Anomalous diffusion in quantum systems with absolutely continuous spectra

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Anomalous diffusion in quantum systems with absolutely continuous spectra

F. Cannata, L. Ferrari

To cite this version:

F. Cannata, L. Ferrari. Anomalous diffusion in quantum systems with absolutely continuous spectra.

Journal de Physique I, EDP Sciences, 1991, 1 (7), pp.1029-1034. �10.1051/jp1:1991187�. �jpa-00246383�

Classification

Physics

Abstracts

03.65 03.40K 72.15R

Anomalous diffusion in quantum systems ^with absolutely

continuous spectra (*)

F. Cannata (~) ^{and L. Ferrari} (2)

(') Dipanimento

di Fisica dell' Universiti and INFN, 1-40126 Bologna, Italy (~)

Dipartimento

di Fisica dell' Universiti and GNSM-CISM, 1-40126

Bologna, Italy

(Received10

January 1991,

accepted

in

final form

4 March

1991)

Abstract. For _a

power-law dispersion

relation

w(k)

_cc k",

corresponding

to _an

absolutely

continuous spectrum of

plane

_waves in

spatial

dimension d~, ^{it is} shown that the condition

a m 2d~ leads to the divergence of the time of permanence in the initial

region

of localization. We define this

regime

_as

quasi-absence of dfljfusion.

Another

interesting

_case, defined _as

quasi- d#jfusion,

turns out to be d~ < a < 2d~, ^{for which} the _mean time of permanence is finite, but with

divergingly

large statistical fluctuations. These quantum

regimes

are discussed in connection with classical anomalous diffusion, Anderson localization in disordered systems ^and van Hove

singularities

in

crystals.

1. Introductiton.

Different criteria have been

suggested

to define the absence of diffusion

(AOD)

in quantum systems. ^Anderson's criterion

Iii

_was the

vanishing

of the

probability

of

detecting

the

particle

far _away from the initial

region

of localization after _a

long

time. The

non-vanishing

probability

of

detecting

the

particle

in the initial

region

after

arbitrarily long times,

_was also assumed _{as a} criterion for

localization,

I.e, _{as a} sufficient condition for AOD

[2].

^A more

elaborated

approach

considered both the

probability

of _return R and the time of permanence Tin the initial

region

of localization

[3] by identifying

the

following

three _{cases :}

T - cc and R _~ 0

(A)

T - cc and R

= 0

(B)

T< _cc and R

=

0.

(C)

Case

(A) corresponds

to solutions which remain localized in space and

satisfy

Anderson's AOD criterion. Case

(C)

refers to extended

solutions, giving

rise to diffusion. Case

(B)

is

intermediate between the

preceding

two ones, and is the

starting point

^{of the} present

analysis.

In reference

[3]

_case

(B)

is associated to _a

singular

continuous

spectrum,

^and to _non square-

integrable

solutions

(called quasi-extendettl

with

power-law decay.

^Case

(B)

is shown _to

apply

(*)

This work _was

partially supported

by the Italian

Ministry

of Education.

1030 JOURNAL DE PHYSIQUE I M 7

to

weakly

disordered quantum systems ⁱⁿ

spatial

dimension two, for which AOD is

expected

to occur, at least in the _sense of _a

vanishing

electric

conductivity _[4].

In the present note we use

essentially

the _same formalism _as in references

[3, 5].

From _a formal

viewpoint,

_our most

important

result is that _{even an}

absolutely

continuous spectrum, with

truly

extended

solutions,

_may

correspond

to case

(El

and

give thereby

rise to a

critically

slow diffusion. In

particular

_we consider _a

dispersion

relation like

w(k)oz

k~

(la)

(k

=

[k[ ),

in

spatial

dimension

d~, leading

to _a

density

of states

(DOS)

N(w)oz wd~/"-i, _(ibj

from the relation N

(w ) dk

oc dk. For any non

integer

value of d~

la, expression (16) displays

a

singularity

in _w

=

0, yielding

_an

integrable divergence

for

a ~

d~.

^{We shall} prove that the condition

a m 2

d~ (2a)

is _necessary and sufficient for the _mean time of permanence ^to

diverge,

with _a

vanishing probability

of return, as for _case

(B).

We shall define

quasi-absence of diffusion (QAOD)

the

transport regime corresponding

to

equation (2a).

Another

important

condition will be

d~w«<2d~, (2b)

yielding

a finite _mean time of permanence, as in _case

(C),

but

divergingly large

time fluctuations. The

regime corresponding

to

equation (2b)

will be defined _as

quasi-df- fusion (QD).

We will compare the two

quantum regimes QAOD

and

QD

with those studied in the

classical

theory

of anomalous diffusion

(for

_a recent

review,

_see Ref.

[6]).

The result that the standard free

particle displays QAOD

in _one dimension and

QD

in two dimension will be discussed _{as a} kinematical

pre-requisite

for Anderson localization in the presence of disorder. As _a first

application

to solid-state

physics,

we outline the

possibility

of _a

«

quasi-solitonic

motion of _wave

packets,

related to _van

Hove-type singularities. Second,

_we recall the connection between

equations (la, b)

and the fractal dimension

di

of the

quantum trajectory [7-9].

2.

Quasi

absence of diffusion

(QAOD).

Following

the definitions in reference

[3],

the time of permanence T and the

probability

of return R in the initial

region

of localization read

(h

=

1)

T

= lim

T( El fl

_lim ^~ _dt

e~ ^~~(

(

_ll~o ^{e~ ~~«}

Wo)

^(~

,

(3a)

e -

0+ e -

0+

0

R

= lim

R(e )

^~~~ lim

eT(e)

,

(3b)

e -

0+ e - 0+

for a

particle evolving

with the Hamiltonian

H~

₌

A~ p~

_~p

=

[p[

is the modulus of the

momentum and

A~

is _a

constant),

whose

stationary

solutions _are

plane

waves, with _an

absolutely

continuous spectrum ^described

by

the

dispersion

relation

(la).

Instead of

using

the definitions

(3) directly,

let _us calculate the distribution

p(r,t)

=

[W(r,t)[~

in r=0.

Introducing

the Fourier transform

q~

~p)

of the initial state

W(r,

0

(that

_{we assume} to be

spherically symmetrical),

_one

has,

in the momentum space of euclidean dimension d~ :

W

(0,

t

)

=

dp

_q~

~p)

e~~~«~~

oc t~ ~~~

j~

q~

( (x/A~ t)~/~) ^x~~~

e~ ^~~ dx

(4)

0

Assuming

_a

~

d~

^and _q~

(0)

#

0,

the

integral

in

equation (4)

_converges, and _:

W(0, t)

_oc

t~~~~

_for

_large

_t

₍₅₎

By

_means of

equation (5),

_{we can} calculate the

probability P(r)

that the

particle

is still in _a

small volume &a around the

origin,

after the time _r has

elapsed

_:

P r

)

= W

(0,

_r

)

^~ a _{oc r} ^~~~'~

for

large

_r

(6)

From the distribution

theory

_{one can} check that

equation (6)

holds for each

«. The

quantity P(r) equation (6)

turns out to be

proportional

to the

probability

of retum in the initial volume at the time _{r, as} calculated in reference

[10]. Taking

the limit

r - cc one has

ROzP(cc)=0, (7a)

in

agreement

^{with the extended} nature of the

eigenstates.

The function

P(r) equation (6)

also

gives

the

probability

that the time of permanence is

larger

than _r.

Thus, setting

w ~p

P(r)

=

dr'

,

~

dr'

the function

[dP/dr[ _represents

the distribution of the times of _permanence in the initial

region.

The _mean time of permanence

is, thereby (v)

=

j~dv( $

v = cc if _«

m2d~

~

<

~

if « < 2 d

~

~~~~

It _can be _seen

that,

for _a

power-law expression

of the time

distribution, equation (7b)

is

proportional

to the limit

(3a)

used in reference

[3]

to define the time T of permanence. In

conclusion,

the condition _am

2d~

is _necessary and sufficient for T= _cc. We define

a m 2

d~

as the

regime

of

quasi-absence of diffusion (QAOD).

We stress that this quantum

regime

has _a classical counterpart. In reference

[6]

the distribution of

waiting

times _v for site-

to-site transitions _{on a} classical

lattice,

is assumed to behave like

v~(~+?),

_{i-e- like}

[dP/dv (Eqs. (6)

and

(7b)), provided

_one takes _p

= 2

d~la.

Thus it is

immediately

_seen that the

QAOD

condition

(2a)

is

equivalent

to the so-called

subd%fusive regime,

in which the

classical diffusion

length

_L~j scales with time _as

[6]

:

L~jOZ

t~/~

p < I

t 1/2

°° In

(t)

^~

(the

_{case L~j oc} t~/~

corresponds

to the normal Brownian

diffusion).

An

important

difference between classical and quantum diffusion is that L~j

strongly depends

_on the

waiting-time distribution,

while the root mean-square deviation

L~

^{of the}

position

of _a quantum

particle,

JOURNAL DE PHYSIQUE I -T I, M7, JUILLET 1991 41

1032 JOURNAL DE

PHYSIQUE

I M 7

obeying

_a

dispersion

relation like

(la),

is

asymptotically proportional

to the time t,

irrespective

of _a and d~

(see

also

Eq. (9)).

This result is

explicitly

obtained in reference

[8]

^for d~ ₌

3, but

_can be

readily generalized

to any

spatial

dimension. The

surprising

result is thus

the coexistence of _a

diverging

time of permanence with the

general

quantum

expression

L~

oc t.

3.

Quasi

diffusion

(QD).

Another condition of

physical

interest _can be found from the convergence of the root mean

square deviation AT of the time of permanence

Av~

=

j~

^dv

$ _{v~- (v)~}

= cc for _a

md~

0 ^~

< cc for _a

<

d~ (7c)

The relations

(7b, c)

indicate that for d~ w « ~ 2

d~

^the mean time

(v)

of permanence is

finite,

but with

divergingly large

fluctuations AT. In this _case it is

possible

to define _{a mean}

time of permanence, but there is _no

operational

definition for the maximum time of

permanence, I-e- for _a time scale above which the

probability

of presence in the initial

region

is

negligibly

small. We define d~ w « ~ 2

d~

as the

regime

of

quasi-d%fusion (QD).

Even the

QD regime

_can be related to _a classical

regime

of anomalous

diffusion,

in which the diffusion

length

_L~j reads

[6]

_:

L~joc 2Dt+ct~/? _(I

~p

~2)

oz

42

_Dt

+ ct In

(t)

_p

=

2),

where _p

=

2

d~la

and

D,

_{c are} suitable constants.

4. Discussion.

A first consequence of the

preceding

results is that in _one and two dimension _even the standard

free-particle

Hamiltonian

H~

_oc

p~

_{generates the}

critically

slow

regimes

of diffusion mentioned above. In fact

H~ yields QAOD

in _one dimension

(«

= 2 d~

N(w

_oc

w ~'~),

and

QD

in two dimension

(«

= d~ ^N

(w

_oc const.

). By comparing

these arguments ^with

the result that Anderson AOD _occurs in dimension d~ w 2 for any disorder

[I I],

_{one may} be

tempted

to argue that whenever

QAOD

_or

QD

_occurs for the kinetic _energy, the introduction of _an

arbitrarily

small disorder makes

(almost)

all the

eigenstates

localized d la Anderson.

This would

suggest

that the critical dimension below which Anderson AOD _occurs for _an

arbitrarily

small amount of

quenched

disorder is fixed

by

the

exponent

a

determ1iling

the

dispersion

relation in the

free-particle

_case. In this context, results have been obtained

by

Bouchaud for

time-varying

random environments

[12].

The influence of the

dispersion

relation _on the localization

properties,

in the presence of

disorder,

has also been indicated

by particle-particle

interactions with suitable

power-law decay [13, 14].

We stress,

however,

that the role of disorder is crucial for the true

localization,

since

QAOD

and

QD

are in any case

associated _to extended solutions

(Eq. (7a)). Furthermore, t(ere

are cases of

non-periodic (but

not

disordered) potentials

in _one

dimension, showing nf

Anderson localization

[15].

The present results should be relevant also in the framework of the so-called

equivalent

Hamittonians in solid-state

physics [16],

which _may

provide

_a

physical

realization of _non- standard

dispersion

relations like

equation (la). Indeed, an'expansion

like

Hecluivalent "

H0(Pc)

+ ~4

a P

Pc1~ (8)

is

possible,

for values of the momentum close

enough

to certain critical

points

_{p~, as,} for

example,

the _van Hove

singularities

in the DOS

[16].

The

preceding

results indicate that if

a m d~ ^is

satisfied, wave-packets

formed

by plane

waves in the

neighborhood

_{of p~} should tend to _a

quasi-solitonic behaviour,

in the _sense that

they

should _move with _average

momentum p~, but with

critically

slow

spreading.

In fact

QAOD

_or

QD

in the

moving

reference frame

p'

= p p

~,

in which the centre of the _wave

packet

is at rest,

just implies

the

quasi-absence

of

spreading

for the

wave-packet moving

in the

crystal

frame.

Now _we notice that the exponent a does

actually

coincide with the fractal dimension

di

of the

quantum trajectory [7-9].

More

specifically,

if _one takes the root mean square

deviation of the

particle's position

L~(t)

₌

~

oc

t[Ar(0)]~

~~

(9)

(for large t),

as the

length

of the

quantum trajectory,

and expresses it in units of the initial localization

length Ar(0),

_one finds _a self-similar

relationship

in fractal dimension

di

= a

[8].

We

finally

recall the connection between the exponent a and the so-called

sprectal

dimension d~ ^{of the}

system.

^{The definition of}

spectral

dimension

depends

_on the reference

dispersion

relation which may be chosen _{as w oc} k

~phonon-like)

_{or w} oc

k~ ~particle-like).

^In

the former _case

[17], equation (16) yields

_d~

=

d~la, by setting N(w)

_{oc w} ^~~ In terms of the

spectral dimension,

the conditions

(2a)

and

(2b)

become

d~~1/2

and

1/2~d~<1 respectively.

In the latter _{case, one} should define _d~

=

2

d~la,

since

N(w)

_oc

w

~~~~~

We

notice that this last choice

yields d~=

_p, as defined _at the end of section 3. The

conditions

(2a)

and

(2b)

_now take the _more

appealing

_{form d~ <} I and I

< d~ _~ ²

respectively.

5. Conclusions.

In

conclusion,

_we have shown that _even for _an

absolutely

_{continuous spectrum} of

plane

_waves

in

spatial

dimensions

d~,

^with a

power-law dispersion relation,

a

diverging

time of permanence in the initial

region

is

possible, provided

the exponent _« ^{in the} power law is

large enough («

_m 2

d~).

The condition

« m 2

d~

^{is defined} as the

regime

of

quasi-absence of d%fusion.

Another

important regime,

defined

quasi-d%fusion,

is found for d~ w « ~ 2

d~,

which

yields

_a

finite _mean time of permanence, but

divergingly large

time fluctuations. These

quantum regimes

_can be related to the anomalous diffusion _on classical

lattices,

due to

long waiting

times

[6]. QAOD

and

QD bridge

between standard diffusion and AOD in the Anderson

sense.

Thus, they

have been discussed in connection with Anderson localization in low

dimensional disordered systems ^and motion of _wave

packets

close to a van Hove

singularity.

In

particular,

it is

suggested

that the critical dimension d~ ^below which all quantum ^states become localized d la Anderson for _an

arbitrary

small amount of

disorder,

is determined

by

the condition

d~/«

_w

I,

I.e. d~ ₌ «.

Recalling

that _a is the fractal dimension of the quantum

trajectory,

the _cross-over to a

regime

of

complete

Anderson localization should be marked

by

the

prevalence

of the fractal dimension _on the euclidean dimension d~ ^of the

embedding

space.

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