Theory of random multiplicative transfer matrices and its implications for quantum transport

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Submitted on 1 Jan 1990

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its implications for quantum transport

J.-L. Pichard, N. Zanon, Y. Imry, A. Douglas Stone

To cite this version:

Theory

of

random

_{multiplicative}

transfer matrices

and

its

impli-cations for

_{quantum transport}

J.-L.

_Pichard(1),

N.

_Zanon(1),

Y.

_{Imry(2,3 )}

and A.

_Douglas

_Stone(4)

(1)Service

de

_Physique

du Solide et de Résonance

_Magnétique,

CEA

_Saclay,

91191 Gif-sur-Yvette

Cedex,

France

(2)Department

of Nuclear

_Physics,

Weizmann Institute of

Science,

Rehovot

76100,

Israel

(3)I.B.M.

Thomas J. Watson Research Center Yorktown

_Heigths,

NY

_10598,

U.S.A.

(4)Applied

Physics,

Yale

_University,

New

_Haven,

CT

_06520,

U.S.A.

(Requ

le

_{27 septembre}

1989,

accepté

le 8 décembre

₁₉₈₉₎

Résumé. 2014 En

_{régime quantique}

_cohérent,

la conductance g d’un

système

désordonné

possèdant

N canaux de diffusion est déterminée _par N

_paramètres

réels

_{("niveaux") {03BBi}}

_qui

caractérisent sa

ma-trice de transfert M. La mesure

_adéquate

_pour

_M,

combinée avec une

_{hypothèse d’entropie}

maximum

"globale",

conduit à une loi de distribution de ces niveaux

_{identique à}

celle des ensembles de matrices aléatoires

_classiques,

si l’on _excepte la

_présence

d’un _comportement nouveau de la densité de niveaux

03C1(03BB),

qui

est loin d’être

_uniforme,

et

_dépend

de

_façon

_{significative}

des

_paramètres

du

_système.

Nous étudions en détail cette densité et ses

_implications

_pour les fluctuations de _conductance, montrant

que le _comportement

_spécifique

de cet ensemble résulte de la loi de

_composition

_{multiplicative}

de _M.

Pour N donné et

_quand

la

_longueur

Lz du

_système

_augmente, ce sont les 03B1i ~

_{(1/2Lz)~n(03BBi)}

(qui

convergent vers les

_exposants

de

_Lyapunov

de

_M)

_qui

ont une densité essentiellement uniforme.

Uti-lisant cette

_densité,

nous

_développons

une

_analogie

coulombienne pour

expliquer

le

_changement

des

statistiques

des niveaux et de la conductance

_qui

_accompagne la transition du

_{régime métallique}

au

régime

isolant. Le

_régime

_{métallique correspond à}

une

_"phase"

de haute densité du _gaz de

_coulomb,

avec une

_{statistique identique}

à celle des ensembles

_classiques

avec interactions

_{logarithmiques,}

si l’on _excepte la

_présence

d’une "interaction" avec des

_charges

_{images près}

de

_l’origine;

ceci

_implique

une loi normale _pour

_p(g)

avec une variance universelle. Des mécanismes

_possibles

_{d’apparition}

de

queues

lognormales

dans le

_{régime métallique}

sont discutés. Le

_régime

localisé

_{correspond à}

une

"phase"

de faible densité où les niveaux fluctuent de

_façon

_{quasi-indépendante}

et où la conductance

est distribuée de

_façon

_lognormale

avec

~(03B4~n g)2~ ~

_2/g0,

_go =

N~/Lz

étant la conductance

_ohmique

classique

et ~ le libre _{parcours moyen}

_élastique.

Les

_{conséquences}

de cette

_analogie

coulombienne

pour la distribution des fluctuations de niveaux et de conductance sont _{confirmées par des} calculs

numériques indépendants.

La distribution

_p(g)

_{n’étant fonction que} de

_03C1(03BB)

dans notre _théorie, nous

étudions dans la seconde

_partie

de ce travail les

_propriétés

d’échelle de

_03C1(03BB)

afin d’examiner la

vali-dité d’une théorie d’échelle à un

_paramètre.

_D’abord, nous

_{généralisons}

_pour chacun des niveaux 03B1i

les lois d’échelle

_déjà

obtenues _pour la chaine désordonnée. Puis nous montrons _que la _convergence

de chacun des 03B1i vers sa limite

_{quasi-unidimensionnelle (N}

donné,

Lz ~

oo)

ne

_dépend

_que de

la conductance _moyenne

_~g~.

Nous en concluons _que

(g)

satisfait à une loi d’échelle à un

_paramètre

dans le domaine de

_paramètres

examiné.

_Cependant,

nous ne _pouvons exclure _que la

_dépendance

en

fonction de i des lois d’échelle obtenues n’introduise des

_paramètres

d’échelle

_{supplémentaires}

pour

03C1(g).

Classification

Physics

Abstracts

71.30 - 71.55J - 72.10 - 72.15R

Abstract. 2014 The

two-probe

conductance, g,

of a disordered _{quantum system} with N tranverse

scat-tering

channels is determined

_by

N real

_parameters

_{("levels") {03BBi}}

_{characterizing}

the transfer matrix M. The

_appropriate

measure for M combined with a

_"global"

maximum _entropy

_hypothesis,

leads to a

_joint

distribution for these levels of the same form as the standard random matrix _{ensembles, except} for the occurence of a novel behavior of the level

_density

_03C1(03BB),

which is far from

_uniform,

and

_depends

importantly

on the _{system parameters.} We

_study

this

_density

and its

_implications

for conductance fluctuations in

_detail,

_showing

that the novel behavior of this ensemble stems from the

_{multiplicative}

composition

law for M. For fixed

_N,

as the _system

_length

Lz ~ oo it is the _{03B1i ~}

(1/2Lz) ~n

_03BBi

(which

converge to the

_Lyapunov

_exponents of

_M)

that have an

_{approximately}

uniform

_density.

Us-ing

this

_density

we

_develop

a Coulomb _gas

_analogy

to understand

_analytically

the

_change

in the level

and conductance statistics which

_accompanies

the transition from metallic to localized behavior. The metallic

_{regime corresponds}

to the

_{high-density "phase"}

of the _gas, with statistics similar to

stan-dard,

logarithmically-correlated

ensembles _except for the _appearance of an "interaction" with

_image

charges

near the

_origin;

this leads to a normal distribution

_p(g)

with a universal variance. Possible mechanisms for the occurrence of

_lognormal

tails in the metallic

_regime

are discussed. The local-ized

_regime

_corresponds

to a

_low-density

_"phase"

in which the levels fluctuate

_{independently}

and the conductance is

_lognormally

distributed with

_{~(03B4~n g)2~ ~}

_2/g0

where g0 =

N ~/Lz

is the classical

con-ductance and ~ is the elastic man free

_path.

The _consequences of the Coulomb _gas

_analogy

for both the conductance and level statistics are confirmed

_by

_independent

numerical calculations. In our

the-ory, the distribution

_p(g)

_depending

_only

on

_p(A),

we

_study

in a second

_part

of this work the finite-size

scaling

properties

of

_03C1(03BB)

to address the

_question

of

_{one-parameter}

_scaling.

First we

_generalize

for each level _03B1i the

_scaling

law obtained for the disordered chain. Then we show that the _convergence of each 03B1i towards their

quasi-one

dimensional limit

(N

fixed, Lz ~

oo)

_{depends only}

on the _average

conductance

_(g)

in the _range of

_investigated

_parameters.

_However,

we cannot rule out a

_dependence

on the index i of the

_scaling

functions which would introduce additional

_scaling

_parameters for

_p(g).

Introduction

It has been known

_for

some time that a

_study

of the

_eigenvalues

of the matrix

Mt M

_(where

M is the transfer

_matrix),

as the

_system

_length

Lz

--; _oo,

gives

useful information

concerning

the

localization

_length

and

_{scaling properties}

of the conductance

_{g(Lz )}

in disordered

_systems

_{[1- S].}

More

_recently,

it has been understood that since the

_two-probe

conductance and the transfer

ma-trix are

_{simply-related}

for

_systems

of _any size

_[6],

such an

_approach

is also _very natural and useful

in

_studying

the statistical fluctuations

_{of g}

in the metallic

_regime

_[7],

where the behavior

_{of g}

can

be _very different from the

Lz

--· oo

_(localized)

limit. Such statistical fluctuations have now been

widely

observed and studied in

_experimental

_systems,

_typically

small

_{("mesoscopic")}

disordered microstructures

_[8].

A

_striking

result of these

_experiments

is that the variance of the conductance

is

_always

of order

_{unity (in}

units of

_e2/h),

_independent

of the _average

_conductance,

when the

conductor is

_{phase-coherent,}

i.e. measured on a scale smaller or

_equal

to the inelastic

_scattering

length.

The transfer matrix

_approach,

when combined with a maximum

_entropy

_{hypothesis (as}

discussed

_below)

and certain

_concepts

from the

_theory

of the standard random matrix ensembles

[7, 9,11],

provides

what is

_probably

the

_{simplest general explanation}

for these "universal

conduc-tance fluctuations"

_[12].

At the same

_time,

the

_discovery

of observable conductance fluctuations has

_reopened

the

_question

of the

_validity

of the

_{scaling theory}

of

_quantum

conductance

_[13],

as

applied

to the whole

_probability

distribution of g,

p(g) [14 - 16].

In this work we extend the

_"global

maximum

_entropy

_approach"

introduced earlier

_{[9 - 10]}

to address the

_origin

of universal conductance fluctuations. In

_particular

we

_exploit

a Coulomb gas

analogy

which arises

_naturally

in this

_approach

to understand the

_{large change}

in the statistical behavior of the conductance which

_accompanies

the transition between the metallic and localized

fact that it is determined

by

the

_eigenvalues

of an ensemble of

_{multiplicative}

random matrices whose combination law determines a definite evolution of the

_{eigenvalue density}

with the

_length

of the

_system

_(number

of factors in the

_product).

In contrast to our earlier work

_{[9 - 10],}

which

emphasized

that this ensemble exhibits statistical behavior in the metallic

_regime

characteristic of the standard random matrix ensembles

_(e.g.

the Gaussian

_{orthogonal ensemble),}

here we also

emphasize

that these ensembles have novel and different

_properties

because its members are

ran-dom matrix

_products,

and must be characterized

_by

a certain

_density

of

_{self-averaging}

_lyapunov

exponents

as

Lz

- oo, In the Coulomb _gas

_analogy

we show that the metallic and localized

regimes

correspond respectively

to

_{high density (strongly-correlated),}

and low

_density

(weakly-correlated) phases

of the _gas; and the former

_phase

leads to normal fluctuations

in g,

whereas the

latter leads to

_lognormal

fluctuations. We thus arrive at a unified

_picture

of the statistical behavior of

_quantum

conductors of fixed width as the

_length

is varied. We test _many of the

_implications

of this

_picture

_by

numerical simulations. In addition we

_study

in detail the

_scaling

_properties

of the

eigenvalues

of

M~M

both as a function of width and

_length

in order to address the

_question

of

one-parameter

scaling.

We consider here electronic

_{quantum transport}

_through

disordered

_systems,

invariant under time reversal

_symmetry

_{(no applied magnetic field),}

with

_{spin-independent hopping (no spin-orbit}

coupling).

We consider the

_{simplest measuring}

_geometry

in which the current is

_supplied

to a

finite disordered

_region

of

_length

_kz

and width

_Lt

_by

two semi-infinite ordered leads and the

driving

voltage

is measured between two electron reservoirs

_feeding

the

_sample

via the leads. This

corresponds

to a

_two-probe

measurement in which the

_voltage

difference is measured between the

current source and sink

_[17].

It now seems

_likely

that a

_complete

_theory

of

_mesoscopic

conduction

will have to take into account the nature of the actual

_measuring

_geometry,

which often involves distinct current and

_{voltage probes}

_{[18 -19].}

For the

_general

statistical

_questions

addressed here

however,

it should be sufficient to consider

_only

the

_two-probe

case. _{In this case, the}

_system

is characterized

_by

N

_propagating

momentum channels in the leads and

_M(Lz)

denotes the 2N x

2N

_{multiplicative}

transfer matrix

_giving

the flux

_amplitudes

to the

_right

of the disordered

_region

(henceforth

referred to as the

_sample)

in terms of the

_incoming

and

_outgoing

_amplitudes

on the

left of the

_sample.

For

_example,

linear

_response

_theory

_[20]

for this

_system,

with the

_assumption

of a uniform electric

field,

yields

a

_many-channel

Landauer formula

_[21]

which states that the conductance

_{g(Lx ),}

measured in unit of

_2e2/h

_(where

the factor 2 comes from

_including

_spin

degeneracy),

is

_equal

to the total

_probability

for electrons at the Fermi surface to be transmitted

through

the

_sample.

_{Thus g}

for a

_given

realization of the disordered

_potential

can be

_expressed

exactly

as the sum of all the interchannel transmission coefficents

_Tij;

or

_{equivalently, simply}

_using

the definitions of the

_scattering

matrix S and of

M,

and

_taking

into account the

_symmetry

of M

(current

conservation and

_{time-reversal),}

in terms of the N

_eigenvalues

_{mi (L z)}

of the hermitean matrix

_{Mt (Lz ).M(Lz ).}

One finds for

_{g(Lz )}

A derivation of the fundamental relation

₍₁₎

can be found in references

_[6,7].

The

_advantage

of

_studying

the formulation in terms of the transfer matrix is that in

_general

one can divide the full

_sample

into a series of shorter disordered slices such that M is

_given

as the

product

of random transfer matrices which describes each

_slice,

whereas a much more

_complex

composition

rule holds for the transmission matrix or for the

_scattering

matrix.

For the usual Anderson

_{tight-binding}

model with

_diagonal

disorder that we have

_employed

different

_slices,

which can be obtained

_simply

_{by rearranging}

the

_{time-independent Schradinger}

equation

for the

_sample.

The

_degree

of disorder is measured

_by

the

_{parameter W}

which is the width of the

_rectangular

distribution taken for the

_diagonal

elements of the Hamiltonian. For

simplicity

we

_study

two-dimensional

_{samples (each}

slice is

_just

a

_row)

for an _energy E = 0

(band

center)

where

_{(with rigid}

transverse

_boundary

_conditions)

the number of

_propagating

channels N in the leads is

_{just equal}

to the transverse width

_Lt.

_Taking

_Lt

fixed and

_Lz

_{varying (quasi}

one-dimensional

_geometry),

Oseledec’s theorem

_[1]

for random

_{multiplicative}

matrices tells us that the N

_{a; ( Lz )}

_self-average

in the

_large

Lx -limit

to N "inverse localization

_lengths"

_0152i(oo).

Thus,

the inverse localization

_lengths

_{characterizing}

the

_Lt

_parallel

channels of the disordered

_sample

in the

large

Lz -limit

do not

_depend

on the

_particular

realization of the random

_potential.

If one could

simply

substitute into

₍₁₎

the

_{self-averaged}

_{ai (oo)}

for the random

_{sample-dependent quantities}

o;,(L~),

one would find that

_{g(Lx )}

_just

measures the number

_Neff

of active transmission channels

[7]

of the disordered

_sample

for which

_{a¡( 00 ).Lz}

1.

The

_{approximate validity}

of such a substitution is not obvious : one

_might

think that there would be substantial

_sample

to

_sample

fluctuations in the

_ai(L,)

in the metallic

_regime

_(l

Lz

~,

where is the elastic mean free

_path

_{and ~}

the localization

_length

of the disordered

sam-ple,

defined as the inverse of the smallest

_positive

_{0152i ( 00 ))}

and that this substitution would make

sense

_only

for

_Lz

_{> ~ .}

_Fortunately,

numerical studies show that this substitution is

_{approximately}

correct, and the

_approaches

based on random matrix

_theory

_justify

in more

_depth

the observed smallness of the fluctuations of the

_a¡(Lz)

in the metallic

_regime

_by

_introducing

the fundamental

concept

of

_{spectral rigidity}

_{[7, 9 -11],}

which is

_intimately

related to the

_universality

of the

con-ductance fluctuations.

Figure

1 shows the ai

(Lz )

of a

_{randomly-chosen sample, arranged}

in order of

_increasing

mag-nitude,

and the

_{ensemble-averaged}

_spectrum.

The difference is

_small,

as is the difference between

this

_averaged

_spectrum

of

_a¡(Lz)

and the

_{self-averaged}

_limiting

_spectrum

of inverse localization

lengths

as

(cxJ) .

We shall

study

in detail these small differences

below;

however we wish to make

the

_preliminary

observation that the data shown in

_figure

1

_justify

the

_concept

introduced in reference

_[7]

of an effective number

_Ne$

of channels which control the metallic conductance

_(as

long

as it is understood that these "channels" are associated to the

_eigenvalues

of

MtM,

and not to the initial momentum channels of the leads whose the correlations have been

_recently

studied in Refs.

_{[22, 23] ).}

Since the

_0152i(Lz)

are

_{sample-dependent,}

we

_analyze

in

section

1 their

statistics,

for the first

Neff

_open transmission channels

_{(metallic regime,}

_{ai (Lz). Lz}

«

_1),

for the N -

_Neff

closed

trans-mission channels

_(a¡(Lz).Lz

»

_1),

and for those at the interface between those two sets

_(which

are crucial for the conductance

_{fluctuations).}

The

_analysis

is based on the

_global

maximum

_entropy

ansatz

_proposed

in reference

_[9],

which has been

_{recently proved}

_[24]

to be consistent with the lo-cal maximum

_entropy

_{approach developed}

in reference

_[11],

and which

_yields

a certain

_prob4bility

distribution for the

_{0152i ( L z ).}

This distribution leads us to introduce different formal

_analogies

with ensembles of classical

_negative

_charges,

free to move on a

_line,

and

_interacting

between themselves and with a

_{positive "jellium",}

at a

_given

_temperature.

_Comparisons

with the results of numerical studies of the Anderson model are

_presented.

In section

2,

the

_implications

of the level statistics for the conductance fluctuations are consi-dered. For metallic

_quantum

conductors

_{( N~,}

»

_1),

the bulk of

_p(g)

is found Gaussian with a uni-versal variance

_(universal

conductance

_{fluctuations),}

while

_possible

mechanisms for

_having

non-universal and

_non-gaussian

tails are

_suggested

_[14].

For

_quantum

insulators

_{(ai(Lz ).Lz}

»

1,

V

_i),

a

_lognormal

distribution is obtained. The

_importance

of the

_averaged

_{spectral density}

for this maximum

_entropy

_approach

is underlined : a one

_parameter

_{spectral density}

of

_{Mt (Lz ).M(Lz )}

yields

a one

_parameter

distribution for

_g(Lz).

Fig.

1. -

az (Lz, Lt)

as a function of the index

_{i, arranged}

in order of

_increasing

_magnitude

for _Lt = ₂₅

and W = 2. The crosses

_give

_{ai(Lz =}

_{25, Lt} =

25)

for a

_given

25x25 disordered _square, while the diamonds

_(o)

are obtained after ensemble

_averaging.

_{The squares}

_(D)

are the inverse localization

_lengths

(aa (Lz

= _{oo, Lt} =

25)).

increases for a fixed value of

Lt

(quasi-one-dimensional

geometry)

is

_analyzed.

We

_generalize

the

scaling theory

of the Id-chain

_{[4, 24]}

to the _many channel case. In

_addition,

our numerical data agree

reasonably

with the

_assumption

_that,

for each

_channel,

this _convergence is

_only

controlled

by

the _average conductance

_{(g) .}

This

_yields

the existence of

_{one-parameter}

_scaling

for

_(g),

in the

range of

investigated

parameters.

In section

4,

we

_numerically

_study

the _convergence of the

_{ai (L)}

towards their limits for

squares of

increasing

size L

_(L

=

_Lz

=

_Lt,

two dimensional

_geometry).

We sketch

_briefly

how the

scaling

works,

without

_having

a limit

_{given by}

Oseledec’s theorem on

_large

random matrix

prod-ucts. Sections 3 and 4 show how

_{large insulating samples}

and their small metallic elements are

simply

related,

using

scaling

relations which are

_{obeyed by}

the cei. For a

_given

_{channel i,}

the

_scaling

relation involves

_only

a

_single

_parameter,

but we do not know for the moment if the visible i-

de-pendence

of the

_scaling

functions does not introduce new

_physical

_parameters.

The

_{compatibility}

between our

_approach

and a one

_parameter

_scaling

for

p(g) depends

on this issue.

1. Random transfer matrix

_theory

and level statistics.

As illustrated

_by

_figure

_1,

the fluctuations of a~

(Lz)

from one member of an ensemble of metal

samples

with the same

_macroscopic

characteristics to another are

_surprisingly

_small,

_typically

of

the order of one _average

_spacing

between

_neighbouring

ai . As discussed in references

[9 - 10],

this

_{"spectral rigidity" gives}

rise to the universal fluctuations

_{[25 - 26]}

of g

in the metallic

_regime.

The second

_important

_property

of the as shown

by

numerical

studies,

which have been observed

for random matrix

_products

in different contexts

_{[27 -}

_29],

is a

_tendency

to have a more or less

uniform density.

distribution

_[30,31],

two "maximum

_{entropy" approaches}

were

_proposed

for the

_{joint probability}

distribution of the variable

Ai

(Lz ),

related to the ai

(Lz )

_{by :}

We denote the two

_approaches

as

_"global"

versus "local". In the

_{global approach}

_{[9 - 10],}

the distribution of the real

_positive

variables

Ai

_corresponds

to a statistical ensemble of matrices

M t (Lz ).M(Lz ),

which is as random as

_{possible (maximum}

_entropy

_{ensemble), given}

a

_density

P L z

(A)

which is assumed to contain all the

_dependence

on

_{physically-relevant sample}

_parameters

such as the size or the elastic mean free

_path.

In the local

_{approach [11],}

it is assumed that the statistical ensemble of transfer matrices which characterize slices of small

_length

_{6 L z}

is as random

as

_{possible, given}

a mean free

_path

~.

_{6 L z}

is taken small

_compared

to

_Lz,

but

_{large enough}

so that the chosen statistical ensemble could be

_{ultimately justified}

_by

a central limit theorem.

Then,

the evolution with

Lz

of the distribution of

Ai

is shown to

_satisfy

a diffusion

_equation.

This

_approach,

which is _necessary a

_single

_parameter

_{theory, gives}

_[32]

for the variance of

_P(g)

in the metallic

regime

the same value as the one

_previously

obtained

_by

_microscopic

_perturbative

calculations

[25 - 26],

in the

_{quasi-1d-limit (Lz}

»

_Lt).

The

_global

and local

_approaches

are identical

_{[33] (in}

the

_large

_N-limit)

if the

_input

_density

_PL.

_(A)

in the first coincides with the

_{one-parameter}

_density

implied by

the second. This later result adds

_something

_very new in

_comparison

with the classic

ran-dom matrix ensembles: the ansatz chosen in the

_global

_approach

is not

_arbitrary,

but can be

_justified

by

the

_{underlying multiplicative composition}

law

_generating

the ensemble.

The distribution of the

_aa,

_resulting

from those two maximum

_entropy

_approaches,

can be

formally

written as :

where

_Ca

is a normalization constant and :

The

_probability

distribution

₍₃₎

is

_formally

identical to the

_{thermodynamic equilibrium}

distribu-tion of N identical

_{point charges}

at

_positions

_Ai

and

_temperature

kT =

11,3, interacting

via a

logarithmic repulsion (as

in a 2d electron

_gas),

but free to move

_only

on the real

_positive

axis in

one-dimension,

with a

_{neutralizing "jellium"}

_background

described

_by

the term

_involving

the

density

p~& {a).

The

_logarithmic

"interaction" of the

Ai

and the effective

_temperature

were de-rived

_{using only}

_symmetry

considerations

_{[9 -11] ;}

_,Q

= 1 for our case

_(real

Hamiltonian which

hence can be

_diagonalized

_by

an

_orthogonal

_{transformation).}

This Coulomb _gas

_analogy

is

fa-miliar in random matrix

_theory;

the crucial difference between the distribution

₍₃₎

and the well-known Gaussian

_Orthogonal

Ensemble

_(G.O.E)

introduced

_by

_Wigner

is that the

_density

p~

(A)

in our case is

_apparently

far from the "semi-circle law"

_{characterizing}

the

_eigenvalue

_density

in the G.O.E. This can be seen

simply

_{by noting}

that for a G.O.E. distribution the

_density

near the

origin

is

_{approximately}

uniform whereas in our case, it is the variable a=, related to

_as

_by

_(2),

which has an

_{approximately}

uniform

_density,

as shown in

_figure

1. Let us

_point

out also that the

one-parameter

density

PL.

(A) implied

by

the local

approach

has been shown

[33]

to

_correspond,

in the limit N - oo, then

Lz

- _{oo, to} a uniform

_density

of inverse localization

_lengths

_{(Xi ( (0)}

between 0 and

£-1 .

In order to use our

_physical

intuition to _guess the statistical

_properties

of

₍₃₎

and related

approximately

uniform.

_Thus,

let us consider the

distribution

_{hL= (111, ..., vN )}

of the variables

v;(Lz)

=

2a¡( Lz ).Lz.

corresponding

to a "Hamiltonian" :

The external

_{potential VL~ (vi)}

contains the

_physical

information on the considered

_sample

through

the

_density

_(J’L"

_(v)

which is now

_essentially

uniform near the

_origin.

The interaction term

_{u(vi, vi),}

which is

_just

_{yielded by}

flux conservation and time reversal

symmetry

in absence of

_spin-orbit

_scattering,

is no

_longer

_logarithmic:

To

_analyze

the

_consequences

of the random transfer matrix

_theory

for the statistics of the

v-variables,

we need now an

_analytical

_expression

for the external

_potential

_VL.

_(v)

_{yielded by}

the

_{positive jellium.}

We shall use for

_simplicity

the

_{one-parameter}

_density

obtained

_(in

the

_large

~z-limit)

in reference

_{[33] :}

This substitution

_neglects,

in addition to the variation of the

_averaged

_spacing

_(a)

between consecutive vi

_(only

noticeable far from the

origin, Fig.

1 and Ref.

[5])

an additional

_Lz

depen-dence of this

_density,

which will be studied later

_(Sects.

3 and

_4).

From

₍₇₎

and

_(9),

we obtain a

simple quadratic

external

_potential,

as shown in

_appendix

7V~

go =

Nt

_being

the classical ohmic conductance.

L,

Let us first consider a metallic

_{sample (.~}

Lz

~).

We remind ourselves that the first

0(Ne$~ N

go

_{"levels" vj}

such

thai vj

Vet! ,..., 1 are the ones

_contributing

to the conductance g

(open

transmission

channels)

since the contribution of "levels" with vj » Vetl is

exponentially

small

_(closed

transmission

_channels),

as

_implied

_by

formula

_(1).

The two-level interaction

₍₈₎

can be

_simplified

as follows :

- for the

open

channels

( vi ,

Vj « Veff ~

1 ),

_expanding

cosh vs - cosh _{v~ ,} one

_{gets :}

i.e.

_again

the usual

_logarithmic

_repulsion,

_except

that now there are

_{"image" charges}

located at

-vj

(for Vj

>

_0).

The second term of the R.H.S. of formula

_{(8) gives}

the interaction of vi with - vj ;

- for the closed chantlels

(v; ,

v3 »

_veff),

the two-level interaction reduces to a

non-interac-ting

potential

term,

plus

a

_remaining

_logarithmic

interaction if two

consecutive vi

are _very close.

This leads us to

_distinguish

between the closed channels of a metallic

_sample

and the closed

chan-nels of a localized

_system.

In the first case, the

_spacing

_(a)

between consecutive

_equilibrium

po-sitions of the levels is small

(a) - 2 1

and the

_{logarithmic remaining}

interaction term still

B

go

/

g g

involves

_{0 ( Ne ff )}

levels. In the localized case,

(a)

»

1,

so that the two-level interaction

_disappears

and

_just

leaves the

_{non-interacting}

_potential

term :

When 2 >>

1,

the

_logarithmic

interaction

can

_only

be

_probed

_by

a

_large

fluctuation of the levels

90

around their

_{equilibrium positions,}

so that two consecutive levels become _very close.

1.1 THE SYMMETRIC COULOMB GAS FOR THE OPEN TRANSMISSION CHANNELS - As we have obtained a

_symmetric

set of

_{negative charges}

from the two-level

interaction,

we have after the

same

_expansion

from

₍₇₎

a

_symmetric

_jellium

of

_{positive charges}

near the

_origin

_(va

«

₁₎

,-which is

_essentially

uniform.

_Thus,

the

_only

difference between the statistics of the

levels vi

« 1

and the usual random matrix G.O.E. statistics is due to the exact

_symmetry

around the

_origin

sa-tisfied

_by

the locations of the

_charges.

However since the force exerted on one of these

_charges

by

the others is screened

_by

the

_{positive jellium (the Debye screening}

_length

_[34]

is of the order

of

_~a~),

one

_expects

that this exact

_symmetry

does not matter _very much.

Thus,

the fluctuations

of the

_spacing

a, between the consecutive vi, measured in

averaged

spacing

units,

must be

des-cribed

_by

the same distribution as those characteristic of the

G.O.E.,

which is

_given

in a _very

_good

approximation

by

the so-called

_Wigner

surmise :

The

_{spectral rigidity (statistics}

_A3

in

_Dyson-Mehta

_[35])

must also be the one of the G.O.E.

But,

the

_symmetric

Coulomb _gas that we have obtained for the vi

implies

in addition

something

very

specific,

which does not _appear in the classical Gaussian ensemble : because of the

_repulsion

essentially

different from the other

_spacings.

Therefore we

_expect

that vl

itself,

which is related to the transmission coefficient

T1

of the best

_transmitting

channel

_{by :}

also follows the

_{Wigner-Surmise,}

as

_supposed

in reference

_[7].

This

_specific

result has been chec-ked

_by

a numerical calculation of 10 x 10 metallic

_squares,

defined as

_explained

in the introduction

(W

=

2),

which _very

_clearly

confirms this

_{prediction (Fig. 2).}

Numerical studies of

_larger

metallic

squares

(50x50) give

the same result. The

_spacing

distribution between the other nearest

neigh-bor v;

corresponds

also to the

_{Wigner-Surmise}

for a metallic

_sample.

Such a result was found for the

Ai

in references

_{[9 - 10],}

but

_{only by}

_"unfolding

the

_spectrum"

i.e.

_{by numerically}

_scaling

to a

variable with uniform

_{density (which}

was

_{approximately equivalent}

to

_considering

the

_v;).

The

_{spectral rigidity}

of the unfolded

_spectrum,

as

measured

_by

the

A3

statistic

_[35]

was also found

to

_{correspond exactly}

to that of the G.O.E. in the metallic

_regime.

_Now,

we are no

_longer

_obliged

to advocate the

_independence

of the local statistics to the detailed form of the

_{confining potential,}

we have recover for

the vi

the classical G.O.E.

_{quadratic potential.}

Distributions of v2, v3 and v4

around their

_averaged

values are

_given

in

_figure

3. If one

_compares

the actual results to Gaussian distributions of same mean and

_variance,

differences in the tails are

_noticeable,

as well as an

asymmetry

around the

_top

_(mainly

for

_v2)

which are reminiscent of the

_asymmetry

_{of p(vl)}

_(Fig.

2 -

_Wigner

_Surmise).

Fig. 2

Fig. 3

Fig.

2. - Small

metallic squares

Lz = Lt =10, W = 2.

Probability given by

an ensemble of 6000

_samples

for the first level vl, measured in units of

(vi)

= 0.39

_(circles),

_compared

to the

_Wigner

Surmise

_(continuous

line).

Fig.

3. -

Probability

of v2, v3 and v4,

compared

with Gaussian distributions of same mean and variance.

((v2)

= _0.86,

(6v2)

= _{0.07, circles;}

_(v3)

= _1.35,

~w3 ~

= _{0.09, squares;}

(v4)

= _1.85,

At the other

_extremity

of the

_spectrum

_(closed

channel of a metallic

_sample),

each _{level viz}

(v~

»

₁₎

_experiences

a force of

_unity

to the

_right

from each level _viz « _v~,

_plus

an additional

logarithmic repulsion

from

_0(7Veg)

_{levels vi}

so that

_{IVi - Vj}

_«

1

_(Eq.(12)).

When

Neff.

»

_1,

we

expect

that the local level statistics are

_{governed by}

the

_{remaining logarithmic}

_{repulsion, yielding}

a

level

_spacing

statistics in

_agreement

with the

_Wigner

Surmise. This is confirmed

_by

our numerical results

_{(Fig. 4).}

Fig.

4. - Closed channels of _a metallic

system : distribution of the last

_spacing

vlo - v9, measured in

aver-aged

spacing

units and

_compared

to the

_Wigner

Surmise

_{(continuous curve).}

Lz = _Lt = 10. W = 2.

1.2 THE LATTICE WITH QUASI-INDEPENDENT FLUCTUATIONS IN THE LOCALIZED REGIME - In this

_regime,1

« vl « v2 ~ ... and the

_spacing

betwen successive

_equilibrium

_positions

of the levels is

_large.

The

_{logarithmic repulsion disappears,}

and we see from

_{(6), (10)}

and

₍₁₃₎

that the

potential

U felt

_by

_{level vj}

is :

-This means that the _vj » 1 have

_equilibrium

lattice

_positions

_v?

_{given by :}

Fig.

5. -

Probability

of

v9 and vlo,

compared

to Gaussian distributions

_(v9)

=

5.4,

(5v) )

= 0.4, diamonds

-

(~10)

= 6.9,

(b V2

= 0.7,

octogons).

Lz = _Lt = 10. W = 2.

Note that the last closed channels of the 10 x 10 studied metallic

_squares

are characterized

by

Gaussian distributions of their

levels,

with variances of the order of

_{2/go (Fig. 5).}

_They

are in

a cross-over

_regime

where the

_{logarithmic repulsion}

remains

_important,

since

_{(vlo - v9~ N}

_1.5,

but we have obtained

_{(6V2) -}

0.4 and

_{(8vla) -.J}

_0.7,

to _compare with

_{2/go N}

0.55. Note also that substantial deviations from the G.O.E. behavior were

_reported

in reference

_[9]

for the

_As

statistics of the unfolded

_spectrum,

for

_{(g) -}

1 where the

_system

_approaches

localization. The

analysis developped

in this section shows that these deviations are real and that even the unfolded

spectrum

does not exhibit the standard G.O.E.

_behavior,

in contrast to the

conventional

view that after

_correcting

for variations in

_{spectral density,}

all random matrix ensembles are

_expected

to show such behavior. The reason that conventional wisdom fails here is the decrease in the

level

_density

with

_{length, again}

_emphasizing

the novel features of

_{multiplicative}

random matrix ensembles. Let us mention that the

_tendency

of the

_joint

_probability

distribution of

the vi

to

approach

a distribution of

_independent

random variables in the

_large

_Lx

-limit has

_previously

been obtained in the

_study

of multimode

_waveguides

_by

Thtubalin

_{[37] (see}

also

_{Ref. [38]).}

We summarize in

_figure

6 the different Coulomb

_systems

_{yielded by}

the

_theory

of random

multiplicative

transfer matrices. For a metallic

_sample,

the level

_spacing

is small and the set of levels can be divided in series of

_0(Ne~)

levels which behave as a Coulomb _gas with

_long

_range

regularity.

The

_part

which is closed to the

_origin

of the levels of the _open transmission channels differs

_only

from the classical

_{Dyson’s "wobbly crystal"}

_[34]

_by

an

_image

effect. For an

_insulating

sample,

the _gas is so rarified that the

_logarithmic

_repulsion

is almost

_completely

lost

_(except

when

two levels

_attempt

to

_cross),

and instead the

_charges

near the

_origin

create a linear

_potential

for

the j th

charge

which fix its _average

_position

to be at a lattice

_position

_V9

=

2 j /go = j ~a},

with

Fig.

6. -

(a)

Schematic

_picture

of the v-levels for a metallic disordered

_sample

with

_Neff

_open transmission channels.

_((g)

~

_{1). (b)}

Schematic

_picture

of the v-levels for an

_insulating

localized

_sample

_((g)

«

_1).

2. Random transfer matrix

_theory

and conductance fluctuations.

We startwith the

_strongly

localized

_quasi-ld

case

_{(L ~>~).Here~i}

=

21go

=

2L/~

where ~

=

Ni,

as first obtained

_by

Thouless

[39].

In this case, the

_typical

conductance is determined

by:

while the conductance fluctuations are

_given

_by

the fact that

_{-it 9}

= _v, and that the first level _v,

has

_{quasi-independent}

Gaussian fluctuation

_(vl

»

_1).

So

_that,

_by

₍₁₈₎

and

_(19),

the fluctuations

of in 9

are distributed

_normally,

with a variance:

This is as if each

_piece

of the wire of

_{length - I}

fluctuates

_{independently,}

_having

a conductance

~91y N

1 and

_{(bg2) _}

1.

₍₂₁₎

is the

_{many-channel generalization}

of the

_scaling

idea of Anderson et aL

_[24]

for the

_{purely ld-system.}

The latter result was also obtained for various models and

cases in one dimension

_(see

_Refs.[15,

40,

41]

and references cited

_therein).

The

_{generalization}

to

quasi-ld

is in _agreement with the

_general

idea that

_{In g}

is additive in

_strong

localization,

and was

Fig.

7

_Fig.

8

Fig.

7. -

Probability

of -~n

_g(circles),

_compared

with a Gaussian law of same mean

_{((In g) =}

_-6.5)

and

variance

_«6in2g)

=

7).

Lz = _{Lt =10.} W _{= 12.}

Fig.

8. - Conductance distribution

compared

with a Gaussian law of same mean

_{(g) -}

1 and variance

(62g) -

0.17. Lz = Lt = _25, W = _4.2.

We believe that this demonstration is

_simple

and

_systematic,

and illustrates the _power of the

ran-dom transfer matrix idea in

_{quantum transport.}

Statistics of 2000 10 x 10

_strongly

disordered

squares

have been

_numerically

calculated. When localization is

_strong

_(W

=

_12,

(.~n g) - -6.5,

Fig. 7),

in g has the

_{expected gaussian}

distribution with a variance

_(~

₇₎

in

_agreement

with

_(21).

For weaker

localization,

a

_lognormal

distribution

_{for g}

does not _agree

_accurately,

but we have noted that relation

₍₂₁₎

is still valid for W = ₈

_(Lt

=

Lz = 10,

(.~n g) = - 2.55,

~b.~n2g~ ~

2.6).

If

we continue to decrease the

_potential

fluctuations,

we find for W =

_4.2,

(g) N

1 and

_{61n g -}

bg.

Therefore,

if we continue to look at

_p(g)

from the localized channel view

_point,

the

lognor-mal distribution would reduce to a

_simple

normal one. This

_prediction

turns out to be

cor-rect, as illustrated

_{by figure}

8,

however the variance

_{(6g2) f’J}

0.17 does not _agree with

_(21),

but rather with the

_{2d-perturbative}

result for the

_amplitude

of the universal conductance fluctuations

(~bg2~u.c.F. ’~

0.1857

_{for g}

measured in units of 2

_e2/h).

This shows us that the

_logarithmic

inter-actions

_{(formula (11))}

are now efficient and reduce the

_amplitude

of the conductance fluctuations.

Then,

it is more correct to look now at

_p(g)

from the

_open

channel view

_point.

In the metallic

_regime,

_~bg2~

does not

_depend

on

_(g)

and takes a universal value. This can

be obtained

_by

_assuming

the

_logarithmic

interactions for the 0

_(Neff)

levels and

_using

the

Dyson-Mehta G.O.E. results

_{[7] :}

_any linear statistic Z

_{roe 1 f}

_(v1)

has a variance which does not

depend

on the _average number

_Neff

of relevant

levels,

when

_f

is a smooth function. The deriva-tion of the whole distribuderiva-tion of conductance fluctuaderiva-tions

_requires

a more subtle

_analysis.

The

genericity

of G.O.E.

_predictions

for the level correlations has

_recently

been studied in references

matrices does not

_change

the level correlations.

Then,

going

back to the

ai

and

_assuming

that the level

_density

_p(A)

is smooth

_enough,

it was

_argued

_[44]

that

_p(g),

as _any linear statistic

de-fined on a random matrix

ensemble,

is

_{gaussian. Figure}

9,

where a statistic

_including

6000 small metallic

_squares

is

_reported,

_supports

such a

_conclusion,

and the variance of this numerical si-mulation

_{(8g2) ’-}

_0.1923)

does not

_{significantly}

differ from the

_{2d-perturbative}

result.

But,

the

universality

of G.O.E.

_predictions

for level statistics and conductance

fluctuations

breaks down in the localized

_regime,

as

_previously

_shown.’,

and even in the metallic

_regime

where non Gaussian corrections to

_{p(g )}

were first found

_by

Altshuler et al.

_[14]

from a calculation of

_higher

cumulants

of _g. We note that our numerical simulations do not show _any visible deviations from

_gaussian

tails for the

_large

fluctuations when

_{(g) >}

1. It is

_possible

that the

_non-gaussian

tails arise from

extremely improbable samples

which

_practically

do not occur in the finite ensemble used. The

difficulty

to see tails

_{corresponding}

to an

_exceedingly

small fraction of the statistical ensemble is known in others

_physical

_{contexts, e.g. :} the Griffiths

_singularity

in dilute

_Ising

models

_[45].

Fig.

9. - Conductance distribution in the

metallic

_regime,

_compared

with a

_gaussian

law of same mean

((g) - 3.62)

and variance

_~S2g~

= _0.19. Lz = _{Lt = 10,} W = _2.

’Ib

_push

further our

_{investigations}

of the conductance fluctuations in the metallic

_regime,

let

us return to the v-variables.

We can obtain the Gaussian distribution within the

Neff

_{approximation, using}

a method intro-duced in reference

_[46]

for

_calculating

the fluctuation

_~N(E)

in the number of levels in an _energy

interval E. The idea is to estimate the

_{energy A}

needed to take

6NetI

_{uniformly spaced}

v-levels from the

_segment

_[1,1

+

_E]

into the

_segment

_{[1- ~,1],}

and to use:

were 3

= 1 for the

samples

with

_{spin-independent hopping}

matrix elements and site

_energies,

note that the total force

_acting

on each level is

_equal

to zero at

_equilibrium,

so we have

_just

to

take into account the two-level interaction between the translated levels

_{("negative charges")}

and

the "holes" which are created after the fluctuation at the

_non-occupied

_{equilibrium positions.}

Then the

_{energy A}

_required

to form additional

_charges

of ~

6 Ney

in the two

_segments

is:

Considering

In

_{Bch x -}

ch

_yB

instead of

_{In )z -}

_g~

will

_just

_change

the value of the _constant, and

will

_give

also a Gaussian distribution with a variance

_independent

of

_parameters.

In order to continue this discussion _very

_{qualitatively,}

we note that the fluctuation of

_~6Neff

uniformly spaced

v-levels around 1 could be

_{energetically}

favourable for small

&g,

but another

kind of excitations could have a lower _energy

_price

for

_larger

conductance fluctuations. Numerical studies of

_{highly unprobable v-configurations}

lead us to

_envisage

on a uniform fluctuation of the level

_spacings.

Let us take the

_spacing

a of the "effective levels" to

_slightly

differ from the _average

(a)

=

2~90 ~

In this

_{hypothesis, g}

is

_{given by:}

which

_yields

that

_tn(g)

will be distributed in an identical fashion to veff :

since

_{~s ’ ~ ~}

_~eS

and

_Neff’

a - 1.

The fluctuations of

_{6 vetr}

_{being approximately}

_Gaussian,

we find that in g has an

_{approximately}

gaussian

distribution both in the metallic and localized

_regimes.

_However,

note as

_long

as

_8g

«

~g~,

the

_{gaussian approximation}

fits the

_lognormal

one.

One _may include non-uniform fluctuations for

_having

a fuller

_picture

of

_p(g),

for

_example

_by

expanding

the

_spacing

_aj =

v3 +1- v~ in Fourier series. There are

_grounds

to

_believe,

_following

reference

_[46],

that fluctuations which are uniform on different scales are

_independent.

Thus one

may express the conductance fluctuations as a sum over a

_large

number

_(of

order

_N~$)

indepen-dent contributions. Such a sum is known

_[46]

to

_produce

a normal distribution on the main

_part,

with correction in the

_higher

moments or in the tails. We

_hope

to return to a fuller treatment of

this issue in a future work.

In conclusion of this

_section,

let us recall that the conductance distribution

_{yielded by}

our

maximum

_entropy

_approach

could be written as:

This

_expression

is not _very useful for

calculations,

but makes clear that the number of

inde-pendent

parameters

necessary to characterize the

_spectral

_density

of the

Ai

will also

_specify

the