Dynamics of manifolds in random media II: long range correlations in the disordered medium

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correlations in the disordered medium

Harald Kinzelbach, Heinz Horner

To cite this version:

Harald Kinzelbach, Heinz Horner. Dynamics of manifolds in random media II: long range correla- tions in the disordered medium. Journal de Physique I, EDP Sciences, 1993, 3 (9), pp.1901-1919.

�10.1051/jp1:1993220�. �jpa-00246838�

Classification Physics Abstracts

05.40 61.40

Dynamics of manifolds in random media II:

long _range correlations in the disordered medium

Harald Kinzelbach and Heinz Homer

Institut fur Theoretiscl~e Pl~ysik, Philosophenweg 19, D-69120 Heidelberg, Germany

(Received

22 March 1993, accepted 19 May 1993)

Abstract In _a previous publication, _we investigated the dynamical behaviour of _a fluctuat- ing manifold in the framework of _a Hartree type approximation. We _now extend this calculation to the _case of _a medium with long _range correlations in disorder. In the static limit, _{we recover}

solutions known from

a replica calculation with hierarchical replica symmetry breaking. In the

dynamical formulation this replica symmetry

breaking

is related _to broken ergodicity _{on an} infinite hierarchy of diverging time scales. We give the phase diagram for the model and discuss the dynamical behaviour in the different regimes.

1 Introduction.

The behaviour of manifolds in random media is _a

topic

in statistical

physics

which has raised considerable interest in the last _years. It is related to _a great

variety

of

problems

from _very different fields in

physics. Applications

_range from the treatment of the

pinning

of _vortex lines in disordered

superconductors

_[I] to interfaces in bulk media and random field systems [2],

surface

growth

_[3],

randomly

stirred fluids [4].

Furthermore,

_one finds _a

striking similarity

to the

theory

of

selfinteracting polymers

_[5] and

spin glasses

_[6].

The

problem

is defined

by

_a static Hamilton function for _a manifold with intrinsic

rigidity, interacting

with _a disordered external

potential

[8],

HIP)

=

/ _d~s Ii

(lll~1~

⁺

^v(P(S)

^{S) +}

_I»Q(S)~ ^{h(S)Q(S) (ii)}

Generally,

is _an N component vector

field,

the dimension of the manifold is

given by

D.

The components of _are the N transversal coordinates of _a manifold embedded in _a N + D- dimensional _space.

The quantity

h(s)

is _some external field

coupled linearly

to the vector Q. Whenever needed,

we assume the

given

Hamilton function to be

regularized

at short distances

by

_some lattice

cutoff. The disordered medium is modeled

by

the

randomly

chosen

potential V(g, s).

In _a previous

publication

_[9],

quoted

as

(I)

in the

following,

_we discussed the

dynamical

behaviour of such _a manifold in the _case of _a medium with

short-range spatial

correlations in disorder. The

following

_{paper now} deals with the

complementary

_case of

long-range

correlated media.

The _papers extend the static results of Mdzard and Parisi [8] derived within _a

replica

cal- culation framework.

Following

their notation, _{we assume} the

potential

_V(Q>

s)

to be _a Gauss

distributed

quenched

random variable with _{mean zero} and correlations

V(Q>

s)V(g', s')

₌

-Nf

^~~

/~~~

&(s

s') (1.2)

The

N-dependence

is chosen such that the

large

N limit of the

theory

is well defined. For

large

values of the argument, ^{the function}

f(z)

is assumed to be

given by

_{a power}

law,

for small values it has to be

regularized.

We _{use an}

explicit expression

f(z)

₌

~~~[ ~~(ao

⁺

lzl)~~~ (l.3)

with

positive coupling strength

_g > 0 and _a

regularization length

scale

given by

_ao.

The model behaviour

depends

_on the value of the exponent _~. One finds two different classes

given by ~(l D/2)

_> 1 and

~(l D/2)

< 1. For

~(l D/2)

> 1 the correlations in disorder

are called

"short-ranged",

this _case is treated in

(I).

In the present paper, we shall discuss the

opposite

_case

~(l D/2)

_< 1 which models _a medium with

"long-range

correlations" in

disorder.

For dimensions D > 2, the whole

theory depends strongly

on the lattice cutoff

implicitly

assumed in

(I.I).

The calculation in this _case

basically

is the _{same as} for D _< 2, but becomes

more cumbersome. So for

simplicity

_we restrict the discussion to the _case D < 2 in the

following.

In their static

replica approach

to the

problem,

M4zard and Parisi _[8] used _a

quadratic

variational method to obtain _a formulation which is

equivalent

to _a selfconsistent Hartree

approximation.

It becomes exact in the limit of

large

number of components, N _{- cc.} In the

case p = 0

they

find _{a power} law behaviour for the transversal fluctuations _on

large length scales,

(lg(s) g(s'))~)

r- Is

s'l~~ (14)

where the exponent

(

is different for the two classes of disorder correlations.

To obtain stable

solutions,

in their

approach replica

_symmetry has to be broken. For

long

_range

correlations

(~(l

D

/2)

<

1)

_a full hierarchical

replica

symmetry

breaking

scheme turned out to be _necessary. The exponent

(

in this _case is found _{to be}

given by

(

₌

(2 D/2)/(1

+

~) (l.5)

In _a

dynamical

formulation of the

problem,

_one mimics the

coupling

^of the manifold to _an external heat bath. The equation of motion for the N components _pa

(s, t) (a

₌ I

N)

of the vector field

Q(s, t)

is

given by

_a

Langevin equation

°~ll>~~

⁼

-fl~lll~ll~

⁺

^{(a(s t) (16)}

with inverse temperature

fl

=

(kBT)~~

and _a Gaussian white noise

(~(s, t)

^with zero mean and

i(«(S>t)(m,(S'>t'))

= 2&(S

S')&(t t')&««, (1.7)

For

large

times, the field distribution

approaches

a static Boltzmann-Gibbs distribution

rw

exp(-flH).

If _{one starts} the

dynamics

at _a time to - -cc the system ^{is in}

equilibrium

_on all finite time scales.

In the

dynamical approach,

it is

possible

to average over disorder without

introducing repli-

cas. As discussed in

(I),

it is also

possible

to formulate _a selfconsistent Hartree approximation

equivalent

to the static

approach

_[8], which becomes _exact in the limit of

large

number of

components, ^N - cc, the so-called

"spherical

limit". In the context of

dynamics, replica

symmetry

breaking

shows _{up as a} violation of _a

fluctuation-dissipation-theorem (FDT)

and

the _occurrence of anomalous

long-time

contributions. As is well-known from the

dynamics

of

spin glasses (cf. [10]),

^such _a

breaking

of the FDT appears if the system ^becomes

non-ergodic.

In

t#

_case,

depending

_on the initial

conditions, only

parts of the

phase

_{space are} accessible

in/the thermodynamic

limit. Relaxation within these parts

("ergodic

components" remains

ergodic,

but transitions between the components can no

longer

_{occur on} finite time scales. If

one

regularizes

the

long

time

dynamics by solving

the

problem

for _a

large

but finite system, _one finds time scales which _are associated with transitions between those

regions

in

phase

_space.

These time scales

diverge

with system size and _are infinite in the

thermodynamic

limit. As

we shall _see, the static hierarchical

replica

symmetry

breaking corresponds

to the

assumption

of _a

hierarchy

of such

diverging

time scales

again quite

^similar to the

dynamics

in the

spin glass problem

_[10]. This

necessity

to introduce _an infinite

hierarchy

of

diverging

time scales is the main difference to the calculation in

(I).

It determines the static limit of the

theory

which

now is different from the _case of

short-ranged correlations,

whereas the treatment of the finite time behaviour is very similar.

The _paper is

organized

_as follows. Section 2 summarizes the

dynamical

formulation and the

corresponding

Hartree approximation. Section 3 discusses the

dynamical

behaviour in the

ergodic phase

and the

corresponding phase diagram.

Section 4 treats the

dynamics

of the

non-ergodic phase.

It introduces the

hierarchy

of

diverging

time scales which is _{necessary to} obtain _a

(marginally)

stable

solution, gives

the results for the behaviour _on finite time

scales,

and the solution _on the

diverging

scales. Section 5

finally

summarizes

briefly

the main results.

2.

Dynamical Hartree-approximation.

In _paper

(I),

_we discussed in detail how _one derives _a Hartree type

approximation

for the

dynamics given by

the

Langevin equation (1.6).

Since the whole derivation is

completely

identical for the

problem

to be discussed in the

following,

_we do not repeat the steps once

more, but refer the reader to

(I). Here,

_we

only give

_a short summary of the equations which

serve as

starting point

for the further discussion.

The

dynamical

behaviour of the system ^{is described}

by

_a difference correlation function which _measures fluctuations of the manifold in space and time.

Using appropriately

rescaled

variables,

it is

given by

Q(x x',

T

T')

_.=

( j _{~ (lg«(s,t)} g~(s',t'))~) ^(2.i)

°

~i

The brackets _{< > mean}

averaging

_over the stochastic

dynamics

_as well _{as over} the

quenched disorder-potential.

Note that

Q

is _a function of variables _x and _T, which _are rescaled _space and time coordinates

given by

T :=

t/t* (2.2)

x :=

s/s*

,

(2.3)

with time and

length

scales t* and

s*,

^defined as

fit*

_"

(g~i~~~)~~/l~~~~~~~/~~~ (2.4)

s* _.=

(fit*)~/~ (2.5)

where _g and _{~ are} the parameters ^{of the disorder} correlation function

(1.3).

For the variable1Y _we find

~y .= ~~

g(i-D/2) /(1-~(i-D/2)) ~(2-D/2)/(1-~(i-D/2))

_{~~ ~~}

ao

being

the

regularization length

scale of the disorder correlation function

(1.3).

The rescaled counterpart ^{for the}

coupling

constant p of the external

quadratic potential

is called _po in the

following.

It is

given by

po .=

fit*p (2.7)

Correlation and response function of the field

Q(s, t)

_are defined

analogously,

~ ^N

C(x x',

_T

T')

_.=

j p L _{iomis, t)} gals', t')) (2.8)

Rix x',

_T

T')

_.=

())

^{~~'~ ~~~~}

j jj _)[j)lj ⁾⁾ _12.9)

~~

The

dynamic Hartree-approximation

discussed in

(I)

is _a selfconsistent

approximation

whicl~

becomes exact in the limit of

large

number of field components, ^N - cc. It is _a

dynamical equivalent

to the static variational

replica approach by

M4zard and Parisi [8].

They

discuss the static solution in the limit _{po -} 0

and,

in the _case of

long-range

correlated

disorder,

_{ao -}

0,

which _{means 1Y -} 0 in _our notation

(-

but 1Ylao finite of

course).

In

(I),

_we derived the

equations

of motion for the _response and correlation function

using

this

approximation.

We define _a Fourier transform with respect to the space variable _x,

R(x,T)

_=:

r(k,T)e~~~~

C(x,T)

_=:

c(k,T)e~~~~

,

(2.10)

where the

integral f~.

stands _{as an} abbreviation for

f@.

The

equation

of motion for

c(k, T)

_assumes a more symmetric

shape

if _one introduces _a function

q(k, T),

g(k, T)

_It 2

(C(k, T#0) C(k, T)) (2.ll)

(Note

that

q(k, T)

is not the Fourier transform of

Q(x, T),

but of

Q(x, T) Q(x, 0).

The

special

case

Q(0, T)

is

given by f~ q(k, T).)

For _{T >} 0 the equations ^{of motion then read}

jar

₊ k2 ₊

p(o)j r(k, T) l'dT' ^{~(T T')r(k,T')}

= o

(2.12)

and

ja,

₊

^k2

₊

p(o) j q(k, ~) l'd~' ^>(~ ~')q(k, ~')

m 2

2jI(k, ~) I(k, o) j

,

(2.13)

where

I(k, T)

stands for

1(k, ~)

_:m

f"d~' (w(~

⁺

~')r(k, ~') >(~

₊

~i) q(k, ~i)) ^(2,14)

As discussed in

(I), c(k, T=0)

in _{turn can} be

expressed by

^the condition

c(k,

_T

=

0)

₌

~~ ^{l +}

I(k, 0)

,

(2.15)

+ po

so no information is lost

using q(k, T)

^instead of

c(k, _T).

The functions

~(T)

and

w(T)

result from the disorder

averaging, they

_are

given by

W(T)

_{= (lY +}

q(T))~~ (2.16)

1(T)

= 2~ _(lY +

q(T))~~~+~~ r(T) (2.17)

with

r(T)

and

q(T)

^defined as

r(T)

₌

/ _{r(k>T) (2.18)}

q(T)

=

/ _{q(k,T) (2.19)}

The constant

p(0)

is the _T

= 0-value of _a function

»(T)

_{.= »o} +

/~dT' ^{I(T') (2.20)}

3.

Dynamics

in the

ergodic phase:

the

FDT-regime.

For

ergodic

systems, one

generally

expects ^the response and correlation function to be related

by

_a

fluctuation-dissipation-theorem (FDT).

For the

given dynamics

it reads

8,q(k, T)

₌

2r(k, T)

for t 2 0

(3.1)

and induces _a similar connection between

w(T)

and

~(T),

8,w(T)

₌

-~(T)

for t 2 0

(3.2)

If the FDT is

valid,

the

integral

expression

I(k, _T) I(k, 0)

in the

equation

of motion for

q(k, T), (2.13),

vanishes and the

equations

^for _r and _q,

(2.12)

and

(2.13)

become

equivalent.

The derivation of the solution for the FDT

regime

is

completely analogous

to the _one

given

in

(I).

So

again

_we

only

summarize it and discuss the results.

one _assumes the FDT to be valid and looks for selfconsistent solutions of the

problem

in this

regime. Following

the _same steps as in

(I),

_we find _a

long

time limit go

(k)

of the form

~~~'~~

~ ~°~~~

po k2 ~°~ _{~ ~ "} ~~'~~

and

correspondingly

^for _go

go =

/ _qo(k)

= 2CD

»i~~~~~~~ (3.4)

k

with

CD ~=

(4x)~~/~ r(I D/2) (3.5)

A stable FDT solution has to fulfill the condition

~c(q) ~c(qo) ^{> 0} for 0 < q < go

(3.6)

where the function _K is

given by

_1/,1-D/2)

(3.7)

lG(g) ~"

())

~~ ~ ~~ ^~

Figure

I shows the

typical

behaviour of ~c(q) ~c(qo) for a

sufficiently

small value of _po. For values of1Y smaller than _a critical

value1Yc(po)

the function has

negative portions

between 0 and go

violating (3.6),

in this

regime

the FDT-solution is unstable.

K(q) K(q~)

'

' 2* 0

' ' C

0.02 ' o =0

~

^:.'

'_ ^~ ;"'

,,

0.5 * 0 ;"'_-

"'1'

0.01 ^~ ["'

0.25*0 ^"

~

~ '

0

,~ ,,

0.01

.,

'"'---

-.-

"'(:"

0 =o ;.'

0.02 ', ,:"

0.03

0A 0.6 0.8 1.2 q'q~

Fig. 1. Typical behaviour of K(q) K(qo) for different values of d.

Approaching1Yc(po)

from

above,

the function

develops

_a local minimum _qm. In the

marginal

case d

=1Yc(po)

this minimum reaches _go the

corresponding

value of K(Qm) K(qo) vanishes,

so the

instability point

of the FDT-solution is determined

by

the

equation

tt'iqo)l~=p~

= 0

13.8)

It _can be solved

explicitly for1Yc(po),

1 1- 2

~~ _~~ ~ _~~

~j~

D

/2))~/~+~~

_l~o^{~~ ~~~~~~~~} _{~~~ ~°} ^~~

~~ ~~

For _/1o

larger

than _a maximal value

/1?~~,

^the FDT-solution remains stable for all values of1Y It is

given by

J1?~~ = 12 CD)~~~~~~~~~~~~~~~

l~li D/2)ji/<1-~<i-D/2)) j3.io) Figure

2 shows the

corresponding phase diagram

for two different values of _~. The

nonergodic phase

is reached for

appropriately

^{small values} of1Y and _/1o

Expressed

in the

original

variables

fl,

_g, and _/1, it _means

sufficiently

low temperatures, strong

coupling

_g and weak external

potential coupling

_/1. In the _{case /1o}

- 0 the value 1Yc(/1o)

diverges,

_so in this _case the FDT- solution is unstable for _{any 1Y.}

°c

(~o)

(D=i)

1.6 y= ^1.5 7= 0.5

~

l.2

ergodic

0.8 ',

,',

0.4 ,,

nonergodic

,

o

0 0.05 0.1 0.15 0.2 0.25 0.3 ~~

Fig. 2. Phase diagram: dc(~l0) _versus the coupling strength _~0 of the external quadratic potential

for ~f = 0.5 and _~f

= I-S-

In

figure

3 the numerical solution for the correlation function

q(T)

is

given.

The behaviour

one finds is

quite

similar to the _one described in

(I)

in the

vicinity

of the second order transition line: for

sufficiently large

values of1Y _one finds the

(trivial)

diffusive _power law

(with

_exponent

(I D/2),

"free

behaviour")

for small times, and _{a crossover} to an

exponential

approach to go for

large

times. For smaller _{1Y, an} intermediate power law

regime

with nontrivial exponent

emerges.

Lowering1Y futher,

the

corresponding

time scale of this

regime

becomes

larger,

at the critical

point

_{1Y =} 1Yc(/1o), ^it

diverges.

The

exponential

tail then

vanishes,

_{a pure} nontrivial

power law

approach

to go remains.

If _we choose

1Y

=1Yc(po) (I

+

e)

with 0 < e « 1

(3.ll)

the calculation

yields

_a behaviour

~~~~ ~

Iii

^{~ ~}

^~~~~~~ _[al ⁱ _]11 ~~'~~)

q(~)

7*1.5, WI

__,...=

~'°

~~ = 0.1 :"

2.5 ,"'

2.0

1.5

o% ₌ i-o

i-o

~o%~=

_i-i

""""'o%~

= i.5

o.5 ^~

o-o

io~ io-2 io° io2 1o4 io6 _~

Fig. 3. Numerical results for the correlation function

q(r)

at different values of d. For

d/dc

# 1

one reaches the transition line in the phase diagram.

with _some constant _c > 0. For _T » _Ti the

approach

to go becomes

exponential.

The intrinsic

time scale _Ti

diverges

when the critical

point

is

reached,

Ti r~

e~~/~ (3,13)

The critical exponent u > 0 is determined

by

the

equation I~(I

_U)~

r(1 2u)

^{" P ,}

(3.14)

where p is

given by

~ ~ i ~~ -i/<i-D/2) _~

~

~(2

D

/2)

_2cD

~~~~°~

~

°~

~~'~~~

Using (3A)

and

(3.9)

_one

finally

_gets

p =

~ +

j2

_~~

~(i

D

/2)j~/h+i) ~ji-~(i-D/2)j/h+1)

~~ ~~~

~(2

D

/2)

^°

The solution for _u

depends

_{on po.} Since _po and 1Yc(po) are related

uniquely,

_{one can express}

u as a function

of1Yc(po). Figure

4

gives

this

dependence

for two

special

_{cases, ~}

=

3/2

and

~ =

l/2 (both

^for D

=

1).

4.

Dynamics below1Yc:

broken

ergodicity.

For1Y < _1Yc the solution discussed above is

unstable,

the

dynamics

has _to violate the FDT.

As is well-known from the

theory

of spin

glasses

_[6,

10],

such solutions _appear, if the system becomes

non-ergodic:

^the

phase

_space

splits

into _a

large

number of

dynamically

closed

ergodic

v

wo.5

0.4

-'~ y~15

0.3

,~

(D=1)

0.2

0 2 3 0

Fig. 4. Critical exponent _u for _~

= 0.5 and _~

= l-S- For d > dc the exponents are constant and

given by the values at d ₌ dc. For d < dc they depend _on d.

components. Within eacl~ of these components relaxation is

ergodic,

it respects the

FDT,

but transitions between the components are no

longer possible

on finite time scales.

As discussed in the last section, _one finds _a time scale _Ti in the

ergodic phase,

which

diverges,

when 1Y

approaches1Yc.

The

divergence signals

that the

phase

_{space start to}

split

_up into closed components.

Ergodicity-breaking

^of course is

only possible

in the

thermodynamic

limit. In _a

large

but finite system, ^all time scales also remain finite. But below1Yc _{one expects} to find time scales which increase with system size and

again diverge

in the infinite system ^limit. In this limit the nontrivial contributions from those

divergent

time scales violate the FDT.

They modify

the

dynamics

^for finite times such that

(marginally)

stable solutions _are

possible.

In the _case of

short-range

correlated disorder discussed in

(I)

_we had to introduce _one

single diverging

time scale _Ti in the

nonergodic phase

below _1Yc and _a

generalization

of the FDT for times _T » _Ti. Let _us call this scenario _a "one step

ergodicity breaking".

In the

corresponding

static

replica

calculation [8],

replica

_symmetry had to be broken in _one

single

step.

The situation becomes different in the _case of

long-range

correlations in disorder. If _one repeats ^{the "one} step

ergodicity breaking"

calculation of

(I)

for this _{case, one} finds that the solution indeed becomes

(marginally)

stable _on finite time scales but _on the

diverging

scale it still violates the

corresponding stability

condition. So here it is not sufficient to _assume the

existence of _one such

diverging

time scale.

Again

the situation is similar to the static

replica

calculation: in the _case of

long-range

correlated disorder the

replica

symmetry ^has to be broken in

infinitely

_many hierarchical steps.

In

dynamics

_we have to introduce _a

hierarchy

of

diverging

time scales _Ti < T2 < < TM with M _{- cc} and

Tk/Tk+1

_- 0 and the

appropriate generalization

of the FDT for each hierarchical level Tj ^< T < Tj+I This scenario is

quite

similar _to the

Sompolinsky

solution in the

theory

of

spin-glass dynamics

_[10]. Let _us call it "l~ierarchicaf' _or "full

ergodicity brealdnl'.

The

dynamics

_on

finite

_{time scales}

T < Ti

samples regions

in

phase

_space which form the

single ergodic

components in the

thermodynamic

limit. For times _T » _Ti there _are transitions between those

regions

whicl~ have _a

hierarcl~ically organized

structure.

To

investigate

the situation in

detail,

_now let _{us assume} the time scales _{Ti < T2 < < TM to} be finite for _a moment.

According

to the discussion

above,

this

regularization

_can be achieved

if the system is

large

but finite. In the next sections _{we are}

going

to derive

equations

for the

dynamics

in the different time

regimes. Eventually,

the

thermodynamic

limit is taken

again.

4. I DYNAMICS _{oN FINITE TIME} scALEs, T < Ti The time

regime

T < Ti is characterized

by

_a relaxation within the

ergodic

components. In this _{case, one} expects ^{the FDT} to hold.

The discussion of the behaviour _on these finite time scales

again

is

completely

identical to the

one

given

in

(I).

In the

non-ergodic phase,

the correlation function

approaches

_a

stationary

value for times

T '~ Tl ~ CC

~~~~~

(Ti~+

k2 ~~'~~

and

qi =

/ _qi(k)

= 2c~

»(Ti)-<i-D/2)

,

(4.2)

k

with

p(Ti) given by

~

jL(Tl " /L0 +

/

_dT~

~(T~)

(4.3)

n

for _{Ti - cc.} As _a consequence of broken

ergodicity, f~dT' ~(T')

does not vanish in the

thermodynamic

limit

(where

_{Ti -}

cc).

As discussed

above,

this nontrivial contribution is necessary ^to render the

dynamics (marginally)

stable _on finite time scales. The value of

p(Ti)

has to be determined

selfconsistently

^from the

dynamics

_on time scales _T » _Ti As in

(I),

the

assumption

that _one has

non-vanishing

contributions _to the _response function _on the

diverging

time scale _Ti leads to the condition

~G'(qi) = o

,

(4.4)

where ~c(q) ^is defined

by (3.7). Equation (4.4)

determines _qi

(-

and

p(Ti)

via

(4.3)) uniquely

for

any1Y

<1Y~. We shall rederive this condition in the next section in _{a more}

general

context.

Again

identical to the result in

(I), q(T) approaches

its

stationary

value _qi

algebraically,

g(T)

_{= gl} ^CT~~

(4.5)

with _some constant c > 0 and _an exponent u > 0 determined

by

-1/(1-D/2)

~

r(i u)~

(4.6)

~

~j

~~~

())

~~' ^~ ~~~

~(~

_~~~

The exponent u

depends

_{on 1Y,} its values in the _{cases ~}

=

l/2

and _~

=

3/2

_can be found in

figure

^4.

4.2 DYNAMICS ON DIVERGING TIME SCALES. As discussed above, _{we now} introduce

time-scales which

diverge

with system size. In the time domain

given by

these

scales,

one

has transitions between the

regions

in

phase

_space which form

ergodic

components in the

thermodynamic

limit. Within this

section,

let _us first _{assume to} have _a finite number M of such

diverging

time

scales,

_Ti < T2 < < TM-

Furthermore,

the scales should be well

separated

in the

thermodynamic limit,

_{we assume}

~

- 0 for

j

= I. M 1

(4.7)

Tj+1

Eacl~ time domain Tj ^{< T <} Tj+i defines _one level of

hierarchy.

The

dynamics

at this level is

governed by

the time scale Tj. We _{assume a}

scaling

form

and

analogous expressions

for _w and

~,

WIT)

_{" WJ}

(T/Tl)

,

~(T)

_#

T~

~j(T/Tj

_{for Tj} « _T < Tj+1

(4.9)

To derive the

equations

of motion, it is convenient to introduce _some short hand notation first.

We write

qJ

(k)

_{~" qJ (k>TJ)}

qj ."

/ _qj(k,Tj)

k

Wj .= W(Tj =

(d

_{+ gj}

)~~ (4.10)

Starting

from the

original

equations for

r(k,T)

and

q(k,T), (2.12)

and

(2.13),

it is easy ^to derive the

corresponding equations

valid for times Tj ^< T < Tj+I

Analogous

to the situation in the

short-range

disorder _case in

(I),

_one finds that these

equations

allow for solutions for the correlation and _response function which _are related

by

the

simple

extension of the FDT

termed

"quasi-FDT" (QFDT)

there. But here _now the parameter m can be different for each hierarchical level. So here _one has

mj

8,q(k, T)

₌ ²

r(k, _T)

_{for Tj} < _T < Tj+i

j

₌ 1. M

(4.ll)

The free parameters _mj have to be determined

selfconsistently.

The

QFDT implies

_a similar relation between I and w, lllj

8,W(T)

=

-~(T)

_{for Tj} « T < Tj+I, ) " I. M

(4.12)

One derives the

following equations

valid for times Tj ^« T < Tj+i>

j

_" I.

M, (k2

+

p(T~))f~(k,1t~) /

u,

dy Ij(1t~ y) f~(k, y)

1

j-I

=

jIJ(~IJ)

qi(k)

+

L

mu

(qu+i(k) u(k)))

(4.13)

u=1

and

lk~

+

»(Tj)) «j(k>~tj) /

u,

_{dy Ii~tj y) «j(k,v)}

=

I J-1

" 2

L _(lllU-1

_lllU)

_{ilbJ(lLj) '°Ul} qulk) 14.14)

u=1

with

ltj .#

T/Tj (4.15)

It is easy ^to check that the equations indeed _are

equivalent

if the

QFDT (4.11)

is fulfilled.

Both

equations

are valid for

j

= 1...

M,

for

j

₌ 1 the _sum

drops

out. In the

equation

for

qj

one

formally

has to set _{mo i=} I.

Taking

M

= I, _{one recovers} ^the

expressions

for the

single

step

ergodicity breaking

discussed in

(I).

An

integration

_over the

QFDT (4.ll)

for

j

= I. M I leads _to the relation

p(Tj+i) P(Tj)

_{= lily}

(Wj+1 Wj)

for

j

= I. M

(4.16)

For the

highest

time

hierarchy

_T » _{TM one} has the relation

P(T

_-

cc)

_{= po}

(4.17)

In this _case the

QFDT yields

PO

P(TM)

_{" lllM} (Wco WM)

,

(4.18)

where _we defined

Wco # llm W(T)

r-m

= (1Y +

qco)~~ (4.19)

and

q« .= iirn

q(~) (4.20)

Again

the limit _means that _one has _{to start} with _some time _T » TM,

keeping

_TM fixed.

The

equations (4.13)

and

(4.14)

describe the

dynamical

behaviour for times Tj ^< T < Tj+i,

j

₌ I...

M,

_or,

equivalently,

I < iJj ^<

Tj+i/Tj

_{- cc.}

The continuation of the solutions to the _cross-over

regions

_{ltj -} I and ltj - cc has to match with the solutions of the next lower _or

higher

level of the

hierarchy respectively. Especially,

one has

qj(k,ltj

-

I)

rw

qj(k)

and

qj(k,ltj

_-

cc)

rw

qj+i(k).

Starting

^from

equation (4.14),

the limit ltj - I

(j

= I.

M)

leads to _a set of conditions for the distributions

qj(k),

(k~

+

P(Tj)) qj(k)

₌

j-1

= 2

£ _{(m~-i m~) (wj w~) q~(k)}

_for

_j

= I. M

,

(4.21)

u=I

again

with _mo

." I. For

j

= I _one gets

expression (4.I)

back.

Performing

the limit ltj - cc is

technically

_more

complicated,

but

again

_one derives the _same set of

equations

for

qj(k), j

= I. M.

Using

the

QFDT-result (4.16), equation (4.21)

after _some

lengthy

but

elementary manipu-

lations is transformed into

~J+i(k) ~J(k)

₌

I

l~~

₊

^i~~+~~

~~

li~~~

for

J

= i M 1

(4.22) and, correspondingly

*~~ _~~ " ~~~

)

(~(fi+~)~~~~~~~~ P(TJ)~'~~~~~~

^for

j

₌ I. M 1

~~ ~~~

The

highest

level of the time

hierarchy,

_T » TM,

again

has to be treated

separately.

The relation

analogous

to

(4.22)

here reads

-(i-D/2)

)-~~~~~~~

_{~~ ~~~}

~~ q~ = 2cD

~

^{l~° ~ ~~}

with _{qco as} defined in

(4.20).

Finally,

from the

equations

^for

fj(k,ltj), (4.13),

in the limit ltj - I _one gets

r(k> Tj)

_{= ~ (lY} +

qj)~~~~~~

qi(k)

+

I

mu ~~

~~~~ (qu+i(k) q~(k))

,

~ ~ ~

(4.25)

valid for

j

₌ I. M. For

j

₌ I

again

the _sum

drops

out.

With

(4.22) inserted,

_an

integration

_over k

yields

_a set of equations ^for

r(Tj),

which has _a

nonvanishing

solution

only

if

2cD

(1 D/2)

_~

(1Y +

qj)~~~+~~ p(Tj)~'~~~/~~

₌ l for

j

₌ I. M

(4.26)

These relations _are the conditions that _one has anomalous

long-time

contributions to the response function _on

diverging

time scales. Note that the _case

j

= I is identical _to the condition

~c'(qi)

₌ 0,

(4.4),

used in the last section _to fix the value of _qi

4.3 THE _LIMIT M _{- cc: FULL} HIERARCHICAL ERGODICITY BREAKING. The equations

derived in the last section determine the

quantities p(Tj),

_qj, and m~ for j = I.. M. A _more detailed

investigation

^shows that

breaking

the

ergodicity

_on M _{< cc} hierarchical levels is still not sufficient to get a stable solution for the

dynamics.

One

additionally

has to construct the limit M _{- cc.}

To this

end,

first let _us introduce _a

(discrete)

parameter

(

E

[0,1]

to

parametrize

the M time scales _{Ti T2}

, TM

by

_a function

T((). By

construction, _we choose the M discrete values of

( equally spaced

in the unit

interval,

(E I,1-~,l-),. ,~,0 _(4.27)

~l ~l ~l

with distance

At

=

j

and define

The

quantities p(T~),

_{q~, w~} as well _as the

QFDT-parameter

_mj valid for the time level T~ <

T ~ Tj+i &re Wrltten &S fUnctlons Of

f,

P(T(f))

_=.

A(f)

qj =

q(Tj)

=.

lit)

wj =

wiTj)

_=:

iii)

mj =:

yin(a (4.28)

For M _- cc,

(

becomes _a continous variable, differences between

quantities

_on

adjacent

hierarchical levels _{now are} written _as derivatives with respect to

(.

Equations

(4.16)

and

(4.26)

then read

fi'lf)

"

ill) *'lf) _14.29)

and

Alf)~~~~~~~

" 2CD _~

ll D/2)

16'J +

4(f))~~~~~~

= 2cD _~

(l D/2) 4(()~~+~~/~ (4.30)

For

equation (4.23),

in this limit _one gets

With

(4.30)

it

turns

out

that

independent conditions

(4.29)

_and (4.30) survive.

~(2

D

/2)

d(f)

₌ Bi

'(f) ^~(~ ~/~) _(4.32)

~ + i

fi(f)

= 82

'(f) ~(~ ~/~~ _(4.33)

-(2 D/2) j~j~

₌

B~ j~ j~1 ~(i

D

/2)

_~

~~~~~

with constants

Bi>

82> and 83 which

depend

_on D and _~.

The relation between _qi and

p(Ti)> (4.2),

_now is written _as

#(1)

= 2cD

#(1)~~

^~/~~

(4.35)

Since _qi

"

#(I)

is

already

fixed

by (4A), equation (4.33)

for

(

= I determines the value of

1h(1),

~ l

~(l D/2)

j(1)

₌

j2~~

_~

(i

D

/2) j

^{~ + l ~ + l ql}

^(~

⁺

l)(1 D/2)

(4,36)

~(2 D/2)

2cD

The behaviour _at the

highest

level of time

hierarchy

has to be treated

separately.

In the continous _case the relations

corresponding

to

(4.18)

and

(4.24)

read

»o

#(0)

₌

*(0)

_{(lY +}

q«)~~ d(0) (4.37)

and

qco =

410)

+ 2CD

w I»o-~~-~/~~ A101'~~/~~ (4.38)

Taken

together,

these

equations

fix the value of

fli(o): starting

with

(4.37),

_one

replaces ji(o),

d(o),

and

#(0) by

the

expressions given

in

(4.32), (4.33),

and

(4.34)

and inserts the

expression

for _qc~

jven by (4.38)

into

(4.37).

It leads to _an

implicit

equation ^for

fli(0)

which _can be solved

exactly by

_a

legthy

but

elementary

calculation. We find

1-

~(l D/2)

lit(°)

_"

CXPO ~ ~

(4.39)

with

~

o := 2 _cD

(I

D

/2)

^{~ ~}

~~

^{+ l}

(~

+

l) (2

D

/2)~~ (4.40)

In the

nonergodic phase,

_we have _po <

p?~~ (with p?~~ given by (3.10)),

_so

(4.39) implies fli(0)

< 1. For po - 0 _one also gets

fli(0)

_- 0.

Equations (4.36)

and

(4.39) explicitly

fix the

endpoints (

₌ I and

(

₌ 0 of the function

fli(() (- and, by (4.32)

to

(4.34),

^of the

remaining

quantites ^of

interest).

The behaviour in between those

endpoints

is at least to _some extent

arbitrary:

the

equations

which determine this behaviour show _a remarkable invariance under

reparametrization

which is called

"galtge

invariance ^" in the context of

spin glasses.

It is _a consequence of the fact that the choice of the parameter

(

introduced above is

completely arbitrary

_as

long

_as the relation between the

original

time scales _{Ti>T2, TM} and the function

T(()

is unique. As _a

result,

the equations derived in this section _are invariant if _one

replaces ( by

a monotonous function

((J~)

of _{a new}

parameter _J~ E [0,J~c] ^with some J~c > 0. Let _us choose

f(°)

" 0 and

j(ilc)

=

(4.41)

The

quantities

4,

ji,

4 etc. now become functions of this _new parameter _J~, we write

#(f(~))

=:

#(~) >

(4.42)

and nalogously for

the remaining

_{functions of} (.

Since 4(()

is

monotonously decreasing,

j(~)

₌

^m(0)

+

(m(I)

_m~°~) ^~

^(4.43)

llc

where

m'°)

_and

m'~)

_{stand for the}

endpoints

of

4(() given by (4.36)

^and

(4.39),

I-e-

m'°)

.=

4(0)

=

fit(0) m'~)

.=

Ail)

= fit(J~c)

(4.44)

As _we shall _see in the next

section,

this

special "gauge" corresponds

to the static

replica- symmetry-breaking

scheme M4zard and Parisi [8] used. Via

(4.33),

it also leads to

explicit

expressions for j1(J~),

~

ji(J/)

₌ 82

m~°)

+

(m'~) m'°))

^{~ ~ ~ ~ ~}

(4.45)

For po - 0 _one has

m(°)

- 0, _so in this _{case a pure power} law behaviour remains, j1(J/) _rw i1~~~~~/~~~~~~~~/~~~

4.4 STATIC SOLUTION. In section 2 _we introduced the

dynamical

difference correlation function

Q(x x',

_T

T').

If _one takes _T

= T'

, one recovers the static correlation function. _It

is

given by

Q(x x', ₀₎

=

/ _(1- e~~~"~~'~ c(k, 0) (4.46)

k

where

c(k, 0)

can be

expressed using

condition

(2.15).

M4zard and Parisi discussed the static difference correlation function in their

replica

calculation for the

special

_{case po -} 0.

Furthermore, they

set the

regularization length

for the correlations at short distances to zero, ao = 0 in

(1.3),

which is

possible

in this _case. It

corresponds

to the

choice

1Y = 0 in _our calculation. From the results derived

above,

_{we get} the

following

results in the static limit.

(I)

Cased

>1Yc.

For1Y > 1Yc(po), ^{the FDT holds.}

Equation (2.15)

then

yields

the _same result _as in the _case of

short-range

correlated

disorder,

c(k, 0)

₌

(i

^{~ wo}

k~ _{+ Po} k2 _{+ p~}

(4.47)

with _{wo 1"}

(1Y +

qo)~~

and _{go as} in

(3A).

(ii )

Case1Y _{< 1Yc.} In the case1Y < 1Y~(po)> there _are nontrivial contributions from the

hierarchy

of

diverging

time scales introduced in section 4.2. With the assumptions

given

there

we find for

I(k, 0), (2.14),

M-1

2

1(k, 0)

# Wl gi

(k)

+ Wco

gco(k)

_WM

gM(k)

+

~ _lllu(Wu+I

gu+1 Wu

gu)

,

u=0

(4.48)

which after _some steps,

using

the results

given

in

(4.16)

and

(4.22),

_can be written _as

21(k,0)

=

(p(Ti) Po)

l

qi(k)

₊

/~"

~mi

_{k +}

po

i»(Tu+i) »oi llll~~l~

~~ ₊

~~~+i~ _(4.49)

Constructing

the limit M _{- cc as} discussed in

4.3,

_we

finally

_get the

expression

1 I _wc~>

I(k, 0)

=

(ill) Po) j

^-1 _{~2 ~}

_jji)

^~ k2 _{+ ~~} ^~

~

~~~ (i)~ ^~~~( _J(~

~~ ~~~

Again

it is obvious that this result is invariant under the type ^of

reparametrization ("gauge

transformation" discussed in the last section. We _{now use} the

"M4zard-Parisi-gauge"

defined in

(4.43)

and extend the

quantity

_j1(J~) to the

region

J~ > _J~c

by

the definition j~~~

by j4_45)

if

J~ < llc

(4.51)

p(J/)

_:=

)~))

=

~n(t.

~~ ~ ~ _~~