A novel field theoretical approach to the Anderson localization : sparse random hopping model

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A novel field theoretical approach to the Anderson localization : sparse random hopping model

Yan Fyodorov, Alexander Mirlin, Hans-Jürgen Sommers

To cite this version:

Yan Fyodorov, Alexander Mirlin, Hans-Jürgen Sommers. A novel field theoretical approach to the

Anderson localization : sparse random hopping model. Journal de Physique I, EDP Sciences, 1992, 2

(8), pp.1571-1605. �10.1051/jp1:1992229�. �jpa-00246642�

Classification

Physics

Abstracts

Ii.30Q

71.30 71.55J

A novel field theoretical approach to the Anderson

localization

_:

_sparse random hopping model

Yan V.

Fyodorov (I, 2),

Alexander D. Mirlin

(I)

and

Hans-JUrgen

Sommers

(2)

(1) St.

Petersburg

Nuclear

Physics

Institute, 188350 Gatchina, St.

Petersburg

District, Russia (2) Universit&t Gesamthochschule Essen, Fachbereich

Physik,

D-4300, Essen,

Germany

(Received 14 February 1992,

accepted

24

April

1992)

Abstract. We

develop

_a novel

supersymmetric

field-theoretical model

describing

_a motion of _a

particle

in a system with

long

_range

percolative

type

off-diagonal

disorder. The model is

investigated

in _a saddle

point approximation,

The delocalization transition manifests itself _as spontaneous breaking of _a symmetry ^{of the} effective

Lagrangian.

Both

phases

of the system can

be described

by

_means of _an

orderparameter

function

having

_a clear

physical meaning

related to

statistical

properties

of Green functions. We calculate the inverse

participation

ratio and correlation functions in both localized and extended

phases.

The found critical behaviour agrees with results obtained in the framework of effective medium

approximation

for the

supermatrix

_a-

model. Some

speculations

about the value of upper critical dimension consistent with the obtained results _are put ^forward.

1. Introduction.

In the course of

development

of the

theory

of second order

phase

transitions mean field

(MF)

models

played

_a

distinguished

role.

They

allow _{one to} describe a transition in terns of _an

order

parameter reflecting

_a

phenomenon

of spontaneous

symmetry breaking

and

giving

_a

possibility

_to label

phases

of _a system. ^A

physical

_nature of _an order parameter ^{extracted from} the MF consideration is the _most robust MF result which survives _even when MF

predictions

are

quantitatively incorrect,

_e.g. when the

spatial

dimension d is below the upper critical dimension

d~.

Usually

_a

partition

function of _a real system can be written in form of _a functional

integral

over « local

» order

parameters

q~ attached to lattice sites I _:

5

"

In _dqi

~XP

(~

£

lqil) _(i)

The MF

equation

for _a

«global»

order parameter q can be obtained _{as a}

saddle-point

condition for the

Landau-Ginzburg Lagrangian £[q~].

The

saddle-point approximation

becomes exact when the range ro of interactions in the system

(e.g.

the range of

spin-spin exchange)

tends _to

infinity.

Corrections _to MF results _can be obtained in the form of _an

expansion

in

ri~.

At

dwd~

the corrections

diverge,

that necessitates the _use of _a renormalization group method

[Il.

Since the field theoretical formulation tumed _out to be _a very useful tool of

investigation

^of

thermodynamic phase transitions,

it is

quite

natural _to try to

develop

_a similar

approach

_to

critical

phenomena

of _a different

(non-thermodynamic) origin.

The Anderson localization is

one of the most famous

phenomena

of such _a type. ^A quantum

particle moving

in _a random

potential

tums out to be localized if the disorder is

sufficiently

_strong. To

investigate

this

phenomenon

Anderson

[2]

introduced disordered

tight-binding

model characterized

by

^the Hamiltonian _:

3ll

=

£

_{u~a~+ a~} +

£

t~~ a~+ a~ ^;

t~(

= t~~

(2)

, ,j

where site

energies

_{u~ are} assumed to be

independent identically

distributed random numbers and the

hopping

matrix elements

t~~ are

equal

to a constant value if sites I,

j

are nearest

neighbours

and _zero otherwise. In the _more

general

_case

t~~ ^can be considered _as random _as well.

The _most

developed

field-theoretical

approach

to the

problem

_was formulated

by Wegner [3]

who introduced the so-called A'-orbital model. This model is _a

generalization

of the

Anderson _one to the _case of A' electron states per site.

By using

_a bosonic

[3-4]

_or fermionic

[5]

version of the

replica

trick the

problem

was

mapped

onto a field-theoretical model of

interacting

matrices

Q~.

The effective

Lagrangian

for the

n-replicated

_system tumed out to be invariant with respect to transformations

forming

a group O

(n,

_n

) [3, 4]

or

Sp (4n [5],

with _a

frequency

_w

playing

the role of extemal symmetry

breaking

field.

However,

in contrast to usual situations the symmetry was found to be

spontaneously

broken whenever the

density

of states is _nonzero.

Namely,

_{an average} value

(Q)

of the matrix

Q~ conjugated

to the

frequency

_w and _so

expected

to

play

the role of _an

order parameter ^is

nonvanishing everywhere

in the band and therefore insensitive _{to a}

localization transition

[3-6].

In view of this fact _a MF

approximation

in such _an

approach

fails to describe _a localized

phase

and there is _no transition _on the MF level _at all.

To succeed in

analytical investigation,

the limit A'- _{w was}

exploited

in references

[3, 4, 6].

Such _a

procedure

results in

discarding

site-to-site fluctuations in

eigenvalues

of matrices

Qi

that

physically corresponds

to

neglecting

local fluctuations of the

density

of states. These fluctuations _are

expected

to have _no influence _on the critical behaviour. The obtained model

belongs

to a class of nonlinear matrix «-models. Its

perturbative

treatment

provides

a correct

description

for the _case of weak disorder

(the

so-called weak localization

region).

A nontrivial localization transition emerges in the «-model framework _as the result of _a renormalization group ^treatment in 2 _{+ e} dimensions

only.

This

gives

a

power-like

critical behaviour of

relevant

physical quantities (conductivity,

localization

length, etc.) [3].

Thus,

_contrary to the _common

phase

transitions

theory,

the «-model

approach

does not

provide

a

description

of _a delocalization transition _{on a} MF level.

Mean-while,

such _a

description

is

highly

^{desirable in order} to

study

the transition in the dimensionalities d _m 3.

Indeed,

for

high

d the localization transition _occurs in the

region

of

strong

^disorder

where

nonperturbative

effects _can

play

an

important

role

[8].

The

only nonperturbative

treatment of the Anderson transition available _so far _was achieved in the framework of the Bethe-lattice version of either the «-model

[9-1Ii

_or the Anderson model

[12, 28, 31].

The Bethe lattice is _an infinite tree-like

graph

with fixed

coordination number of all sites. This hierarchical lattice _structure makes the

problem

analytically

tractable. In view of the

exponential

^{increase of} number of sites with the distance from the

origin,

the Bethe lattice cannot be embedded in _a d-dimensional space for any finite d and is

commonly interpreted

as

corresponding

to infinite

dimensionality.

It is known that statistical models defined _on the Bethe lattice

usually display

_a critical behaviour

coinciding

with that obtained in MF

(infinite-range) approximation. So,

for the localization

problem,

where _a traditional MF

approximation

is not

informative,

the Bethe lattice formulation is _a natural

starting point

for

studying

the transition.

However,

it is not evident how to

proceed

in calculations of finite-dimensional corrections to Bethe lattice results. A way ^to overcome this

difficulty

_was

proposed by

Efetov

[13]

in _a context of the

supersymmetric

«-model

approximation.

He

managed

to construct a field-theoretical reformulation with _an effective

Lagrangian having

_a nontrivial

saddle-point coinciding

with the solution of _a basic

selfconsistency equation

for _a

corresponding

Bethe lattice model.

However,

Efetov's method

seems to be

quite

artificial.

Meanwhile,

in the recent papers

[14, 15]

it _was shown that in th _i so-called sparse ^random matrices

(SRM)

^model a

Lagrangian description

of localization tran ~ition emerges in _a natural way. The SRM model

[16]

introduced

originally

in _a context of

spin glass theory,

combines

features of both infinite range

(MF)

^and infinite dimensional

(Bethe lattice)

models.

Namely,

in this model any site I has the _same

probability p/N

to be connected with any other site

j,

where N _» I is the total number of sites and _p is the _mean

connectivity parameter.

^{On the} other

hand,

for any

particular

realization of connections between sites

they

form _a structure

locally equivalent

to _a Bethe lattice with random coordination number.

In the

present

paper we derive _a novel field-theoretical formulation of the localization

problem starting directly

from _a

microscopic tight-binding

model of the type

equation (2)

and

using

the SRM model _{as a}

guiding example.

^{It allows} us to describe the localization transition

as a spontaneous

breaking

of

symmetry

^and to reveal _a functional nature of the

corresponding

order parameter. ^{A discovered}

physical meaning

of the order parameter ^function

[12, 15]

relates _our formulation to the _common

qualitative picture

of localization transition

[17].

The outline of the paper is the

following.

In section 2 _we define the model and derive the field-theoretical formulation for correlation functions. In section 3 _we calculate the

two-point

correlation function in _a

saddle-point approximation.

In sections

4,

5 _a spontaneous symmetry

breaking picture

of the localization transition is

developed

^and the

physical meaning

^of the order

parameter

^{function is}

explained.

Sections

6,

7 _are devoted _to the

investigation

of the obtained

expression

for the correlator in the localized states

region.

In section 8 the critical

behaviour of the diffusion _constant is determined. Section 9 contains a discussion of results and _some

speculations conceming

the value of upper critical dimension for Anderson

localization.

2. The definition of the model.

Let _us

investigate

_a

tight-binding

Anderson model defined _{on a} d-dimensional

(cubic)

lattice.

Diagonal

elements of the

Hamiltonian,

_« site

energies

_{» u~, are} assumed _to be

independent

random numbers

identically

distributed

according

to a

probability density y(u,).

The main

peculiarity

of _our model

making

it different flom those considered

by

other authors is _a distribution of

(real)

_«

hopping

matrix elements _»

t~~ ⁼ tj~. ^Our

particular

choice for such _a distribution is

f(tip)

₌

(I p~~) 8(ti~)

+ p~~

h(t~~) (3)

where

h(z)

is _an

(even)

distribution function

nonsingular

at z =

0,

_p~~

=

fi«( jr,

_r~ with

fi

₌

po/ £

_«

( jr ) being

a normalization constant, po m I and the function _«

( jr,

r~

) being

of the form

l

0

~

[r(

_wro

«((r(

=

(4)

0; [r(

_~ro.

Thus,

in such _a model

hopping

elements between lattice sites

separated by

a distance

exceeding

_{a «}

hopping

_{range » ro} are

identically

_zero. On the other

hand,

two sites located within _a

hopping

_range flom each other have _a

probability fi

to be connected

by

_a nonzero

random

hopping

element

tq

distributed

according

to a

probability density h(tq)

and _a

probability

I

fi

to be disconnected.

If the

hopping

_{range ro is} of the order of the lattice

spacing

_a the connected sites form _a

percolative

structure and the present ^model can be considered _as

describing

«quantum

percolation

_»

[18].

In th<

opposite

limit row a we arrive at _{a «} sparse random matrix _» model

properties

of which _wer,i studied

analytically

in

[14-16]

and

numerically

in

[42].

The form

(3)

for _a distribution

f(t~~)

^of

off-diagonal

matrix elements _was chosen

by

_{us on}

the

following grounds.

An introduction of such _{a «}

percolative

_» structure

(I)

allows _us to

apply

to the Anderson model technical tricks

developed

in the _course of

investigation

of correlation

properties

^of _sparse ^{random matrices}

[14, 15]

and

(it)

introduces in the

theory

_{a new}

large parameter

ro that makes the

problem analytically

tractable

justifying

_a

saddle-point approximation.

Let _us

mention,

that in the

theory

of

magnetic phase

transitions in ideal

crystals

the range of

exchange

interaction between

spins

is

just

a parameter

justifying

_a mean-field

approximation.

It is _an

exploitation

of _a similar

parameter

^{that allowed}

Sherrington

and

Kirkpatrick [19]

to

develop

_a mean-field model of disordered frustrated

magnets spin glasses

which

appeared

to

display

_a rich and

quite

unusual behaviour

[24].

We

hope

that the introduction of

a parameter ^{of similar} nature into the

theory

of Anderson localization proves ^to be useful _as well.

It is worth

mentioning,

that if the parameter ro _~ a, the saddle

point approximation

used in the present paper is

expected

to be valid _as well

provided

the space dimension d _» I.

The main

object

of interest in the

theory

of localization is the

following

correlation function

K~j,k)=lj( )k ^{) j}

,

e-+0

(5)

E+ie-fl

~

E-ie-fl

where the bar stands for

averaging

over the disorder. In

particular,

a

frequency-dependent conductivity

_«

(w )

is related to

K~j, k) by

^the Kubo formula

[20]

which _{at zero}

temperature

in the low

frequency

limit

w = 2 ie _- 0 takes _a form

[7]

«(w

_-

0)

=

~ lim

e~ £

_(r~

r~)~ K~j, k)

=

~ _~

lim

e~

^~

k(q~)

~

V~ (6)

e~o _k

~~E~o _aq~

_~"

where v is _a volume per site, d is the

dimensionality

of space, E is

equal

to the Fenni energy and

k(q~)

_{stands for}

a Fourier transform of

K~j, k).

In the

conducting phase

_«

(w

-

0)

has _a

nonvanishing

value due _{to a} diffusion form of the correlation function

k(q~)

=

~~"~ ⁽⁷⁾

Dq

I _w

where _p is the

density

of states and D is the diffusion constant. On the other

hand,

in the localized

phase K~j, k)

₌

e~~ f~j, _k)

_with

a function

f~j, _{k) having}

a finite limit when

e _- 0. A

physical quantity

which is

frequently

used to

distinguish

the _two

phases

is the _so- called inverse

participation

ratio P

(E)

related to the value of

K(k, j )

_at

coinciding points

k

=

j by

the

expression [21]

P

(E )

=

lim eK

~j, j )

=

f~j, _{j (8)}

'rP

e ~ o "P

To calculate

K~j, k)

_we introduce at every site

j

two

four-component

supervectors

@~j; p=1,2:

~~j ~~~~~ ~~~~' ~~' _~P~j

~~~

with real

commuting

_components R=

(R~~~,R~~~)

and Grassmannian

(anticommuting) components

X~ ₌

(X *,

_X

).

We denote

by

_a

dagger+

Hennitian

conjugation

and

by

a star*

complex conjugation

defined for Grassmanians _as follows

(I)

_:

(X)*"X*, (X*)*"~X, (XiX2)*"XIX?'

The Grassmanians _are normalized

according

to :

ldxx

=

j dx

^* x ^* =

~~~

( lo) (2 gr)

Then it is

possible

to rewrite the

expression

for

K~j, k)

_{as a}

superintegral

_:

Kj, k)

=

is~~i(X?Xi)j (X2Xt)k exp(- Soi~i) (11)

where

[5l~

=

fl [d~ i,jl [d~2, jl, [d~p, j1

_~

_dl~~~/ ~~f/ _{~X~j ~XP,}

J

~'~~

,

So i~i

₌

~j+ (LE

_{+ I}

8

) 8j~ LHj~) ^{4~ (12)}

j,k

Here _we have introduced the

8-component

supervector

@j,

_@j+ =

(WI, it

)~ and the

diagonal

matrix

I

_with

eigenvalues

I for p =

I and I for p =

2. The

positive

infinitesimal _e

ensures the convergence of the

integral

_over the real

components

^of the

supervectors

~j.

Averaging

^the

exponential

ⁱⁿ

equation (I I)

over the disorder _we have

exP

sj~

m exp

soj~ j

=

(fl F(~~)) exp~£

«~~

r(~~+ £~~)) ⁽¹³⁾

; >«j

with

F(@)my~( @+i@) exp( ^{~E@+I@ -I@+} _@) ⁽¹⁴⁾

(I)

To fmd the definition of supervectors and

supermatrices

^and

simplest

miles of

handling

them _see, e-g. [22, 23] and also [43].

and

r(x)

win

(i

₊

_ji(h~(x) -1)) (15)

Use has been made of the fact that

«q=«([r;-rj[)

takes _on

only

the values

«~~ =

0,

1, and

y~(x)

and

h~(x)

_are the Fourier transforms of

y(z), h(z)

_:

y~(x)

=

dzy (z )

e~ ^~~~

,

h~(x)

= dz h

(z e~'~~ (16)

Due _to

h(z)

=

h(-z),

the function

r(x)

is real and _even function of _x. Furthermore

0=r(0)mr(x).

In the limit _{we are} interested

here, fi

is small _so that

r(x)m fi(hF(X)

l

),

To

proceed

further _we

decouple

the

supervectors

_@~ attached to sites

j by

means of the functional

generalization

of the Hubbard-Stratonovich

identity [14] (see Appendix A)

:

exp( j£«~~r(~~+£~j))

= ,~

=

s~g

_exP

£ _«j ld~ ijd~i g~(~ )

r~

~(~

+

£

_~

₎ g~(~ ⁾

+

£ g~(~,) (17)

,~ ,

r~~(@+i~)

_{is meant as the inverse}

integral

kemel of

r(@+i~).

_{The functional}

integration

_goes over a space of _even functions

g~(@ )

₌

g~(- )

with

g~(0)

₌ 0 in view of

corresponding properties

of the kemel

r(x).

It is convenient to represent

g~(@)

in the

following

form _:

gi

~ )

_"

gf _{~ )}

₊

g)( _{~ )}

₊

_{fi ( ~ )}

₊

_{ii (~ )}

*

(i 8)

Here

g)(@

is the _«

longitudinal

_»

part,

^{that is the class which} we expect ^{the saddle}

point

solution

belonging

to :

~~(~) "~~1+~~2XlXi+~~3X?X2+~)4XlXiX?X2 ₍₁₉₎

g)(@)

is the «transverse» bosonic part, ^{which will be}

important

for the correlator

K~j,k):

~)(~ ₎

~

~)i

_lli + ~~2112 + ~~3113 + ~~4114

(2°)

with

~~

f

^{~~ ~ ~}

^~~

_' ^~~

f~

~~ ^~

~~

_~~

U3~~(X?Xl~XlX2)~ U4~~(XIX~~XiX2) (21)

f~*(@ )

^{is the} part ^with fennionic coefficients

fl(~)"filXl+f~X2+flXiX?X2+f~X2XlXi. (22)

All coefficients _are functions of the four-dimensional _vector R

=

(Ri, R~).

The _measure

4

slg

will be

proportional

to

fls~g;

with s~g~ ₌

fl (s~g[s~g(s~f~(s~f~~).

The function

, s= I

g~(@), equation (18),

looks

formally real,

however the

integrations

have to be deformed

appropriately

in the

complex plane

in order to ensure convergence of the Gaussian functional

integral (Appendix A).

Substituting equation (17)

into

equation (13)

and then into

equation (I I)

and

changing

the order of

integrations

_over

s~g

and [5~@ we

get

K~j, k)

=

j s~g exp(- £~g]) ~~~

~~ ~~

~~~~J

^{~ ~}

(X~ XII (23)

j

(X2 Xl)

k j # k

where

(d)~

₌

) _{ld~ i o(~)F(~)} e°'~'~ _jdwi

=

jd~iijd~~i

_;

ji)~

= i.

(24)

With

equations (19-22)

_we find

(Xi

_{Xi X2 X1*}

)

=

Z/ l~~~

~

F

~lt )

e°~~~~

(2

_gr

)

(X?

_Xi

)

j ^~

Zj ~)~~~~

F

(R) e~~~~(- _{g)2(R )}

+

lg$(R)

₊

_{ffl fji} ) (25)

aT

(X2 Xl)

k ^~

Zk l~

^~

2

F

(R) e~~~~(g~2(R)

₊

_{lg~3 (R)}

₊

fk2 it )

2~gr)

Here we

integrated

out the Grassmannian

components

X. The function

F(R)

is

equal

to

F(@ putting formally

_X

~ X ^* = 0.

The effective

Lagrangian £lg]

ⁱⁿ

equations (23)

is

given by

t~gi=(z«j~ id~i id4rigi(w)r-~(w+£4r)gj(4r)-zin idwif(w)e°'~'~

~~

(26)

Equations (23-26)

constitute the basis of further considerations.

3. Saddle

point ajproximation

_{and Gaussian} fluctuations.

Going

into the momentum space we

represent

^{the effective}

Lagrangian

in the form

t~gi

₌

)z ^{I- I} _idwi id4rig-q(~) r-~(~+ L4r) _gq(4r)

₊

zti~>igii (27)

q

"q

"o

,

with

t~~>igii

=

£ _{id~ i id4r1gi(w}

_r-

_~(~

+

L

_4r

gi(4r)

in

id~ IF (~ )

_{exp gi}

(w ) (28)

For

completeness

we may write

~'J / _i

_~~~~~ ~~

~q ' ~i

~'~ /~/2 I ~q~~

~~~~

'

~~~~

q q

where q runs

through

the allowed values of the rust Bdllouin _zone and N is the number of

sites. At small momenta

(q(

we have

1/«~ l/«o

_m

)q~

_with

«1

«~=£r~«([r()/2dccr(+~, «o=£«()r()car(.

, r

Taking

into _account that

r(x)

cc

fi

_m

I/«o

_{we see} that the first _tern in the

Lagrangian (27)

is

proportional

to a

big _parameter r(»a~

_{that suppresses} fluctuations with

q»ri~.

Longwavelength

fluctuations

corresponding

to wavevectors q

ri

are small _as well

provided

we consider _a

macroscopic sample

of the characteristic size

f~»ro. Therefore,

_{we can}

evaluate the functional

integral

in

equation (23) by

the

steepest

descent method.

A

homogeneous

solution _g~

(@ )

=

g~(@ )

^{of the saddle}

point equation

^~~

=

0 satisfies

8gi (~ )

now the

following

condition :

~~~~~ ^~

~~

~ ~~~

~

~~~

~

~ _~~ ~~~~~ ~~~~~~~~~~~

^{~ ~~~~}

This

equation

_can be rewritten _as

g~(~>)

=

«o(r(~>+ ^£ ~)) ₍₃₁₎

where

(.. )

~

is defined _as in

equation (24)

with _g~

(@ replaced by g~(@ ).

^We are

looking

for its solution _as a function of _two

superinvariants

_:

gs(~ )

₌

g~°~(~

⁺ ~,

~

⁺

£

_~

_{) (32)}

because

F(@)

is of this type,

equation (14),

and r _commutes with all rotations

f

_{which leave}

_@+ ii

_invariant

_forming

a

graded

Lie group UOSP

(2,2( 2,2) [23]

f+ if

=

I ₍₃₃₎

Rotations

(

_{which leave both}

_@+ ii

_and

_@+

_invariant

_belong

_{to the}

_subgroup

UOSP

(2(2)

_x UOSP

(2(2),

for which

(+ (

= I

,

(+ I(

=

I ₍₃₄₎

holds. We

expect

^that a stable solution has the

symmetry

^{of the}

Lagrangian (26)

that is

g~((@)

=

g~(@).

On the other hand for

e-0,

see

equation (14), F(@)

has the full

symmetry, equation (33),

so there

might

be a spontaneous symmetry

breaking

from

symmetry, equation (33), g~(f@ )

₌

g~(@ )

down to symmetry

equation (34)

with _a usual

interpretation

of the

frequency

_{w =} I e/2 _{as a}

symmetry breaking parameter.

What makes _{us sure} that the saddle

point

solution has indeed at least symmetry,

equation (34),

^is that _{we are} able to attribute _a definite

physical meaning

to the function

g°(x,

_y

_).

_In fact _as it will be discussed

later,

the function

g1°~(x,

_y

)

can be related _to the

joint probability density

of real and

imaginary

_parts of the one-site Green function. As it would be

clear

later,

this

physical meaning implies

that

g~°~~(X>

Y)

"

g~°~(x,

_y

). (35)

A nice feature of the ansatz,

equation (32),

is that _we

immediately

have

Z~

₌

id4

F

(4 )

_exp

g~(~ )

₌ ^l

(36)

applying

_a theorem which _we may call

Parisi-Sourlas-Efetov-Wegner

theorem

(PSEW) [34, 43]

and which in _our

particular

case states that the

integral

_{over a} function

depending only

_on

the

length

of _a supervector ^and

vanishing

at

infinity yields

the function at ₌ 0. This leads in the _case of Parisi-Sourlas _to dimensional reduction

[35].

From

r(0)

=

0 and

equation (31)

we see that

g~(0)

₌

0,

flom

y~(0)

_{= we} have F

(0)

₌

1,

which proves

equation (36). Putting

the Grassmannian

components

^{of in}

equation (31)

to zero we reduce this

equation

to the

form

which remains _an

integral equation

for the function

g~°~(x, y)

of two variables.

By partial integration

with

respect

to

(Ri

^and

(R~( using r(0)

=

0, r~~~(0)

= 0 we obtain

d~Ri

~~~

gs(R')=«o (~ (R(Ri)jF(R)expg~~R)(R~-o+ R(Ri

"

RI

+ «~

j ^~~~~

r

(i>(Rj R~) ~' ~~

F

(R )

_exp

g~(R )

~ ~

2 gr

Rj

«~

~~~

~

r(2)(Rj _{Ri RI R~)} ^~' [~ ^~' (~

F

~R)

_exp

g~(R ) (38)

(2

_gr

) Ri R2

where

r~~~(x), r~~~(x)

_mean first and second derivatives with

respect

to the

argument

x.

From

equation (37)

_{we see} that

g~(R)

- «o

r(±

_w

)

= «o In

(I fi)

for

(R

_- w and from

equation (38)

_we see that for

(R'(

- 0

In order to calculate the correlator

K~j, k)

to lowest order _we should

merely

substitute

g~(4)

for

g~(@ )

ⁱⁿ

equations (23)-(25). Using

notations introduced in

equations (18-22)

we

make _sure that

g~(@ ) belongs

_to the class

g)(@ _).

In view of this fact

K~j, k)

vanishes for

j

# k and for

K~j, j)

we obtain

K~j, j)

=

l~~~

~

F

(R)

_exp

g~(R ) (40)

(2

_gr

)

For _a

nonvanishing K~j, k), j

# k _we have to consider small fluctuations around the saddle

point

solution.

Expanding

the effective

Lagrangian £~g]

_up to second order with respect to small fluctuations &g~

( )

= g~

) g~(@

we obtain

sl~

=

z _«~j

ⁱ

j

id~ iid~ri sgi(~ )

r-

I(~

+

£~r) _sgj(~r) z ( (sgi(~ _)2)~ (&g;(w ))])

(41)

Substituting

in this

expression &g;

₌

&g)+ &g)+ _&g(

with the fennionic

part &g)

₌

f~

+

fj*

we obtain _a

corresponding decomposition

~£

=

~£~ ₊ ~£~ ₊ ~£~. The last _tern in

equation (41)

contributes

only

for

longitudinal

fluctuations

&g).

For the main contribution to

K~j, k), j

# k we have _to take into account in

equation (25) only &g)

_up to linear order and _can

neglect

it in the normalization

Z~

=

Z~

₌

I, equation (24).

Let _us recall that due to

supersymmetry

9~g

exp

(-

£

~g])

₌

(42)

and therefore the lowest order contribution to

K~j, k), j

#

k,

is

simply

l~~~~

^~~P

§

^~

~~) ^{(Xl XII (X2 X1)}

~~~ ~~

&~~

exp

(-

&~£~)

~~~~

2

with

(X?

_Xi

)

^°

"

~~~

~

F ~R

) e°'~~~(- &g)

+ I

&g§)

J

(2

_gr

)

(X2 Xl )

"

~~~

~

F

(R ) e°'~~( _&g(~

₊ i &g(~

)

~~~~

(2

_gr

)

Here the fennionic contributions from

fj[ f~i

^and

f~~ it

_can be

neglected.

Since the

eigenfunctions

of r in the _transverse group are

proportional

to ui, u~,

u~, u4 these four contributions

decouple

as well and _we may restrict in

equation (43)

to fluctuations

&g)

= &g)~ ^{+ i} &g)~ ^{which appear in}

equation (44). Omiting

_now the upper index

T we may write the relevant part ^{of the} fluctuation

Lagrangian

_as

The I in the exponent means matrix

(integral kemel)

inversion.

Defining

the bracket

(...) by

the average with the

weight

in

equation (43)

we obtain

formally

the correlator

(&g~*(R) &g~(R'))

which

obeys

the

integral equation

(&gj*(R') &g~(R"))

=

«~~ r~~~(R'iR")

£

_«ji

^~~~ _~F (R)

e°'~~~

rl~~(R' iR)(bgi*(R) _{&g~ (lt")) (46)}

I

(2 gr)

Going

into _momentum

representation

_we obtain for

j

_~ k

using equations (43)-(44)

K~j, k)

₌

£

e~~~5^~~~~

~~~

F(R)

e~'~~~

l~~~'~ ^F(R') e~'~'~(&g*~(R &g~(R'))

N

~

(2gr)~ _(2gr)

(47)

Defining

an

eigenvalue problem

which is

symmetric

with respect to the _measure M

(R )

m F

(R )

_exp

g~(R )

_:

l~~R

~

Jf(R) r(2>(~> £~) _p~(~)

~ ~

~

y~~(~>) (~g)

(2 gr)

we obtain from

equation (46)

ld4~ M(R) 1l'~(R)(&g*~(R) &g~(R'))

₌

^{«~ A~} 1l'~~R'). (49)

(2 gr)~

^l

"q Au

Therefore,

for

j

_{# k we} have

KQ, k~

-

i

^{~~~'~ ~~~}

i

i

"~i iv

Cv

~5°~

With the

help

of the scalar

product

(fi, f2)

=

)~)~~M(R)

fi(R) f~(R) (51)

the coefficients c~ take the form

(1, +v)2

~v = ~~

(w~, w~)

Note that

by completeness

and

equation (40)

£

_c~

=

l~~~

~

M

(R)

₌ K

~j, j ) (53)

v

(2

gr

)

Combining equations (50)

and

(53)

we find the full Fourier transform of

K~j, k)

valid in the

leading

order _:

~ C~

~(q )

~

i1

~ A

(54)

v q v

Note that for _a stable saddle

point

and definiteness of the

integral

kemel _we need

(1/«~

^A

~ ~ l.

To

proceed

further and to

get

^the

explicit dependence

of

K~j, k)

_on the distance _we should

study

in _more detail the structure of

eigenfunctions _{1l'~ (R).}

As it will be shown in

subsequent sections,

this structure is

qualitatively

different in two

possible phases

of the system : localized and

conducting.

To understand the

origin

of this distinction _we should understand the

physical meaning

of the function

g1°~(x, y).

4. Delocalization transition _{as a}

spontaneous symmetry breaking phenomenon.

Before

explaining

how it is

possible

to describe _a transition from localized to delocalized states in _terms of the present model, let _us

briefly

recall _a

profound analysis

of

analytical properties

of Green functions in both

phases following

arguments

by

Thouless

[17].

Let us consider the

one-point

Green function

j,

^I

~l~ ^101°ll~ ₍₅₅₎

~JJ~~~~~~

_~

E+ie-fl

«

~~~~

~"

where the

eigenvector

_icy

)

of

fl corresponds

to the

eigenvalue E~.

For _a finite

system

^{of N} sites the function

G~~(E

^is an

analytic

function of energy with

poles

_on the real axis _at

points

E~

and residues

a~(E~)= (Q(a)(~

whose _sum is

unity.

For extended

eigenstates

la )

all residues

aj(E~)

are random numbers of the _same order of

magnitude ^N~~

When

N - w the

spacing

between

subsequent eigenvalues

shrinks _{to zero as}

N~~

_and

we can

replace

the _{sum over}

eigenvalues

in

equation (55) by

an

integral.

For the

imaginary

part ^{of the} Green

function, V~,

^that

gives

_:

Vj=ImG~~(E+ie)=NIm jdp ~~~)~~~~--grNp(E)a~(E),

_e-+0

₍₅₆₎

E+ie-p

with

p(E) being

the

density

^of states.

Therefore,

_a

typical

value of

imaginary

_part

V~

(E )

is of order of

unity

if the energy E lies within the range of delocalized states. This value

can be related to the

probability

for _a

particle

to escape ^to

infinity

^{in unit} time

[17].

In contrast, within the

region

of localized _states the residues

a~(E~)

^remain finite _even at N - w. If the

eigenstate la )

has _a characteristic range L and is located around _a lattice site r~ then

a~(E~ )

cc L ^~~ for those sites

j

that

obey (q

_r~ w L and _are

exponentially

small

otherwise. Then, as it follows from

equation (55),

the

quantity

_V~ is of order _e if _a

given

site lies at a distance of the order of L

(E )

from the _centre r~ of _an

eigenstate

_a

) corresponding

to an energy

E~

within the band

(E E~(

_{w e} and of order of _e otherwise.

Taking

into

account that the total number of

eigenstates

in _a band of width AE _e is N

~

Np (E

_e and

their centres r~ are distributed _at random

throughout

the

sample,

we conclude that

V~ ^{cc e~ with a small}

probability

of order

ep(E)

L ^~~ and V~ ^{cc e in all other cases.}

In

previous publications [12, 15]

_an intimate

relationship

between _a traditional

picture

of the two types of

eigenstates

in the Anderson model

presented

above and _a behaviour of the

function

g~°~(x, y) satisfying

the

saddle-point equation

_was revealed. This

relationship

stems from _an

identity

which for the present ^{model takes} on the

following

form

(details

of the

derivation _are

presented

in

Appendix B)

:

2

~~°~~x, Y)

- «o

dzrf~z)

all

d~fE~ll »)

_exP

i _{(»x illY} ¹ ₍₅₇₎

where

f~(u, v)

is the

joint probability density

of real

(u)

and

imaginary (v

w

0)

_part of the

one-site Green function

G~~(E+ie),

^the function

r~(z)= dwr(w)e'~~/2gr

and

x =

R)

₊

Rj,

_y

= R

Rj.

_The

qualitative

difference between

analytical properties

of Green functions for localized and extended

regions

manifests itself in the different

limiting

behaviour of the function

g~°~(x,

_y when _e

- 0. If the energy E lies within the

region

of localized states the

argumentation presented

above suggests ^{that the function}

f~(u,

~j has _a

pronounced peak

of order of _e~ for _{~ S}

e and has _a

magnitude

~

e~

for

e « ~ S _e ~. Then the function

g~°~(x,

_y

)

related _to

f~ (u,

~

) by equation (57) depends

on x for _{x a~ e} and _{x 5~ e}

only

and is

actually independent

of _x in between. When _e

- 0 the function

g1°~(x,

_y

)

tends _{to a} function

g)j)(y)

_{for any}

_given

_{x with the}

_point

_x

= 0

being

_a

singular point.

This

singularity provides

_a

nonvanishing

value of the

density

of states in view of the

relationship

iiz ?~*(l'~~ ⁾

=

(° r~~>(o) _{arp(E) ~5g)}

following

flom

equation (57)

^{and the usual}

expression

for the

density

^of states _:

grp

(E)

₌ + Im G

j~

(E

+ I _e

)

~ ~ ⁼

lim du d~

~f~(u,

~

) (59)

~ e~o

In contrast, ^when the energy E lies within the range of extended states, the function

g~°~(x, y) displays

_a nontrivial

x-dependence

at x~ I

falling

off at _a characteristic scale

x a

A,

with A

being

determined

by

the inverse of _a

typical

value of the

imaginary _part

of the Green function.

We _see that the function

g~°~(x, y)

can be used to label the two

phases

of the model

conducting

and

insulating. Moreover, by using

this function it is

possible

to describe the transition from one

phase

to the other in _terms of _a spontaneous

breaking

of _an intemal symmetry ^{of the}

Lagrangian (26).

To understand this

point

let _us recall that when

e =

0 the

Lagrangian

is invariant with

respect

to _a rotation _g~

(@ )

- g~

(ii ₎

_with

I

~ UOSP

(2,2 2,2), equation (33).

This results in

an invariance of the

saddle-point equation, equation (31)

with

respect

to the transformation

gs( 4 )

~ gs

(f _{~ ).}

_Let

US recall that gs

( 4 )

"

g~" 4

^~

~, ~

^~

£ _{~ )}

m g^~°~

(X,

_y

).

AS it _Was

demonstrated

above,

the solution

g~°~(x, y)

of the

saddle-point equation (31)

in the localized

states

region

tends _{to a} function

gf)(y)

_{when e-0 for any}

x~ I. Since the variable y =

+

ii

_is

_apparently

an invariant of the transformation

-

ii,

_such

a transformation renders the solution

g)$)(y)

invariant

(this

fact

justifies

the lower index notation for this

function).

Thus _we conclude that in this _case the symmetry ^{of the}

equation

and the

symmetry

of its solution coincide.

(The

term

e@+

in

F(@ )

breaks the symmetry

explicitly

but the

symmetry ^{is restored when} _{e -}

0.)

In contrast, in the delocalized _states

region,

when _a

dependence

of the function

g~°~(x,

_y

)

on the variable _x

=

+ remains _even for _e

-

0,

the group of symmetry ^{of such} a

solution cannot be the whole group UOSP

(2,2(2,2)

_any

longer

since x'

=

+

I+ ii

_#

x.

Rather,

such _a solution is invariant

only

with respect to transformations

(, _{equation (34),}

_that

render both ⁺ and ⁺

ii

_invariant

_forming

_{the group UOSP}

_(2[2)

x UOSP

(2(2).

We

see that the symmetry ^{of the solution is lower than the} symmetry ^{of the}

equation,

that is

nothing

else but _a

spontaneous

symmetry

breaking phenomenon.

It is worth

mentioning

that in _a traditional field-theoretical formulation of the Anderson localization

[3-6]

_severe difficulties _were encountered when

trying

_to describe both types ^of

states localized and extended

by

a

single

order

parameter

^similar to those used for _a

description

of

ordinary thermodynamic

critical

phenomena.

In that

approach

effective

Lagrangians

of

interacting

matrices _were

derived,

with _an

averaged

value of such _a matrix

being expected

to

play

the role of _an order parameter. ^{However it} was found that the value of such _an order parameter ^is

proportional

to the

averaged density

of states and thus _nonzero in both

phases.

This fact led authors of cited papers ^to the conclusion that the

symmetry

^{of the}

Lagrangian

is

spontaneously

broken in both

phases, insulating

^and

conducting. Moreover,

localized states

appeared

to be inaccessible at the level of _a mean-field

(saddle-point) approximation

and _were claimed _to be described

only

as instant _on solutions

[6].

In view of this fact _we feel that _a mean-field

picture

of

properties

of _a disordered solid based _on such _a kind of

Lagrangian

formulation cannot be considered _as

satisfactory

for

sufficiently high degree

of disorder. The

Lagrangian

formulation

developed

in the

present

_paper seems to be free from all these drawbacks.

Moreover,

the

origin

of the described difficulties in

quite

clear to us now

they

stem from

neglecting

the functional _nature of _a proper order parameter

3j°~(x, y)

function.

Figuratively speaking,

it is _as if _we try to

interpret

the derivative

ax

~ =~ o