Charge density waves and the replica method

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Charge density waves and the replica method

Giorgio Parisi

To cite this version:

Giorgio Parisi. Charge density waves and the replica method. Journal de Physique I, EDP Sciences,

1993, 3 (2), pp.579-584. �10.1051/jp1:1993151�. �jpa-00246742�

Classification

Physics

Abstracts

02.50 67.40

Charge density

_waves

and the replica method Giorgio

Parisi

Dipartimento

di

Fisica,

II Universitk di

Roma,

Tot

Vergata,

and

INFN,

Sezione di Roma, Tor

Vergata,

Via della Ricerca

Scientifica,

Roma 00133,

Italy

(Received

3

July

1992,

accepted

in final form 26

September1992)

Abstract In this note I will present some

analytic

studies of the

properties

of

charge density

waves near the

depinning

point.

Computations using

the

replica

method _are done in _a

simplified

model.

1 Introduction.

The

subject

of this note will be the

study

of _a

simplified

model for

charged density

_waves

_[1-4]

using

the

replica

method. Before

discussing

the

pinning

of _a real

charge density

_wave in the

simplified

model it _may be convenient to recall _some of the

properties

of real

charge density

waves. Some

possible justifications

of the

simplified

model _are

presented.

2.

Charge density

_waves.

In the model for

charge density

_waves I consider

here,

the

dynamical quantities

_are the fields

#;(t),

which

satisfy

the

following

differential

equation

_[5]:

ii

_" E +

hi;

_cos

(#I h;), (2.1)

where

A#I

is the lattice

Laplacian

and I denotes _a

point

of _a d dimensional lattice. We _are interested to know the time

dependence

of the _current

J(t)

₌

Li#I(t),

when _we start with _a

V

simple boundary

condition at time

equal

to _zero

(e.g. #I(0)

₌

0).

For small values of

E,

J goes ^to zero at

large times,

while for

large

values of E J goes ^to a non zero value at

large

times. We

expect

therefore the existence of _a critical value of E

(I.e.

Ec)

which

separates

^the two

regimes.

It is reasonable to believe

that, just

at the critical

point (I.e.

E ₌

Ec),

the current

decays

_{as a power} law:

J(t)

~-

t-~

(2.2)

580 JOURNAL DE

PHYSIQUE

I N°2

One of the

goals

^of a

theory

would be to

compute

^{the value of} uJ, and _more

generally

the behaviour of

J(t)

at

large

times in the

region

where E is _near to

Ec [5].

^{This is} a rather difficult task and _we limit ourselves to the

study

of the

properties

of the model for E less but

near to

Ec.

In this _{case we} do not need to consider the

dynamical equations

in

detail,

because

we know that _at

large

time

#

^{will tend} to _a

(stable)

solution of the static

equation:

0 = E +

A#I

_cos

(ii h;) (2.3)

If _we consider _a

given sample

in _a volume V we can define _as

N(E, V)

^{the number} ofsolutions of

equation (2.3).

Usual

arguments suggest

^that

equation (2.3)

should not have solutions for E >

Ec

while it should have _many solutions in the

opposite regime

where E _<

Ec.

In other

words it is convenient to define the function

f(E)

_as

f(E)

= lim

N(V, E)/E. (2A)

This function should be

positive

for E _>

Ec

and vanish

just

at the critical

point.

In

principle

_we could

try

also to compute ^the

expectation

value of _any functional

A[#]

^defined

as

(All])

=

~ _{A1~2l/N, (2.5)}

where the _{sum runs over} the N solutions of

equation (2.3).

However _we cannot

identify

the

expectation

value in

equation (2.5)

with what _we obtain

when _we start with _a

given configuration

of

#

at time _zero, because in this _{case we} select the

solution of

equation (2.3)

which is

mostly

_near to the initial condition

(moreover only

stable solutions would be

relevant).

However

just

because _near the critical field the total number of solutions becomes

small,

_{we can}

conjecture

that in the limit E

-

Ec

the

expectation

value in

equation (2.5)

tends to the

dynamic expectation

value defined

by solving equation (2.I)

with fixed

boundary

conditions.

These

arguments imply

that the

computation

of

f(E)

and of the

expectation

values in

equation (2.5)

^would tell _us not

only

the value of the critical

point

but also the

properties

of

charge density

_waves

just

at the

depinning point.

Using

^standard

manipulations

_one should consider the

partition

function

Z(")

=

/ _fl d#ad~adladh

~"~' _"

(2.6)

exp

/ _{(la (- Ala}

+ cos

(#a h)

₊

E)

+

~a (-A

+ sin

(#a

h

~a ))

_,

where the index _{a runs} from I to n;

$,a

^and

~a

_are Fermionic

(anticommuting)

fields.

However the result is not

quite

correct because in this _{case one} would _{sum over} all the solutions with _an extra factor

equal

to

sign (det (-A

+ cos

(#a h))).

This

difficulty

_may be

removed,

_as

suggested

in [7]

by introducing

2m

replicas

of the _n

Fermions;

the final results _are obtained

by taking

the

analytic

continuation to the

point

_m

=

1/2 starting

from _m

integer.

Similar formulae _may be obtained if _we start from _an

equivalent

version of

equation (2.3),

I-e-

ai arcos

(E

+

A#i)

₌

-#I hi mod(2~), (2.7)

where the variables _{ai are} introduced in order to take into account the multivalueness of the function _arcos.

Although

this last _{case may} lead _to

a

simple integration

_over the variable

h,

_{a more} careful

study

should be done in order to obtain _some

prediction

_on the models. It would

certainly

be

interesting

to

firstly

do the

computation

on a random Bethe lattice of fixed coordination number _z. In the _z

going

to

infinity

limit _one should obtain the _same result of _a

simple

minded field

theory

_[6]

(see

Sect. 5 for _{a more}

precise

definition of the

model).

3. The

coarse-grained

model.

We _{can argue} that the main effect of the fields

hi

is to

pin

the values of

#

_{more or} less

strongly (strongly

if the values of the h's in _a

given region

are

similar, weakly

if

they

differ of about

~ so that their effects cancel _ones with the

others).

We

finally

arrive at another model _11, 2]

which is

supposed

to describe the

large

scale behaviour of

charge density

_wave:

ii(t)

=

fR(f) f

₌

A#i(t)

_{+ P+ ni,}

(3.1)

where

R(z)

₌ 0 for _{z <} 0

R(z)

= I for _{z >}

0,

_q is _a random uncorrelated Gaussian noise and p is _a control

parameter

which

plays

the _same role _as E. One _can

easily

convince ourself that

the critical value of _p is

just

_{p =} 0.

Our task is _now to compute ^the

large

time behaviour of

J(t)

at p = 0. Let _us

assume that

the

previous equations

have _a solution in the continuum

limit,

where

they

become

I(z, _t)

=

fR(f) f

₌

A#(z, t)

+ _P +

n(z), (3.2)

where the correlations _among two

q's

_are

given by

n(z)n(Y)

₌

b(z

_Y).

(3.3)

In this _{case we can use} dimensional

analysis.

We _{can measure}

everything

in dimensions of

length ([xi

₌

I).

If _we do _{so we} find:

[VI =

-d/2 _Ill

= 2

Ill

=

(4 d)/2 lJl

=

-d/2. (3.4)

Dimensional

analysis implies

that

uJ =

d/4. (3.5)

The crucial

assumption

is the existence of the solution of

equation (3.2)

in the continuum. In

many cases stochastic differential

equations

with _a dimensionless

coupling

_are not well defined

directly

in the continuum limit. If _a lattice

spacing

is introduced at _a

preliminary stage,

one may find _an additional

dependence

of the observables _on the lattice

spacing.

Naive dimensional

counting

cannot be used and the renormalized

operators (which

have _a finite

expectation

value in the continuum

limit)

have _an effective dimension which differs from the naive dimension.

The

previous analysis

_assumes the absence of anomalous dimensions and it

corresponds

to _a mean field

approach.

The absence

(or

the

presence)

of

divergences

in the continuum limit should be studied

using

renormalization _group

techniques.

This has _not

yet

been

done;

however accurate numerical simulations in I and 2 dimensions show that the

prediction

_{uJ =}

d/4

is satisfied with _an accuracy less than

I§l,

_so that it _seems rather reasonable

(although

_we lack _a formal

proof)

that

equation (3.5)

is _an exact statement.

582 JOURNAL DE

PHYSIQUE

I N°2

4. The

replica study

of the

coarse-grained

model.

In this section _{we are}

going

to

study

the static solutions of the

equations

for the

coarse-grained model,

I-e-

f

< 0

f

=

A#I(t)

+ _p + vi.

(4.1)

In this _case the static

equations

_are

given by

_an

inequality,

not

by

_an

equality,

_so that the

quantity

that is

interesting

is the volume of the

configurations

of the field

#,

which

satisfy

the

inequality (4.I).

Therefore _{we can} define the

partition

function for _n

replicas

_as

Z~~~(P)

₌

f

dl#lR(-f))~

,

(4.2)

and the

corresponding

free _energy

density

_as

Fn(p)

₌

limv-m

In

(Z(~)) ^(4.3)

The

logarithm

of the _most

likely

of the volume in

#

_space is

given by

F(p)

=

lima_ofn(p). (4.4)

We

expect

that

F(p)

_goes to -oo when _p

-

pj.

It is

relatively

_easy to set up the

computation

of

Z(~)(p).

We

firstly

introduce _a lattice

spacing

in order to avoid

divergences. Using

the usual

techniques,

_we

get

z~~~

"

/ dV fl

_{d l~ald}

_[>al

_d

_{l§'al ~XP(~F)}

~"~' ~

(4.5)

F = I

/

_dz

_la

(A~aa

+ _p

la

+

q)

,

where _we have

neglected

_a

simple multiplicative

factor

(here

the Jacobian is constant and Fermions

are not

necessary).

^The

integral

over the variable t goes from 0 to

infinity.

The

integral

_over the field

~a is trivial: it

implies

that I is _z

independent;

the

previous

formula

simplifies

to

Z(~)

=

/ _{dq fl}

_d

[ta] dlad

_[~aa]

exp(-F)

₌

/ _dla

exp

(-VF(p, 1))

~~~ _~

~~P(~F(P, >))

_"

/

d~l

/ _fl

dla

_~XP

i ~ _{la (la}

_{+ +}

_{P) V~/2}

"

~~~~

~

°~

a=I, n a=l~ _n

fl _I/laexp 12 ~j _{la /2}

_+I

~j _la

_+p.

a=l~ n

a=l~

n a=I, _n

In the limit where the volume V goes ^to

infinity,

^{the saddle}

point

^method _may be used and it

gives

_a saddle

point

at

la

=

-I/p

for _p

negative.

The

corresponding expression

for

F(p)

is

given by

F(p)

= I +

In(- p), (4.7)

which

diverges

at p =

0,

as was

expected.

A

simple computation

shows that the correlation function of

#

in momentum space is

given by I/k~,

which _agrees with the results of dimensional

counting

at p = 0.

We have _seen that the

approach suggested

in the

previous

section _can be used in this

simplified

case. The value of the threshold field and of the correlations _near the critical

point

confirm the results

coming

from _a naive

analysis.

5. On the

justification

of the

coarse-grained

model.

In

principle

_one could write

a different form the _coarse

grained model,

I-e-:

I(z, _t)

=

iwR(I), (5.1)

where

f

is the _{same as} in

equation (4.I).

In other words _uJ controls the _response of _a

system

at threshold. The choice _{w =}

I,

that _we have done should be

justified

_more

carefully.

It would be natural _{to use} for _uJ the value

coming

from _mean field

theory.

However in the infinite range

charge density

_wave

model,

where

a form of _mean field

theory

is correct the value of uJ is

equal

to I

or to

3/2 depending

_on the ratio in the Hamiltonian of the coefficients of the kinetic term and of the

pinning

term (uJ = I and

Ec

= 0 in the weak

pinning region,

while

uJ =

3/2

and

Ec

> 0 in the

strong pinning region.

In this situation _we do not have _{an easy} time to

justify

_our choice _w

= I. However _mean field

theory

is very

simplified

in that it does not t,ake into account the fact that the force of other

spins

_{on a}

single spin

fluctuates with time. It _seems reasonable that this

sharp

distinction between weak and

strong pinning

is _an artefact of the infinite _range model and it should

disappear

in finite dimensions where the threshold field is

always

different from _2ero.

Computations

in finite dimensions _are

always

difficult _so it may be

interesting

to consider _a model with infinite range, but finite coordination number _z. The

equation

^of motion is

ii

= E +

B/z ~j _Ci,

k

(#, #k

_cos

(#i hi ), (5.2)

k

where the matrix C is _a random

symmetric matrix,

whose co1~lponents are

always

0 _or

I,

which

satisfy

the constraints

~j _C,,

k =

~ _Ci,

; = z

(5.3)

k _i

In other words the connection matrix C defines _a random lattice iii which each

point

has

exactly

coordination number

z. When _z goes ^to

infinity

_{we recover mean} field

theory

and the value B

= I is the

boundary

_a1~long weak and strong

coupling.

In this model each site feels _a

fluctuat,ing

field

coming

from the other

points

_so that it is unclear if the

analysis

done in

simple

_mean field

theory

still holds. In

priiiciple

_one should be able to control this kind of models with

analytic argu1~lents,

but in

pratice

it is not _so easy. It is rather

likely

however that

Ec

is

always

different from _2ero and that _w is

independent

of B.

The distinction between

strong

and weak

pinning

should be _an artefact of the existence of _an infinite number of

neighbours

in the naive _mean field iiiode.

Preliminary

numerical simulations confirm this

expectation: Ec

si different from _zero also in the weak

pinning region.

More careful

computations

_are needed to

compute

^{the value of} w and _we

hope

that this

problem

_can be

soon clarified.

584 JOURNAL DE

PHYSIQUE

I N°2

An

analytic study

of the dilute model would be _very

interesting

and it could _open the

possibility

of

performing

detailed

computation

in the finite dimensional models. This

problem

is rather

difficult,

however it _seem to be not out of reach of

present day

theoretical

techniques.

References

[1] MIHALY

L.,

CROMMIE M. and GRUNER

G., Europliys.

Lett. 4

(1987)

103.

[2] WEBMAN I.,

Phyl. Mag.

B 56

(1987)

743.

[3j ^{PARISI G. and PIETRONERO}

L., Europliys.

Lent- 16

(1991)

321.

[4j ^{PARISI G. and PIETRONERO}

L-, Pliysica

A 198

(1991)

666.

[5] FUKUYAMA H. and LEE

P.A., Phys.

Rev. B 17

(1978)

535.

[6] ^ERZAN A., VEERMANS A., HEIJUNGS R. and PIETRONERO L.,

Physica

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(1990)

447.

[7] FISHER

D-S-, Phys.

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1396.

[8] ^KURCHAN

J.,

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Prepint

_to be

published.