~ S~

^where ^the _a;~ are functions of the

A

random

couplings,

and _are thus time

independent

for _a

given sample.

The time evolution of this system, ^like its

thermodynamic properties,

is easiest treated in the

eigen

basis of J. The low temperature

phase

is characterized

by

_a

macroscopically large

amplitude

^of the

spin

mode

corresponding

to the

largest eigenvalue

A

= 2J. The critical

temperature corresponds

to that at which the

susceptibility

associated with this mode

diverges,

at T~ = J. In the _mean

spherical model,

it _can be shown that in _zero

field,

the order

parameter ^is

simply

the

expectation

value

(S( ~j) IN

= q =

(T~ T)/l~ (where

the

angular

brackets denote the thermal

average).

It is this

quantity

that has been demonstrated _to possess very

large

fluctuations in the static _mean

spherical

model. The strict

spherical model,

on the contrary, does not exhibit

anomalously large

fluctuations in the condensed

phase.

Assuming

a relaxational

dynamics

for the

system,

we

begin by considering

sufficient time to have

elapsed

that initial conditions have

decayed

_away

(the

time scale _on which this

happens

will be _seen

below).

In this _case the

equal

time correlations _assume their

stationary

values

(5)

given by

the effective Hanldltonian

H~~= £(2z- PA )S(

where _z is the _« chemical

A

potential» conjugate

to the total

spin length.

It is chosen such that the weak

spherical

constraint is

satisfied, namely

£ ($~(t)

=

N

(I)

;

The average denoted

by

the

angular

^brackets from this

point

on is to be taken _over the random noise

appearing

in the stochastic

spin equation

of motion written below

$(t)

=

ro)

+

1~;(t) (2)

with the Gaussian random noise

obeying (Y~;(t)Y~y(t')) =2ro3;y3(t-t'). ro

is _a

microscopic spin flip frequency.

The

simplest (site-,

_as well _as disorder- in tl~is

case) averaged spin

correlation function in this

asymptotic

time

translationally

invariant

regime

is

C(i)

=

( _£ _(s;(ii) _s,(ii

+

i))

=

£ c~ (i) (3)

where

Ci (t)

= N~

(Si (0) Si (t))

_are the correlation functions in the transformed variables.

In the

stationary regime,

the

equal

time

spin expectation

value in

equation (I)

becomes

C

(0)

=

£ Ci (0)

=

£ (2

_z

PA )~

=

l

(4)

~

using

the

equation

of motion

(2),

after

transforming

to the

diagonal

^basis ^of J.

Equation (4)

is

just

the

spherical

constraint relation obtained in the static

problem,

and is used to determine

the parameter z as a function of the

temperature

^T. ^The ^solution ^for z in the

high temperature phase

is

[6]

2 _z

= I +

p ~J~.

In the condensed

phase,

the solution is 2 _z

=

2

pJ+ N/q

where

q is the order

parameter

^defined

already.

The solution for q represents an average

value, by

which _we _mean here that the

spin expectation (fj)

_can _take _on

a wide _range of

values,

and its distribution is

given by

the Kac

density

function _v. The definition of this function and its

explicit

form _are

given

in the section _on the low temperature

phase.

A note on the time

dependence

of the correlations _: the functions

Ci (t)

in this model

decay

to _zero at

long

times _even in the condensed

phase,

because the

magnetization

remains _zero in the absence of any extemal field. This _can be _seen

by applying

_a uniform

field, solving

for the

magnetization subject

to the

constraint,

and

letting

the field tend to _zero

ill].

Thus _our calculation refers

only

to the condensed

phase

for the

spin

model that is

analogous

to the

superfluid phase

in the Bose gas. The relaxation times and the time

dependence

of the various modes will _now be considered in the various

temperature regimes

of interest.

Spin

correlations above the transition.

In the

high temperature phase (Tm J~,

_a

straightforward analysis yields

C

(t)

=

j

CA

(t)

=

& ^~

_dx

^fi(f _x)

e

~ ~~~°_~~ ~~

(5)

using

the

Wigner

semicircular distribution

ii 2]

in

doing

the _sum _over

eigenvalues (rescaled

to

(6)

lie in

[- I,

I in the second

step (I

=

z/pJ~.

For

sufficiently long

times 2

pJro

t »

I,

_one

finds for

example

at

temperatures

^much greater than the critical temperature ^that

C(t)

m

(4 gr)~

^~'~

(roJ/Tt)~~'~e~~°~ _(Recall

_that _t _is defined _as the difference in times of measurements of the

sfins

in the correlation function

equation (3),

whereas the time tj

elapsed

^since

letting

the

system

^relax

freely

is

always

assumed to be

greater

^than any of the characteristic relaxation times of the

system,

^which ^therefore ^is

independent

of the initial

conditions).

The onsite

spin

correlations thus

decay

with the

expected

rate of

ro

^for

temperatures

T» T~.

Close to the transition and

just

above it _we may

expand

in

=

(T- T~)/l~,

and for

sufficiently long

times it is found that

C(t)

m

$~(2 _{grro t)~}

_~'~ _e~~°~°~

(6)

which

gives

the time scale

goveming

fluctuations

r

(T)

= I

fro

^~ _near the transition. We _see that _as in the

Ising model,

_zv

=

2

(meaning

here

by

_z the

dynamical

critical

exponent).

If _one

assumes a mean field correlation

length exponent

as in the

Ising model,

this

yields

z ₌ 4 in accord with the

expected

Van Hove result for relaxational

dynamics

for

Ising-like

models

[13].

In

fact, equation (6)

for the

spin

correlation is the _same result _as found for the

Ising spin glass

model close to the AT line and

just

above it

(see

Ref.

[9],

p.

880].

At

l~ exactly,

in the

spherical

^model the correlations

decay

_as t~ ~'~ This is in accordance with results from _a

variety

of models with disordered

Hamiltonians,

such _as the random axis

model

[14],

the SK model

[15],

the random

anisotropy

model

[16]

and in numerical studies of the _± J short range

spin glass [17j.

Thus in the

paramagnetic

as well _as critical behavior the

spherical

model

reproduces

_some familiar results. Now _we turn to the low temperature

phase,

where it is necessary ^to consider _more

carefully

the various averages that _are _to be

performed.

Spin

correlations in the low

temperature phase.

When T

< T~ ^{it is} necessary ^to consider the contribution of the critical mode

separately

from that of the

remaining

modes. Above T~ ^the fluctuations in the _mean

spherical

model _are

negligibly

small and do not

play

_any role. However

they

do _so for T<

T~,

^where ^the fluctuations in

expectation

value of the condensed mode

6~j

_are

important.

In the low

temperature phase,

_we have _seen,

phases

that _are

macroscopically

different contribute to the

thermodynamic properties.

In the low

temperature phase,

the

equation

of motion for the condensed mode contains information

only

about the average relaxation _rate of correlations in

Si

2J, which is

ro(2

z 2

pJ).

The

equal

time

expectation

value

(S)j(t))

which is time

independent

is

given accordingly

to be

(2

_z 2

pJ)~

=

Nq.

This is the

expected

_average

value. However there _are

large

fluctuations in this

quantity.

The basic

quantity expressing

the

physics

of the _mean

spherical

model is the Kac

density

_v

(p T),

which

gives the weight

of the contribution of _an ensemble of

spins

with different

spin length normalization, namely

of _an ensemble that satisfies the condition

£ ⁱⁱ

=

pN.

The

thermodynamic properties

of the _mean

,

spherical

model _are

weighted

_averages with respect to the measure v, I.e.

Am(T~

=

dp v(p; T) Aj(T)

in which A ^~ is _a

thermodynamic

observable calculated in the _mean

spherical model,

for which the normalization is fixed _as in

equation (I),

while A~ is the _same observable evaluated

using

(7)

the

strictly

constrained

ensemble,

with _a value of _p not

necessarily equal

to

unity.

This is the Kac relation

[18]

relevant to the

spin problem.

We _now take the

point

of view that _v

represents

a

probability

of

finding

_a

given

canonical ensemble of

spin length density

_p _{# I} in the _mean

spherical

model

(where

_we have set p =

I).

We then

write,

in

effect,

_a

generalization

to time

dependent properties

of the Kac

relation _as follows

G

(t T)

=

dp

_v

(p

_;

T)

G

~(t T) (7)

where G

might _represent,

for

example,

the

spin

correlation function. The relation above would hold

provided

the system ^has ^attained ^the ^time translation invariant

asymptotic

_state,

and also

providing

that the time t is small

compared

with the characteristic times

r(p )

= I

fro (2 z(p )

2

pJ),

after which the various

p-ensembles

would

begin

to mix. Thus the time scales of observation for which the observations hold is

necessarily

short except rather close to the

transition,

where the characteristic times

diverge.

For

sufficiently

short times

then,

_we _argue, the evolution of the

spin

mode will be _a

weighted

_average of the

evolution of states of different

spin length density.

As these have _a

macroscopically differing

value of free energy

they

would not be

expected

to be connected

by spin

fluctuations for _very short times. At

longer times,

the energy differences may be

bridged,

_so that many ensembles

are

averaged

_over, and for times

longer

than the system

size,

_one would obtain behavior

averaged

_over all ensembles and thus described

by

the

single

_average time scale

(2

_z ²

pJ)~ '.

The

weight corresponding

to _a

spin

ensemble whose

length density

_» is _p is

[5]

)

_(p _po)

~~~

~

^~ ^~ ^~~ ^~^°~ ^~~~

(x) being

the theta

function,

and with po =

T/J.

It remains _now _to calculate the correlation function

C(j

for _a

given

value of

p in the

strictly

constrained model, and

integrate

with

respect

to the _measure in

equation (8).

The calculation of

C(j

is done _once

again

with reference to _a stochastic

equation

_as in

equation (2),

however this

time,

in

place

of the effective

Hamiltonian,

it is _necessary to substitute the

appropriate quantity.

This is done

by recognising

that the relaxation rate which is

simply given

in the _mean

spherical

model

by ro

3^~

H~~/36~

has _now _to be

computed

for the

case of the

strictly

constrained model. The relaxation time obtained

by looking

at the energy

change

due to fluctuations is _a factor N down from that obtained for the _mean

spherical

model. This follows from

calculating

the free energy

change

due to small fluctuations about

the _mean value

occurring

in the

strictly

constrained

model,

to get

r~'(p)cc

N(2 z(p

2

pJ),

_a

quantity

of order _one.

Using

this in the

Langevin equation

one finds

(t

<

O(N))

Cjj(t)

₌

(p

_p

o)

e~ ^~°^~'~~ ^~°~

(9)

The statistical average over ensembles is defined

by (temperature

T _< T~ ^understood ^fixed and for t _< O

(N))

c~j(t)

=

dp

_v

(p) cjj(t)

=

fi(2 _{qro t)3'4} _K~,~( _{ro t/2 q)} _(io)

= q

t/r~~

+ i e~

fi

(8)

where _an average time

r~~=q/2ro= @/2ro

_was introduced

(as

well _as the

identity K~,~(x)

=

fie~~

₁₊

). ^Thus,

^for ^times

^longer

^than ^the ^time ^scale r~~ this

yields

x

C~j

~

t~'~exp (-

~),

while for short times it is _a stretched

exponential

without the _power law

prefactor.

Note

again

that this is

expected

to hold for short times

(times typically

accessible to

experiment)

in _a time window which

expands

_as one

approaches

_{T~ from} below.

This _non

exponential

relaxation

essentially

due to the

averaging

of

dynamical

_response _over all states

permitted

thus arises in _a natural fashion in the _mean

spherical

model. At very

long times,

_t _cc

N,

there is _a _crossover _to the average

exponential decay given by

C~ fit)

_- ^e~ ^~° ^"'~°

(l I)

The full _response below T~ contains contributions from the other modes _as

well,

C(1)

=

C2J(t)

₊

Crest (t)

with

(for

_t _< O

(N))

~rest

_~~~

"

J« ) J)

~~~~

The inverse square ^root

dependence

_appears in the condensed

phase

in other systems such

as the random axis model

[14]

and the random

anisotropy

model

[16],

however in the present model this

dependence

holds

only _upto

times of the order of

N,

^after which the

decay

becomes

exponential

as in

equation (11).

To

reiterate,

the above low

temperature

formulae _are

expected

to hold

only

for times shorter than the time scale

expressed by

_r~~. As _we remarked

before,

in the

long

time limit

length

fluctuations would get

arbitrarily large.

Then _one would _use the average relaxation rate in

equation (2)

instead of

performing

an average over ensembles

evolving separately

_as in

equation (7).

Discussion.

In

conclusion,

_we have

presented

_one

possible

formulation for the short time

dynamical

response of the _mean

spherical model,

in _an extension of its known static

properties.

We Stress here that this

represents

a

possible

but

by

_no _means

unique

formulation of the

spin dynamics

^within the context of _a

spherical

constraint. It is based _on the

plausible assumption

that the

long

time

dynamics

differs from

dynamics

at short

time,

since the

energetic

cost

renders

large spin

fluctuations

improbable

at short times.

Here _we have considered _zero field correlations

only.

As _we mentioned

earlier, application

of _a finite uniform field does not result in any dramatic

change

in the

properties

of the

model,

as far _as the

large

fluctuations _are concemed.

We comment _now _on the

applicability

of these results to

experiments

_on the

dynamical

response of

spin glasses.

In

spite

of the

oversimplified

character of the

spherical approxi-

mation

itself,

it _can be

argued

that the model _as

presented

here allows for _some intrinsic features of

spin glasses

in _a

fairly

natural and tractable fashion. One feature is of _course the

non

selfaveraging property

^of the

spin

correlations in the

spin glass,

and the second

(related)

is the existence of _a

spectrum

^of time scales in the system. These

admittedly

have their

origin

in the _way that the constraints _are

imposed

in the

system,

but it _can be

hoped

that their

consequences share

something

in _common with the models that have been

proposed

for

random systems, ^and with the measurements made _on real

spin glasses

where it should be

JOURNAL DE PHYSIQUE T I, M 5, MAT 199> 28

(9)

recalled the

longest

times of observations _are

quite small,

and

only

the very shortest time scales of response are

explored.

^The correlations studied here

could,

for

example

be looked for in _zero field _muon

spin

relaxation

experiments,

which

probe

local

spin

correlations

(a

review of such

experiments

_can be found in

[19]).

In the above _we have not studied the

interesting dynamical

effects in

spin glasses

that _are due to so-called

waiting

time

dependence.

In this paper it is assumed that the initial state is

one in thermal

equilibrium.

Effects due to

switching

_on extemal fields and

studying

the relaxation of the

partially equilibrated system

^have ^been ^studied

by

several _groups

[20],

and

are not considered in this _paper. Here _we note

only

^that in

equilibrium,

_we

predict

_a _non

exponential decay

of the

spin

autocorrelations at short

times,

with _a

temperature dependent

time scale

given by

_r~~ _cc @.

Finally,

_a comment _on the

interpretation

of the spectrum ^of relaxation _processes embodied in

equation (10),

which is _a

superposition

of

exponential decays

with different characteristic

decay

times. There exist _a number of models for

glassy

relaxation which _assume the existence of _a distribution of relaxation times. These _are often taken to arise due to _a distribution of

energy barriers that the

system

^is

supposed

to

leap

_over. A hierarchical

dynamics

based _on activated processes is found for

example

in

[21].

Unlike these

models,

the relaxation times in this paper refer to characteristic times associated with small oscillations around the

multiple

allowed states of the _mean

spherical

model. Therefore there is _no

Vogel-Fulcher

law for the relaxation time r~~ in the

spin decay.

^The short time behavior _crosses _over

(the

model has

nothing

to say about the _crossover

region)

at very

long

times to _a

simple exponential decay

with _a

macroscopically large

time scale which is associated with

macroscopic

excursions in state space.

Acknowledgment.

I thank J. Rudnick for useful

discussions,

and the Centre de

Physique Thkorique

at Ecole

Polytechnique

for its kind

hospitality.

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