Periodic and Non-Periodic Band Random Matrices: Structure of Eigenstates

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Periodic and Non-Periodic Band Random Matrices:

Structure of Eigenstates

Felix Izrailev, Luca Molinari, Karol Życzkowski

To cite this version:

Felix Izrailev, Luca Molinari, Karol Życzkowski. Periodic and Non-Periodic Band Random Ma- trices: Structure of Eigenstates. Journal de Physique I, EDP Sciences, 1996, 6 (4), pp.455-468.

�10.1051/jp1:1996224�. �jpa-00247197�

Pçriodic and Non.Periodic Baud Random Matrices: Structure of

Eigenstates

Felix M. Izrailev (~~~>*), Luca

Molinari

_(~>**) and Karol

Zyczkowski _(~)

(~) ^{Università di}

Milano,

sede di

Como,

via Lucini 3, 22100

Como, Italy

(~) ^{Budker Institute of Nuclear}

Physics,

Novosibirsk 630090, Russia (~)

Dipartimento

di Fisica and

INFN,

Via Celoria 16, 20133

Milano, Italy

(~)

Uniwersytet Jagielloùski, Instytut Fizyki

M.

Smoluchowskiego,

uI.

Reymonta

4, 30-059 Krakôw, Poland

(lteceived

5 October 1995,

accepted

4

Jan1Jary1996)

PACS.05.45.+b

Theory

and mortels of chaotic systems PACS.72.15.Rn

Quantum

Iocalization

Abstract. The structure of

eigenstates

for the ensembles of standard and

periodic

Band

Random Matrices

(BRM)

is

analysed.

The main attention is drawn to the

scaling properties

of the inverse

participation

ratio and other _measures of Iocahzation

Iength.

Numerical data

are compared with analytical results recently derived for standard BRMS of _very

large

bond

size. The data for

periodic

and standard BRM allow _us to exhibit the influence of

boundary

conditions

on the

properties

of eigenstates.

1. Introduction

Recently,

much attention has been drawn to the so-called band random matrices

(BRM) (see,

e-g-

[1-19] ).

^In contrast with the traditional Gaussian ensembles of full random

matrices,

which

enjoy

rotational invariance and have _no free

parameter,

^BRMS

depend

_{on a} control

parameter,

related to their band structure. Since the

typical

localization

length

of _an

eigenstate directly depends

_on the effective band _size b, statistical

properties

both of spectra and

eigenvectors change significantly

with this

parameter. Moreover,

for limite

matrices,

the localization

length

can be

compared

to the matrix size N and the transition from localized _to extended states can

be studied.

The

increasing

interest in nlatrices of this kind is

explained by

^their broad

application.

In

particular,

much _progress has occurred in connection with

"quantum chaos",

the

subject

that deals with

dynamical _systems exhibiting strong

chaos in the classical limit

(see,

_e-g.

[20-23]

It _was shown

that,

in the semiclassical

hmit,

the generic structure of Hamiltonian matrices _in

some basis _is banded

and,

if the

corresponding

dassical motion is

chaotic,

nlatrix elements _can be assumed

pseudo-random,

_see details in

[2, 21, 22].

Another

important application

_is related to lD and

quasi-lD

solid state models with random

potentials.

In the lD

geometry,

the size of the band b > 1 has the

meaning

of

long

_range

(*)

e-mail addresses:

izrailev©vxinpb.inp.nsk.su; izrailev©physics.spa.umn.eau

(**)Author

for

correspondence le-mail: molmari©milano.mfn.it)

random

hopping

in the

tight-binding representation.

In the

quasi-lD

_case, the

parameter

^b

plays

the role of number of _transverse

channels;

this

analogy

has been

proved rigorously iii.

Other

applications

_are

currently investigated

in the statistical

approach

to

complex

atoms

[24, 25]

^{and nudei}

[26].

An ensemble of random matrices with _non-zero matrix elements inside _a band _was introduced for the first time

by Wigner [27].

^{In his}

model,

the main

diagonal

of the matrix contains in-

creasing elements,

the other entries in the band

being

random and drawn

according

to the _same

probability

distribution. After the

pioneering

work of

Wigner,

the model has been

forgotten

for

a

long time, mainly

because of mathematical dilliculties in

getting rigorous results,

in favour of the _more accessible ensembles of full random matrices

[29].

^{Matrices with} a band structure

have been considered

only recently,

in

application

to chaotic

dynamical systems il, 2,17, 28].

After extensive numerical studies

[3,4,6, 8-12,14,17,18,22], revealing

the remarkable

scaling properties

of

eigenvectors

of

BRMS, important

_progress has

recently

been made in the

analytic approach,

based _on the

powerful supersymmetric

method

ii,13, là,16,19]. Many

_new results about the structure of

eigenstates

were

obtained,

in the limit

b,N

_{- cc} with finite ratio

= b~

IN.

In this paper, we present numerical data for

periodic

band random matrices

(PBRM),

which

are still _open for

analytical

treatment. The structure of these matrices is

slightly

diiferent from that of standard

BRMS,

and it relates to a

periodic

structure of all

eigenstates. They

_appear

in a natural _way ~n.hen

considering periodic

lD _or

quasi-lD

disordered models. In _a

descriptive language,

if the model of standard BRM may be associated with _a chain of N

interacting sites,

PBRMS

correspod

to the _same chain of

interacting

sites

joined

into _a

ring.

An

important application

has been discussed in

[30],

^{where the relation between level} cur-

vature

(second

derivative of the energy with respect ^{to the}

flux)

and conductance is studied

by varying

the

magnetic

flux

through

a

quasi-lD

_ring. In

particular,

it _was found that the

mean absolute value of curvature _is

proportional

to the _mean conductance both in localized and metallic

regimes;

other

important

conclusions _were drawn for the distribution of the level

curvature. The _same model of PBRM has been

recently

considered [31] ^to

analyze

the

velocity

autocorrelation

C(çi)

and to exhibit its

singularity

in ç§ = 0. Since the

sensitivity

of levels to _a small

change

of _an external

perturbation

is related to the structure of

eigenstates,

it is

important

to compare the statistical

properties

of the

eigenstates

in the _{two versions} of BRM.

This paper is

organized

_as ^follows: in Section 2 _we describe the

model,

refer to the known results _on the

scaling

of the mean localization

length,

present new data for standard BRM and demonstrate the diiferent

scahng properties

^of PBRMS. Section 3 is devoted to

analyze

various measures

characterizing

fluctuations of localization

lengths

of individual

eigenvectors.

We

study

the distribution of inverse

participation

ratio

(IPR)

for both models of BRM. To better visualize the diiferences between the linear

system (standard BRM)

and the circular

system (PBRM),

_we introduce the

concept

^of

barycenter

of _an

eigenvector.

^{It is} a

complex quantity.

The distribution of the

phase

allows _us to show the eifect of

boundary conditions,

while the distribution of the modulus of the

barycenter

_proves it to be _an alternative _measure of the localization

length.

2.

Scaling

of the Mean Localization

Length

The ensemble of standard real

symmetric

BRMS is

specified by

the band width b and

by

the matrix _size N. Inside the

band,

matrix elements

H~m

_are distributed

according

to the Gaussian

distribution with _{zero mean} and fixed variance

iH~m)

=

°, iHim)

=

~(j)j~~~lljj~ ^il)

With this

normalization, eigenvalues (in

the limit of

large N)

_are located in the interval

(-2, +2), independently

of the value b. The

delta-symbol

is introduced _{to assure} normalization of the Gaussian

Orthogonal

Ensemble

(GOE)

in the limit of full matrices 16 =

N).

Non-

symmetrical

hermitian BRMS have

recently

been studied in

[32].

^Since

analytical

results for BRMS _were obtained in the hmit N _> b > 1

ii,13, là,16],

_we

numerically study large matrices,

their _size

being only

^restricted

by

the computer.

For infinite BRMS

(IV

-

cc)

with _a finite band b >

1,

all

eigenstates

_are found to be

exponentially

localized

[4, 7,8,17,18,22].

This _means that the

amplitude

_çJ~ of _an

eigenstate decays exponentially

fast _away from _{a center} of

localization,

_çJn

r-

exp(- in

_no

Îlicc)

The

quantity icc,

known _as the localization

length,

is of

great importance

in

describing

the

global

structure of

eigenstates.

Based _on

scaling _arguments,

_in

[4,22]

^it was

predicted

that the

localization

length icc

is

proportional

to b~. _{The exact}

expression

was obtained in

iii

and

applies

also to BRMS where the band structure is achieved in full matrices

by decreasing

the variance of matrix elements away from the main

diagonal by

a suitable

envelope

function:

im

₌

CP~IE) b~, 12)

where _c is _a constant of order 1

depending

_on the

envelope

function and

p(E)

_is the

density

of states given

by

the semicircle

law,

PIE)

=

~

~

E~

R~

=

8b(H(~). ₍₃₎

Dur interest is in BRMS of finite but

large

size

N;

such matrices _can be associated with

physical systems

in _a finite reference basis. To

study

the structure of

eigenstates

in the finite

basis,

_{we use a} definition of localization

length

based _on the information

entropy 7iN

of _a

normalized

eigenstate

of

components

_çJ~,

~iN

=

1~7~i~

lni1~7~i~), 14)

Since the maximal value of the information

entropy

is

N,

in references

[22, 33]

^it was

suggested

to define the localization

length ij~l

as

~Î~

^~

~~P((~N) IÎGOE ). (à)

In this

definition,

the localization

length

is normalized in such _{a way} that for the most extended

eigenstates,

which _appear in the limit b

=

N,

the value of ij~~ is

equal

to the _size of the

basis, ij~l

= IV. The

quantity 7iGoE

is the average entropy of random

eigenstates

of GOE and _can

be evaluated

analytically

7iGoE =lfil ) _{+1) -lfil()}

m

in1(), ₁₆₎

~l(x)

is the

digamma

function

(see

details _in

[18, 22]

^In

equation là) (...

has the

meaning

of ensemble average over

eigenstates

^{of the} same energy. The entropy localization

length ij~l

defined

by equation là)

varies from 0 to N

depending

on the

degree

of localization.

The essential

point

of the

previous

studies

[4, 6,13,15, 22]

^{is that the} statistical

properties

both of

eigenstates

and

spectrum depend

_on the ratio of the localization

length icc

to the size of the basis N. Since for _a fixed energy the localization

length icc

is

proportional

to

b~,

the

scaling

parameter in the limit N _> b _{» was}

suggested

to be of the form

~2

=

~. ^I?1

In

particular,

in [4] ^it was

numerically

found that the localization

length _ij~l

scales with _as follows:

~~

~Î

l

ÎÎIÀ'

~~~

with the estimated parameter ^value ci G3 lA. A detailed

analytical study

_[16] has shown that the above

scaling

_expression is

approximate, although

_very close _to the _{correct one.} For the

comparison

with the results of

[13]

^{it is convenient to} use the variable

~~ l

~fll

~~~

in which the

scaling (8)

reads:

In yi " In _{+ ail ai}

" In _ci,

(loi

This formula _{is exact}

only asymptotically,

with _two diiferent values for the the coefficient _ai

il

_Si:

aÎ~~

m 1.58 for

strongly

localized states (À <

1)

and

aÎ~~

m 1.79 for extended states (À >

ii.

Though

the absolute values of the

asymptotic slopes a[~~ depend

on the normalization used to define the

scahng parameter,

^{their diiference Aa} =

a)~~ aÎ~~

= 0.21 _con be

directly compared

with the numerical results. Due to its

smallness,

the shift Aa could

hardly

be detected in the first numerical data [4] and

yet

ⁱⁿ

Figure la,

where the

scaling

^{is exhibited} in the variables In yi, In

using

_new data of _{a more} careful

study

of BRMS of

type Ill.

The _existence of _two limit parameters

a/1

can be settled from the _same data of

Figure

la, using variables

mapping

the

dependence (loi

in the

simplest

form

In(yi/À)

= ai This is done

in

Figure 16,

where the two

straight

lines _are lits

yielding aÎ~~

m 0.18

(lower line), al~l

_m 0.30

(upper hne),

obtained from the data _in the _{two regions.}

Although

the data confirm the existence of two diiferent limit values in the

expression (loi,

the obtained shift

Aae,p

m 0.12 is about half the theoretical

prediction.

The

discrepancy

_is not

clear,

and _may be related to not

sulliciently large /small

values of

pi (we

recall that the bond _size has to be

large,

b _>

1,

_in order _to expect a correct

scaling).

The

property

of

scaling

for BRMS _was derived

analytically

_[15] under the

assumption

b _< N.

For the

numerically

accessible values of

N,

this condition _is violated for

large

values of the

parameter

À, and in

Figure

one can observe deviations from the

dependence (10)

for > 3 and _an

expected

failure of

scaling

_in this

strongly

delocalized _regime.

In order to improve

statistics,

we collected

eigenvectors

_{in an energy} window at the center of the

spectrum including

half of all

eigenvectors since, by

relation

(2), they

have similar localization

lengths.

For each chosen

pair

of parameters ^{b and}

N,

_an additional average was

performed

_{over an} ensemble of 20 500 matrices.

in _y1

4

~°~~

2

/

z~~

À'

Qt~

~Nn400

#

o N=600 é

x woo

/

~=1000

-2 s °

a

~

~~

*

_3 -2 -1 0 2 3 4 5 ~

a) ^{in À}

0

In(yi/À)

0.5 (1)

o-o (2)

-o.5

~ Î

a N+400 o N=600

x NM00

-1.5 _{Ô N=1000}

Q

~

2.0

-4 -3 -2 -1 0 2 3 4 5 6

b) ^{In À}

Fig.

I.

Scaling

properties for standard BRM.

a) Dependence

of the entropy Iocalization

Iength 1(~

on the

scaling

_parameter

= b~

IN,

_usmg the variables _m equation

(10). b)

The _same data

as m

la)

in

other

variables,

with evidence of the difference in values of _ai The two

straight

Iines _are the fits

(10);

the upper Iine

Il)

_{gives ai}

" 0.30 for the range 0 < In < 2.5 and the Iower Iine

(2)

_{grues ai}

" 0.18

for the _range -3.0 < In < 0.

To

study

the influence of

boundary

conditions _on

eigenstates,

we consider _an ensemble of

periodic

BRMS. To have all

eigenstates periodic,

_çJ~+N

" §7~, one adds _non-zero matrix

elements

(which

_are assumed to be distributed in the _same way as inside the

band)

to the lower left and upper

right

_corners of band matrices in such _{a way} that the number of _non-zero

elements in each _{row is} constant and

equal

to 2b

[30].

^{To obtain the} same

spectral radius,

the Gaussian distribution of matrix elements has _{zero mean} and variance

H2 Il

+

ô~m

~~ 2b 1

Iii

In this

ensemble,

the hmit _case of GOE is recovered for _an odd value of N and b

=

IN +1) /2.

Due to the

periodic

structure of

eigenstates,

_{one can}

expect

a diiferent

scaling

behaviour of

8

În y1

6

4 (2)

2

0

~~

~Î~~

Ô N=1000

-4 _* N=1200

~

-6

-6 -5 4 -3 -2 -1 0 2 3 4 5 6

~j ^{in ~}

2.0

Î~ (~li/À)

~

Î 5 & x ~

*

& x

~ *

Î.Ù ^~~*

(Î) ^~

0.5

~~ ~

~Î~~

Ô N=1000

0,Ù N=1200

D N=1400

-0.5

-6 -4 -2 0 2 4 6

bj ^{in ~}

Fig.

2. As in

Figure

1, for

periodic

BRMS, with

fitting

_parameters dl ₌ 0.22, _ai " I.o

(hne (1))

and dl

" 0.70, _ai "1.2

(hne (2)).

the localization

length 1(~,

^which may be

important

in view of the

application

of PBRMS to

quasi-ID

disordered models with _a

magnetic

flux

through

a

ring [30, 31].

Numerical data for the rescaled localization

length _1(~

for the ensemble

Ill)

of PBRMS _are

presented

in

Figure

2a-b. From these

figures

_{one can} conclude that the

scaling

of the form

(8)

holds

only

in the region of

strongly

localized _states

(In

<

-2).

For

larger

values of the

scahng

parameter À, ^the

scaling

has _a diiferent form: the fit of the data for In > -2 gives the _expression

In yi " ai In ₊

dl, (12)

with _{ai "} 1.2 and

dl

_" 0.70. In the variables

pi,

this leads to the law

pi

D~a

"

,

D

=

exp(di _), (13)

+ DÀ"

which still _watts for _a theoretical

explanation.

For

large

values of In À, the fluctuations _are

extremely large;

this does not allow for _any

phenomenological expression.

In the above

study,

_we used _a definition of localization

length

based _on information _en-

tropy (4).

Another

definition,

often used in solid state

applications,

is based _on the inverse

participation

ratio

Pp~

,

where

N

P2

_"

~

_Χ7~Î~

₍₁₄₎

n=1

By normalizing

this localization

length

with reference to

GOE,

_one introduces the

quantity fl2 (II, làj

~~

~ ~2~~~~)~

^~

~~

~~~~

where

P2(GOE)

is the

participation

ratio for GOE random matrices and tends _to

N/3

in the limit of

large

matrices.

In reference

[15]

^the

quantity p2

for standard BRM _was found to scale in the _{same way as} fin

fl2

=

1l~l~~ ⁱ¹⁶⁾

or, in the variables _y2 À,

~~ _{Y2 °2} În ₊

d2

, y~ =

Ù2 Ù2

(li)

In _contrast to

pi,

the

expression (16)

is exact in the whole range of À. Numerical data for

the

scaling

of

p2

_are

presented

in

Figure

3 for standard and

periodic

BRMS.

Figure

3a shows

a

quite good correspondence

to the

predicted scaling (16)

^for standard

BRMS;

the fit

gives

the

slope

_one and the value

d2

+ In _{c2 m} -0.15. One should remind that the

parameter

c2

depends

_on the

shape

of the

envelope

function for the variance of matrix elements

(for general expressions

_see

iii).

The main result of the numerical

study

of

periodic

BRMS is that the

scahng

of

p2

_seems

to have the _same law _as for

pi (see Fig. 3b).

Another

important

observation is that the fluctuations for the localization

length illl

_are much

stronger compared

to those for ij~~. ^This fact does non allow to obtain a more accurate estimate for the factor _o; the data show

that,

approximately,

_{a2 m ai m} 1.2.

3. Statistical

Properties

of

Eigenstates

Ii is well known that conductance of finite

samples generically

shows anomalous

sample-to- sample

fluctuations. In

particular,

in the metallic

regime

the so-called universal conductance

fluctuations have been

discovered,

with the remarkable

property

that the ratio of the _variance of conductance to its _mean value does not

depend

_on the _size of the

sample.

In this

regime

the fluctuations of the conductance _are Gaussian. In the other hmit of

strong localization,

the fluctuations _are of _very

specific form, namely,

the

log-normal

distribution for the conductance holds if the localization

length icc

_is much less than the _size of the

sample.

Since these

properties

of conductance _are

directly

related to the structure of

eigenstates,

it is

important

to

investigate

the statistical

properties

^of

eigenstates.

A first

study

of the distribution of the

entropy

localization

length

in

eigenvectors

of standard BRM is described in reference

[loi.

In papers

[16,19]

other characteristics of

eigenstates

_are studied for standard BRMS both in the localized _< 1 and delocalized _{> regimes.} One should note

that,

in the

description

as a

quasi-ID

disodered

model,

the _size of the band b _can be

8

În _y2

o 6

4 _a ^°

o 2

0

O N=100

_~ & Nz200

D Nn400

~

-4

-4 -2 0 2 4 6 8

ai ^{in ~}

8

Îny2

.a O +

6

4 .

(1) 2

0

° N=100

_~ > N=200

. Nn400

+ NM00

-4

~

-4 -2 0 2 4 6 8

bj ^{in À}

Fig.

3.

Scaling

property for the Iocalization

Iength t(~,

_see

Ils),

for standard and periodic BRMS:

a)

Standard

BRMS,

the

straight

Iine _is the fit

(17)

with d2 _" -0.15 and _ce2

" 1.o.

b)

Periodic

BRMS,

the hnes

(1,2)

_{are given}

by

d2 # 0, 02 " 1.o and d2 _" 0.3, 02 " 1.2

,

correspondingly.

associated to the _mean free

path. Therefore,

the metallic regime

corresponds

to the condition

«

/Î

_{< < N. In what}

_follows,

we

study

fluctuations in the structure of

eigenstates

^for standard and

periodic BRMS,

which _may be relevant to the statistical

properties

of conductance

in

quasi-lD

models

las

well _{as in} lD-models with

long

_range

hoppings).

Since for standard BRMS the inverse

participation

ratio

P2

has often been studied _ana-

lytically

_in the

hterature,

_we

investigated numerically

fluctuations of

P2

_in diiferent _regimes.

We _start with _a

comparison

^of the IPR distributions

W(P2)

for standard and

periodic

BRM.

Figure

4a

presents

^results obtained for both models for small values of the

scahng parameter

À.

Diiferences between the distributions for the _two models _are

clearly

visible: fluctuations of

P2

_are smaller for

periodic

BRMS. Notice that the IPR distribution of standard BRM with b

= 9 (À m

0.20) corresponds

to the data of PBRM obtained for b

= î

land

therefore with

m

0.12).

W(P~)

12

$

D b=9

. bu

> bn4

8

~

4

0

a)

~~ ~~ ~~ ~~

1~2

~~

w(~)

i.o

0.8

0.6

off

0.2

Ù-Ù

ù-ù 0.5 1-ù 1.5 2.0 2.5 3~0

bj

^Îl

Fig. 4.

a)

Distribution

W(P2)

of the

participation

ratio P2 for N

= 400 and different values of b, for standard BRMS

(open symbols)

and for

periodic

_ones

(fuit symbols); b) Comparison

of the scaled

distribution of IPR

W(~)

obtained for standard BRM with b

= 3, N

= 400, with the expression

(18).

On the other

hand,

in the

regime

of

strong

localization < 1

(in practice

the band width b

was set to 3 _or to

4)

the distribution

W(P2)

does not differ for both ensembles. In this _regime the

probability

distribution has the

peculiar _property

of double

scaling. First,

it scales _as other

quantities characterizing

band

matrices,

with the

parameter

₌ b~

IN. Second,

in the _case

< 1 the distribution becomes universal in the rescaled variable _~

=

P2/(P2).

In

Figure

4b _a

hystogram

of IPR in rescaled variables for standard

BRM,

with

=

0.04,

is sho1A~n. The sohd line represents the distribution

recently

obtained

by Fyodorov

and Mirlin

[19]

C° ~

~2~2

W(/t)=~j -k~7r~/1-2k~7r~

_exp

--/t

(18)

~~~9

3

Though

it _was derived for

complex

Hermitian

BRMS,

it also describes the _case of real sym- metric random matrices discussed in this work. The

analytical

form

(18)

_{gives a}

satisfactory

2.5

ÎÎÎ(P~)

o N=100

2.Ù ~~~~

NW00

1.5

1.o

o.5

o-o

2.0 2.5 3.0 3.5 4.0 4.5 5.0

P2N

Fig. 5. Distribution of the inverse

participation

ratio P2 for fuII random matrices 16 =

N).

correspondence

to the numerical data.

Discrepancies

_nlay be

explained by

a not

sufliciently large

matrix size N since the

analytical

results _were obtained

[19]

^{in the limit} b, ^N - cc with

=

b~/IV

- 0.

With _an increase of

b,

the distribution

W(P2)

becomes _{very narrow} around the _mean value

(P2)

m

3/N.

Numerical data obtained in this _{case are}

presented

in

Figure

5. The width of the fluctuations of IPR around the _nlean decreases _as

1IN _[19].

The finite _size eifect of

asynlnletry

of the distribution

W(P2)

is

clearly

visible.

A

scaling

property ^of the inverse

participation

^ratio can be _seen in the

dependence

of the

mean square deviation à

=

((Pi _{(P2 )~)~/~}

_on the _mean value

(P2

In the

theory

of disordered

solids,

such _a relation _is

important

to characterize the

type

^of fluctuations. In

[16,19]

it _was

shown

that,

for

non-synlmetrical

Hermitian BRMS and in the hmit of

strong localization,

the ratio

à/(P2) approaches

the _constant value

Ill.

The

dependence

of à _on the _mean

value

(P2)

for standard and

periodic symmetric

BRMS _is

given

in

Figure

fia. It _was obtained

by

direct

computations

on matrices of types

il)

and

(11),

with sizes N

=

100, 200,400,800

and diiferent values of band size b. The

scahng

holds

only

in the _very localized

regime,

for

(P2)

> 0.1.

Unexpectedly,

for smaller values of the _mean

participation

ratio

(0.03

<

(P2

<

o-ii,

there is _no diifereiice between standard and

periodic

BRMS. The ratio

à/(P2) being

constant for locahzed states reflects

extremely large

fluctuations in the width of

eigenstates

(see also, [16,19]);

such strong fluctuations have also been found in the

dynamical

locahzation of

eigenstates

^{of the kicked} rotator

model,

_in momentum space

[22].

The behaviour of the ratio

à/(P2)

_~vas studied

analytically

in

[19].

^In

particular,

it _was shown that the convergence ^to the limit value

1Il

is not monotonic and very slow. These

peculiarities

_are reflected in

Figure

6b where the _same data _are

plotted

in other variables.

Only

for

extremely large

values À~~ >

20,

the ratio

à/(P2) approaches

the hmit value

(again;

for small values of b there _is clear

deviation).

A

comparison

^of the ratio

à/(P2)

for standard and

periodic

BRMS shows that in both hmits of

strongly

^localized or extended

states,

^the

diiferences _appear to be small. The distribution of IPR is diiferent in the two models in the intermediate _case where the locahzation

length

is of the order of the

sample

size

(see Fig. 6b).

2

à

(1) 1>~

5

~

A N=200

D Nn400

Ô NM00

o N=100

~ ^> N=200

. Nn400

+ NM00

Î~~

3 4 5 6 75 2 3 4 5

~j <1~2>

1-o

ôllP~>

_o

_j=jjj

0.8 a

o NWo0

~

0.6 ~~~~---

~~

(Î)~

+ N=lo0

> N=200

. N#400

+ NMo0

0.0

0 5 10 15 20

~~

25

b) ^~

Fig.

6.

ai

Mean deviation à _{~ersus mean} value

(P2)

of the _inverse

participation

ratio.

Open

symbols correspond to standard BRMS, fuII

symbols

the

penodic

_ones. The dotted

straight

Iine

(1)

is the theoretical hmit à ₌

fi(P2)

for very Iocahzed states.

b) Dependence

of the ratio

à/(P2)

_on the

scahng

_parameter À~~ extracted from the _same data.

This indicates that the

large

fluctuations of the inverse

participation

ratio for standard BRMS

(read,

fluctuations of the localization

length)

_are increased

by

the

eigenstates

localized in the

vicinity

of the

edges (see

discussion _in

[19]).

In order to

study

the diiference _in the structure of standard and

periodic

BRMS in _more

detail,

_we introduce the concept of

barycenter

B of _a normalized

eigenstate

of components _çJ~

in _a

given

basis

B

=

pexp(1ç§)

=

(

(çJn_(~ exp

1~~" ₍₁₉₎

~=i

N

It is _a

complex

number in the unit circle. Its radius p is related to the localization

length:

for

strongly

localized

eigenstates

^it is of order

1,

^{and for extended} states it is very

small,

of order

1IN.

The

phase

_{ç§ can} be associated with the

position

^of the center of the

eigenstate

in the

0.25

P(4)

0.2

o

o

a

a ~ ~

a~ _~ o

o°

a ~ o

~15 °

~ o

~ o o

~ o a o

o

o-i

o.05

o-o

~'~ ~'~~ ~'~~ ~'~~

4

^~'~~

Fig.

7.

Dependence p(#)

for standard BRM

(crosses

and broken

hne)

and

periodic

BRM

(open circles)

with b

= 10, N

= 100 The dashed fine represents ^the mean value of _p.

given basis. Therefore the

barycenter

contains information both about localization

length

and location of

eigenstates.

This parameter ^{is convenient} for the

description

of

global properties

^of

periodic eigenstates.

In

Figure

7 the

dependence _p(ç§)

is shown for the intermediate _case of moderate value

=

1,

for standard and

periodic

BRMS of _size N

= 100 and band size b

= 10. In

particular,

_{one can}

see how the locahzation

length (vertical axis)

of _an

eigenstate depends

_on the

position

of the

center of localization

(horizontal axis).

The data show that for standard BRM there is _a

strong

decrease of localization

length

for _states which _are localized close to the

edge. Contrary,

for

periodic

BRM the centers of locahzation _are

uniformly

distributed.

As above

indicated,

the

quantity

_{p may} be treated as an alternative definition of localization

length.

In

Figure

8a _{one can see} how the distribution

P(p) depends

_on the

scaling

parameter ^À for standard BRMS of size N

= 400. In

particular,

the most remarkable property ^of

eigenstates, namely, extremely large

fluctuations of the localization

length

^{is well} reflected. The distribution

P(p)

tutus out to be useful for the

comparison

of standard and

periodic

BRMS. The _main

diiference _in the structure of

eigenstates

_can be

easily

extracted from

Figure

8b where the distribution

P(p)

is given for standard and

periodic

matrices with _same values of b and N

(compare

with

Fig. 7).

The two

representations (Figs.

6 and

8b) together provide

relevant

comparative

information about the structure of

eigenstates

^of the _same matrix.

Acknowledgments

We thank G. Casati and I. Guarneri for the

hospitality during

_our

stay

in

Como,

where this work _was

started,

and also Y.

Fyodorov

and A. Mirhn for discussions and valuable

suggestions.

F. M. I. wishes _to

acknowledge

the

support

of Grant ERBCHRXCT ₉₃₀₃₃₁ Human

Capital

and

Mobility

Network of the

European Community

and also the

support

of Grant No RB7000 from the International Science Foundation. K-Z-

acknowledges

financial

support

^{from Polish KBN}

grant "Quantum

chaos".

~

~

~~~~ _~=Î

3 ^{~ ~"~}

2

2.0

P(p)

5

1-o

o.5

Ù-o

o-o 0.2 0.4 0.6 0.8 0

b)

^~

Fig.

8.

a)

Distribution

P(p)

for standard BRMS of size N ₌ 100 and dilferent band _sizes b. The

change

in the distribution _is

clearly

_seen when

passing

from Iocalized (À = b~

IN

_{< 1) to} delocahzed (À » 1) eigenstates.

b)

Comparison of the distribution

P(p)

for standard

(open circles)

and

periodic (fuII circles)

BRMS; ^ail parameters _{as m}

Figure

7.

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