Statistical properties of one-point Green functions in disordered systems and critical behavior near the Anderson transition

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Statistical properties of one-point Green functions in disordered systems and critical behavior near the

Anderson transition

Alexander Mirlin, Yan Fyodorov

To cite this version:

Alexander Mirlin, Yan Fyodorov. Statistical properties of one-point Green functions in disordered

systems and critical behavior near the Anderson transition. Journal de Physique I, EDP Sciences,

1994, 4 (5), pp.655-673. �10.1051/jp1:1994168�. �jpa-00246939�

Classification

Physics

Abstracts

71.30 71.55J 72.15R

Statistical properties of one.point ^{Green functions in}

disordered systems ^{and critical behavior}

near

the Anderson transition

Alexander D. Mirlin

('. *)

and Yan V.

Fyodorov

(2. ^*,

**)

1')

Institut für Theorie der Kondens~erten Materie, Universitàt Karlsruhe, ⁷⁶¹²⁸ Karlsruhe, Germany

(2) Department of

Physics

^of

Complex Systems,

Weizmann Institute of Science, ^Rehovot 76100, Israel

(Received J Oc.lober 1993, accepted 9 Feb;uaiy J994)

Abstract. We investigate the statistics of local Green functions G(E,x..r)=

(>. (E ù

)~'j x),

_in particular of the local density of states p cc Im G(E, _.r, x), with the Hamiltonian

ù

_{describing trie motion of}

a quantum

particle

_{in a} d-dimensional disordered system.

Corresponding distributions _are related _{to a} function which plays the rote of _an order parameter ^for the Anderson metal-insulator transition. When trie system can be described

by

_a nonlinear _«- model, the distribution _is shown _to possess a

specific

_« inversion » symmetry. ^We present an

analysis of the critical behavior _near the

mobility

edge that follows from the abovementioned relations. We explain trie origin of trie non-power-like critical behavior obtained earlier for effectively infinite-dimensional models. For any finite dimension d

< cc trie critical behavior _is demonstraled _la be of trie conventional power-law _type with d _cc playing ^the rote of _an upper critical dimension.

1. Introduction.

The

phenomenon

of the localization of _a quantum

partiale

_{moving in a} random media _is still far from

being completely understood, despite _great

efforts

spent

^{after the Anderson's}

original

paper

[Il

The

scahng theory

^of localization

proposed by

Abrahams _et aÎ.

[2] predicted

_a

transition from the

conducting

to an

msulating phase

with increase _in the disorder for any

spatial

dimension d~ 2. The

Lagrangian

formulation of the

problem

_is based on ideas of

Wegner [3]

who introduced the so-called N-orbital model.

Developmg

further this

approach,

_a

(*) Peinianent addiess

Petersburg

Nuclear Physics Institute, 188350 Gatchina, St.

Petersburg,

Russia.

(~~) After

September,

1994 _: Fachbereich Physik, Uni-GH Essen, Essen, Germany.

number of authors

[4-6] mapped

the

problem

onto an effective non-linear «-mortel

(NSM)

with

making

_use of the

rephca

trick. The

corresponding renorrnalization-group analysis

in 2 + _e dimensions

[5-8]

put ^the

scaling

ideas on a

quantitative ground, leading

to an

e-expansion

for the critical exponents

describing

the behavior in the

vicinity

of the

mobility edge.

At the _same

lime,

the

following pecuharity

of the Anderson localization _was realized

[3, 9]

: the

one-point

Green

function,

which

plays

in the

Lagrangian

formulation _a rote

analogous

to that of the order parameter ^{of usual second order}

phase transitions~

does net exhibit any critical behavior. This

fact _{seems to} break the Goldstone theorem and leads _{to a} failure of the usual mean-field

approach.

As _a consequence, the order parameter ^{and the} upper critical dimension of the

theory

remained unidentified.

The

supersymmetric approach,

devised

by

Efetov

[loi

_as a

rigorous

alternative to the

mathematically

ill-defined

replica

trick~ _gave a new boost _to the

investigations

of the

problem.

The

validity

of the

predictions

of the

scahng theory

of localization _was

questioned by

Efetov

Il and Zimbauer

[12]

who studied the Anderson transition _m the

supermatrix

NSM _on the

Bethe lattice

(BL).

A very unusual

non-power-like

cntical behavior

(namely,

an

exponential

decrease of the diffusion constant and _a

jump

in the inverse

participation ratio)

_was found.

which seemed _to be in contradiction with the

scaling hypothesis.

It _was

emphasised

in

[12]

that the whole function _{on a} super ^coset space

plays

the rote of _an order parameter ^{for the}

theory

of Anderson locahzation.

However,

the

physical meaning

of this function defined _{on a} formai level in

iii, 12]

_was

completely

unclear.

Further

insight

into the

problem

was achieved _{m our} earlier paper

[13]

where the

phenomenon

of the Anderson localization _was

mvestigated

^{in the framework of the} BL version

of the usual

tight-binding

model rather than the NSM. In addition _to

reproducing

the

unconventional cntical behavior obtained in

il1, 12],

_a clear

physical meaning

_was attributed to the order parameter ^function : it was shown to be related to the distribution of the _one-

particle

local Green

function,

the

imagmary part

of which is

equal

to the local

density

of states

(LDOS).

That confirrned the

general

fundamental idea

suggested by

results of the

study

of mesoscopic fluctuations

[14]

in the

theory

of Anderson localization the whole distribution function _{is to} be considered in order to extract the relevant

physical

information. The

discovered

physical interpretation

of the order parameter ^function

[13] provided

the link between the formai

description

of the Anderson transition in terms of _a spontaneous symmetry

breaking

which

naturally

emerges in the framework of the supersymmetnc

approach

and the

commonly accepted

view _on the nature of localized and extended stores

[15].

A _new field-theoretical reformulation of the

problem

was constructed in

[16]

for the

nonlmear «-model and in

il 7]

for the

tight-binding

model.

Corresponding Lagrangians

have _a nontrivial saddle

point providing

a kind of the mean-field

approximation

which _was called the

effective medium

approximation (EMA)

in

[16].

Because of the way the

approximation

is

constructed,

this saddle

point reproduces

the solution of the basic non-linear

integral

equation of the Bethe

lattice, leading

to the _same

non-power-hke

cntical behavior. The authors of the

recent paper

il 9]

_put forward the

hypothesis

that such _an exotic cntical behavior _is the intrinsic

feature of the Anderson transition and should be

expected

^{for real}

physical

_systems in

arbitrary spatial

dimension d

~ 2. Some

physical

picture ^{of the transition}

advocating

this point ^{of view}

was

suggested.

On the other hand, _a number of

physical

arguments supporting ^the

opposite

point ^of view was

given

in

[17].

In the present paper we

complete

our

argumentation presented

ⁱⁿ

[17]

and put it onto a

quantitative

basis. For this purpose, we denve

general

expressions for the distribution of one-

point

Green functions m a framework of the

supersymmetric

forrnahsm. We would Iike _to note that

investigation

of the

mesoscopic

fluctuations of one-point ^{Green function}

lin particular~

of

LDOS)

is of

special importance

because this

quantity lin

contrast to, e-g-,

conductance)

is _a

local _one and its distribution _can be defined _not

only

for _a finite

sample

but _remains nontrivial

m the

thermodynamic

limit _as well.

The outline of the paper is _as follows. In section 2 _we

explain

the concept ^{nf the order} parameter ^{function in} terms of both nonlinear «-model and

tight-binding

mortel. We derive the

general expressions

for the distribution of the local Green function. A nice symmetry property

relating

the behavior of this distribution function _at

large

and small values of its argument follows from the obtained results. In section 3 _we calculate

explicitly

the distribution function of the local Green function within the NSM

approach

in the

simplest

case of the unitary ^NSM.

From this

expression

_{we extract} the distribution of LDOS which is considered in section 4.

Finally,

in the _section 5 _we

apply

these results _to the investigation ^{of the Anderson} transition.

We show that the

non-power-like

critical behavior obtained in

il1-13, 16, 17]

is _an artifact of the

pathological

_space structure of the BL or of the EMA

imitatmg

this

spatial

structure. These

models

correspond effectively

to the _case of infinite

spatial

dimension which _{tums eut} to be _a

singular

point ^{in the} case of Anderson localization. We argue thon for any finite d the critical behavior at the

mobility edge

is of the

power-law

type ^with

d-dependent

critical

exponents, ^d = ce

playing

the rote of an Upper critical dimension. As d

- ce some of the

exponents ^tend to zero or to

infinity matching

the

non-power-hke

behavior _on the BL and in EMA.

Most of the results of the present paper were announced earlier in

il 8].

2. Order parameter ^{function and the} distribution of

one~point

^{Green function.}

We _start from the disordered

tight-binding

mortel defined

by

the Hamiltonian

Ù

=

£

_p,

a)

_{a, +}

£

t~~

a)

_a~

_{Il )}

' <'J>

where the site energies p, ^or the

off-diagonal

matrix elements t,~ ^are

supposed

to be random variables. The _main

abject

of

investigation

_m this paper is _a distribution of the local _one-

particle

Green functions

GR

=

(jÎlE+t~ -Ù)~'j j)

~~

G~

=

(jj(E-i~ -Éi-'j j).

Here

subscripts

^R and A stand for the retarded and advanced Green function

respectively

E is the

(Fermi)

_energy and _J~

~ 0 is the level

broadening (an imaginary frequency).

In order to find this distribution _we consider the

following

set of correlation functions _:

~X't,m

=

(G[ Gj)

,

j3)

where the

angular

^{brackets denote} the averaging over the disorder.

TO calculate

lft,

~~

we use the

supersymmetric formalism,

which allows _{one to} express the

disorder-ai<eraged products

of Green functions _{in terms} of correlators of _a certain supersymme- tric model. The details of the formalism _can be found in

[10, 20].

We _are gomg to consider

throughout

the paper the _case of broken lime reversai invanance

corresponding

to the unitary symmetry ^of the effective action.

Physically

this consideration _is relevant for the system

subjected

to a

magnetic

field. However ail formulae of the present section can be

immediately

rewntten with _miner

changes

for the

orthogonal

_symmetry

(unbroken

lime reversai

invanance)

case as well.

According

_to the supersymmetry

method,

_we introduce _at any site i of the lattice _a 4- component supervector ~P, =

(R,

j, x, _,~

R,

~, x~

~)

with 2

complex commuting (R)

and 2

complex

Grassmannian

lx

components. The effective supersymmetnc ^{action then} reads

[20, 13]

e~ ^~ l*1

=

exp ₁

£ 4~) (E

_e~ L ₊

1~

_{~P, i}

£

t~~

~P) ~P~~

,

(4)

, <,j>

where L

=

diag (1,

1, 1,

1),

and the correlation function

(3)

can be written _m the form

n> f

~~>

~ f

m

~

~

~l ~Ù' _~J.

'

~

~~JÎ2 ~j,

2

)~

_~ ^{~ ~}

(5)

,

In _view of the fact that the

preexponential

factor in

(5) depends only

on the field at the site

j,

the field variables _m ail other sites _t #

j

can be

integrated

eut. This allows _{us to rewrite}

equation

(5)

in the

followmg

_way

'~f, ~m-f

m ^"

q

^{d lfi}

lR1

R _)~

lR? R2 )~

G1lfi

)

,

16j

where the function

G(~P

is defined _as

G(~P~)

=

Îfl

d4~~

e~~l*1 ₍₇₎

,«~

We will call

GI

~P

)

the order parameter ^function ; in context of the supervector formalism this

notion was introduced in

[13].

Different

analytical properties

of this function _m locahzed and

extended

phases

allow _{one to} describe the locahzation _{transition as a} spontaneous symmetry

breaking phenomenon.

This

justifies

the _use of the _{term «} order parameter »

(see

Sect. 5 for _a

more detailed

picture).

In view of the invanance of the action

S(~P)

with respect to the transformations

~P -T~P _; Te

U(111)

_x

U(1il _),

_{the function}

_G(4~)

_is

_actually

a function of 2 scalar variables, ^which are the mvanants of these transformations

G(4~)mG 4~~

^~~

4~, 4~~ ^~

4~) ⁽⁸⁾

2 2

In _view of this property, ^the Grassmannians _in equation

(6)

can be

easily integrated

_eut and equation

(6)

_con be reduced _to the form

~~

~~

~ ~ ~

~~~~ ^~~~~

~~ ~~

~

é

R~~Î(RÎ

~~~

Having

at our

disposai

ail _moments uX't,,,~ we can restore the whole distribution of the local Green function

(2).

Let _us consider for this purpose the _inverse

Fourier-Laplace

transform of the function

G(R(, R()

_:

G

(R(, R()

=

~

du

Î~

du e~"~~~ ^{~~~ ~~~~~~~~} _w lu, v

). (10)

-

ce 0

Substituting

equation

(10)

mto equation

(9)

and

performing

the integration over

R,

and

R~

we get

'~t,

n< ^~

d~ du

~l'

(l1)

~oe Î~

(U +

lÎ _(~

l v )"'

According

to the definition

(3)

of

lft

~,

this _means that w(u,

v)

is the

joint probability

distribution of real and

imaginary

_parts of the inverse Green functions

Gj', Gj'

Gi'=u-iv; Gj'

=u+iv.

(12)

Consequently,

the distribution function P

(g,,

_g~ ^of g, =

Re

G~

=

u/(u~

₊

v~)

and

g~ =

Im

G~

=

v/(u~

₊

v~)

is

equal

to

P(gj,g2)=

~

~~wl~~'~, ^~~

~

(131

(~> +

~2)

_{~'+ ~2 ~'+ ~2}

We _now suppose that _{we can}

perform

the derivation of the non-linear «-model

starting

from

some

microscopic

formulation. The

following

situations are known when this

procedure

is

ngorously justified

:

ii)

N-orbital

Wegner

model with N w states per site

[3-5, 20]

_;

(ii)

weak disorder

(metalhc)

limit

[6, 10]1

(iii)

_quasi ID models with _a

large

number of transverse channels

[21, 22]

_or with _a

long

range

hoppmg [23]

(iy)

_a system ^of small metallic

granules [1il.

To find details of the denvation of the

graded

nonhnear

«-model,

the reader is

again

referred to the literature

[10, 20].

The brief sketch of the

procedure

_{is as} follows _:

Iii

_{averagmg over} the disorder _m

equation (4)

_;

(ii) decoupling

^of

resulting quartic

terras m the action

by introducing

composite 4 _x 4

superrnatrix

variables

Qi conjugated

to the

dyadic product ^L'/~

_4~,

4~) L'/~ (the

so-called Hubbard-Stratonovich

transformation)

_;

(iii)

Gaussian integration over the supervectors _~P, ^and calculation of the remaining

Q- integral by

the steepest ^{descent method.}

As _a

result,

the matrices

Q,

_are restricted _to the

saddle-point

mamfold of the form

Q

_" r i AT

LT~'

,

(14)

where _r and A are real numbers, A _~ 0 and T is _a certain _set of transformation

forming

the coset space

U(1, 1/2)/U Il 1)

_x

U(1 1).

^It is convement to use a standardized set of

Q-matrices

:

Q

=

ITLT~

',

_so that

" r +

AQ.

The expression

(5)

for the correlation functions

(3)

_can

now be written _in the form _:

~n>-t

~~'f,

m ^~

fi fl

~À~

(QI

e ~ ~~~ d~fi

(~ Î ~')~ _{(~Î ~2)~}

X 1

x exp

(14~~ L'/~(E

i

AQ~ L'/~ 4~) ⁽¹⁵⁾

Here S

(QI

is the action of the

resulting

nonlinear «-mortel _:

S

(~'

=

i

£

_{Str Q~ Q~} + i ar

(p )

_~

£

Str

LQI l16)

~

<,J>

and

(p)

is the average

density

of states defined below.

In fuit

analogy

with equations

(6), (7),

_{we can integrate} eut m equation

il 5)

ail variables Q~ ^with k #

j

and define _an order parameter ^function m terms of the

Q-fields [12]

_:

F

joj)

₌

fl

dp

joi)

e~ ^~ (QI j17j

1#j

Then equation

(15)

is reduced to the

following

forrn

~ri,

~, ^t

~

d_p

iQ

F

iQ

d4~

iR i Ri )f (Ri _R~)m

_x

x exp

(14~ L'/~ (E

_r

AQ L'/~

_4~

il 8)

Calculation of the

integral

_over 4~ in

equation il 8)

_can be

easily

done with _use of the Wick theorem, and _we gel

Jft

~ ⁼

'~'

~

dp iQ i

F

(Q i

_x

ii _{k)! im k)!}

kl

] ^f-k ] m-k ] ]

~

~E-i

-AQ Îii ÎE-r-AQ133 ÎE-r-AQ ¹³ ÎE-r-AQ ³¹

~~~~

In _view of

Q~

=

-1 _we have

_~ E-r

~

A

Qm«+ar(p) Q. l~°~

lE-1-~Q~

"

jE-ri~+A2 lE-1)~+A~

Using

the defimtion of the correlators

Kt,

~

it is easy ^to show that the

coefficient

m front of

Q

in

equation (20)

is _ar limes average

density

of states

(p )

=

) _{(GA G~)}

=

~~)

~.

(21)

"1

(E r)

₊ A

Using (20)

and

performmg

_a

simple

transformation _{we may} present

equation (19)

in the

followmg

forrn

~~

G~

_« f

G~

_{« m}

~~~~ ÎÎ

"(PI "(Pi

Î ii

k

)ÎÎÎ~ _k)!k!~ ^{~~ ~~~}

^~

^{~~~ ~~~}

^~

^~~

^~

^{~~~ ~~'}

^~~~~

We will _use the standard

parametrization

of

Q-matrices il 0]

-iÀj

pje'~'

Q

~

(U ^~lÀ2

^Àl2~

~~~

U~~

V

pje~'~' _iÀj V~~ ~,

^'

p~e~~~ iÀ~

ÀI~Î+À~Î> À2~Î~ÀIÎ>

0~vi~oe; 0~H2~i; o~4j,42~2ar;

U

= exp

°

(~

_; V

= exp1

°

~~

^;

⁽²³⁾

£Y fl

with Grassmannians _{a, a} *, fl, fl * The

corresponding

measure is then given

by

dp (Q)

"

~

dÀ

j

dÀ~ d4

j

d4~

da da ^*

dfl dfl

^*

(24)

IAi À

2)

It _can be shown

[12]

that _m view of the

U(i Î1)

x

U(1 1)

invanance the order parameter function F

(Q) depends only

on «

eigenvalues

_»

j, À~

(cf. Eq. (8))

_:

F

loi

=

F

iÀ,,

À

~i 125)

Thus~

_we defined the order parameter ^functions m bath _vector and «-model formahsms

(G (4~

^{and F}

(Q), respectively).

The connection between them follows

immediately

from

(6), Ii 8)

_:

GI

m

)

=

dp iQ i

F

iQ i exP1-

4~

L'J~IE

_r

AQ i L'/2

m

j 126)

Let _{us now}

shghtly

^{redefine the function}

GI

^~P

)

;

Gjm)

=

e-1(~-'~*~L~ _{GolA'/~~P) 127)}

so that

Go14~ )

₌

dp iQ )

^F

iQ e-'*'~"~

Q~'~~ *

128)

Then, according

to

equations (22)

and

(28)

~X'iÎÎ ~m-1

=

~j

_~,

d~°lR? Ri)~ lR? R~)~ Gol~P)

m f

~ ~ ~ ~ _~

é2G~

Î!

_m!

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

éjR() à(R()

^~~~~

On the other hand,

expanding

the exponent ⁱⁿ

equation (28)

in power series _we gel

m t

Go(R(, R()

=

£ ~q R( R("'lf)°( ₍₃₀₎

t,n<

~'

It _is easy to see from

equations (29)

and

(30)

that

Go(R(, R()

_satisfies the

followmg

hnear eqUat1°n

Go(R(, R() é2~~

=

d(S() d(SÎ) _{Jo12 Ri Si ) Jo12 R2 52~}

~

j~2~

~

jS()

^{' ~~~~}

l

where

Jo

is the Bessel function.

Furthermore,

_{summmg up} the _{serres in}

equation (30)

we obtain

Go(R(, R()

=

)dz

dz * P

o

(z

_exp

(- iR(

z ^* +

tR(

z

(32)

lm_z

~ 0

Here P

o(z)

m P

o

lu,

v is the

probabiiity

distribution of the real _u and

imagmary

v

pans

^{of the} norrnahzed Green function

_G~-«

~

G~-«

~

"(PI

^{~~~~~ ~}

"(PI

^{~ ~~} ~~

The relation

(31) implies

the

followmg

symmetry property ^{of he} distribution function

Po(z):

Pol- z-'1= izi4Poiz). 1341

Therefore,

the distribution of the one-point ^{Green functions} is invariant with respect to the inversion _z

- -1/z. This property ^is exact

only

in the nonhnear «-model

(saddle-point)

approximation.

The denvation

performed

in this _section for the unitary symmetry case con be

repeated

with

miner

changes

^{for the}

orthogonal

case as well. The symmetry property

(34)

has

exactly

the

same

form, independently

of the symmetry ^{of ensemble.}

3. Distribution of one~pomt ^{Green function} in the _case of

unitary

symmetry.

In this _{section we}

perform

_an

explicit

calculation of the distribotion function

Po(z ),

starting

from

equations (32)

and

(28)

of the

preceding

section. Our aim is _to express

Po(z

in _terms of

the order parameter ^function

FIA,, À~). Taking

^for the sake of convenience

4~

=

(R,,

0~

R~, 0)

with real

R,,

_{R~ in}

equation (28),

we hâve

~0(~Î> ~Î)

~ dl~

(Q)

~

(Q)

_eXP

(~ l~Î _Q''

₊

Î~Î

_Q33

_{~l ~2}

_Q>3 ~

~2 Q3') (35) Substitutmg

here the

parametnzation (23)

for the elements of

Q-matrix

and

integrating

over ail

variables except

À,,

_{À~, we} get

Go(R(, R()

=

~~' ~~~

F

(À,,

~

)

e~^{~'~~~ ~~~~} _x

À1 ₂

x

(2 Io(2

_pi

Ri R~)

À

R( R( -1, (2

_p,

R, R~)

_pi

R, R~(R(

₊

R( )) (36)

where

Io, Ii

_are the modified Bessel functions. The distribution function P

o

(z

can be obtained

by

mvertmg ^the transformation

(32)

which is of Fourier type ^with

respect

to the variable y =

R( R(

and of

Laplace

type ^with respect ^to x ₌

R(

₊

R(

_:

Po(u,

v =

) _{j~ dy} ) ^~

^{~ ~~} _dx

_Go %, j

_exp

_(iuy

₊

_{vx) (37)}

"

ce

"1 -<ce

For this purpose, we should make _an

analytical

continuation of

Go(R(, R()

_to the whole region Re

(R(

₊

R()

~ 0

(onginally

it is defined in the region

R(

~

0, R(

_~ 0

only).

This can be done

by

_using the identities

io(2

_P,

R, R~)

₌

j'~'°' ^{dp p~j+R(R]~~}

~"l

,-~~

p~

'

~~°

Ii (2

_p,

R, R~)

₌ ^~'

~

~ ~~

dp e~'"~(e~~

^{~~~ l}

).

~"l~l~2

<->ce

Substituting (38)

in

(36)

and then _m

(37)

_{we get}

j

oe 'C°

^dÀ

dÀ~ <

+<ce

~~, ,,,, ~,,

~°~~~~~

(2ar)~ -~~~ -ioe~ ~l~À2~~~'~~~~ <-ioe~~~

^{~ ~}

x À

j

(x~ y~

^{e~'~~~ ~~}

~~~~~~'-

p

lx

_e~'~~(e~~~^~~~~~~' l

ii ⁽³⁹⁾

2p

Performing consecutively integration

over x, y and p, we obtain

j

dÀ, dÀ~

Po(u,

v ₌

FIA

j,

À~)

x

2 _ar

Ai

À~

~l~l[+Sl +i~Î-1>ùlijôi2vÀ-U~v~-1). 140)

Two short comments _are

appropriate

here. At

first,

_we observe that the _« compact » variable À~

plays

an

auxiliary

rote _in

(40). Namely,

the distribution

Po(u, ^v)

^{is determmed}

by

the

function

Fo(À,)= ~~~

F(Àj,À~) (41)

4 " À>

À2

depending only

on the _« non-compact »

eigenvalue

,.

Secondly,

P (u~ v is determined

by

the value of

Fo(À, )

and its first and second derivatives in the

point

~'

^~

~/v~~ ÎÎ~~ Î~ ÎIrÎz~ImÎ~'Î

^~~~~

Therefore,

two variables

À,,

À~

equally

introduced in _a very formai way as

eigenvalues

of

«retarded-retarded»

land

«advanced-advanced» _as

well)

blocks of the

supermatrix

Q (see Eq. (23)) play essentially

different rotes _: À~ tums ont to be _an

auxiliary

variable and

Ai acquires

a

physical

_meaning determined

by equation (42).

After substitution of

equation (41)

_mto

equation (40)

the remainmg

mtegration

_over

À,

can be

explicitly performed

due _to the presence of the &function. After _a

straightforward, though quite lengthy calculation,

the result _can be

expressed

in _a rather compact ^form :

Po(u,

vi

=

~ IA

1)

^~

Fo(À, )(

,,2 ₊ ~2₊

(43)

v~ ôÀ

j

ôÀ,

Ai

~_~

It is easy ^to

venfy

^that the distribution function

(43)

does

satisfy

the symmetry property

(34).

This is the immediate consequence of the fact that

Ai

m equation

(43)

^is an mvariant of the transforrnation _z

- z~ ^' v _- VI

(u~

+ v~)~ _{u -} u/

(u~

+ 1>~

).

Let _us note that ail calculations _in this section _were done

explicitly

for the «-model of

unitary

symmetry. ^{In the} case of

orthogonal

_symmetry the calculations become much _more

cumbersome and _we have _not got a relation

analogous

_to

equation (43)

in _a

manageable

form.

4. Distribution of local

density

of states.

In two

preceding

^sections we

investigated

the

joint probabihty

distribution

Po(u,

_v of the real and

imaginary

parts ^of the normalized one-point ^Green functions. In this _{section we} consider the distribution of the imaginary part v ₌ p/

(p) (LDOS) only

P

,

Iv )

=

du P

o

lu,

v

j44)

For the _case of unitary symmetry we can substitute _into equation

(44)

the distribution

Po lu,

v calculated in the _section 3

lit

is _{more convenient}

actually

to use for this purpose the

expression (40)

rather thon the final form

(43)).

Then after _some

algebra

_{we get}

~

>/2

p (v

j

^{à' "} dÀ

,

Fo

IA

,

~ ~

,

(45)

'

év2 Ù

1>~ + l

2L

Àj

2 v

where

Fo(À j)

is defined in

equation (41).

The LDOS distribution function

P, Iv

can also be obtained in _{a more}

straightforward

_way,

by

direct calculation of LDOS _{moments :}

~~~~

j21

_jn

_~Î ^~~

~

ii

ii'~'

)1

'~Ù~

_~~~~

For the unitary case

lf)°j~

is

given by equation j22)

and _we find

~~~~

j21),'

^~

^Ii k)1 in'~'i

_k

_)lklkl ^{~~ ~~~}

^~

^{~~~ ~~' ~~~ ~~~ ~~' '~~~~}

Equation j47)

_can be rewritten in the form

(V")

=

2~ "

Îdjl(Q)

^F

^{(Q Ill Q" lQ33}

⁺

^Q'3

⁺

^{Q3')~ (~~)}

It _can be

easily

shown that ail terms in the expansion of

equation j48)

which contain different

powers of

Qj~

and

Q~j

^{vanish in view of the} invanance of

F(Q), equation j25),

_so that

j48)

is indeed

equivalent

to equation

j4?).

That allows _{one to restore} the distribution function

Pi lui

in the

following

form _;

~

lIV) ₌

dJ1(Q)

F

(Q)

à

V

(1Q> 1Q33 +

Q'3

_{+ Q3>}

)j ⁽⁴⁹⁾

Using

_now the parametnzation

(23)

^and

performmg

the

integration

_over ail the variables except

À,,

À~, we

reproduce

^{the result}

(45).

The

procedure leading

to an

expression

for the LDOS distribution in the form

(49)

_can be

straightforwardly repeated

for the

orthogonal

symmetry case. That

gives

m fuit

analogy

with

(49).

However, the actual calculation of the

integral (50)

is very cumbersome and _{we are} not able to present ^{the result} m a

simple

form,

analogous

to

(45).

Let us note in conclusion of this section that the LDOS distribution

(45)

_possesses the

following

symmetry property

P, Iv

^'

= v~ P

,

Iv1 151)

In the

asymptotical regions

of small _{v «} and

large

_v w values of

LDOS,

it can be wntten in

the form _:

P, iv)

~

,fi

_v- 3/2

j"

^~

^)~~,~~ $ _{i~foiÀ ii,}

v »

(52)

~/2

lÀ

v À

P, lui >,fi

v~ ^~/~

j~

^~

^~)

~~~

~~ (ÀFO(À )),

v « (53)

,/~~

jÀ- Iv)

dÀ

5. LDOS distribution _near the Anderson transition.

The relations obtained above _are of

general

nature and do _not

depend

_on a

spatial

dimensionality.

Ail information about the distributions of local

quantities

is contained in the order parameter function. In the

simplest

^{0 D and}

quasi

D _cases this function _can be found

exactly,

that allows one to

study

the distribution of LDOS and

eigenfunction

_components for _a

small

partiale [25

_or a wire

[24, 26].

In the present paper we

study implications

of the obtained relations for _a system ^of

higher

dimension in the

vicinity

of the Anderson metal-insulator

transition. It has been known for _a

long

time

il 5]

that the LDOS distribution function has _a different behavior in the localized and extended

phases

of _a

system. Namely,

m the delocalized

phase

the LDOS distribution has _some finite

limiting

forrn when the extemal _« level width

»

J~ tends _{to zero.} In contrast, in the localized

phase

the LDOS distribution is

singular

in this

limit,

so that

keeping

_~ ~0 is necessary m order _to

regularize

this

expression.

In

reference[13]

_we found _{a one-to-one}

correspondence

between the symmetry

breaking

mechanism which govems the Anderson transition

(with

_an order parameter

being

_a function _m the intemal

superspace)

and this

physical description

of the localized and extended

phases.

This _was

possible

due _to relations of the type ^of

equations (10)

and

(13)

which

provide

_an

intimate connection between the behavior of the order parameter ^function in the internai superspace and the distribution of LDOS

reflecting

fluctuations of

eigenfunctions

in the usual coordinate space. Below _we present ^the

analysis

of _a cntical behavior _near the Anderson transition point ^based on the _same kind of relations. In

fact,

_we find it _more convenient to work

within the NSM

approach using

relation

(45).

We would like to stress however that the

analogous

considerations _can be

straightforwardly

carried _out for the

tight- binding model,

_I.e.

in a supervector ^forrnalism as well. Our conclusions

conceming

the critical behavior _can be obtained in this way, too, and _are therefore of

general validity

for the disordered systems under

consideration.

We staff our

analysis by reminding

the reader of the spontaneous symmetry

breaking

description

of the localization transition _in the framework of the

supersymmetric

formalism

il1-13].

The first term in the nonlinear «-model action

(16)

^is invanant with respect to the

global

U

(1,

2

)

transformations

Q,

-

TQI

T~ ^' The second terra breaks this invariance down to

U(111)

x

U(1il

_{). One}

_might

_{therefore expect} that in the limit _~ -0 the function F

(Q)

defined

by equation (17)

becomes U

il, 12)

_invariant. The

only

function

possessing

such _a property is a constant ;

furtherrnore~

it _is easy to prove that

FIL

= l~ so that the

U(1, Ii 2)

invariance

implies

that

F(Q)=1.

In

fact,

it tums out that the condition F

(Q)Î~

_o ^-

defines the localized

phase

of the system.

However,

the convergence of the

function

F(Q)

to

unity

is non-uniforrn in

Q.

More

precisely,

it _can be shown that _as

J~ - 0

F

jQ )

>

Fi j2

_ar

jp )

_~

j

) j541

with

Fi ix

~

Il i i ₁₅₅₎

Therefore,

when _~

- 0 the

typical

scale of the variable À

j is J~~ _~, i e. is defined

by

the symmetry

breaking

_{parameter ~.} In contrast~ in the delocalized

phase

the function F

jQ)

=

F

jÀ

j~

À~)

_{remains a} non-trivial function of À

j~ À~ ^{in the hmit} ~ =

0 and thus the abovementioned symmetry is

spontaneously

broken. However~ in the

vicinity

of the transition point ^{the function F becomes}

practically independent

^of À

~ F

IA

i>

À~

)

> F

dIA

),

In

addition,

a

large

scale A

(diverging

at the transition

point)

emerges which governs the

typical

values of

the variable

Ai

^{This scale}

plays

_a rote

analogous

to that of _~ ^~' _in the localized

phase.

The main difference is that _~ ^' is determined

by

the symmetry

breaking

terra in the

action,

and the appearance of A is _a consequence of the spontaneous

breaking

of symmetry. ^The described mechanism

explams

the _use of the terni _« order parameter ^function »

conceming

F

(Q),

The similar

picture

of the localization transition _was found in

[13]

for the _case of

tight-binding

model _in the framework of the supervector formalism. In this _case the order parameter ^function

is defined

by equation (7)

_; it is _a function of _two scalar variables

(see (8))

which _{we now} choose in the formx

=

~P) 4~j

+

4~j

_4~~

m

4~'

_4~,

y =

4~) 4~, 4~j

_4~~

m

4~'L4~. _Again

_the

action

(4)

is

U(1, 112)-invanant

_{at ~}

=

0 _{; ~} ~0 breaks this _mvanance down. For the

function

G(x, y)

such _{an mvanance means}

being independent

of _x. The localized

phase

is

again

the

phase

where the invanance is _not

spontaneously

broken. This _means that _as

~ - 0 G

lx, y)

^tends to a function which does _not

depend

_{on x.} This convergence is _not

uniform _{m x;} in

fact, G(x, y)=Gt(i~x, y)

when _~ -0. In the delocalized

phase

the

dependence

of

G(x, y)

on x remains in the limit _i~

=

0. However, in the

vicinity

of the

transition

point

this

dependence persists only

in _a region of

large

^{values of} x. The

corresponding

scale A in the space of the variable _x

diverges

at the

mobility edge. Therefore,

the variable _x

plays

_a rote

completely analogous

to that of the _« non~compact _»

eigenvalue À,.

We _now

proceed

in the

following

_way. We shall substitute the order parameter ^function obtained for the NSM _on the Bethe lattice

il1, 12]

into

equation (45)

and

subsequently analyze

the

corresponding

distribution

Pi (v

). We demonstrate that

although qualitatively

such _a form of the LDOS distribution function should hold for d

~ oe as well, _some

quantitative

^features of the solution _are inconsistent with the d-dimensional lattice structure and reflect the structure of BL. Guided

by

a

physical reasoning,

we find the correct behavior of

P, Iv

and observe how it

modifies, according

to equations

(45), (52), (53 ),

the behavior of F

(À,,

À

~).

^{This leads} to a

drastic

change

in the critical behavior _at the

mobihty edge.

We start our

analysis

from the extended _state

phase.

As _was shown in

il1, 12],

close _to the

transition

point

on the Bethe lattice the function

F(À,, À~)=F~(À,)

^has the

followmg

behavior _:

À '/2

gr (n À

F

d(À

=

Î _a În À _SIn À W A

(56)

A În A

À >/2 À 1/2

F~j(À

cc exp b In

A~

exp c In _A

ÎÎ

w A

(57)

with the

large

_«

symmetry-breaking»

scale Ah1 and some numencal constants a, b, _c.

Substituting equation (56)

into

equation (45)

_we find

P, (v

_cc v~ ^~/~ A~ ^'/~

ln~

_A

(1

^{+ cos " ~~}_{In A}^{~~~~ ; A~ ' « v « A}

⁽⁵⁸⁾

Furthermore, the function

P,

Iv is

exponentially

small when either v w A _{or v «} A

',

in view of the relations

P, Iv

_cc

F~(v/2

,

v « A ^'

(59)

P, lui

cc

F,j(1/2 vi,

v w A

(60)

and the

corresponding

asymptotic ^behavior

(57)

of the function

Fj(À

(we have omitted the

preexponential

factors which _are of _no importance ^for our consideration).

According

to

(58)

and

(59), P,(v)

has a maximum in the region

v~A~'

_The

qualitative

behavior of

P,

iv is shown in

figure ('),

We _stress that the normalization

integral

^of

P, iv

^{is dommated}

by

the region v

A~' (in

view of the v~ ^~/~ decrease in

P, lui

at

larger vi.

Therefore, not

only

the _most

probable,

but also

typical

values of _{v are} of order of

A~'.

_{In contrast,} the _main contribution to ail _moments of the distribution

P, lu

_comes from the region v

A, yielding

~~"Î~~P,lv)dvccAq-,

,

q»1, j6j~

~~~ p/<p> ^~

Fig.

l. The

shape

^{of the} distribution function Pi (v) of the norrnalized local density ^of states v

p/(p)

in _a vicinity of trie Anderson _transition point. ^Trie most

probable

and typical value

v A ^' _is much less than trie average value

(v)

=

The critical behavior of the scale A is discussed in trie _text.

Let _us show that such _a

qualitative

form of

P, iv )

is in fuit agreement ^{with the}

predictions

of the

scalmg theory

for the multifractal behavior _as d

- ce

[28].

In the

scaling theory,

the

following

cntical behavior of the

averaged

moments of _wave functions

P~(E) (P~ being

^the inverse

participation ratio)

in both localized and delocalized

phase

can be found

[29, 28]

P~(E

_cc

E~

^{E ~«} cc f ^~'«

(toc. phase) (62)

R~(E)

cc N~ ^'

P~(E )

_cc

(E _E~

^"~ _cc f ^~/~

(deloc. phase) (63)

where N _is the total number of _sites (the

volume)

of the system~ f is the correlation

(localization) length, foc (E-E~(~~

_at the

mobility edge.

The cntical indices ar~,

p~, x~ satisfy

the

following

relations _:

p~ =

du

(q 1) p~, x~

=

ar~/v ⁽⁶⁴⁾

It _was shown _m

[28]

that ail x~ ^{should tend}

simultaneously

to zero:

x~~1/d

when

d

- ce.

Thus,

the exponents

p~,

^{which determine}

according

to equation

(63)

the behavior of

(')

Let _{us notice} that the distribution function

Pi

(v)

suggested by

trie results of trie renormalization-

group treatment in 2 _{+ p} dimensions [27, 14] has trie _same qualitative features and satisfies trie exact relation (51),