37? Mm

no. HU<]

ON DELOCALIZATION EFFECTS IN

DISSERTATION

Presented to the Graduate Council of the

University of North Texas in Partial

Fulfillment of the Requirements

for the Degree of

DOCTOR OF PHILOSOPHY

By

Anna Bystrik, B.S., M.S.

Denton, Texas

May, 1998

p.

Bystrik, Anna, ''On Delocalization Effects in Multidimensional Lattices",

Doctor of Philosophy (Physics), 1998, 128 pp., including 3 pages with 3 illus-

trations, 15 pages with 33 graphs, references, 58 titles.

A cubic lattice with random parameters is reduced to a linear chain by the

means of the projection technique. The continued fraction expansion (c.f.e.) ap-

proach is herein applied to the density of states. Coefficients of the c.f.e. are ob-

tained numerically by the recursion procedure. Properties of the non-stationary

second moments (correlations and dispersions) of their distribution are studied

in a connection with the other evidences of transport in a one-dimensional Mori

chain. The second moments and the spectral density are computed for the various

degrees of disorder in the prototype lattice. The possible directions of the further

development are outlined.

The physical problem that is addressed in the dissertation is: the possibility of

the existence of a non-Anderson disorder of a specific type. More precisely, this

type of a disorder in the one-dimensional case would result in a positive localiza-

tion threshold. A specific type of such non-Anderson disorder was obtained by

adopting a transformation procedure which assigns to the matrix expressing the

physics of the multidimensional crystal a tridiagonal Hamiltonian. This Hamil-

/

tonian is then assigned to MI equivalent one-dimensional tight-binding model.

One of the benefits of this approach is that we are guaranteed to obtain a linear

crystal with a positive localization threshold. The reason for this is the existence

of a threshold in a prototype sample. The resulting linear model is found to

be characterized by a correlated and a nonstationary disorder. The existence of

such special disorder is associated with the absence of Anderson localization in

specially constructed one-dim«nsional lattices, when the noise intensity is below

the non-zero critical value. This work is an important step towards isolating the

general properties of a non-Anderson noise. This gives a basic for understanding

of the insulator to metal transition in a linear crystal with a subcritical noise.

Key words: multidimensional disordered lattice, continued fraction expan-

sion, quantum localization, Green function, spectral density of states, recursive

relations, Lanczos method, Mori technique, projection technique, correlations.

37? Mm

no. HU<]

ON DELOCALIZATION EFFECTS IN

DISSERTATION

Presented to the Graduate Council of the

University of North Texas in Partial

Fulfillment of the Requirements

for the Degree of

DOCTOR OF PHILOSOPHY

By

Anna Bystrik, B.S., M.S.

Denton, Texas

May, 1998

COPYRIGHT© BY

Anna Bystrik

1998

111

ACKNOWLEDGMENTS

It is my pleasure to express my gratitude to Professor Paolo Grigolini, my advisor, who initiated this project and who encouraged and supported me throughout the preparation of this work, I benefited greatly from his inspiring and original scientific approach. It was my privilege to work with him.

I am grateful to Dr. Luca Bonci for all the effort and talent he brought to the computational part of this project.

The scientific discussions with Professor Bruce J. West were most helpful to me and provided me with the general perspective on the subject.

I would also like to thank Professor Samuel E. Matteson, Professor William D. Deering, Professor Jacek Kowalski, Professor Richard Mauldin, Professor Paolo Grigolini and Professor Bruce J. West for serving as mem-bers of my dissertation committee.

The constant encouragement of many members of the faculty of the Physics Department was truly invaluable to me. In particular, I want to thank Professor Kowalski for guiding my first steps at UNT and Professor Deering for his moral support.

I also thank the PIC Coordinator Jeff Hill for creating most friendly and productive working environment during my teaching experience at UNT.

Finally I wish to thank A. D.. Without his constant support, both mental and spiritual, this work would never be possible.

IV

CONTENTS

1 Introduction j

1.1 Abstract . . . . . . . . . . . . l

1.2 Overview / 3

2 Theoretical foundations 5

2.1 Models and phenomenon: Lattices and localization 5

2.1.1 Anderson localization 5

2.1.2 Multidimensional lattices 9

2.2 Mathematical Tbols 13

2.2.1 Projection technique 13

2.2.2 Recursion method 20

2.2.3 Green's function, density of states and memory function . 23

2.2.4 Continued fraction expansion (c..f.e.) 26

v

3 Statistical properties of Mori chain parameters and delocaliza-

tion 33

3.1 Statistical approach . . . 33

3.1.1 Statistics considered 33

3.1.2 Quantities of interest 35

3.2 Numerical calculations 42

3.2.1 The method, advantages and disadvantages 42

3.2.2 Graphs 44

3.2.3 Discussion g7

4 Conclusions and plans for the future research 76

4.1 Conclusions 76

4.2 Plans for the future research 80

5 Appendixes 82

5.1 Central limit theorem 82

5-2 Proof of the fundamental recurrence formulas for continued frac-

tio*1 85

VI

1. INTRODUCTION

1.1. Abstract

A cubic lattice with random parameters is reduced to a linear chain by the means

of the projection technique. The continued fraction expansion (c.f.e.) approach

is heron applied to the density of states. Coefficients of the c.f.e. are obtained

numerically by the recursion procedure. Properties of the non-stationary second

moments (correlations and dispersions) of their distribution are studied in a con-

nection with the other evidences of transport in a one-dimensional Mori chain.

The second moments and the spectral density are computed for the various de-

grees of disorder in the prototype lattice. The possible directions of the further

development are outlined.

The physical problem that is addressed in the dissertation is the possibility of

the existence of a non-Anderson disorder of a specific type. More precisely, this

type of a disorder in the one-dimensional case would result in a positive localiza-

1

tion threshold. A specific type of such nan-Anderson disorder' was obtained by

adopting a transformation procedure which assigns to the matrix expressing the

physics of the multidimensional crystal a tridiagonal Haxniltonian. This Hamil-

tonian is then assigned to an equivalent one-dimensional tight-binding model.

One of the benefits of this approach is that we are guaranteed to obtain a linear

crystal with a positive localization threshold. The reason for this is the existence

of a threshold in a prototype sample. The resulting linear model is found to

be characterized by a correlated and a nonstationary disorder. The existence of

such special disorder is associated with the absence of Anderson localization in

specially constructed one-dimensional lattices, when the noise intensity is below

the non-zero critical value. This work is an important step towards isolating the

general properties of a non-Anderson noise. This gives a basic for understanding

of the insulator to metal transition in a linear crystal with a subcritical noise.

Key words: multidimensional disordered lattice, continued fraction expan-

sion, quantum localization, Green function, spectral density of states, recursive

relations, Lanczos method, Mori technique, projection technique, correlations.

1,2. Overview

In periodic systems, a consequence of the validity of the Bloch's theorem is that

one-electron eigenstates are always extended. This is not the case if the system's

periodicity is violated. The latter can be done by applying an external field or by

modeling a transport process as an asymmetric one [1], One of the popular tools

for introducing a non-periodicity in the periodic system by using random variables

as a system's parameters [2]. This is going to be our approach as well. Once the

periodicity is violated by any means, an extension of eigenstates is not necessarily

an order. The wavefunction's localization becomes a possibility. Randomness is

associated with the phenomenon of Anderson localization [3J.

The recently introduced [4] approach links possible correlations between the

random parameters of the system of interest to the enhancement of the localization

length (and the possible breakdown of localization). Essentially, we are dealing

with two competing tendencies: Uncorrelated randomness is known to induce

localization, whereas correlations tend to destroy localization. The presented

dissertation is another step in the direction to confirm the latter.

Instead of adopting a traditional approach of generating correlated energy

fluctuations either by the means of dynamical map-driven disorder [4] or by

"man-made" symmetry requirements [5] we investigate a tight-binding Hamil-

tonian obtained by means of a special reduction of the prototype system to the

one-dimensional model. The reduction is done by means of the projection tech-

nique.

This mathematical procedure results in both correlated and site-dependent

noise. The role of a short-range correlations is noted. An appealing new feature,

nonstationarity of correlations and dispersions associated with the evidence of

transport, is emphasized.

2. THEORETICAL FOUNDATIONS

2.1. Models and phenomenon: Lattices and localization

2.1.1. Anderson localization

Localization phenomena in periodic structures have been a subject of fascination

in the scientific community since their discovery by Anderson in 1958 [3]. Ander-

son studied a single realization of a tight-binding Hamiltonian with site energies

uniformly distributed with width W (a "diagonal disorder"). This model was

used by Anderson to investigate the dependence of the electronic diffusion in the

random crystal potential on the ratio *7. It was shown that in a one-dimensional

system with site coupling strength V electron states are localized for any positive

noise intensity W. There is an extensive literature devoted to the physical and

mathematical nature of this phenomenon [6j. In his Nobel speech [7], Anderson

discusses the crystal Hamiltonian

H ~ ]C€m lm) (ml + ^nm ̂ n,tn

where cw denotes the energy of the site |ra) and Vnm denotes interaction between

crystal sites \n) and (mj, which are not necessarily nearest neighbors. The crystal

considered by Anderson is not necessarily one-dimensional. Later in this disser-

tation, the attention will be concentrated on a one-dimensional sitnation[8]. The

energies of the sites are randomly distributed with the distribution of width 2s

and of a general bell-shaped or box-shaped form.

The density of states D (E) is introduced to describe the number of those

eigenstates of (2.1) that correspond to an energy interval from E to E + dE.

This quantity changes with increase in the width of the distribution E from a

smooth curve D (E) to a graph characterized by a sharp spikes to a collection of

delta-fimctions ("discrete spectrum"). According to Anderson, when the energy

spectrum is discrete, the eigenvectors are localized.

Let us consider the eigenstate t>p (r), where the space variable £ ranges over

the entire space of the Hamiltonian (2.1). Let us concentrate our attention on

the specific case when £ coincides with the position of a given site. The density

of states at a given position is given by the expression

D(E)~*£\%(r)\26(E-Ep) (2.2)

(we omit normalization to unit crystal volume).

Let us now define a Green's function as the matrix element of a resolvent ^

and let us express it in terms of position-dependent eigenstates:

G(L,Z!,E) = (L E-H

L ) = (2-3)

) b 3 E < I P p

By utilizing the property

ton 1 = PP— iirS (x - XQ) B—^0 X XQ *4" ^ X XQ

(2.5)

the Green's function can be re-written as

G CcrT, B-¥ii)=PP'Z%(E)%(t') W W s ( E - ^ '

(2.6) V P

where PP denotes the principal part. Compare the latter result to (2.2), from

which we conclude

D(E) = -~ImG{rir,E), (2.7)

so the density of states can be expressed as a diagonal matrix dement of the

resolvent.

7

Let us denote the initial state of an electron in our lattice as (0)) = \m)

and define the correlation function as the overlap integral

(t) — e xP ( - i f f t j jm), (2.8)

where t is the waiting time and exp {-iHtj is an evolution operator based on the

Hamiltonian H. This quantity is related to the probability of finding the electron

in its original state after a time t (so-called survival probability):

p. = | ( m j * m ( i ) ) | 2 H < M t ) | 2 ' (2-9)

On the other hand the Laplace transform C ($m (£)] of a correlation function

OO -(s) = C [3>m (t)] « J exp (-st) (t) dt = (m| — — j m ) (2.10)

0

can be expressed in terms of the Green's function

(-iE) = iG (m, m, E). (2-11)

Assuming that the crystal is infinite by large, it can be shown [9], [8] that there is

no diffusion when lim ps is finite, t-~>CO

lim p„ {t)^p^ 0, (2.12) £—+OQ

indicating localization (for an extended all over the crystal, but initially local-

ized, wave function the probability of the electron presence at its initial position

8

becomes eventually zero). Additional discussion on survival probability can be

found in [11], [8] and [12}.

2.1.2. Multidimensional lattices

Localization of the wave function may be induced when we introduce randomness

into either diagonal or non-diagonal (or both) elements of the system's Hamil-

tonian. However, the localization may be destroyed if the random energy fluctu-

ations are correlated (examples may be found in [4]).

We will address the nature and source of this randomness in Section 2.2 and

also in Chapter 3.

Consider a one-dimensional tight-binding Hamiltonian

Ha, = V) am jtn) (m| + b £ (|n) (m\ + |m) (n\) (2.13) m n>m

fn-m|=l

with fixed nearest-neighbor coupling strength b and site energies am. Assume

that the site energies are mutually independent random variables with a given

distribution, and that the parameters of this distribution axe fixed once and for

all. Then it is well-known that no transport of electrons can occur in this case

[3], [13].

It is notable that any degree of randomness localizes the wave function of a

one-dimensional delocalized system. That is, there is no threshold below which

9

noise could be disregarded. We will refer to the width (square root of dispersion)

of the discussed random distribution as the noise Intensity. This noise intensity

can be infinitesimally small, but still will inevitably induces localization.

The same effect of the localization of the system's wave function can be ob-

served if the site couplings are uncorrelated random variables.

Let us consider a three-dimensional (cubic) lattice described by a Hamiltonian

similar to (2.13). We write i, j and k for discrete spatial site coordinates in x,y and

z Cartesian directions. The lattice is infinite in all directions and the coupling is

non-zero only between the sites whose multiindec.es a = (i, j, k) are a unit distance

apart, commonly referred to as nearest-neighbor (n.n.) coupling. The value of the

coupling is assumed to be constant and site energies e(«) are randomly distributed

around a common value ec:

c (a) = ec -j- s (a) (2-14)

We will refer to these energy fluctuations e (a) as noise. Noise intensity, S, is the

parameter responsible for spatial localization/derealization of the system's wave

function \1>.

In a perfect crystal we would observe long-range correlations! and perfect con-

ductivity. In addition, the wave function would be delocalized. If uncorrelated

noise is introduce to the system, we observe a "metal-insulator" transition: local-

10

ization characterized by quenched correlations. But if we introduce correlated

fluctuations they may allow an electron to undergo a process of quantum diffu-

sion by tunneling from site to site, thus increasing conductivity by extension of a

large percentage of eigenstates over the whole crystal. This phenomenon, known

as derealization, we also found to be associated with the site-dependence of the

intensity of energy fluctuations in the equivalent lattice (to be addressed later).

The value of the noise intensity is connected to the value of localization length

I. The existence of a so-called threshold value S = ay* is well-established in the

literature [14]. When 3 exceeds the localization takes place. If S is below

3th, the initially localized wave function may eventually become non-zero on an

arbitrary site.

Note that in our model the random fluctuations of the energies of any two

sites are mutually uncorrelated. The extreme subcase of S = 0 would correspond

to totally correlated noise, when no randomness is present in the system and no

localization phenomena can be observed. The reader may also think of the case

of large energy fluctuations (E > 3th) that are highly correlated in such a way

that all site energies e(a) are close to each other, providing good conditions for

tunneling and delocalization in the system. Thus one can ask whether a specially

correlated noise of intensity 3 above the threshold value is sufficient to produce

11

localization. Artificially correlated nontrivial noise was introduced in [4] and was

found to suppress localization . Although a breakdown of the localization was not

found, a strong increase in the localization length was observed. Now the way to

proceed would be to establish the connection between specific characteristics of

correlated noise and the suppression of localization. If successful, the connection

would enable us to predict localization/ delocalization of the wave function based

on the type of noise and vice-versa.

In Section 2.2 and in Chapter 3 we introduce a one-dimertsional lattice in

which site energy fluctuations are correlated in a special way. rhe system with

these special fluctuations possesses a non-zero localization threshold. For the

values of energy fluctuations below the threshold, the wave function is delocal-

ized. Delocalization is preceded by an increase in the localization length that

can be detected. Enhancement of the transport process will be manifested in

the behavior of various quantities (correlations, dispersions, energy level density,

transmission coefficient) described in Chapter 3. We will base a construction of

this fictitious one-dimensional lattice on some three-dimensional prototype lattice

with below-threshold uncorrelated noise. The latter lattice is known to have a

wave fiuiction. We employ a projection technique described in the

following Section and then analyze the character of correlations present in the

12

resulting one-dimensional lattice.

2.2. Mathematical Tools

2.2.1. Projection technique

This subsection contains the procedure for replacing an infinite multidimensional

lattice with an equivalent semi-infinite one-dimensional lattice with noninteraction.

Different versions of this procedure are known as Lanczos method, Mori technique

and the projection technique.

The Nakajiama-Zwanzig projection method deals with time-independent phys-

ical variables and their time-dependent probability distributions like the Schrodinger

picture in quantum mechanics.

The Mori approach deals with time-dependent physical variables and time-

independent probability distributions, like the Heisenberg picture in quantum

mechanics. Accordingly, a "Schrodinger representation" and "Heisenberg repre-

sentation" terminology was adopted in [15] and [13]. The continued fraction proce-

dure used herein is founded on the projection technique applied to the Heisenberg

picture.

The aforementioned method will be used by us as it is described in [16], [13],

13

[8].

The Lanczos scheme replaces the original Hamiltonian with a Hermitian tridi-

agonal matrix. That matrix can be interpreted as the Hainiltoman for a fictitious

one-dimensional lattice:

O O O O " o -

A0 Ai A% As An

Figure G1

Site emerges and n.n.-couplings of this one-dimensional chain are given by

the diagonal and off-diagonal elements of the tridiagonal matrix respectively. We

denote a vector of diagonal elements by

A = [A-]i=o,nc-i (2.15)

and a vector of off-diagonal elements by

B = (2.16)

where subscripts are reserved for site indices. Here Bi+1 is the coupling between

the site i of energy A, and site i + 1 of energy Ai+j. The notation fi is reserved

14

for the states and will be addressed later in this Section. The number of sites of

the one-dimensional chain equals nc (i.e. it is given by the number of diagonal

elements). Ideally, this number would be infinitely large, thus representing a

semi-infinite Mori chain. Of course, for computational purposes we are forced to

introduce a finite value of the linear size nc the choice of which will be farther

discussed in Chapter 3.

This transformation preserves many important qualitative features of the orig-

inal multidimensional lattice, such as localization of the wave function on the finite

number of sites. This fact is of particular interest for us.

Let us describe the projection method according to [8], [13] . The method re-

places the multidimensional Anderson lattice with the semi-infinite one-dimensional

chain. We will use a cubic lattice as a prototype lattice, but the approach is

applicable to different models, including Cayley tree with an arbitrary number

of nearest neighbors of a given ate. Let us make a choice of a particular site

fa = (rni,rfl2,ffl3) and let us consider this site to be a departure point of a semi-

infinite one-dimensional chain,

l/o) = !*B> • (2.17)

We shell not assume the states of the three-dimensional lattice to be nor-

15

malized. As we will see, even if |/0) obeys a normalization condition it will not

automatically provide for the rest of the states j /$) to be normalized. The formal

solution of the Schrddinger equation

j t !/.(«)> = - i f f |/o (t)> (2.18)

provides the evolution of the wave function |/o (£)) starting with an initial condi-

tion j/o):

|/o (*)) — exp (-iHtj | / 0 ) . (2.19)

The initial condition refers to the moment when time t = 0. Now let us rewrite

the equation for the correlation function (2.8) as

$ U\ _ (/O l/o (*)) / a nft\ ( 2 " 2 0 )

where normalization is explicitly included. We define the projection operator

p _ l/o) </o| f n 0 - v

p " ~ M m ( 2 ' 2 1 )

on the Hilbert space spanned by )/o) and the complementary projection operator

Qo = 1 ~ Po- (2.22)

The definitions of projection operators [P,], projected Hamiltoniaw [Hi], orthog-

onal states {j/o), | /i), I/2), • • •} and correlation functions {$*} are introduced

16

below:

Pi

Hi

I ft)

M * )

m ifi\

Villi)

( f i (1 " pi)}&

j=o

ftl/i-i)

( M f i ) '

(2.23)

With the help of (2.22) and (2.23), one can re-write (2.18) as

£ dt I fo (t)) = —iAo \fo (t)) - iexp ( - i F t ) | / i ) , (2.24)

where

A) . {fo H\f0)

(2.25) (/o|/o) *

Then, using inverse Laplace transform of Dyson identity and properties of the

projection operator, we can see that SchrcJdinger equation (2.18) is equivalent to

Langevin-type dynamics expressed by the equation

l £ dt |/„ (t)> = - i 4 ) l/o « > - B? J l/o (t - r)> (t) ir - i | / , (t)>, (2.26)

where

B? ( f i l / i ) _ {fo H\fo)

(/o|/o) (/o i/o)

This result is related to Mori's approach, which can be found in [17].

17

(2.27)

Note, that we can using the properties of the projection operator, the following

holds true:

(/» | g j / . ) - (/o | g j / . ) • ( / . |A[ /») Bi (2.28)

</ol/o>

Prom the definition (2.25) and the property (2.28) it is clear that Aq measures

a mean quantum mechanical energy (h^ of the state |/i) and B\ measures a

quantum mechanical variance (uncertainty AH) of this quantity. If we imagine

that (for some reason)

J3i == 0, (2.29)

then

AH = 0 (2.30)

and the state j/e) is an eigenstate of the Hamiltonian. Similarly, if the coefficient

Bi vanishes, it would mean that a projected state |/<_i) is an eigenstate of a

projected Hamiltonian Hi-i-

The iterative procedure that corresponds to the dynamic equation (2.26) leads

to the integro- differential equation

dt

where

X

I Si (<)) = ~iAi I Si (*)) - Bi+1 /1 Si (t - r)) $»+1 (r) dr - i j/ i+1 (t)), (2.31)

A = {/<!/<>

18

(2.32)

and

. </.!/.> </i-i l-gj /«> (2.33)

( / w l / w ) (A-ilA-i) "

Thus the parameters of the one-dimensional chain (shown on Figure [Gl]) are

defined.

It can be shown [13] that, based on (2.23), the following is true:

- H l / < > - 4 l / < > - B ' i l / i . , ) . ( 2 . 3 4 )

Then if we set in the recurrence relation (2.34) the coupling coefficient B?

to be equal to zero, the state j/»+i) would be formally defined by its immedi-

ate predecessor \fi) only. The truncation of the iterative procedure takes place:

the sequence [|/o),..., j / i ) ] exhaustively describes the dynamics concerning j / o ) .

Additionally, this would mean that the projected state | / i_i) is an eigenstate of

the projected Hainiltonian Hi-i. The latter follows from the fact that (by anal-

ogy with (2.30)), the condition of vanishing ith coupling leads to zero quantum

mechanical uncertainty:

A Hi„i = 0. (2.35)

Note, that while the state |/j) is not (necessarily) an eigenstate of the original

Hainiltonian, those eigenstates of the original Hamiltonian that overlap with \fi)

19

can be found by diagonalizing the three-diagonal Hamiltonian obtained by means

of recursion method (described in the next subsection). Often the simple trunca-

tion is not satisfactory. We will address this delicate question later.

2.2.2. Recursion method

The approach we are about to introduce describes a multidimensional lattice of

interest as a collection of shells. The central shell consists of an initial site only.

The energy of this site is not modified and is declared to be the energy of the

initial site of an equivalent one-dimensional lattice. In the nearest neighbors

approximation, the next shell consists of the nearest neighbors of all sites on

the previous shell (excluding those that were already taken into account). The

shell number i defines the radius of the iV-sphere in the space of the original N-

dimensional lattice. All the sites on this sphere contribute to the "energy" of the

i**1 site of an equivalent one-dimensional lattice. Their contributions are combined

to construct a set of orthonormal states {|/o), | / i ) , . . . , | /»),.. .} according to the

following formulas:

!/o) (2.36)

Bi\h) = H\fo)-Ao\fo)

20

ft|/2> = H\h)-Al\f1)-B1\fo)

Bi+i \fi+i) = H \f,) - Ai\fi) - B t i f a ) ,

where

A = f2-37)

B, =

This procedure is known as recursion method. It can be shown [13] that the

recursion scheme defined by (2.36) and (2.37) is equivalent to that defined by

(2.34), (2.32) and (2.33), with the only difference that the normalization require-

ment is already build in for (2.36). In the set the diagonal Green's function

matrix element can be expanded [8], [13] as

Goa (E) = ( f o E-H

fo) = W • (2.38)

' , ' B? E-Ai 2

r-. . B\ E-Ai ...

The expansion of the type (2.38) is called a continued fraction expansion (fur-

ther abbreviated as c..f.e.). The detailed description of the c.f.e. will be given in

another subsection of this Chapter.

21

We generate a large number of copies (configurations) of the original lattice.

Each configuration is characterized by a different realization of a random distrib-

ution of vertex energy fluctuations. For each configuration we transform (by the

means of recursion method (2.36) the physical system s Hamiltonian into a tridi-

agonal matrix, which serves as a representation of some one-dimensional tight-

binding system: a semi-infinite linear lattice with n.n. coupling (Mori chain).

This coupling, unlike the site coupling in the prototype lattice, is not necessarily

constant (in fact, in our case, we found it to be site-dependent). Additionally,

each different realization of the original noise produces different realization of

the parameters of the equivalent chain. We study statistical properties of these

parameters A and B.

A comprehensive review of the recursion procedure used to reduce a number

of many physical systems to one-dimensional description can be found in [18] and

in [13].

We perform averaging over an ensemble of systems subsequent to the recursion

procedure. This Gibbs approach is different from the original Anderson perspec-

tive when a single realization of the random energy distribution is considered. The

interplay between these two perspectives Is investigated in a recent paper [19].

22

2.2.3. Green's function, density of states and memory function

As mentioned already, if the strength of sate energy fluctuations exceeds a cer-

tain value compared to the coupling constant V, then the wave function of the

original system becomes localized. In this case a localization threshold must be

present in the equivalent system as well, which should be attributed to the ex-

istence of nonzero correlations. Therefore the behavior of the one-dimensional

system described by the tight-binding Hamilton!ail in the presence of the corre-

lated fluctuations is strikingly different from that in the presence of uncorrelated

noise.

The relationship between the density of states and the Green's function was

discussed in details in the first section of this chapter (see (2.7)). In the same

section the asymptotic behavior of the survival probability is discussed in con-

nection with the localization phenomenon. Below we will see how the projection

technique and continued fraction expansion can be applied to our study of these

quantities. In the next Chapter other quantities of interest, generated with the

help of the projection technique and an ensemble averaging, will be discussed.

A preliminary study of the density of states D (E) as a function of the energy

E gives us an opportunity to see its modification when we exceed the threshold.

23

The density of states (further referred to as d.o.s.) can be calculated as

D (E) = — lim Im Goo (E + iS), v 7T5-*+0

(2.39)

where the Green's function is given by the cJF.e. (2.38).

Let us assume for a moment that our original crystal is ideal (i.e. with no

noise). Then the parameters of an equivalent one-dimensional crystal (the coeffi-

cients A and B of the continued fraction (2.38)) are determined uniquely. If we

truncate an expansion (2.38) in such a way that an equivalent one-dimensional

lattice has nc sites, then its Hamiltonian is represented by an nc x nc matrix.

This coefficient matrix is of the tridiagonal or Jacoby form in which its dements

= 0 if \i — j\ > 1. That is,

Aq B\ 0 0 . . . 0

Hid

Bx Ax Bi 0 ••• 0

0 /?2 A3 Rj • • • 0

0 0 B$ A4 . . . 0

0 0 0 0 A ric

(2.40)

The original three-dimensional site energy disorder is a product of random

number generator. This randomness models energy fluctuations in the real phys-

ical system.

24

The ratio of the intensity of energy fluctuations to the coupling constant p

serves as the quantitative measure of the noise intensity. For ^ < 1 the effects of

randomness can be disregarded.

Numerical computations are capable of producing the values of the Green

function Goo {E + iS) for our equivalent one-dimensional system on the discrete

set of energies only . Additionally, we have to use a small constant S > 0. The

consequence of 6 being positive is an artificial "widening" of the spectral lines.

Thus, it is imperative to use an energy step that is smaller then 6. That will

provide the resolution that is high enough to enable us to see all the spectral

peaks.

The actual output of our calculations is the d.o.s. given by (2.38).

For example, if our "energy" range is from —10 to 10 and the number of

discrete energy points on an abscissa axis is 1000, then the energy step is 0.02 and

the constant <5 we used was 6 = 0.05 > 0.02. Plots for the numerically computed

density of states [G14] indicate that number of peaks in d.o.s. decreases as the

noise increases. For example, for the fictitious linear chain of nc — 10 sites we

observe only 9 peaks, for nc — 20 in the presence of noise we observe only 17

peaks, and so on.

Let us consider the set of integro-differential equations (2.31). By multiplying

25

(from the left) this expression by | f i ) , the following can be obtained:

l (t) = -tAfy (t) - Bf+1 J $i(t- r) $ f + 1 (r) dr, (2.41) o

that the function 5>;+i can be interpreted as memory function of the correlation

d_ dt

function

By performing a Laplace transform (defined by (2.10)) of the equation (2.41),

the following can be shown (with the help of convolution theorem):

a. • ( s ) - 1 = - i M * ( s ) - B f h ( • ) ( s ) . (2.42)

Iterating the equation (2.42), it follows;

|>0 (S) = — — |j2 '

$ 4 iAo + - " "Jg

s 4- iAi 4-

2

Bi 8 4" iA.% 4 s + iA$ + • • •

which is consistent with (2.11) and (2.38).

2.2.4. Continued fraction expansion (c.f.e.)

Mori chain energies and couplings may be interpreted as coefficients of a continued

fraction expansion (c.f.e.) of the corresponding Green's function (2.38) (or of

the corresponding correlation function (2.43)). Continued fraction expansion is

26

a mathematical tool designed to provide better convergence compared to other

commonly used expansions (e.g. Taylor series). This property is one of great

practical importance, since we are often forced to use truncations of infinite series

for the purposes of executing numerical calculations.

For more information about mathematical properties of the c.f.e. one may

refer to [20], [21]. The relationship between power series and continued fractions

is discussed in details in [13], Certain c.f.e. associated with power series can

converge outside the domain of convergence of the series.

If the c.f.e. of the Green's function (2.38) is truncated at ith level, then the

corresponding recursion procedure is truncated and the projected state i/i-i) is

an eigenstate of the projected Hamiltonian Hi~\ (see (2.35)). The Mori chain is

cut and the energy spectrum exhibits a transition from continuous to discrete,

the property that, according to Anderson, indicates localization. However that

discrete behavior of the density of states only mimics Anderson localization. One

may suggest, based on the continued fraction approach, that for some reason

(related to the fluctuations of the expansion coefficients caused by the disordered

distribution of energies) a given Bi vanishes.

But often it is not so (in our case numerical calculations specifically indicate

that it is not so). The sharp peaks in the energy spectrum must be attributed to

27

the effect similar to the "decoupling effect" introduced in [22], when (instead of a

decoupling from the remainder part of the environment) a quenching of dissipative

properties induces by environment takes place, More on this subject is presented

in the Discussion section.

Note, that some works that we refer to adopt the same continued fraction

approach applied to the classical Liouvillian L. The connection between quan-

tum SchrSdinger dynamics and the classical dynamics within the framework of

the Heisenberg picture can be established by correspondence L <-»• iH.

Projection method is a very general one and can be appUed to any "reasonable

operator, not necessarily H or L. Recursion method can be used in order to obtain

spectral density of this operator. It can be used to diagonalize matrices (Lanczos)

and can be adopted for non-Hermitian operators (([13]), ([23])).

Now let us focus on the tail of order i, that would be disregarded by truncating

ci.e. at i th level:

B\ k [ E } ±

E-A B2

(2.44)

E-A+l i+2

E-A+2- i+-E-...

28

so that

Goo (E)= R — Aq * ~ u2 '

B-* ^ - a E-A2-

3 2 d 2

E - • *~l-E-A-i- U (E)

Following [13] we can assume that asymptotic values of A and B? exist and equal

A and B2, respectively:

A = lim AI (2-46) i~-*oo

JB2 = lim BF. %—+OO

Similarly, we introduce

t (E) ± lim ti (E), (2.47) t—*00

called terminator. Since the exact relation

holds true, the terminator satisfies the self-consistent condition:

t(E)~ E-A-t(EY (2.49)

which leads to

t(E) = \[E-A-J(E-Af-W>\, (2.60)

29

were the choice of sign is the consequence of » dissipative requirement discussed

below.

Let us denote, without making any assumption at this stage about the tail

assuming real or complex values,

r(A') = (2.51)

T(E) = \JiB2 -(E- A)2.

Then (2.45) takes the following form:

Goo(B) = ' B?

E — AQ E — AI , S

E-A2 S BL i

E-A-i~r(E) + iT{E)

(2.52)

So, in the special case when coupling B is strong enough, the energy of i

can be effectively replaced with (A-i + r {E) — ir (E)), were the quantities r (E)

and r (E), defined by (2.51), are real. Consequently, the time evolution of the

projected state can be expressed as r (E)t

J/»-i (t)} = exp [~i (Ai_i + r (E)) t] • exp \fi-i), (2.53) 2

so that we have a dissipative decay that dictates the choice of sign in (2.50). The

interplay between the dissipation and external perturbations has an interesting

30

connection with real (i.e. not induced by a truncation of a c.f.e, and corresponding

Mori chain) Anderson localization and with the problem of irreversibility. This

connection is discussed in [?].

The existence of the limits (2.46) is hard to prove analytically for most of the

realistic cases of interest. But numerically "asymptotic" values of A's and B's

can be computed, roughly speaking a stable stationary value (if any) that the

coefficient of c.f.e. reaches can be accepted as such. The generalization of the

(2.49) for the case of ensemble averaging procedure performed on isotropically

disordered original lattice was done by [25] and is discussed in [13].

Partial numerators and partial denominators of the c..f.e. (2.38) are expressed

as [1, - B \ , . . . , -Bl , . . . ] and [E - Ao,..., E - An,...], respectively. By setting

the partial numerator Bn equal to zero the c.f.e. can be truncated at the n*h

level, the truncated c.f.e. is called nth approximant (or nth convergent). The 11th

approximant can be written as a rational expression , where nth numerator Pn

and n01 denominator obey the fundamental recurrence formulas [13], that in our

case can be written as

Pn+1 = (E - Ar>) Pn - (2.54)

- f t n - f l = {E An) Rfi BnRn-l,

31

> where n — Q,1,2,...,Bq — 1.

The initial conditions for nth numerator are

= 1 (2.55)

Po - 0

and the initial conditions for nth denominator are

i?_i = 0 (2.56)

Ro — 1»

Hie formulas (2.54) can be proven by induction. Since the proof is omitted in

[13], we performed the calculations in the appendix 2.

32

3. STATISTICAL PROPERTIES OF MORI

CHAIN PARAMETERS AND DELOCALIZATION

3.1. Statistical approach

3.1.1. Statistics considered

In our model the site energies c (a) of the original three-dimensional lattice are

random variables distributed with a probability density / . The different options

for / are described below.

1. Energies e can be uniformly distributed on the interval IX1X2]. A "flat"

probability density is given by / / ^ = X2-Xl • Then the expectation of the

energy over this distribution is

mr "f" -t \

M e = — - — (3.1)

33

and the variance of the energy (dispersion) is

12 D e - ^ e - M e f ) ^ ^ (3.2)

We will refer to the width of the non-zero energy interval \x2 — X\ | as to the

noise intensity E/j.

2. Energies can also have a normal (Gaussian) distribution with probability

density

f a "

( 3 - 3 )

so that m gives the expectation value and d gives the dispersion. We refer

to the half-width of our distribution as to the noise intensity E©. This

quantity is defined by the square root of the dispersion (3.6).

Other options for energy statistics may include the binary distribution, the

Cauchy distribution, etc. One may associate various reasons for the presence of

energy fluctuations (for example, interaction with thermal bath or the presence

of impurities or coupling with a chaotic system) with specific distributions. In-

teraction with a bath of phonons that models thermal vibrations of lattice ions

produces Gaussian energy fluctuations, impurities may result in "flat" ("box") or

Cauchy distribution, both studied by Anderson, and so on. However the quali-

tative effect should not depend on the specific type of distribution (while critical

34

values of parameters may vary, so we cannot, strictly speaking, compare intensities

of a flat noise with that of a

Gaussian noise).

Statistical properties of an equivalent tmidimensional lattice are evidently dif-

ferent from those of original three-dimensional lattice and are to be investigated.

Their behavior and its impact on the localization of the system's wave function

is the subject of this dissertation. .

3.1.2. Quantities of interest

All of the quantities of interest are computed as averages over different realizations

of noise. All of the quantities of interest are dependent on the type of statistics

and are affected by the fact that total number of realizations nr and the length

nc of the truncated Mori chain are finite.

So far we introduced two random vectors (2.15) and (2.16):

A — 1

and

B = [£<]*=!,

They were obtained in Chapter 2 from the set of mutually independent random

parameters of the original three-dimensional lattice by the means of projection

35

technique . Since each component of the vectors A and B is derived by vising more

than one site of the original lattice, they are no longer necessarily independent.

Denote by

(3*4)

their common distribution function. Here a* and represent deviations of values

assumed by AI and from their mean values M [A] and M {!%]. It is said that

ai and bi are centered around their average values.

Quantities

WAB(̂ 1, • • • , ^2nc-l) = J ' • • J ^1) • • • , An,.-1, &1, • • •, K c ~ l )

(3.5)

where k\,..., > 0, are called mixed central moments of the random vari-

ables AQ, . . . , Bnc_ i of order k — ki + ... k2nc-i • They may be expressed as e$pec-

tations M

Mixed moments of the first order are simply expectations M of the individual

random variables. Moments of second order play a special role. If we fix jp,

0 < p < 2nc — 1 and set fcj = 26ip for all I, we obtain the dispersion D of the

corresponding random variable £. The square root of the dispersion

(3-6)

36

is called the mean square deviation of the variable £. When all ki are 0 except

kP = 1 and kq »= 1 (0 < p < q < 2nc — 1) we get covanance of two random

variables from our set:

M [(£ - M£) (rj - M7?)j = cm (£, t])

Correlation (or correlation coefficient) of two random variables is determined from

R, / ,-x ( 3 " 7 )

We will mention a few properties of the correlation coefficient:

1. —1 < RitV < I. If variables £ and rj are such that R^n = - 1 they are called

anti'CorrelatecL If variables £ and r) are such that — 1 they are called

completely correlated.

2. if = 1, then with probability 1

^ = + (3.8)

i.e. in this case the relationship between £ and rj is linear. Thus RiiV may

be viewed as a measure of linear dependence between variables £ and r).

3. If variables £ and 77 are such that R^v = 0 they are called uncorrelated.

37

We will perform a large number of numerical experiments that will generate

random energies of the original lattice. For every experiment a new realization

of noise will be used. Based on their realizations, coefficients A and B will be

calculated for an equivalent one-dimensional lattice.

Let vlf'and S f ' be specific values of the coefficients A and Bt with rcspcct to

the rtft realization of noise, r = 0 , n t .

The total number nr of realizations1 would have to be infinitely large for a per-

fect statistical analysis. Our upper bound is limited by computational capabilities.

The subject of computational errors is to be addressed later in this Chapter.

In the special case when our random variables are discrete, the equations (3.5)

and (3.7) assume the following form.

• Dispersion (variance) is given by the formula

D A « MA2 - (MA)2 , (3.9)

where

n r r=l

is the mean value (expectation) of the energy fluctuation of the ith site of a

one-dimensional lattice and

M A 2 = - | - i : ( 4 " ) 2 - (3.1J) r r=l

38

Similarly,

DBj = MB? - (MBi)2, (3.12)

1 ^ = £ X X ' >

m«,2 = ̂ E^r)2. r=l

We study averages MAj, MS, and dispersions DAi,DBi as functions of site

number i for various noise intensities. Additionally, we study dispersions for

a fixed site i = (nc — 1) as functions of the intensity of energy fluctuations

in the prototype lattice.

Note again that we average over different realizations of noise.

Alternatively, it is possible to introduce 'time" averaging over one-dimensional

chain, for example

i nc-l -I ne-1

DtimeA ~ ~ v4? — ( j p Ai)2, (3.13)

nc i=o nc i=0

however the latter kind of averaging procedure might be beneficial (i.e. corre-

spond to the Gibbs averaging used by us) only for stationary processes. For a

stationary process the "ergodic* situation takes place: "time" averaging coin-

cides with averaging over ensemble. We found that our case, on the contrary,

exhibited strong non-stationary behavior. The need to introduce ensemble aver-

39

aging aroused originally from the consideration of correlation functions (discussed

below) as quantities of interest.

• R*n denotes a correlation coefficient between energies of the Ith site and of

the (i + n)th ate (i.e. n is the distance between these two sites along the

Mori chain).

For the special case n — 1 we write Rf for Rft. We will refer to coefficients

R^n as to diagonal correlations.

Similar equations are used for the coefficients R?n, which denote correlations

between coupling of the ith and (i +if1 sites and that of the (i + n)th and

(i + n 4-1)—sites . In the next section we conduct a numerical study of a

dependence of R£„ and Rfn on i for various intensities of energy fluctuations

in the original (three-dimensional) lattice. We refer to coefficients Rfn as

off-diagonal correlations.

Note that both diagonal and off-diagonal correlations are symmetrical, i.e.

Ri,n — Ri+n,~tf

• Rf£ denotes a correlation coefficient between an energy of the ith site and

a coupling of the (i + n)th and (i + n + l^sites. Likewise, a coefficient

40

Rf^n »s introduced below. We refer to coefficients RfJ? and Rf * as cross-

correlations:

ffAB _ J_ 1 — MA) (-fij-fn — MBt-fn) ^ «i>n - n r ^ y c ^ . D ^ + n ' 1 '

p M _ 1 A (B,M - M B , ) ( 4 5 „ - M 4 . ) k " n - h y/OBt • DAj+„

The cross-correlations are also symmetrical, i.e.

• We will also study i?p>n (for both As and B's), R*g and R^ as functions

of i = p + n for j? fixed.

Other quantities of interest are introduced below.

We define a relative coupling strength as

c ' - j r k i <316>

and a transmission coefficient as

j7. — C3i-7\

1 A-Am ( 3 - 1 7 )

The values assumed by random variables of equations (3.16)and (3.17) during

the Vth numerical experiment will be written with the superscript [r]. Finally, we calculated d.o.s.

41

D (E) ~ lim ImGoo (E + i6), 7T $ —*-fO

as it was described at Chapter 2.

3.2. Numerical calculations

3.2.1. The method, advantages and disadvantages

We implement a realization, then calculate vectors A and B based on this realiza-

tion. Then we calculate MA and MB— averages over different realizations. We

will subtract these averages from individual realizations of coefficients to obtain

centered coefficients.

Our graphs are to be described in the next subsection. The error bars are not

shown to preserve the clarity of the figures. Nevertheless, there is an unavoidable

error of the order of magnitude ^==. It is caused by the fiiriteness of the number of

realizations. The number of realizations nr is limited by the time of the program

execution and by the amount of available memory. The execution time (as well as

an amount of required memory) increases linearly with nr. It is possible to save-

some memory by directly utilizing disk space but that increases time needed for

the operations of data storage and retrieval.

42

Another problem is associated with the necessity to truncate c.f.e. Hie largest

nc we used was equal to 64 and in many cases it was even smaller.

In the study (25] of application of the recursion method for disordered sys-

tems, the authors investigate the convergence of a ci.e, of spectral densities.

They investigate various two- and three-dimensional models and had find that

the number of shells under consideration (i.e. continued fraction coefficients, or

the length of the Mori chain) 11 < re < 15 is sufficient for realistic d.o.s. Our

numerical calculations lead us to the same conclusion with regard to the density

of states for our model.

If uncorrected noise results in a localization length I' that is known to be

finite, we migjht face the possibility that correlated noise will result in a greater

localization length I, I > V. If the localization length exceeds the size nc of a one-

dimensional crystal we consider the system to be devocalized. It is a common way

to deal with boundary effects. If we study our system for sufficiently high values

of noise we may ensure that states are strongly localized and boundary effects

can be neglected. There is also another way to deal with this problem. Our plots

can be extrapolated beyond nc to mimic an infinite crystal. Continued fraction

expansion used in our numerical algorithm is designed to provide better conver*

gence compared to other commonly used expansions. This aspect is discussed

43

in detail in[13]. The recursion method, employed by us, provides a reduction of

the cubic lattice with random vertex energies to a semiinfinite linear chain. The

advantages of this method were pointed in [18]. Our motivation was to establish

a priory existence of positive localization threshold.

We developed several algorithms for the purpose of conducting numerical com-

putations outlined earlier in this work. The following references were used in the

design of the algorithm: [43], [40], [41], [42].

3.2.2. Graphs

Obviously, the random vectors A and B are different for different noise realiza-

tions. Every sample of a random realization of one-dimensional site energies for

different sites is unique, only average characteristics are (almost) reproducible.

First, the figures that correspond to box ("flat") distribution of energy fluc-

tuations in the prototype lattice are going to be discussed, then we proceed to

the description erf graphs for the case of Gaussian distribution. All graphs are

presented in dimensionless units.

For a "flat" distribution of energy fluctuations the results for mean values of

diagonal and off-diagonal coefficients are presented in Figure [G2].

We see in this figure that the mean value of the elements of the vector MA

44

oscillates around zero {as they should by their construction) with an amplitude

small compare to the average element of MB.

We also see in this figure that the mean value of the coupling between sites is

positive and increases slowly along the Mori chain remaining almost constant for

the sites with i > 10.

(Note that if the graph of versus t 'intersects" the i-axis, it does not

imply that the average coupling equals zero. Zero coupling would imply that the

c.f.e. (and the transport process) terminates. Being plotted as dots, the mean

coupling values are mostly located off-axis. Zero mean may also indicate an equal

weight of positive and negative coupling constants, while none of them is actually

zero. The same comment applies to the graphs for the further discussed quantity

Co)

On the graphs presented in this work the position i of a quasi-atom is also

referred to as the level number" or "coefficient number" of the corresponding

c.f.e.; the original noise S is referred to as "est".

We used as a computational tool recursive building of the coefficients of the

continued fraction expansion for the level density. Dispersions of these coefficients

versus a coefficient number (as well as versus the noise strength) were calculated.

Results are presented in Figures [G3], [G10].

45

The error bars are not shown to preserve the clarity of the figures. The finite-

ness of the number of realizations was addressed in the previous subsection.

The three-dimensional original lattice implies cubic growth of the original lat-

tice size for the linear growth of the desired equivalent linear chain. It limits the

maximum number of c.f.e. coefficient accessible to us. ^

We analyzed our results for various nr ranging fromlOG to 8,000.

Data for the dispersion demonstrate an insignificant dependence on nr for

nr > 1000. Even the smallest attempted nr = 100 leads to the graphs with

the qualitative behavior virtually identical to the more precise ones. Graphs for

n r = 2000 to 5000 to 8000 are in fact indistinguishable within our resolution

limits. Examples are presented in Figures [G3], [G4].

Data on averages and correlations (the correlations are depicted in Figure

[G4]) also do not show any strong dependence on a large nr.

Data indicate that both DAtie and DBnc_i (see [G5]) increase steadily with

the increase of the noise intensity E. For a given value of original noise 3/t

(see [G3] )both dispersions decrease "asymptotically" as the coefficient number

increases.

The latter result can be attributed to the nature of a mathematical procedure

we employ. Each linear site dispersion is a result of an effective "averaging" over

46

the corresponding prototype shell. Each shell consists of nearest neighbors of the

original vertices of the previous shell. Thus a recursion procedure is adopted. The

original sites, each characterized by a fixed disorder, contribute to the construction

of a corresponding linear vertex. The number of original sites in a shell increases

with the number of a continued fraction coefficient.

The tails of these graphs can be extrapolated, the resulting curves can be used

for future development that is discussed at the end of this dissertation.

The starting point of DB v. i graphs for all values of E is (1,0) ([03] ). This

fact serves as a test of our computational tools since the procedure employed to

construct B leads to the well-determined B\. Also, for an obvious reason, we

obtained R*0 — 1 and R?0 = 1 {[G6]).

The plot of DB35 and D . B 3 5 v. S (i.e. the dispersion of a coupling of the

highest coefficients calculated) indicates an almost linear increase of variance with

increase of the noise intensity [G5]. This fact is consistent with the other results

of this work, which show the short correlation length for A3 and B3 on the graph

[G6] of R£n and R^n v. i (i = 3 + n) and the negligibly small correlations for

[G7]. The same comment applies to DA. However deviations from a linp r̂

increase prevent us from treating A and B as a vector of independent random

variables.

47

If we imagine that in our model the effects of the Lanczos procedure resembles

the effects of a linear" averaging over the neighboring layer of sites we would ex-

pect a linear increase of "asymptotic" dispersions with the original noise. Indeed,

according to the central limit theorem (see Appendix 1) we would expect DBj to

be directly proportional to 2 and inversely proportional to i for sufficiently large

coefficient number i. The higher is the one-dimensional site number the more

sites of the original multidimensional lattice are involved in its construction, but

claiming that the result just mimics an averaging procedure over many random

variables (corresponding to high n for rjn in Appendix 1) would not be precise for

the reasons indicated above.

The initial growth in the coupling dispersion (the "hump") when i is increasing

persists only for i < 6. For i > 6 we observe a pronounced decrease in DB when

i is increasing for any fixed intensity of the noise. In our calculation i < 35.

The graphical representation of DB dependence on the noise and on the order of

coefficient is given in [G3], [G4].

Unlike AB's, the A's and the B's exhibit a correlated behavior [G6]. Although

rather short as well, the correlation length for A3 is affected by the increase of

the noise intensify. Although no special study was done we can see a greater

correlation length on the graphs of R^n and v. i (where i = 3 + n) for the

48

cases of larger original noise compared to that for the cases for smaller noise.

The results obtained for Rf versus i (for any fixed noise) clearly demonstrate

that the subsequent diagonal coefficients for i > 6 are anti-correlated to a sub-

stantive degree (for i < 6 a positive correlation can be observed). See [G8] ,[G9] .

The magnitude of the anti-correlations [G8] oscillates and exhibits a gradual

increase as we go farther from the initial site. Accordingly, when studying the

dispersion DAX (S) [G5] we would expect some deviation from a almost linear

behavior exhibited by DB35 v. E. Indeed, in the neighborhood of E~ 17 the slope

of the graph changes. The increase in the slope indicates the higher dispersion

(compared to that of the non-correlated variables).

The slope changes again (decreases) as i increases farther. The rate of the

dispersion growth is not constant.

The transition from extended to localized wave function would manifest itself

in the asymptotic behavior of DA v. i for a fixed near-critical noise. For a

wave function to be localized, the graph has to approach a positive constant

asymptotically. We did not reach the value of i large enough to demonstrate an

obvious asymptotic behavior, since an extension of a wave function over the whole

lattice would require an involvement of an infinite number of coefficients. So we

would not be able to use E = S c exactly since it would require a memory capacity

49

that is not available to lis. This (the restricted nc vahie) is a common problem in

this field. See tG3] and [G10]. • •• • — •

We may observe, however, an evident enhancement of the localization length.

For the lower intensity of noise the values of D Ai will get sufficiently close to

zero "earlier",meaning for the smaller values of i (compared to that for the higher

values of iioise). If dispersion vanishes as we go farther away from the initial

site it would mimic the perfect crystal and provide favorable conditions for an

enhancement of the localization length and the eventual extension of the (initially

localized) wave function.

If the dispersion is asymptotically connected with the coefficient number by

the inverse power law, then it will never reach 0. But it may be considered

negligible after we take into account the accumulated numerical errors. An expo-

nential decay would provide a fastest convergence to an asymptotically ordered

one-dimensional lattice.

We also computed c.f.e. coefficients and their dispersions for the case of /

Gaussian noise. The DA and DB graphs showing site-dependence of the dis-

persions presented in Figure [G10]. In this figure we see that qualitatively, the

results are similar to those the case of uniform distributed energies of the original

multidimensional lattice sites. The benefit of using Gaussian noise is that we are

50

able to observe the horizontal shelf ofthedispersion graph clearly. It is an indi-

cation of validity of Anderson theorem for high values of noise. Note, that the

noise intensity EG used must be interpreted as a half-width of the corresponding

distribution, while Sfi for the case of a box distribution equals the whole box

width. It means that the value of SQ effectively corresponds to the larger noise

then the same numerical value of H/j would. Such a strong noise decreases the

localization length enough for us to observe the effects.

The result described above corresponds to clearly correlated subsequent co-

efficients (cross-correlations are still negligible for our range of the parameters).

The site-dependence of the correlations of the subsequent coefficients are shown

at [Gil]. Correlations are shown on [G12].

The correlations are substantial for i > 10 and the both DAt and versus

i graphs become essentially horizontal for the same region. A substantial increase

in the average coupling coefficient [G12] (compared to a almost constant MBi>10

for a "flat" case [G2J) indicates an increase in transport. The elements of MA

are small compare to the elements of MB and oscillates around zero level (the

same features where present for the "flat" case).

We also calculated the spectral density D (E) by the means of c.f.e.., as it was

discussed in the previous Section of this Chapter. For example, we can observe

51

how the number of resonant peaks in this energy level density graph decreases

to 33 when we introduce the noise of intensity H/j = 15 into the system. That

number decreases further to 29 peaks when we increase the noise to S/j = 17 (see

[G14]). The number of coefficients used was nc = 35, the number of realizations

nr = 2000.

We can see sharp random peaks in the transmission coefficients. Because the

mean values of off-diagonal coefficients are of the same (rather limited) order of

magnitude and their dispersions are limited as well, the nature of these peaks

should be mostly attributed to the close values of some random subsequent pairs

of diagonal coefficients. See [G15].

52

Average diagonal and off-diagonal continued fraction coefficients obtained by recursive building

f IS I i

10

i

«f*17 — ttblOx—

• S 10 portion ofquMJ-ttomn Mori chain

12 14 1 i«

4 • a 10 12 poatton ol qutaHrtom m Mori chain

Figure [G2a]

53

Average diagonal and off-diagonal continued fraction coefficients obtained by recursive building

'0 IS 20 B poMcn ofqiM-MMi MorieMn

0.15

mJLm

10 1ft 20 26 portion of qowvom m Mori <*mr\

Figure [G2b]

54

Dispersions

10 12 14 1«

ootfflcfentiiuffllMn 14 W

Figure [G3]

55

Statistical data on continued fraction coefficients

r \7

* «» £ £ m *

i ? » n-\ •; ' /s ;i V ' ̂

OtM.08 • otisia.os ottx22.08 < etls31.A <

10 20 30 40 pociton o< quMi-atom m Mori oncm SO

01

0.05

0

-0.05

-0.1

•0.1 S

-Q.Z

*0l29

•0.3

•0J5

•0L4

RA3nccts1? o RA3n cstsl* -— Rfl3n csis 17 * RB3n csir 17 — i •'

! ;u ** s*\: t: •*

I; i: f i If #: I;

I! fe If

10 15 20 25 pot don or quio-ctom in Mori chain 30 35

Figure [G4]

56

0.9 nc = 35, nr =2000, dislribulion = flat

Figure [G5]

57

Correlations of continued fraction coefficients obtained by recursive building

e 8 10 12 potrtoo of quuHHom m Mort ctiam

osis20 ccisSO

e 8 10 12 poMMn o» quu-nom in Mort cMM

Figure [G6]

58

Cross-correlations of continued fraction coefficients

Otis 16 *•1=17

0015

V 8-0.005 b

I

4 6 8 10 potrton of quttt-«tom in Mori cham

0.03 ccis16 ctis17 C*i=20 OttsOU

6 8 tO 12 poMonofquM-domnMoncfiain

Figure [G7a]

59

QXM

003

0.02

0.01 -

* /\

S 101

5

<02

0 /

/ / ^

A/ LA

r

RAB3J csiz16 o RAB31 Mis 17 — RBA3i*lix16 a RBA3i<teir17—

/ ;

I \

\

!A / \ \ 71 \y

\ i

\f *

fl 10 ooefOoient ntifitMr

-4-12 14 16

•0.025

« I coefficient number

Figure [G7b]

60

Correlations of subsequent diagonal continued fraction

h / v v-

10 IS 20 2S pocrfon of quuHMivi In Mori eftan

-0.1

, -0.15 -

-0 .2 -

i i § *0-25

*0.3 -

; -0J5 -

air 13 1, OffrtS — 1 «isi7 •

A A< /, W A' -.V/\V7 A-' W K U • \

-J /i / / 5t/r \ / \ i *, *•••*** / *-•% a. /

(i * \ i 7 v V

V * » I''

10 15 20 25 porton o( qua**tom « Mon Cham

Figure [G8]

61

csM.08 csi*31.8

| -0.6 3

20 30 40 position of quasi-atom in Mori chain

Figure [G9]

62

Dispersions of continued- fraction coefficients.

wmJLm

• a 10 ooifl̂lifit nuntw i

Figure [G10]

63

Correlations of subsequent continued fraction coefficients

as

07

ae

as

a*

as

as

at

o E c •at

<2

43

,— 1 "1 ' » Wat4 on* 15 —-oiicte Otic!7 , oguia —

\ # A

10 14 16

Figure [Gil]

64

Correlations and cross-correlations

of diagonal and off-diagonal continued,.friction coefficients

Qjttt

0 X 0

0*2

001

4 J 0 2

-O.C3

RAB»<*ts16 o RAB3ICttrt7 — HeA3(ftrt6 o RBAttftfetT —

9

/ \

A A / \ / \ a

W :

n

/ / S J \ \ /,' w

^vrtT^l

1 \

I i I \ f A '

y \*

/! \\

* / Y

• J t

Figure [Q12]

65

Average diagonal and off-diagonal continued fraction coefficients obtained by recursive building

mt*iT CtteU..—

•mf « 4 • • «

pai«ono<qu—-«tomwMortch*n,l

6 "1" 10 12 potitton of quMffttom in lAon chan, i

Figure [G13]

(K>

0.01

0.001

0.0001

1©-05

1e-06

Figure tG14]

67

2000

1600 -

1000

600 -

rtt j-V-\ i \ i \ V \ / v/ \/\

-500 -

-1000 30 3S

Figure [G16]

68

3.2.3. Discussion

Anderson theorem for a one-dimensional lattice states that fluctuations of the

vertex energies and couplings induce localization. Neither we nor the authors of

the earlier works on this topic are trying to "disprove" this well-established fact,

the objective is to set boundaries of acceptability of this theorem. The following is

an intriguing question: Is it possible (and under what conditions) to challenge the

rUim that any positive intensity of fluctuations induces a localization in a one-

dimensional system? We conject that the introduction of noise with some specific

properties will allow a non-zero energy fluctuations to coexist with a transport

phenomenon.

The search for the "non-Anderson" noise that would not result in zero-threshold

localization was conducted in recent years by a number of authors. The compre-

hensive literature review on the breakdown of Anderson localization (caused by

the presence of correlations in the random distribution of the Hamiltonian matrix

elements) can be found in [4]. Both numerical and analytical approaches have

been adopted by different authors. For example, analytical arguments on the

existence of localized eigenstates in the RDM can be found in [26]. Numerical in-

dications of the existence of a broad band of eigenstates extended over the whole

sample can be found in [4].

69

The latter paper investigates the case of a disorder driven by Ldboviteh and

T6th maps and establishes an empirical rule: short-range correlations, similar to

those of the RDM, bring into existence bands of states with localization length

comparable to the sample size, l o n g - r a n g e correlations widen the band.

In the case under consideration we started with an isotropic ^ i n d e p e n d e n t )

distribution of an noncorrelated energy fluctuations in the cubic lattice. As the

graphs for d i s p e r s i o n indicate, the intensity of a one-dimensional noise m a equiv-

alent lattice increases for every given site as we increase the original noise inten-

sity. But "energies" that correspond to vertices of the Mori chain turn out to

be short-range correlated and site-dependent. This significant property, the site-

dependence of the dispersion and correlations, we address as "anisotropic noise

or "non-stationary noise". The former term is reserved for the cases when, ac-

cording to the model, we interpret a one-dimensional index as a special one. The

latter term signifies that if the fluctuations in the Mori chain were to be generated

by a map, the site index would acquire the meaning of a discrete tune.

The idea to use c.f.e. in order to move from multidimensional to one-dimensional

lattice can be already found in [18]. Let us imagine that the resulting linear chain

is truncated simply by setting one of the nearest neighbor couplings to be equal

to zero (or to be sufficiently small compared to the site energy difference). This

70

truncation leads to the behavior of the d.o.s. of the system that mimics the

well-known manifestation of the localization phenomenon: discrete sharp peaks

in spectral density (as opposite to a continuous d.o.s. that characterizes trans-

port) . Accordingly, this truncation of c.f.e. is interpreted by [27] as Anderson

localization. It has been shown not to be the case. In the work [28] authors

demonstrate the natural quantum-mechanical alternative to the truncation as a

localization mechanism.. Namely, the latter occurs when the wave function looses

a small amount of itself at any collapse.

Now we would also like to mention, in the connection with the possibility of a

breakdown of Anderson localization in a one-dimensional chain, studies conducted

in [26], [5], [4] and [29]. Both [26] and [5] consider the dimer model, from analytical

and computational point of view respectively. The third paper [4] introduces a

correlated, noise into the system. The correlation function is characterized by the

property

{WnWn>) = 3w, (3-18)

so the stationary fluctuations are considered. By having short-range correlations

the violation of Anderson prescription can be obtained. Long-range correlations

•Tr- i -Ty, (3-19) \n — n'\

71

can increase the number of extended states [4].

Why is it so important to us to locate the general statistical characteristics

of fluctuations that are responsible for the transport or for the quench of the

transport? In the next paragraph we will elaborate on the topic.

One may think of various justifications for the introduction of special "delo-

calizing" properties in the statistics of parameters of a one-dimensional lattice.

Either an interaction with an environment or an intrinsic property of a system

can be held responsible. Mathematical transformations may result in the special

properties of a transformed model. For example, one may think of an equivalence

of a classical standard map to a quantum tight-binding system known as kicked

rotator [19], (30]; QKR can be "delocalizecl" by introducing external fluctuations

that leads to a diffusion with a classical coefficient. It has to be remarked that

there is an effort to introduce intrinsic sources of fluctuations (see, for

example, [19], [31] and references therein) in a general framework of adopting a

coarse-graining procedure as a source of irreversibility in quantum mechanics. For

us the delocalizing properties are artifacts of the projection procedure (while the

prototype lattice is the one designed to model the actual system).

Applying the basic idea of [18] and [32] we can move from a multidimensional

lattice to a one-dimensional one, which is equivalent (in a ^ven sense that was

72

already discussed in this dissertation) to the prototype lattice. Namely, if a small

noise (below a critical value) is not sufficient to induce a localization in a prototype

lattice, the same effect should take place in an equivalent one. The question is

how to extract the properties responsible for the breakdown of Anderson theorem

in this case. The emphasis in the Aforementioned important work [4] was made

on the plausibility of the fact that stationary correlations are responsible for the

extension of the eigenstates.

But the procedure we utilize naturally leads us to the investigation of a nice

aspect of nonstationarity. Additionally, our noise is inherently correlated. We

speculate that these two aspects interact and that the nonstationary nature of

the noise is one of the key properties ought to be investigated further. We plan

a further investigation of the initial "hump" and the tails [Graphs}. The depen-

dence on the key property, that remains to be found, has to be non-linear: a slight

change in it around a critical value will provoke the transition. Hie dope of the

tail and the initial "hump" might be a good candidates for the role of the key

property.

Let us imagine that the property is the tail. The horizontal positive tail cer-

tainly provides localization. Zero dispersion would lead to a transport. Transition

then takes place for somewhere in between.

73

The case of an exponential decay of the second moment can be interpreted

as the case of an exponential decay of a diffusion coefficient since they are pro-

portional. Then we also have an exponential decrease of an average localization

length. The number of eigenstates that overlap with the initial state of an elec-

tron's wave function reduces accordingly.

Neighboring level spacing as a property that characterizes MIT was investi-

gated in [29] and {33]. Metal region (that corresponds to subcritica! values of

noise) was found to be characterized by Wigner distribution, while the Poisson

spacing distribution was found to be a property of an insulator (that corresponds

to the above-threshold values of energy fluctuations). We find the latter fact

to be a consequence of a level degeneracy for an isolator that is composed of

non-interacting atoms (zero overlap integral V — 0 for the tight-binding model

wen-investigated in solid state). Then we have a non-zero number of energy states

with vanishing spacing s between them, so after proper normalization we obtain

(0,1) point for Poisson distribution as a contrast to (0,0) point for the Wigner

distribution.

We must remark here that the utilization of a tight-binding ilamiltonian leads

to a continuous transition from a metallic to an atomic state as the overlap in-

tegrals that provide for site interaction decrease (which, in a solid state analogy,

74

would correspond to an increase in an interatomic distance). Thus the vanish-

ing coupling cannot be considered to be a natural property that characterizes an

abrupt localization. In solid state an abrupt drop of conductivity to zero, known

as Mott transition, cannot be obtained from tight-binding independent electron

approximation. The full calculations [34] include fermion repulsion and the re-

sulting departure from a tight-binding prediction describes an abrupt transition.

As the stationary noise intensity increases, localization in a cubic lattice occurs

abruptly. So we expect that some property, similar to the level spacing found in

[29] and [33] by a Lanczos diagonalizing of a cubic crystal Anderson Hamiltonian,

must be responsible for the phase transition in an equivalent linear lattice studied

by us. Since we found that this linear chain is characterized by a non-Anderson

noise (that is in our opinion responsible for the deviation from Anderson predic-

tion), it is logical to study statistical properties of these fluctuations.

The paper [27] investigates a truncation of a one-dimensional chain (and of

corresponding continued fraction) as a mechanism solely responsible for the local-

ization. If an exact truncation of a c.f.e. occurs then the resulting finite c.f.e. has

a finite number of poles. But for many reasons, some of which were mentioned

above, truncation is not necessarily a natural dynamical feature of the system

in study. Our approach is more delicate: it may be the site-dependence of the

75

initial "humps" or the non-statiooarity of the tails on DB.versus.n graphs or the

correlations. Effectively, we are indirectly taking into account the "terminator"

of a continued fraction expansion.

Any spacial periodicity implies ballistic motion (Bloch) when any wave packet

that co r r e sponds to a particle (a pseudo-free electron in our case) extends over

the whole crystal (we are not discussing the cases when the period is much longer

then localization length). It means that if an electron in the certain level has a

mean nonvanisbing (group) velocity then that velocity persists forever. A per-

fect conductivity of an ideal crystal (that exists in spite of the scattering) is a

manifestation of a wave nature of an electron (34]. In a periodic, array of scat-

terers a of wave propagation is the coherent constructive interference

of scattered waves. In our model the ballistic subcase is provided by a particular

condition when the intensity of energy fluctuations in the prototype lattice van-

ishes which leads to a constant linear coupling. Nonvanishing fluctuations can be

attributed to impurities, missing ions, thermal vibrations of the ions (elec.tron-

phonon interaction) and other deviations from periodicity in the potential the

electron experiences. The periodicity can also be violated by introduction of an

asymmetric hopping probability as a result of action of an external field and by

other means. Our idea is to utilize projection technique as our way to distribute

76

noise in a one-dimensional system in order to get localization.

Hie localization in the linear chain is guaranteed by the localization in the

prototype lattice. But there is an essential difference in the mechanism respon-

sible for the existence of a non-zero localization threshold. Three-dimensional

noise is an i s o t r o p i c property, random for each particular realization but site-

independent from statistical point of view. The resulting linear noise is essen-

tially anisotropic ("nonstationary", "site-dependent"). While in the first model

the increase in the magnitude of the noise above it's threshold value is solely re-

sponsible for the transition in study, in the latter case an infinitesimally small

constant (site-independent) noncorrelated noise would produce localization, while

a finite nonstationary noise is required to do so.

77

4. CONCLUSIONS AND PLANS FOR THE

FUTURE RESEARCH

4.1. Conclusions

Cubic lattice (with constant site coupling) with both subcritical and above-critical

intensity of site energy fluctuations (noise) is reduced to a semi-infinite linear chain

of quasi-atoms by the means of the projection technique. Application of PT allows

us to generate a one-dimensional crystal based on the various physical models that

exhibit positive localization threshold.

Consequently, we have a way to generate a non-Anderson noise in an equivalent

linear crystal that would lead to derealization phenomena.

An ordered crystal is characterised by extended Bloch states. Even infim-

tesimally small disorder, according to Anderson, leads to localization in a one-

dimensional chain. By "putting some order back into this disorder" we induce

78

an existence of a non-zero localization threshold in a linear system and investi-

gate properties of the resulting non-Anderson noise. In the future we plan to

extrapolate these properties in order to mimick an infinite crystal. This approach

provides a procedure for generating arbitrary long Mori chain (up to computer

limitations).

Construction of Mori chains enables us to guarantee an existence of a non-zero

threshold in a one-diraensional system. Gibbs averaging allows us to investigate

nonstationary statistical properties and to separate localization from the simple

truncation of the c.f.e.

The original cubic lattice exists in either delocalized or localized phase, depend-

ing on the noise level. The phase is an invariant with respect to the transformation

to an equivalent linear crystal.

Linear crystal is found to be characterized by a site-dependent couplings and

energies of quasi-atoms. We performe Gibbs averaging over an ensemble of crys-

tals. Averaged characteristics of quasi-atoms become site-dependent as well (un-

like average energies and coupling of the original crystal). Moreover, an original

isotropic uncorrelated noise transforms into anisotropic correlated noise that is

the property of both energies and coupling of an equivalent crystal. This result-

ing non-Anderson noise is associated with a non-zero localization threshold in the

79

projected lattice: one-dimensional crystal exhibits delocalized behaviour in the

presence of (subcritical) noise.

Ensemble averaging procedure (as alternative to an averaging along an equiv-

alent chain that would resemble time-averaging in a map-driven system) enables

us to see the property of nonstationanty.

Since even relatively weak stationary uncorrelated noise in one-dimensional

system produces transition from extended Bloch states to localized eigenstates, we

hold joint action of nonstationarity and correlations responsible for the existence

of a positive localization threshold in the linear crystal under consideration..

In the absence erf original noise the conditions of Bloch theorem are reproduced.

As energy fluctuations of cubic lattice increase, the overall increase in projected

noise takes place.

No long-range correlations are present in our one-dimensional model. Short-

range correlations in Mori chain are found. They are sufficient for the existence of

transport. We found no drastic change in the functional form ot second moments

of the energy distributions associated with quasi-atoms. The MIT is accompanied

by a smooth transition (for the quasi-energies dispersions) from asymptotic decay

to a constant non-zero asymptote (the latter mimics the conditions of Anderson

theorem).

80

We fwd the delocalization in one-dimensional lattice to be associated with

nonstationary (unisotropic, site-dependent) nature of noise (it may include not

only asymptotic behavior but also transient "hump" in variance for the coupling

stength parameter).

Our work, in a conjunction with [5], [26] and [4], is a pari; of an ongoing

search for the general properties of a non- Anderson noise that are responsible for

delocalization phenomenon. We also refer our reader to the abstract as well as to

the discussion subsections of the first and previous Chapters.

Here we would like to insert a citation from [13, pp. 177-178]: "The importance

of the localization problem has also stimulated numerical studies and computer

simulations [10], [39]. By far the. most useful and most often employed approach

to perform numerical calculations on disordered systems is the recursion method.

It shows the great advantage of a reduction of the multidimensional random lattice

to a semiinfinite linear chain [35]. Haydock [36] first applied the recursion method

to the Anderson model [... ]. In fact, it is still an unresolved problem the critical

study of the coefficients an and bn, of their oscillating behavior, and of the rate of

decrease of their amplitude and a connection with Anderson's localization theory.

We present this work as a step towards resolving this problem.

81

4.2. Plans for the future research

As a minor future development of the model presented we will introduce a random

couplings for the original lattice.

We thirilr to be helpful to conduct a systematic study on the correlation length.

That is the quantity that can be introduced to characterize the decay of corre-

lations between two sites as the distance that separates them along the chain

increases.

An interesting study can be made out of applying different kinds of a random

distributions as a distribution for the random vertex energies and coupling of a

multidimensional lattice.

As a major further step, the original object of our investigation may be chosen

differently and studied by the means developed. In particular, we plan to inves-

tigate a structure known as Cayley tree. The Cayley tree model was studied by

many authors, in particular [37].

The growth of the number of vertices of a multidimensional lattice is polyno-

mial (namely, cubic in the case under consideration) with respect to the size of

the equivalent one-dimensional system. This problem is even more serious for the

Cayley tree when this growth is exponential. For example for the full binary tree

82

the number of vertices that form each layer grows as 2n where n is the height of

the tree.

The computational limitations on the size of the system in study constitute

the common problem in the field. We are considering to employ a calculation

of a Liapunov exponent A as a numerical toed for the explicit calculation of the

localization length I The direct diagonalization of the Hamiltonian of the

equivalent linear lattice is thought to be employed as an alternative tool of an

estimation of a localization length.

The ultimate goal is to enable us to predict the suppression of localization of

the wave function in various systems based on the type of noise and vice-versa.

83

5. APPENDIXES

5.1. Central limit theorem

The term central limit theorem is used for any statement of the following nature

[38]:

upon certain conditions, the distribution function of the sum of indi-

vidually independent random variables converges to the normal

(Gaussian) distribution with the growth of the number of the sum-

mands.

(It must be noted that Gaussian distribution is a particular case of a more

general object: L6vy distribution. Gaussian distribution corresponds to the cases

of a finite second moments, while infinite second moments would lead to Uavy

distribution.)

The exclusive importance of the central limit theorem lies in theoretical ex-

planation of the following common observation:

84

if a random outcome of an experiment is determined by a large number

of random factors (influence of every single one of which is negligibly

small) then such an experiment may be well-approximated by a normal

distribution with its mean and its dispersion chosen appropriately.

Let, {£fc, k > 1} be a sequence of mutually independent random variables with

distribution functions Gk (x) = < a:}. Let those variables have a finite

mean M£fc = a* and dispersions = of. Let the dispersions also satisfy the

conditions

jB® = ^ > 0 for n > 1. (5.1) k=1

The normalized sum of random variables £2>. • •, £n 18 defined as the follow-

ing random variable:

-<"•), (5-2) fc=1

which is characterized by Mr?n = 0, Drjn = 1 for any n > 1.

Let Fn (a:) be the distribution function of the normalized sum r)k and let

*

$ (x) = —L== J e~*?dx (5.3) —OO

be a normal (0,1) distribution function. For finite dispersions, the central

limit theorem sets the conditions for the convergence relationship

lim Fn (x) -$(x). (5.4) ft—*00

85

This convergence is uniform on ( — 0 0 , 0 0 ) .

The most commonly used version of the central limit theorem has to do with

the sequence of the random variables which share a common distribution function.

Theorem 1 (L6vy - Lindebergh). If {£fc,fc > 1} is the sequence of pmrwise

independent random variables with the same distribution function, then the nor-

malized sum (5.2) can be re-written as

(5.5)

(where a == M£fc, <r2 = D£k). Ita distribution function Fn (x) satisfies the limit

condition (5.4).

The most general condition for the validity of the central limit theorem for the

sequence

^ 1} of random variables with finite dispersions is the TAndebRrgh s

condition: for any k > 0

1 f (x-akfdGt(x)= 0, (5.6) n-*co B~ u, J

n k=:1 |s-a*|>kB„

where Gk (&) is the distribution fimction of the random variable £fc.

86

5.2. Proof of the fundamental recurrence formulas for con-

tinued fraction.

Let us consider a general c.f.e.

F = ^ M X l +

x% + J/n+1 a-3 + • • • ;

®n+l + ~ T * * * *

Let us roake an inductive assumption for (n. + l) th approximant:

Fnii = Fn-iyn\i +Fnx„n-

We also can set

F_! = 1, (5-9)

F0 =* 0.

This relationship (5.7) obviously holds for n = 0 providing that the initial condi-

tions are specified by (5.8).

Now let us truncate the c..f.e, (5.7) on L = n + 2 level:

87

= — — h i • <5-10> Xi + ;

X2 + — y^Tx ^3 + • • • "" &»+2

a'n+i + -•K»+2

We can achieve it by setting a partial denominator a:n+2 to be equal to zero.

Let us denote

Vn+l — 2/rH-l^n42> (5.11)

tf'n+l = ^+1^+2 + yn+2-

Now the (n + 2)d approximant 5.10 can be rewritten as:

Fn+2 ^ (5-12) x % + — ^

X2+ ' Vn+lXn+2 a?3 + • • •

®n+l®»+2 ~t* Vn+2

Note, that it has only (n + 1) levels and thus can be denoted Fn + 1 with partial

denominators and partial numerators given by

A

y[ = 2/»»

(5.13)

for all i < (n + 1) and by (5.11) for i — (n + 1). By the means of a general

88

inductive assumption (5.8) the (n + 2)d approximant (5.10) can be rewritten as

Fn+2« K+i - k + i s U i + { 5 1 4 )

But since F t '_x = Fn_i and F,; = Fn and taking (5.11) into account, we get:

Fn\2~ Fn-lVn I l®n 12+Fn (®n! l®n 12 + Vn I 2) = 12 (Fn-lVn 11 + I l)+FnJ/n I 2-

(5.15)

Substituting the left-hand side of the inductive assumption (5.8) for the expression

{Fn—iVn 11 + FnXn 1i), we can see that the relation

F n + 2 = Xn+^Fn+i 4- Fnyn+2 (5.16)

holds, Q.E.D..

89

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