New aspects in the theory of the integer quantum Hall effect

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New aspects in the theory of the integer quantum Hall

effect

J. Riess

To cite this version:

New

_aspects

in the

_theory

of

the

_integer

_quantum

Hall effect

J. Riess

Centre National de la Recherche

_{Scientifique,}

Centre de Recherches sur les Très Basses

Températures,

B.P. _166X, 38042 Grenoble

Cedex,

France

(Reçu

le 5 octobre _1989,

_accepté

sous

_{forme définitive}

le 3

_janvier

₁₉₉₀₎

Résumé. 2014 A

partir

d’une

_approche

de la théorie de l’effet Hall

_quantique

entier

_(IQHE)

proposée

récemment, un modèle

explicite

de désordre faible est étudié. On montre, que la

dynamique

d’un électron conduit à des

_phases

aléatoires dans le

_{développement}

_temporel

de la fonction

d’onde,

ce

_qui

_permet de décrire l’évolution d’un électron en terme de

_{probabilités}

associées à des chemins différents dans le

_plan

de

_{l’énergie-temps}

du processus de diffusion. Cette

analyse

conduit à une

_explication

détaillée du mécanisme

_{microscopique,}

_qui

est à

_l’origine

de

l’IQHE du

_système.

Les

_principaux

_aspects nouveaux de notre

_{approche théorique}

_générale

sont

discutés et confrontés à des théories antérieures basées sur des modèles similaires. Abstract. 2014 In the framework of

our

_recently

_proposed

new theoretical

_approach

to the

_integer

quantum Hall effect

_(IQHE)

an

_explicit

weak disorder model is

_{investigated.}

Here the

_dynamics

of a

_single

electron causes randomization of

_phases

in the time

_dependent

wave function of the

electron, which

_permits

the

_description

of the time evolution of the electron in terms of

probabilities

associated with different

_paths

in the

_energy-time

_plane

of the

_scattering

_process. This leads to a detailed

_explanation

of the

_microscopic

mechanism, which is

_responsible

for the formation of the Hall conductance

_plateaus

in this system. The

_principal

new _aspects of our

general

theoretical

_approach

are discussed and confronted to

_previous

theories based on similar models. Classification

Physics

Abstracts 72.20M - 73.20D - 73.20J 1. Introduction.

It is

_{generally accepted}

that the

_integer

_quantum

Hall effect

_{[1] (IQHE)}

_originates

from a

localization-delocalization process due to a disorder

_potential

in the two-dimensional electron

system.

In the last decade considerable theoretical effort has been devoted to this

phenomenon

and to

_transport

_{properties in the presence of disorder and}

of a

_strong

_magnetic

field

_[2].

Very recently

we have

_developed

a new

_point

of view for

_{investigating}

two-dimensional

systems

of

_charged

_particles

in crossed electric and

_strong

_magnetic

_{fields in the presence of} a disorder

_potential

_{[3, 4].}

We considered the time

_dependence

of the

_{single-particle}

Schroeding-er functions of a

_large

class of 2-dimensional

systéms.

It was found on

_general

_grounds

that adiabatic states are

dc-insulating

and that the

_macroscopic

Hall current is carried

_by

the non-adiabatic states

_[4].

Hall conductance

_{plateaus correspond}

to values of the

_magnetic

field for

JOURNAL DE PHYSIQUE. - T. 51, N° 9, 111 MAI 1990

which the Fermi energy is situated in a

_region

associated with adiabatic states. The

_plateaus

disappear completely

when the Hall field

_(and

hence the Hall

_current)

has reached a

sufficiently high

value,

where all states have become non-adiabatic. As an illustration we have

presented

an

_explicit

model

_system

_[4]

with a substrate

potential composed

of

_homogeneous

disorder and of a _{sequence of smooth} barriers and wells. Here the non-adiabatic states could be calculated

_explicitly

in a weak disorder

_{approximation}

whenever the Hall field is

sufficiently

low. In this case the

_quantum

mechanical

_{scattering process induced}

_by

disorder could be understood in detail. In

_particular

it was shown that those of the non-adiabatic

states, which are intermediate between

_fully

adiabatic and

fully

non-adiabatic

_give

rise to a

special

kind of

_{(disorder induced)}

non-classical

_{particle propagation}

and that this is the

microscopic

origin

of the so-called

_compensating

currents which lead to

_{integer quantization}

of the Hall conductance in this

_system.

In the

_present

_paper we further

_analyze

the nature of the

_{scattering process for the model}

system

considered in reference

_[4],

which will be

_briefly

reviewed in section 2. In section 3 we will show that the disorder induced

_{scattering process}

leads to

_phase

_randomness,

which causes

_additivity

of the

_{scattering probabilities}

_originating

from different

_paths

_{in the} energy-time

_plane

of the

_{scattering process.}

_(In

Ref.

_[4]

this

_additivity

has been assumed without

proof.)

In section 4 we discuss the basic features of the time

_dependent

electronic wave

functions,

which are

_responsible

for the

IQHE

in our

_systems.

With the results of section 3 the

microscopic process of the Hall

plateau

formation in the model

system

of section 2 becomes

transparent

and understandable in detail. In section 5 we stress the salient new

_aspects

of our

general

theoretical framework. We

_critically

examine

_previous

theories of the

_IQHE,

which are based on similar models.

2.

_Description

of the model.

We consider a

_system

of

_{independent particles}

of

_{charge q}

situated on a

_{long strip}

in

x-direction

_(with

a width

_Ly)

and

_subject

to a static substrate

_potential

_{V (x, y ),}

a

_magnetic

field

B =

(0,

0,

B )

and _an electric field E =

(0,

Ey,

0).

We consider

_potentials

_{V (x, y )}

which do

not describe a

_macroscopic

electric

field,

i. e. their

_integrals

over x and over y vanish. The

one-particle

Hamiltonian has the form :

where 0 (t) = - cEy Ly t.

As in

_[4],

we

_{neglect edge}

effects and chose the

boundary

conditions

If 0

in

₍₁₎

is considered as a

_parameter

and if

V (x, y )

has no

_symmetry

(e.g.

in the _presence of

_disorder),

the

_eigenvalues

of H are

_periodic

with

_cp

with

_period

_hc/q

_(cf.

Refs.

[3-5]).

It

further follows

_{[3, 4]}

that in the adiabatic

_regime,

i.e. where

_Ey

is

_sufficiently

small such that

cp

(t )

in

_{H[o (t ) ]}

can be considered as a

_parameter,

the

_{single particle}

currents are

periodic

in

time with

_period

T

=

1 hl (qEy LY) 1 .

. Since r is very small for realistic values of

Ey Ly

(e.g.

T = 4 x

10-12

s for

_Ey

_Ly

= 1

mV),

this _means that adiabatic _{states are}

dc-insulating

and

further,

that any dc-current must result from non-adiabatic states

_(including

states inter-mediate between

_fully

adiabatic and

_fully

_{non-adiabatic).}

Here

_{V 1 (x, y )}

describes

_homogeneous

disorder and

_{V (x )}

is a _{sequence of} smooth barriers and wells

_{(see Fig. 1)}

whose first and second

_spatial

derivatives

_(piecewise)

_{vary slowly} with

respect

to the

_magnetic

_length

À =

(hc/qB)1I2,

which is of the order of 100

Â

_or less in

high

magnetic

fields. In the absence of disorder

_{V 1(x, y ),}

the adiabatic solutions of the

time-dependent

Schroedinger equation

are _{then in very good}

_{approximation}

_{given by}

the Landau functions :

where

_{up, n (x,}

t )

is the

_product

of a Hermite

_polynomial

_{X np (x)}

and a Gaussian

with

where

_{V’ (x )}

denotes

_{dV (x )/dx.}

The

_energies

are

r _T11I__

(úJ =

1 qB 1 / (me».

In the

_following

we consider a

_single

Landau band and

_drop

the band index n.

The adiabatic variation of

_Ep

is

_{essentially given by}

the term V

_{[x. (t ) 1.}

Therefore,

for states

t/p, t/p

with localization centres

_{xp(t), xp. (t )}

on different sides of a barrier

_(or

of a

well)

in

Fig.

1. -

Shown are a smooth substrate

_potential

V

_{(x )}

and the nature of the

_{single-particle}

states in the

case, where in addition to

_{V (x )}

a weak disorder

_potential

_{V 1 (x, y),}

an electric field

_Ey

and a _strong

perpendicular

magnetic

field B are _present. Under weak disorder conditions these states are

characterized

by

the

_position

of the localization

_{centres x,}

of the

_{corresponding}

Landau orbitals

I/Jp

in the absence of

_{V 1(x, y) :}

full line

_regions

on the abscissa

_correspond

to

_fully

non-adiabatic

(classically

conducting)

states, shaded

regions

to _{intermediate states, dashed}

_regions

to

_fully

adiabatic

(dc-insulating)

states. The

_figure

_corresponds

to the _parameter values of reference

[4]

_(Ly

=

0.01 cm, B = 6 T,

Ey

= 2.37 _x

10- 7

V

CM- 1).

The

unperturbed

(V

1=

0)

energies

Ep

are

_{represented by}

V (x) (see (7)).

Outside the dashed

regions

these

_energies

coincide with the

_perturbed

_energies

_(on

the scale of the

_figure).

_xp,

_xP

_{schematically}

show the localization centres of two

_unperturbed

states

figure

1,

the

energies

_{E., [ 0 (t ) 1,}

_{Ep [ 0 (t ) ]}

intersect as

indicated

in

_figure

2. Here the two

different

slopes

represent

functions

_{t/1 p [x, y, p (t ) ], t/1 p’ [x, y, p (t)]}

localized on different sides of the barrier

_(well).

Now consider two levels

_{Ep (4> )}

and

_{E., (0 ),}

which intersect at

0 = 0

*

(Fig. 2).

If the

_perturbation

_{V 1(x,}

_y)

is

added,

these levels anticross

_according

to the von

Neumann-Wigner

theorem

[6].

Fig.

2. -

~-dependence

of the

_{single-particle energies.}

Dashed lines : absence of

_V.1(x,

_{y ),}

full lines : presence of a weak disorder

_potential

_{V 1(x,}

_y).

In the case, where this

perturbation

is

weak,

i.e. where

the

_perturbed

functions

_{w,, (x, y, 0 )}

are in very

_{good approximation expressed}

as a linear

combination of the two

_unperturbed

functions

_03C8p,

_t/lp’

in the

_anticrossing

interval

This is true for _any

_anticrossing

interval of

_length

_{hc/2 q}

_(with

_{I/Jp, I/Jp’}

_{replaced by}

the two

unperturbed

functions associated with the considered

_interval).

_Further,

in this weak disorder

approximation

a

perturbed

adiabatic function

W n (x, y, cp (t » is

equal

to an

_unperturbed

function

_{4rp(x, y, 0 (t»}

_{whenever 0 (t) = 0 * + (hcl4 q) x (odd}

_integer),

i.e. whenever

0 (t)

is

_{exactly halfway}

between two

_anticrossing

values of

0.

As a _consequence, the adiabatic evolution of each of the

_{corresponding}

perturbed

functions

Wn [x, y, .p (t)], Wn + 1 [x, y, .p (t)],

... in

figure 2

describes an

elementary charge,

which

« oscillates » between the two sites

_{xp (0}

_{* )}

and

_{x., (0}

_{* )}

situated on two different sides of the barrier

_{(well) [3, 4].}

This motion is

_periodic

with a time

_period

T =

displacement

of

_charge

is

_completely

non-classical,

since the

_{charge disappears}

at

x = xp (cf> *)

and

_{simultaneously}

_reappears at

_x=z-.xp,(~*)

where it

_stays

for a time

At =

T /2

before it

disappears

and _reappears at x =

xp (0

* )

and so on

_(this

_corresponds

to a

discontinuous motion of a mass

_point

from =

_xp

_{(~ * )}

to =

_x.,

_{(0 * ).}

For more

_details,

see references

_{[3, 4].}

If

_4>

_(t)

varies

_slowly,

i.e. if

_Ey

is

_sufficiently

small,

all states are adiabatic

states

_{wn[x, y, cfJ (t ) ] (These}

states form a

_complet

set in the Hilbert

_space).

If

_Ey

increases some of the

_perturbed

time

_dependent

functions become non-adiabatic. This means that at each

_anticrossing

of two

_energies

_complete

Zener

_{tunnelling [7]}

between

adjacent

adiabatic levels _occurs, and the wave functions

_essentially

behave as

_unperturbed

functions

_{t/I p [x,}

_{y, 0}

_(t)],

i.e. with localization centres

_moving

with the constant

_velocity

v =

cEylB,

and with energy

expectation

values

_following

the

_unperturbed

curves of

_figure

2. For a constant electric field

_Ey

the Zener

_{tunneling probability}

P decreases

_{exponentially}

with

_increasing

V1pp,2 ,

which itself increases

_{exponentially}

with

_decreasing

distance

(xp - xp, )

of the two

_unperturbed

orbitals

_{t/I p (t)}

and

_{t/I p’ (t),}

of which the

_perturbed

adiabatic orbital

_{Wn IX, Y, 0 (t ) 1 ’iS}

_composed

in the

_{neighbourhood}

of the intersection of the two

unperturbed

levels

_{Ep [ 0}

_{(t ) ]}

and

_EP,

_{[cfJ (t ) ] (see}

Refs.

_{[3, 4]).}

As a _{consequence each energy} band

_(with

a fixed Landau level index

_n)

of our model

_system

contains a central zone of non-adiabatic

_{(P # 0),}

i.e.

_conducting

states

_(contained

between -

_Eb

and +

_Eb,

in

_{Fig. 1),}

which is sandwiched between two zones of

_fully

adiabatic

_(P

=

0),

hence

dc-insulating

_states.

Usually

for the

_highest

and lowest adiabatic states of a band condition

₍₈₎

is not fulfilled and the states are more

_complicated

_{superpositions}

of

_unperturbed

functions as in the weak disorder case. But since these states _carry no dc-current their

_explicit

structure is not needed in the

present

investigation.

At the

_edges

of the non-adiabatic zone there is a small zone of intermediate states for which

P :0 0, P =#= 1, i.e.,

which are neither

_fully

adiabatic

_{(P = 0)}

nor

_fully

non-adiabatic

(P = 1 ).

By

writing

P = 1 or P = 0 _{we mean} here P close _to 1 _or close _to

0,

such that in the

course of the

_scattering

_{process the} effect due to _{the very small} deviation from P = 1 or P = 0 is

negligible.

(In

the model

_system

of Ref.

[4],

the

_parameter

values have

been chosen such that all non-adiabatic

_{(including intermediate)}

states can be

_expressed

in the weak disorder

_formalism.)

At each level

_anticrossing

the intermediate wave functions

_split

into two

_{parts :}

one

_part

_{develops according}

to the

_anticrossing

_(adiabatic)

branch with

probability

1 -

_P,

the other

_part

_according

to the

_{intersecting,}

i.e.

_unperturbed

branch

(non-adiabatic

_branch)

with

_probability

P

_(see

_Fig.

_3).

3. Phase randomness.

Let us consider a wave

_function,

which

_belongs

to the

_fully

non-adiabatic _{energy range}

between

_Ea

and -

_Ea

in

_figure

1. At t =

0,

_{we assume} this function _to be _an

unperturbed

Landau function

_{Ji, (x,}

y,

t )

with a localization centre

_{xs (t}

=

0 )

situated e. g. _on the left of _a

barrier,

say between X2 and x3 in figure 1. Since P = 1 in the

fully

non-adiabatic _{energy range,}

the modulus of this wave function continues to have the form of an

_unperturbed

_function,

which moves in x-direction with the

_velocity

v

_(v

is

_positive

if B and

_Ey

are

_positive).

After

some time its centre enters the intermediate zone _{between x3 and X4 in}

_figure

_1,

_{corresponding}

to _{the energy interval}

_[Ea,

_{Eb ],}

where adiabatic _{processes become}

_important

_(we

rember that P decreases

_{exponentially}

_with

_{xp - xp,}

₁

for

_Ep

=

Ep,,

i.e.

_{with xp}

and

_xp,

on

_opposite

sides of a barrier or

well,

see Ref.

_[4]).

During the passage through

the intermediate _{energy levels the} wave function

_undergoes

a sequence of «

splittings

»

(according

to

_Fig.

_3),

as a result of which it is transformed into a sum

E £p’ (t ) 4,p (x, y, t )

of

_unperturbed

functions

_03C8P.

We choose the

_origin

of t such that

p

Fig.

3. -

Splitting

between adiabatic and non-adiabatic time evolution in the

_vicinity

of t = t *

(mod T ).

Pn,n+i

1 denotes the Zener

tunnelling probability

for the non-adiabatic processes

(dashed

lines),

1 -

_{Pn, n +}

_{1 the}

_probability

for the

_{corresponding}

adiabatic behaviour

_{(full lines).}

The

_expansion

coefficients

_{dn (t f), dn + 1 (tf )}

of an orbital at t _{= tf are obtained from those} at t = _to

by

relation

(12).

p (t = 0)

lies

_exactly

between two

_{anticrossings.}

At a time

_{T = mT /2}

_{(m integer)}

the

coefficients

_{cp (t )}

result from a

_unitary

matrix transformation which is

_composed

of

_complex

2 x 2 matrices each of which describes a

_particular

_splitting

_process

_(indicated

in

_Fig.

_3).

As a consequence each

e; (T)

can be

_expressed

as a sum of contributions

_{c [j (s, p ) ]}

due to different

sequences of splittings, i.e.

_originating

from different

_splitting

_paths

_(see

_Fig.

_4),

which we number

_by

the

_{index j (s, p ),}

with :

where

_{(’R [j (s, p )])2}

is the

_probability

for the r-th

_splitting

on

_{the j-th path}

« from s to _p ».

The final

_probability

for the

_occupation

of the

_{component t/1 p}

at the time T

_originating

from the initial function

_{t/1 s (x,}

y, t =

0 )

is therefore :

where

_{J(s, p ) is}

the total number of

_paths

« from s

_{to p}

». We will show below that the

phases

ç[j(s,p)]

are random. Therefore the second term in

₍₁₀₎

vanishes if

_{J(s, p ) > 1.}

Now

functions with the

_highest

and the lowest _{energy in the}

_expansion,

see

_figure

4. But for

ihese

particular

coefficients one has

_{J(s, p )}

= 1

(hence

the second _term in

(10)

does _not

exist).

This means that for

_large

_m, due to

_phase

_randomness,

the

_final

_occupation

numbers

cp*(T)

c§ (T)

do not

_depend

on the

_phases

of the matrix elements but

_only

on the _square of their absolute

_values,

i.e.

_{they only depend}

on the individual

_{splitting probabilities}

(’ R [j (s , p ) ])2 :

The

_phase

randomness of the

_amplitudes

_{c [ j (s, p ) ]}

_originating

from different

_splitting

paths

«

between s and p

» in

figure

4 is

proved

as follows. Consider a

_{single splitting process}

Fig.

4. - Shown _are different

splitting paths,

which contribute to the transformation of an

_unperturbed

function

_t/J

s with coefficient

Cs (0)

at t = 0 into a linear

combination ¿ c; (m T /2) t/J p

at t = p

mT /2.

In the

_figure

m

_equals

6

(whence ch

=

c,

(6)

and

Cf

=

Cp _ 3

(6)).

E,,

...,

Ep, Ep...

denote the

unperturbed

( V 1= 0 )

energies.

Note that in this weak disorder

_{approximation}

at times t =

m-r: /2, m

integer,

each

_perturbed

adiabatic orbital

_{wn (mT /2 )}

coincides with a

_{corresponding}

(Fig. 3)

by

which the

_complex

coefficients

_{dn (to ),}

_{dn + 1 (to )}

are transformed into

_{dn (t f),}

dn + 1 (tf )·

Such a _{non-adiabatic process between weakly perturbed} levels occurs in various areas of

_physics.

A process

analogous

to ours has been studied in the context of

one-dimensional conduction

_[8-10].

Here

_explicit

_equations

for the time evolution of the

corresponding

coefficients

_{dn (t ),}

_{dn + 1 (t )}

have been obtained

_{[8, 10].}

The derivation of these

equations

can be extended in a

_{straightforward}

_way to our two-dimensional

_system

described

by

the Hamiltonian

_(1).

As a

_result,

in the

_expressions

of reference

_[8]

one

_simply

has to

replace x by

y and

a/ax

by

a / ay - i q Bx / (hc)

and to make the

identifications

_{e = q,}

F Ey, L = Ly,

a (t) = -

(q/hc)

4,(t),

Uk,« IL=

t/1p(cfJ),

U-k+« IL=

t/1p’(cfJ),

Uk,-k =

VP, p, .

One obtains two

_coupled

first order differential

_equations

for

_{dn (t )}

and

_{dn + 1 (t ),}

which can be solved

asymptotically

[8, 10].

The result is of the form

[10]

Here _{’P n, n +}

_{1 1S}

the

_phase

of

_{V 1}

_(where

_{t/I p}

and

_03C8p

are the

_unperturbed

functions involved in the considered

_splitting),

and

_{Pn,n +1=ppB,p’}

is the

_{corresponding}

Zener

_tunnelling

probability.

The

_phase

terms _{a n, n + l’}

_Bn,n+l,

_Yn,n+l,

_{sn, n +}

₁

_depend

on the

_energies

Ep [~

(t ) ],

Ep,

[~ (t ) ]

in the considered time interval and

_on

_{1 V;,p’ 1,}

but are

_independent

of

’~ n, n + 1 .

In reference

_[10]

a term

_{An, n +}

₁ is common to all the

_phases

_{a n, n + l’}

_{f3 n, n + l’}

l’ n, n + l’

5 n, n + 1.

It is different for different

_{splitting energies,}

and it was shown to be of order

TT or

_larger

in the

_examples

considered in reference

[10].

_Generically

(i.e.

_except

for very

special cases)

it is not

_equal

to an

_{integer multiple}

of 7r. This means that

_generically

the

products

of the factors exp

_{(An, n +}

i )

originating

from

_subsequent

_applications

of

_equation

₍₁₂₎

along

different

_{splitting paths}

from a

_given

initial

_point

to a

_given

final

_point

in

_figure

4 constitue a

_quasi

random set of values on the

_complex

unit circle

[10].

In reference

_{[9, 10]}

a

_{perturbation potential}

in the form a delta function was

_used,

which means that all the

_phases

_{’~ n, n + 1} are

_equal.

In our case of a random

_perturbation

V 1 (x,

y ),

however,

the

_phases

_{~ n, n + 1} are

_randomly

distributed as a function of n. This leads then

_immediately

to the conclusion that the total

_{phases originating}

from the different

splitting paths

are

random,

since

_they

are

_composed

of

_different

sums of the random

_phases

~ n, n + 1

(plus

a term

independent

of the

~ n, n + 1).

We remark that it is

_{highly probable}

that an

_{analogous phase}

randomness also holds in the

general

case

_(«

_strong

disorder

»),

where each non-adiabatic function

_develops

into a linear combination of more than two basis functions

_{03C8p (x,}

_y,

_{t )}

in a time interval

_T/2.

As we have seen this

_phase

randomness

_considerably

_simplifies

the calculation of the time evolution in x-direction of the total

_{particle density}

of a

_wavepacket

and makes the

_scattering

_process

understandable in terms of

_{probabilities.}

In the weak disorder case numerical calculations of such scattered

_particle

densities are in _progress.

4.

_Origin

of the Hall conductance

_plateaux.

We consider now a

_situation,

where the _{Fermi energy EF} is situated in a _{range of} adiabatic levels such that all non-adiabatic

_{(including intermediate)}

levels below

_EF

are

_{fully occupied}

and all non-adiabatic levels above

_EF

are

_empty.

We suppose as an initial condition at t =

0,

that all

occupied

non-adiabatic,

including

intermediate _states have the form of

non-adiabatic states

_(for

each band with

_occupied

non-adiabatic

_states).

We further assume

t = 0 to be in the middle between _two consecutive

splittings (see

Fig.

4).

In the _course of time

each of these functions

_{03C8s, (x, y, t ) develops}

_according

to the

_multiple

_splitting

process

described above.

_(The

adiabatic functions are not

_split.

This means that

_they

are not mixed

with the

_{functions of}

the non-adiabatic

_zone.)

At a time t = T =

mT /2,

_m

integer,

m >

_1,

the total

_occupation

_number

_{Cp (T ) 2}

of a

_component

_{03C8p (x,}

_{y, t )}

of a non-adiabatic

state of a band is now the sum of

_probability

terms

₍₁₁₎

_coming

from all the

_t/ls

_{which span the} non-adiabatic

_subspace

of that band at t = 0 :

where

1 c, (t = 0 ) 1 ’ = 1.

Here

_{J (s, p )}

is the total number of

_{splitting paths «}

from s

_{to p »}

in

figure

4

_{and j (s, p )}

is the index of such a

_path.

As we have seen,

_equation

(11),

hence

(13)

follow from

_phase

randomness,

as a _{consequence of which only} the

_splitting

_{probabilities}

(’R [ j (s, p ) ] )2

have to be taken into account for the calculation of the final

_occupation

numbers

_{1 cp (T) 12.}

The

_right

hand side of

₍₁₃₎

is obtained

_explicitely

_{by calculating}

in each

_splitting

interval

simultaneously

all the

_occupation

_numbers

_{Cp ( 2}

of the considered band at a time

t =

mT /2

from those of the

previous

time

(m - 1 )

r/2,

by

means of the matrices

_(12),

but where the

_phases

of the matrix elements are

_{suppressed. Starting}

with the initial condition

1 Cs (t

=

0 ) 2

= 1 for all s associated with the non-adiabatic states of the band and

_using

the

fact that the transition

_{probabilities}

associated with the four

_possible

_paths

of an individual

splitting

process

pairwise

add up to one

_(see

_Fig.

_4),

one

_immediately

sees that all the

occupation

numbers

ICp 2

of the functions

_{03C8p (x, y,}

_{t )}

_{which span the}

_subspace

of the non-adiabatic states of the band at t =

mT /2

(m integer)

_are

equal

_{to one :}

Equation (14)

can be extended to the

_general

case, which is not based on the weak disorder

assumption

and on

_phase

randomness. In the

_general

case one obtains

[11] :

Here

1 dn (t ) 12

is the total

_occupation

number of the n-th of the adiabatic basis states

wn (x,

y,

t ),

which span the

subspace

of the non-adiabatic states of the considered band at the time t. Our derivation of

₍₁₄₎

constitutes an immediate verification of

₍₁₅₎

in the

_special

case of our weak disorder model.

_Equation

₍₁₅₎

follows

_directly

from the

_unitarity

of the time evolution

_operator

and from the

_special

form of the adiabatic time evolution. Further it can be shown

_[11]

that these

_general

mathematical

_properties

_(which

_imply

₍₁₅₎

and the

_periodicity

of the modulus of the adiabatic functions

_{wn (x, y, t ))}

have the

_{following physical}

conse-quences

(provided EF

lies in a _{range of} adiabatic

_energies

for the considered value of

Ey

and the

_temperature

is

_sufficiently

low,

such that the non-adiabatic states are not affected

by

inelastic interactions with the

_surrounding

heat

_{bath) :}

a)

The total

_{charge density}

of the

_system

is

_periodic

in time with

_period

T, i.e. it is constant

V (x, y )

in the course of time

_(in

_particular

no

macroscopic

field

_Ex

is

created),

and all adiabatic

_energies

_{En (t )}

remain

_periodic

with

_period

T.

b)

There is no

_dissipation

(the

_{total energy associated with the occupied} non-adiabatic

states remains

_equal

to the

_ground

state _{energy in the} course of

_time).

Hence

c)

0-xx

= Uyy

=

0,

1 y

=

0,

1 x

= constant

(iy

and

I x

denote

dc-currents).

The last

equation

means that there exists a Hall conductance

_plateau

as

_long

as

_EF

is in a _{range of adiabatic}

states.

Let us now return to our

_explicit

weak disorder model. Here

₍₁₄₎

means that all

unperturbed

states

_{03C8p (x, y,}

_{t )}

_representing

the

_fully

non-adiabatic states are

_occupied

with

probability

cp 2

= 1 for all

_{times tN}

=

NT,

N

integer.

This means

_that,

_being

_fully

non-adiabatic states,

they

are

_{fully occupied}

for all times.

_(Note

that the set of

_{indices p}

corresponding

to these states

_depends

_{on t : p}

= s + N = s +

_{tN / T .}

Further we remark

_again

that the

_{probabilities}

_{cp (t ) 2 result}

from the

_projection

on the

_unperturbed

states

03C8p (x, y, t )

of all the different individual

_{single particle}

functions at the time

_t).

These

particular unperturbed

_{functions 03C8 p (x, y, t )}

representing

the

_fully

non-adiabatic states are

spatially

localized in the

_fully

non-adiabatic zones between barriers and wells

_{(see Fig. 1).}

Their localization centres move with

velocity

v. This means that the distance between two

neighbouring

centres

_{xp (t )}

and

_{xp + 1 (t )}

is

_{passed by}

the unit

_{charge q}

in the time interval r.

Therefore in these

_{spatial regions}

_occupied

_by

the

_fully

non-adiabatic states a dc-current

1x

(per band)

flows in

x-direction,

with :

Ix

flows at all times. This

_gives

a contribution to the Hall conductance of

_q2/h

_{per band.} The dc-current of the

_fully

adiabatic states is zero

[3, 4].

In

_figure

1 the localization centres

of these states are situated between the

_{spatial regions occupied by}

the localization centres of the

_fully

non-adiabatic states, which move with the constant

_velocity

v towards the next

intermediate

_region

to their

_right.

Since there is conservation of the total

_charge

in the

_system,

one

_{charge q}

_{(per band)}

must

_disappear

_{per time} interval r in the intermediate

_region

on the left of a barrier

_(well)

and one

_{charge q}

must _{reappear in the} same time

interval

on the

_right

of a barrier

_(well).

Otherwise no dc-current

(16)

could be

_possible

on the

_spatially

distinct

_fully

non-adiabatic

_regions

between the barriers and wells. This non-classical

_particle

_propagation

across the

_fully

adiabatic

_{spatial regions}

in the center of the barriers and wells is the result of the

_{time-dependent scattering}

_{process in the} intermediate _energy zone, which includes

scattering

between states with localization centres on different sides of a barrier

(well).

This

process is

responsible

for the so-called

_compensating

currents

_(see

Ref.

_[4]

for a more detailed

discussion).

If

_EF

is situated within a _{range of intermediate} or

_fully

non-adiabatic

levels,

equations

(14)

and

₍₁₅₎

no

_longer

hold. In this case some initial

_{coefficients cs (t}

=

0 )

1

corresponding

to

states above _{the Fermi energy} are zero in

_{equation (13), i.e.}

the

corresponding

spatial

sites

xs (t

=

0)

_are

empty.

But due to the discussed time evolution some of the

corresponding

coefficients

_{C, + N (0 + N -r)}

will be different from zero after a time interval NT.

_(Further,

occupied

sites at t = 0 may be

ûnoccupied

_{at t} =

NT).

This may lead _to

macroscopic charge

rearrangements,

since xS + N

(N )

is

_equal

to

_{xs (o ),}

which was not

_occupied

at t = 0. Hence an

additional electric field in x-direction _{may be created.}

_Further,

since these new

_occupied

components

03C8s, + N (t

=

NT )

have _an energy above EF,

, states which are

composed

of such

components will be modified

_(their

_{energy will be}

_reduced)

_by

inelastic

_scattering

_(due

to

between two inelastic events. This means there is

_dissipation,

hence non-zero

_diagonal

conductivity.

5. Discussion.

In our weak disorder model each

_scattering

event _{in the intermediate energy} zone

simultaneously gives

rise to an adiabatic

_{splitting corresponding}

to non-classical motion with

change

of the side of the barrier

_(well),

hence either to forward

_(i.e.,

to the

_right

in our

_case)

or backward

_scattering,

and to a non-adiabatic

_splitting

_{corresponding}

to classical motion without

_changing

the side of the barrier

_(well),

hence to forward

_{scattering only.}

Now consider the case of a wave

_{function t/I (x, y, t )}

which at t = 0 is an

_unperturbed

Laudau orbital with localization centre in the

_fully

non-adiabatic zone to the left of a barrier

_{(or well),}

moving

to the

_right

with

_velocity

v. After some time the orbital reaches the intermediate zone and is

_subject

to a _{sequence of splitting processes} as described above. Once a

_component

ipp (t)

belonging

to the

_fully

non-adiabatic zone localized on the

_right

hand side of the barrier

(well)

is

_occupied

with non zero

_probability

_{cp* (T) cp (T),}

it is no

_longer

_split

for some time. It remains on this side and its modulus moves with the classical

_velocity

v _{away from the barrier}

(well)

until it reaches the next intermediate zone. As a _consequence

_(and

because of

individual

_charge

_{conservation)}

the

_probability

for

_occupying

a

_component

with centre

xp

on the

_opposite

side of the barrier

_(well)

increases with time whereas that for

_occupying

a

component

on the

_original

side decreases.

_Finally,

after a sufficient number of time intervals

T /2

practically

the entire wave function

_{ip (x, y, t )}

is localized on the

_opposite

side of the

barrier

_(well).

After the passage of the first barrier

(well)

an

_{originally unperturbed}

Landau function has thus

_developed

into a

_{function t/I (x,}

_y,

_{t )}

which is somewhat delocalized in x-direction. This

delocalization _{is increased after each passage} across a

_subsequent

well or

_barrier,

since each of its

_components

_{cp (t ) 03C8p (X 5 y 1 t )}

is

_{decomposed again}

in the described manner

_by

the

scattering

events in the intermediate zone _{of every}

_subsequent

barrier or well. But in the

_fully

non-adiabatic zones between a barrier and a well each of the

_components

_{cP (t ) 03C8P (x,}

_{y, t )}

of the wave

_{function gi}

_(x,

_y,

_{t )}

has an

_unchanged

modulus which moves in x-direction with the common constant

_velocity

v.

The

_{scattering process which}

_brings

an

_incoming

Landau wave

_packet

from one side of a barrier

_(well)

to the other side

_requires

a time

_typically

of the order of T times the number of

intermediate states

_(in

the extreme,

hypothetical

case where there is

_only

one intermediate

state _{per barrier} or

_well,

a time

_equal

to

_7/2

is

_required,

see Ref.

_[4]).

_During

its time evolution

_through

the intermediate states a

_{single-particle}

wave function is a time

_dependent

sum of non-adiabatic and adiabatic

_components.

This means, that in each time interval of

length

T the wave function contains a

_part,

which is

_periodic

in 0 = -

_{cEy Ly t}

(or,

in the

extreme case mentioned

above,

which at least

_represents

half of a

_period).

To describe this

periodicity

by

an

_expansion

with

_respect

to

₀

or

_Ey

all orders would have to be included.

Hence a linear

_{approximation}

is not

_adequate,

it would

_only

be a

_good

_{approximation}

for time intervals

_appreciably

smaller than

_r/2.

This is

_equally

true for a

_perturbed

one-dimensional

conductor

_(loop),

since the

_{corresponding}

mathematical

_developments

are the same

_(see

Sect. 3).

For the case of one-dimensional conductance it has

_already

been

_emphasized

previously

[8, 12])

that linear response theory fails to describe the time evolution of the scattered states in cases where the Zener

_{probabilities noticeably}

differ from

_unity,

i.e. in cases

_where,

in our

_terminology,

the

_{time-dèpendent}

wave function passes

_through

intermediate states.

adiabatic

_{(i.e. non-conducting)}

and intermediate

_(i.e.

_{non-classically}

_conducting)

single-particle

wave functions is non-linear in the electric field on

_{general grounds}

_[11],

and we have shown that the

_macroscopic

Hall current

_Ix

is

_generated

_by

the currents of the non-adiabatic

states and that the intermediate states constitute a substantial

part

of this _current, since

_they

carry the so-called

compensating

currents

[4, 11].

Further,

in the

special

case, where the

intermediate states are

_{only weakly perturbed by}

the disorder

potential,

we have shown

explicitly,

that the

totality

of the time evolution

_(for

times

_exceeding

_T/2)

of each

intermediate state is needed for the

_proof

of the

_integer

_quantization

of the Hall conductance.

Therefore linear _response

_theory

_(which

is

_applicable

_only

for times much smaller than

T /2)

cannot

_give

a correct

_microscopic

_quantum

mechanical

_explanation

of the

IQHE

in these

systems.

Our

_systems

are characterized

_{by periodic boundary}

conditions in the direction of the

macroscopic

Hall field. Now if the

IQHE

is a bulk

effect,

linear response

_theory

should also fail to describe the

IQHE

in

_systems

with other

_boundary

conditions.

_Recently

work based on

Dirichlet

_boundary

conditions in the direction of the

_macroscopic

Hall field and

_generalized

periodic

boundary

conditions

_(depending

on an

_auxiliary

parameter)

in the direction of the Hall current did succeed in

_deriving

_quantization

of the Hall conductance in the framework of the Kubo formalism

_[13-16].

However the fulfilment of a series of additional

_assumptions

was

indispensable

to achieve this result. On the other

hand,

in our

_theory

no such additional

assumptions

(and

no

_auxiliary

parameter)

are _necessary to obtain our conclusions. We

_hope

that further work will

_clarify

the relation between these different

_approaches.

In our weak-disorder-model the

_expectation

value of the

_velocity

operator

in x-direction of a non-adiabatic state is

_equal

to

_cEy/B

if the orbital is

_fully

non-adiabatic, i.e.,

located in the

full line interval before a barrier

(or well),

and it is

_{again equal}

to this value once it has been scattered

_through

the barrier

_{(or well),}

if one waits a

sufficiently long

time,

as we have seen. In the meantime this

_expectation

value is

_higher

_(in

_{general by}

_{many orders} of

_magnitude,

see

[4]).

If at a

_given

time one sums over these

_{time-dependent velocity expectation}

values of all the non-adiabatic states

_(part

of which are non-linear in

_Ey),

one obtains

_again

the result

_(16),

i.e.,

an

_expression,

which is

_{time-independent}

and linear in

_Ey.

If the electric field

_Ey

is

_{sufficiently high}

_(i.e.

if the Hall current

_Ix

is

_sufficiently

_large),

such that afl adiabatic states become almost

_fully

non-adiabatic,

linear response theory becomes

again

applicable

(in

complete analogy

to the

_{corresponding}

one-dimensional conductance

[8]).

This

_corresponds

to the situation where the Hall conductance

_plateaus

have

_disappeared

(breakdown

of the

_quantum

Hall

_effect).

The results

_presented

in this and our

_previous

[4,11]

articles open new

_perspectives

for the

theory

of the

IQHE.

In

_{particular they}

elucidate the

_microscopic

_scattering

mechanism in the

quantum

Hall

_regime.

The nature of this mechanism has been illustrated

_by

our

explicit

model

system

(Fig. 1)

in the weak disorder case. Our results contrast with certain

aspects

of

_previous

theories,

where similar Hamiltonians as defined

_by

₍₁₎

and

₍₂₎

have been used. Here we

briefly

examine the so-called gauge invariance

_argument.

The essence of this

_theory

is

reviewed in

_{[17] (based}

on a more

_complete

form of the

_original

_argument).

The considered

systems

are

_{topologically}

_equivalent

to our

_systems

defined

_by

₍₁₎

and

_(2).

As in our case the

electric field is created

_by

an

_increasing

_magnetic

flux 0

(t)

(through

a

hole).

The

_arguments

of the _gauge invariance

_approach

are

_entirely

based on the adiabatic wave functions. It is

claimed

_that,

_{if the Fermi energy is in} a

_{mobility gap,}

an adiabatic

_change

of the flux

_by

an

amount

_hc/q

_transports

_exactly

an

_integer

number of electrons across the disordered

_system

in a direction

_{perpendicular}

to the electric field. If we

_apply

this statement to our

_explicit

model

_system

_{represented by figure}

1 this would

_correspond

to the adiabatic transfer of an

(or well)

to the full line

_region

on the

_right

hand side of this barrier

_{(or well).}

_According

to our results such an adiabatic _process is

_impossible

in such

_systems,

since in the _presence

_{of disorder}

the adiabatic wave functions are

_{dc-insulating}

(the

adiabatic

_particle

densities and currents are

periodic

in

_time)

and therefore do not contribute to the dc-Hall current

_{perpendicular}

to the electric field. We have seen that the

_macroscopic

Hall current is carried

_by

the non-adiabatic states, and in

_particular,

that those of the non-adiabatic states, which are intermediate between

_fully

non-adiabatic and

_fully

adiabatic,

carry the

s,o-called

compensating

currents,

which

_compensate

the loss of current of the

_{dc-insulating}

states

_{(i. e. ,}

of the

_fully

adiabatic

states)

and of the

_{dc-insulating}

fractions of the

_{dc-conducting}

states

_(i.e.,

of the

_fully

adiabatic

components

of the intermediate

_states).

Next we

_analyze

another well known

_theory,

where the Hall

_conductivity

is

_expressed

as a

topological

invariant

_[18]

and which _{may be considered} as an extension of the gauge invariance

_argument.

In this

_theory

the

_conductivity

is

_expressed

in a version of the Kubo formula

_{(i. e.}

_by

linear _response

_theory),

which is then

_averaged

over the unit cell of two flux

parameters

(each

of which describes an electric field in one of the two

_spatial

_directions).

This

average can be shown to have the mathematical form of a

_Berry

_phase,

which describes the increment

_(in

addition to the

_{dynamical phase}

_factor)

of the

_{global phase}

factor of the

adiabatic wave function

_along

a closed

_loop

in the two-dimensional _{space of the} two flux

parameters.

It is then claimed that the unit cell of the

_parameter

_space

_represents

a torus.

From this it

follows,

according

to

_general

results of differential

_geometry,

that the flux

averaged

Kubo Hall

_conductivity

times

_h/q2

is a

_topological

invariant

(equal

to an

integer).

The value of this

_integer

is not determined

_by

this

_general

_argument,

but the idea is that it be different from zero.

According

to our results such a

_theory

cannot describe the

_conductivity

associated with the

macroscopic

Hall current, since it is based on the adiabatic wavefunction

_and,

in

addition,

on

linear-response theory.

Moreover,

since the adiabatic time evolution is

_involved,

which in the presence of disorder leads to

_periodic

_currents, as we have seen, we would

_expect

that the

averaged

conductivity

thus obtained is

_actually

zero. From a recent mathematical

_analysis

_[19]

it appears, that the flux averaged Kubo Hall

_conductivity

is not a

_{(non-trivial)}

_topological

’invariant.

This will be the

_subject

of a

_forthcoming

_publication.

We conclude with some

_summarizing

remarks. The class of models which we have considered are free of contacts and do not

_give

rise to

_edge

states. This

_supports

the view that the

IQHE

observed in

_large

two-dimensional

_systems

_[1]

is not a contact or

_edge

state effect.

This is consistent with

_{experiments performed}

without contacts

_[20].

In our

_systems

the

microscopic

origin

of the

IQHE

is due to the characteristic time

_dependence

_{in the presence} of disorder of the

_{single-particle}

wave

functions,

where non-linear effects

_play

an

_important

role. The

_{time-dependent, microscopic scattering}

mechanism can be illustrated in detail

_by

an

explicit

model

_system.

Our results

_provide

a new

_picture

for the

_{understanding}

of the

IQHE.

Acknowledgment.

The author thanks 0.

_Viehweger

for

_sending

a _{copy of reference}

_[15]

_prior

to

_publication.

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