Spectrum of 2D Bloch electrons in a periodic magnetic field : algebraic approach

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Spectrum of 2D Bloch electrons in a periodic magnetic

field : algebraic approach

A. Barelli, J. Bellissard, R. Rammal

To cite this version:

Spectrum

of 2D Bloch electrons

in

a

periodic magnetic

field :

algebraic approach

A.

_Barelli,

J. Bellissard and R. Rammal

_(*)

Centre de

_{Physique Théorique,}

CNRS

_Luminy,

Case _907, 13288 Marseille Cedex 09, France

(Received

on _May 2, 1990,

accepted

on June 13,

1990)

Résume. 2014 Les méthodes

algébriques

que l’on a introduites récemment pour l’étude des électrons

de Bloch sous

_champ

_magnétique

uniforme sont

_{généralisées}

au cas des

_{champs periodiques.}

En

utilisant une

_{approche semi-classique,}

on étudie le cas où la maille

magnetique

est commensurable

avec celle du réseau. En

_general

et selon la

_{valeur 03A6}

du flux

_magnétique

moyen a travers la cellule

616mentaire, deux cas distincts semblent se

_distinguer.

Le

_premier

cas est 0 =A 0, où la structure

en niveaux de Landau est retrouvée

_(cas

non

commutatif).

Dans le second cas 03A6 = 0, on obtient

une structure de bandes non triviale

_(cas

_commutatif).

Nos résultats sont illustrés avec des

exemples simples.

En

_particulier

on montre, sous certaines conditions, que le mecanisme de

stabilisation de la mer de Fermi, avec un _quantum de flux par fermion, se

_généralise

au cas d’un

champ magnétique périodique.

Abstract. -

Algebraic

methods

_recently

introduced for 2D Bloch electrons in a uniform

_magnetic

field are extended to the case of

_{periodic magnetic}

fields.

_Using

a semiclassical

_approach,

we

investigate

the case where the

magnetic

unit cell is commensurate with the lattice unit cell. In

general

and

_according

to the _{value 03A6} of the average flux

through

the

_magnetic

unit _cell, two distinct cases take

place.

The first one

_corresponds

to finite values of 03A6, where the usual structure of Landau levels is recovered

_(non

commutative

_case).

In the second case where 03A6 = 0, a non

trivial band structure is obtained

_{(commutative case).}

Our results are illustrated

by simple

examples.

In

_particular

we show that, under certain conditions, the mechanism of stabilization of the Fermi sea

_by

the gaps

(with

one _quantum flux per

fermion)

holds in the

_general

case of

periodic magnetic

fields. Classification

Physics

Abstracts 03.65-71.25-75.20

1. Introduction.

In view of the

_fascinating

behavior of two dimensional

_(2D)

electron

systems

in a

_strong

magnetic

field,

it is natural to consider more

_{general problems}

where the

_magnetic

field is no

longer

uniform. In this paper, the third in a

_series,

we consider the case of

_{periodic magnetic}

fields

_acting

on fermions

moving

on a discrete 2D lattice. The

_particular

case of a uniform

magnetic

field has been the

_subject

of various contributions

_during

the _{past. Recent progress} and the state of the art have been summarized in references

_[1,

_2].

Part of our motivation for

studying

this

_general

case comes from the

_{recently developed}

methods in non commutative

geometry

to

_investigate

some

_properties

of 2D electrons in a uniform

_magnetic

field

_[1-5].

Among

various

_aspects,

let us mention some basic results obtained so far

_by

this

_{approach :}

justification

of Peierls

_{substitution,}

new differential calculus which allowed the

_{investigation}

of the fine structure of the energy

spectrum,

semiclassical methods for the calculation of the energy gap

boundaries, rigorous proof

of the

Quantum

Hall

Effect,

etc. On

_{general physical}

grounds,

some of these results are

_expected

to survive in more

complicated

situations such as

the

_{periodic magnetic}

flux case. Another reason for

working

this kind of

problem

is the

spectacular

result obtained

by

Novikov et al.

[4]

for the free-lattice case, where the

ground

state has been

_fully

characterized

_[6].

The

_questions

we

_try

to answer in this _paper are two interrelated

_{problems :}

1)

What are the

_appropriate

algebraic

tools to be used in the

_periodic

flux case

_(generalized

rotation

_{algebra, etc.) ?}

2)

How much different is the energy

spectrum

in that case in

_comparison

with the uniform

flux case ?

The paper is

organized

around these two

_questions,

and rather traditional

_aspects

are left

out. In section

2,

we summarize some of the known results for the

_problem

at hand and

give

a

rather consistent set of notation and definitions. Section 3 is devoted to the treatment of two

simple examples :

the

_{strip problem}

and the checkerboard lattice.

_There,

we work out the

algebraic

formulation and the

_{quantization procedure}

needed for semiclassical calculations. In section

4,

the

_stripped

flux lattice is considered in

_general

and the

_{fully general problem}

is worked out in section 5. Some

_applications

of our results are

_given

in section 6 : the increase

of the energy levels

by

the

magnetic

distorsion,

the stabilization of the Fermi sea

_by

the gaps.

Our

_concluding

remarks are summarized in section 7.

2. Periodic

_magnetic

fields.

The most

_important

work on this class of

problems

is

probably

due to the soviet school. To

our

_knowledge,

the

spectrum

of 2D electrons in a

_{periodic magnetic}

field

_{B (x, y)}

has been

investigated

first

_by

Novikov et al.

[6].

Attention has been focused on the lowest Landau level as well as its evolution under the action of a

_{crystal potential}

considered as a

_{perturbation.}

To be

_{self-contained,}

a brief _summary of this work and its extension is described in section 2.1.

The

_{general problem}

of a

_{periodic crystal potential}

and a

_{periodic magnetic}

field is defined in

section 2.2. In this

_respect,

the same notation as in references

_[3,

_4]

will be used.

2.1 FREE PARTICLES IN A PERIODIC MAGNETIC FIELD. - The

problem

considered in

reference

_[6]

is relative to a

_{periodic magnetic}

field

_{B (x, y), doubly periodic}

and directed

along

the z-axis :

F

_being

a solution of

One of the main results of reference

_[6]

is relative to the

_degeneracy

of the lowest Landau level

_(LLL).

In

_fact,

it turns out that the addition of any

doubly periodic

increment to the

homogeneous

field leaves the LLL

_{fully degenerate}

in

_spite

of the loss of

_symmetry.

Furthermore,

the

_magnetic

_{flux 0}

_through

the

magnetic

unit cell

_Tl

_T2

is the

_principal

topological

characteristic. In

_particular,

for

_integral

or rational

_flux,

the wave function of the

ground

state can be obtained in terms of

_elliptic

functions.

_Qualitative

_arguments

have been used

_[6]

to describe the

_magnetic

band structure

_resulting

from a weak

_{potential perturbation,}

periodic

with the same lattice.

These

_surprising

results deserve some comments. First it is useful to recall that in the presence of a

_magnetic

_field,

wave functions exhibit nodal

_points

rather than

nodal lines,

the

latter

_being

a

_generic

occurrence in

_systems

with time reversal

_symmetry.

The fact that the LLL wave functions are

_completely

determined

_by

the locations of their nodes is due to

constraints

_{of analyticity}

and

_periodicity

[7].

Due to the

_{double-periodicity of B (x, y ),}

the

problem

is

_equivalent

to an electron on a torus and the number of nodes of the wave function is now finite and

given by

the

_(integral)

number of

quantum

flux

_rb.

This

_{description provides}

a

_{totally general}

and

_{gauge-independent}

characterization of any

state of the LLL. In

_particular

this

_gives

an

_opportunity

to

_study

the

_boundary

condition

sensitivity

in a rather unusual context. In fact if one defines a random

_potential

over the _torus,

the

sensitivity

of the wave function to

_changes

in

_boundary

conditions can be

_{probed by}

considering

the «

_brainding

» of the zeros. Such an idea has been

exploited

in reference

_[8]

to

probe

the

_sensitivity

of LLL wave functions in disorder-broadened bands to

_changes

in

boundary

conditions. In

_particular

a marked difference in behavior has been found between localized and extended states.

2.2 2D LATTICES AND PERIODIC MAGNETIC FIELD. In this paper we

_investigate

the

tight-binding

model,

in a

_{periodic magnetic}

field. More

_precisely,

we are interested in the

eigenvalue problem :

describing

the

_hopping

of electrons between

_{nearest-neighbouring}

sites

_{(r )}

on a lattice under

a

_magnetic

field.

_{Here 4r}

_{(r )}

denotes the wave function

_amplitude

at site r and t is the

_hopping

matrix

_element.

Without loss of

_generality,

the energy scale t is fixed at t = 1. In

equation (2.4),

the

phase

factor y,,, is

and describes the net effect of the vector

_{potential A (s)}

on the

_hopping

matrix elements.

For the sake of

_simplicity,

we will consider the square lattice case. Extensions to other lattices do not offer a

_{major difficulty.}

The

periodic

magnetic

field is

_{specified by}

a

_magnetic

unit cell _q x q2,

(q 1 _

1,

q2 -- 1) corresponding

to a

_pattern

of _{ql q2}

_magnetic

_{fluxes .0 , 1 , 2}

with

_{1 _ a-1 _ q l,}

1 - c-:::: 0’ 2 --,-

q2.

Using

the notation of our

_previous

_papers, we have

y ul U2 - 2 00 7r 0 Il 0’2

for the reduced

1 2

*

o 1 2

fluxes.

_Examples

of

_periodic

_patterns

are

_given

in

figures

1,

2. For

_{example, ql =}

_q2 =

1,

describes the uniform field _{case ; q2 = 1} and

_{arbitrary q corresponds}

to the

_stripped

lattice,

etc.

Fig.

1.- Notation used in the text for the

magnetic

fluxes :

(a)

simple stripped

case with

q = 2,

(b)

generalized stripped

_case.

The main purpose of the next sections is to

_provide

an

_algebraic

formalism to handle the

present

problem.

Therefore well known

_properties

such as: invariance under

0 11 0’2

cp Ut U2

+ cP 0;

_{4, l 0’2 - cP Ut}

U2’ etc., will be left out.

Also,

special properties

of the

spectrum

(opening

and closure of the gaps,

nesting,

hierarchical structure, gap

labelling,

etc.)

will not be addressed. The reason is

simply

that many of these

_properties,

if

_they

survive in the

periodic

flux case, are natural consequences of the

_{algebraic approach.}

Of course the same

Fig.

2.-

_{(a) simple}

checkerboard lattice,

(b) generalized

case for

_{arbitrary q}

1 and q2.

It is

_important

to notice that the

_problem

worked here is somewhat different from the free-lattice case. In this

_respect

the

_{tight-binding}

model must be viewed as the infinite

coupling

limit of a lattice

_{potential. Accordingly}

we limit our calculations below to the weak field

limit,

where much can

_already

be learnt.

Specific

extensions to other limits

_(e.a.

rational

flux,

etc.)

can be

_{investigated using}

the same

_machinary

that we have

developed

in the uniform

magnetic

case

_[3,

_4].

_

Before

_going

to

_examples

let us

_specify

some of our notations.

The wave function will be written as a vector value wave function in the

_following

_{way :}

Each

_component

_{1/1’ ri}

r2 can be

interpreted

as a wave function on a renormalized lattice whose

elementary plaquettes

are

_{given by}

the unit cell of

_periods

with size _q 1 and q2.

We can now rewrite the

_{Schrôdinger equation}

_(2.4)

as a matrix

_equation

for the

components

of 1/1’. This will be

_given

in many details in the next sections.

However we remark that the

_{Schrôdinger equation (2.4)}

is

_{easily expressed}

in term of the

magnetic

translations

_Ut

_{and U2 [9]}

Indeed

_{equation (2.4)}

for a _square lattice can be rewritten as :

For other

lattices,

similar

_expressions

are found

involving only polynomials

with

respect

to

Ul

and

_U2.

Let us

_give

the

_expression

of these two

_operators

in terms of the vector 1/1’. To do so we

introduce the

_ordinary

translation

_operators

_Tl

and

_T2

_acting

on wave functions as follows :

then we

_{get :}

In the

_coming

sections we will

_compute

various gauge transformations

by simplifying

the

expression

of the

_Schrôdinger

operator

suitable for semiclassical

_expansion.

We also use finite Fourier transform :

3. Two

_examples.

Before

_considering

the

_general

situation with

_{arbitrary magnetic}

_patterns,

we will discuss first

two

_{specific examples.}

The first one is the

simplest stripped

flux lattice where two values of y

q x 1

_patterns.

The second

_example

is the so-called checkerboard or

_staggered

flux lattice.

This

is

actually

a

_special

case of 2 x 2 _patterns, which is

generalized

in section 5.

3.1 STRIPPED FLUX LATTICE.

3.1.1

_{Algebraic formulation.}

- The notation used is

_depicted

in

_figure

la.

Using

formulae

_(2.10)

we find

_{immediately :}

where fi,

i =

1,

2 are the lattice

_position

_operators.

By

the

_following

gauge transformation

which allows us to see

_that,

as far as the

_algebra

is

concerned,

all

_operators

of interest are

expressed

as matrices with elements

given

_by

polynomials

with

_respect

to U and V where :

They

are a

_pair

of

_unitary

_operators

_satisfying

As shown in the

_previous

_paper

_[2]

we can realize U and

_by

means of two

_self-adjoint

operators

KI

and

_K2

with :

Let us introduce the new translation

_{operator :}

thanks to the new non commutative gauge transformation

_{given by :}

3.1.2

_Quantization

and semi-classical calculation.

Using

the above

_notation,

the Hamilto-nian R can be

expressed

as :

and in matrix

_{representation}

H is :

Classically,

Yo = _Y

1 =

0,

the _energy

_spectrum

of H is

easily

calculated as : 2

_(cos

_k2

± cos

_A:i).

The

_quantized

version of H is

_{given by : V}

=

e i ’Y 1/2 K 2, Ur

=

e - i ’Yr K 1 /

y’

/2. r =

1,

2. Here we use

Weyl’s quantization

scheme,

with two

_operators

_K1

and

_{K2 satisfying}

_{[Kt, K2]}

=

i-Within this

_picture,

H can be

_decomposed

into a sum of two terms :

where ’U denotes the 2 x 2

identity

matrix and

_Hl

is

_only

function of

_Kl.

It turns out that

Hl

is a Jacobi-like matrix

(tri-diagonal)

and a discrete Fourier transform is

helpful

to

_perform

practical

calculations. In what

follows,

we outline these calculations in some detail. For a

given

set {f/(r)},

0 :5 r :5 q - 1,

we define the discrete Fourier transform

by :

Under this

_unitary

transformation

_Hl

becomes :

being

the hermitian

_conjugate

of A.

With this formulation

H

=

H1

₊ 2 cos

_(y

1/2 K2). 1]

is _easy to

_manipulate

and

_particularly

for semiclassical calculations. In what

_follows,

we limite our discussion to the lower

_edge

of the energy

spectrum.

More

_precisely

we shall calculate the flux

_dependence

of the Landau _energy

levels of

fi

close to E = 0. For

this,

we use the same

_procedure

as for the uniform field case

[3].

Our

_{starting point}

is Schur’s formula which can be written as :

here P and

_Q

are the

_{eigenprojectors,}

The semiclassical result is obtained

_by

_expanding

Îleff

at low y o and Y1-

Simple

calculations lead to :

for first order in y and this is

nothing

else than the

simple

harmonic oscillator. The

quantization

of

_Beff

_{gives :}

_En

= _y

(2 n

₊ 1

).

The second order result is obtained

through

a

simple

iteration of

_{equation (3.15),}

with E

_{= En}

as an

input

value. The final result reads as :

Using

the known results :

we obtain the

_{energies :}

The

_{physical meaning}

of this

_simple

result will be discussed in section 6.1. Notice that

En

is invariant under the

_{permutation yo H y}

1 and reduces to the known result for

7o=yi’

3.2 CHECKERBOARD FLUX LATTICE. - The notation is

_depicted

in

_figure

2a.

_Proceeding

in a

similar

fashion,

the

_magnetic

translation _operators

_Ul

and

_U2

(after

gauge

transformations)

can be written as :

and the

_{corresponding}

Hamiltonian is

_{given by :}

Equation (3.20)

is

_actually

a

_{starting point}

for

_performing

semiclassical calculations as will be

shown below.

We use the

_quantization

rules : U =

exp(i(y)Ij2 KI), V

=

exp (i (y )1/2 K2) in

the

At low

_fields,

we

_expand

as usual and use Schur’s

formula,

with the

following spectral

projectors :

The effective Hamiltonian

_Heff

is then

obtained as :

with

This leads

_finally

to the

_energies

(n > 0 ) :

up to order

_{0 (y r 3).}

4. General

_strip

case.

In this

_section,

we

_generalize

the

_strip

case

_{with q}

values _{y r} of the flux

_(see

Fig.

1 b for the

notation).

4.1 FORMULATION. As before the

_analysis

sketched in section 2 we

_get

after gauge

transformations the

_magnetic

translation

_operators

_Ul

and

_U2

in the form :

with

and

As

_{for q}

= 2 the Hamiltonian can be written as :

we define :

This

_gives

in

_particular

where we have defined :

This allow us to have

_simple

matrix elements for

H :

4.2

_QUANTIZATION

AND SEMI-CLASSICAL EXPANSION. - As

_quantization

rules we use

For

_instance,

we have

Now the

_expansion

close to _yr = 0 can be

_performed

as before.

_Up

to second order

(in

yr),

the relevant matrix elements are obtained as :

Using

now Schur’s

_formula,

one obtains the

_expansion

of the effective Hamiltonian in _powers

of {y}.

It is convenient to use the

_following

notation:

where f

is an

_{integer .}

To first order in _{y r} one obtains :

This

_expression

can be reduced to a more

_simple

form :

and this leads to the _energy levels :

The next order calculations are more

_involving.

After a tedious

_algebra,

one ends _up with :

Here S is a rather

_{complicated expression, given by}

a sum of 15 different terms each

_involving

complicated

sums over the moments

_{of the {yr}}

distribution. The calculations are too

_lenghty

and tedious to be

_reproduced

here. Careful examination shows that S vanishes for every

pattern {y}.

Thus,

the net result can

_simply

be written as :

where 5 is a

_{positive quantity given by :}

The

_meaning

of 5 is _very

_{simple. y(s)}

is somehow a

_{« magnetic}

form factor _»,

_describing

the

distorsion of the

_magnetic

flux

_pattern,

and vanishes in the uniform flux case _yr = _y

(r

=

0, 1,

..., q - 1

).

The denominator in the

expression

of 5 is the

spacing

energy between

the

_ground

state

_{(s = 0)}

and the excited _{energy levels}

_{(s > 1 )}

of a discrete 1 D

_{tight-binding}

chain of

_length

_q. This form of 5 allows for the

_prediction

of its

_expression

in the

_general

case, worked out in the next section.

5. The

_général

case.

This section will be devoted to the calculation _{of various gauge} transformations necessary to

simplify

the matrix form of the

_magnetic

_translations

_Ul

and

_U2.

We choose here a Landau gauge where

_A2

= 0. Then the

potential

vector _can be written as

The

_expression

for

_Ul

and

_U2

becomes

_{(2.10) :}

We first consider the

following

gauge transformation a

given by

the q 1 q 2 x q 1 q 2 matrix :

Here

_{0 -- ri -- qi -}

1

_{( i}

=

1, 2 )

and

tîi

1 denote the lattice

position

operators.

Moreover :

This matrix is

_computed

in such a way that the new

_magnetic

translations

given by :

with

This allows us to see that every

operator

of interest can be

actually

written as a

So,

as before we will

_represent

U and in terms of

_operators

_KI

and

_K2

as follows :

Introducing

the

_{operators :}

and the

_following

non commutative _gauge transformation R’:

with

and

We

_{get :}

with :

Using

this transformation we can

_perform

the semiclassical

expansion

in much the same _way as in sections 3 and 4. The calculation is rather cumbersome and we

_postpone

it to a forth

coming

paper

[10].

Nevertheless we

_conjecture

that the Landau levels for the Hamiltonian :

are

given by

the

following

formula :

and

6. Discussions and conclusions.

6.1 INCREASE OF THE MAGNETIC ENERGY. - Let us first discuss the

physical

content of the

general

result

_{equation (5.13)}

of the

_previous

sections.

_Up

to second order in the

_magnetic

flux,

the Landau level energy

En

contains two terms. The first one is the usual result

_{[3, 4]}

for the uniform

magnetic

field case

_{with 0}

as

_magnetic

flux. This contribution is

_expected

on

general grounds.

Indeed classical

_{trajectories,}

which

_correspond

to

_energies

near the zero

field band

_{edge (E}

= _± 4

),

have

length

scales much

larger

than that of the

magnetic

unit cell.

Accordingly,

the

_quantization

is controlled

_by

the _average flux

_0,

rather than the detailed

structure cp ap

inside the

_magnetic

unit cell.

_Similarly

the

_degeneracy

of

_En

is also

_preserved

for the same reasons. The second contribution 8 to

_En

is

_independent

of the

_{integer n,}

and

corresponds

to the shift of the zero field band

_edge

_{when cP =}

0. The new term 5 is

_governed

by

the

_magnetic

« structure

_{factor » y}

as defined above and describes a

_narrowing

of the zero

field band width because 5 is

_{always positive.}

This also means that

_{inhomogeneities}

in the flux

pattern

increase the Landau level energy

En,

the increment

_{being independent}

of n. The

reason for this

_general

trend can be outlined as follows. Consider the

general

case of the

one-electron Hamiltonian :

and compare the

following

two cases : the first one

_Jeo

describes a uniform

_magnetic

_{flux .0}

and the

second,

R,

corresponds

to a non-uniform flux but

_having

the same

_averaged

_0.

The

Hamiltonian 36 can be

expanded formally by writing :

Here j

(x)

and p

(x)

are the electron current and

_density

_operators,

and

_{8 A (x)}

is the

_change

in

vector

_potential

_upon

_going

from

_Jeo

to JC. For an infinite

_system

_8A(x)

increases without

limit as a function of x. Finite

_systems

are therefore needed for

_practical

calculations with

equation (6.2).

However,

for our _{purpose, it is sufficient} to consider

_{equation (6.2)}

as a

formal

_expansion.

On a

_lattice,

Je assumes the

_following

form :

where we have used the standard notation. The local current

_density

from

_{node j}

to node i is

given by :

2 Im

(e iAij C t

_{C j»)’}

Therefore in a uniform

_magnetic

_field,

the current

_density

vanishes because

_{eiAij C t}

_Cy)

is

_nothing

else than the energy per bond and then a translation

invariant. This remark

_implies

that in

_equation

(6.2),

the linear terms in &A

_give

a

_vanishing

contribution to Je. The

only

remaining

terms are

_quadratic

in ôA and therefore

6.2 STABILIZATION OF THE FERMI SEA. - The

general expression

equation

(5.13),

for the Landau

_levels,

leads to some

_interesting

consequences

regarding

the stabilization of the Fermi

sea

_by

the gaps. This

_problem

has

_already

been discussed

_by

us in the case of a uniform field.

Here we consider the same

_question

but for a non uniform

_magnetic

flux.

_Actually

two cases seem to

_{appear : .0 96}

0

_{and 4> =}

0.

Case

0 : For non

_vanishing

y = 2

ir -Oo

the Landau levels structure allows for an

CPo

extension of our

_previous

result to the

present

case. More

_precisely

consider N =

xNs

fermions on a _{square lattice made of}

_{Nç sites,}

and let us

_{introduce n}

such that

8 =

y 271 2

is the

global

shift in

equation (5.13) :

The energy per site can then be written as :

when the m first Landau

_levels,

each

_having

a

_degeneracy

-L,

, are

completely

filled. Under

2 7r these conditions we have : m y -

27T’x,

and then

. As a consequence, m

= 1

corresponds

to the minimum

of E

as

_long

as

( 1 -

24

n 2)

:>

0,

NS

which is satisfied

for q

«

1,

i.e. small fluctuations in the flux

pattern.

This result remains true if the semiclassical

_expansion

is extended to next orders. To see

this,

we consider the case of the

_staggered

lattice with two values of the flux :

Yo and y 1. _

In terms of : the energy

En

can be written

and for m field levels :

In the

_opposite

limit

_{where e »}

_1,

m = oo is stabilized and this

_corresponds,

for fixed x, to y = 0 i.e. the

vanishing

flux limit where the discrete level structure

disappears.

Case cP =

0 : This is

_exemplified

in the case _{y = - y o} of the

staggered

square lattice. With the

_previous

notation,

this

_corresponds

to y =

0, e

= _oo, and one obtains a band

_spectrum

(the corresponding density

of states is

_given

in the

_Appendix).

For this

_particular

value of y,

equation (6.6)

reduces to :

which is

_simply

the new lower band

_edge.

At low fermion densities x, it is sufficient to

consider the

_{limiting density}

of _{states p}

_{(E ),}

close to

_Eo

= - 4 ₊

-’.- .

In this limit a

_simple

8 calculation

_{gives :}

In

_particular

for a = 1 we recover the known result in zero field.

_However,

as a function of _a,

E

increases for

’Y’ ;/=

0,

i.e. at a -- 1.

N,

This

example

shows

again

that E increases with the field

distorsions,

even in the case when the Fermi sea is stabilized at zero field.

7. Conclusion.

The content of _{this paper has been summarized in the introduction. Let} us conclude with two

remarks.

i)

In _{this paper} we have introduced a new

_algebraic

formalism which is the

_appropriate

one

for

_solving

the

_periodic

flux

_{problem. Evidently,}

the

_{enlarged algebraic}

structure so

introduced,

calls for further studies. In

_fact,

for the sake of

_simplicity,

we _{have limited our}

attention to the semiclassical

_region.

_However,

it is not difficult to consider other

_problems

where the methods and tools elaborated in our

_previous

_papers can be used.

_Similarly,

for

purely

mathematical purposes, a

_{deep investigation}

of these new

_algebra

is called for.

ii)

As a direct _consequence of our

_preliminary

_results,

we were able to _{extend the range of}

validity

of the mechanism of Fermi sea stabilization. This sheds a new

_light

on this

concept,

which we believe is a result of _very

_{general importance.}

_Furthermore,

its extension to other

physical

problems

is

_certainly

a valuable task.

Appendix.

Staggered

flux lattice.

In the

_special

case of a

_staggered

lattice

_{(Fig. 1 b), with y}

₌

_y’,

_{Y2 Y,}

the band structure can be

_explicitly

calculated. The

_eigenvalues

reduce to two

_coupled

sets of linear

Here we have

_adopted

an alternate Landau _gauge in order to have a

_{simple expression}

for the

coupling

between the two sublattices A and B. Periodic solutions

lead to a 2 x 2 determinant. The solution of the secular

_equation

is

where

_{y’ =}

0

_{(i.e. a}

=

1)

and

y’ =

7r

(i.e. a

=

0) reproduce

two known

limiting

_cases. The

density

of states p

(E)

corresponding

to the above

_dispersion

relation is

_{given by}

where the sum is taken over the first Brillouin zone.

_Making

the

_change

of variables

u =

kx

₊

k y,

v

_{= kx -}

_{k y,}

one obtains :

The calculation of p

(E)

in the

général

case is rather cumbersome. Let us first consider the case

_{kx -}

_0,

_{ky -}

_0,

i.e. close to the lower band

_{edge -}

4

1 + a 1/2 .

(

in this limit one has :

2

)

This allows for a

_simple

calculation of p

(E)

to first order in

p

(E)

assumes the

_following

form :

In the

_general

case, p

( E )

= _p

( - E )

_can be calculated

explicitly starting

from

equation (A4).

and

_finally,

Here are the usual

_elliptic

integrals.

One notices the

_{following properties}

of p

(E)

for a

1 :

i)

p

(E)

= 0 at E =

0 ;

close to E =

0,

_p

(E)

has a _cusp and takes the form :

which becomes

_singular

at a «5 1.

ii)

There is a

_{logarithmic divergence}

of

_{p (E )}

at E = _±

2 ( 1 -

a 2) 12.

iii)

For E =

[8(1 - a)

1/2,

_p

(E)

has _a

jump, corresponding

to the

complete filling

of one of

the two Fermi

_{« pockets}

»,

present

at a =A 1. Indeed the Fermi surface is made of two disconnected

_pieces,

and this as

long

as a:o 1.

As a final remark notice that for a =

1,

x, = - oo and one recovers the zero field result :

Similarly,

for a =

0,

one obtains the known

expression :

References

[1]

BELLISSARD _J.,

_{Operators algebras}

and

_{applications,}

vol. _II, Eds. E. V. Evans and M. Takesaki

(Cambridge University

Press,

Cambridge)

1988.

[2]

BELLISSARD J., The Harald Bohr

Centenary,

Eds. C. _Berg and F.

_{Fluegege (Royal}

Danish Acad.

Science,

Copenhagen)

1989.

[3]

RAMMAL R. and BELLISSARD J., J.

_Phys.

France 51

₍₁₉₉₀₎

n° 17.

[4]

RAMMAL R. and BELLISSARD J., J.

_Phys.

France 51 ₍₁₉₉₀₎ n° 17.

[5]

BELLISSARD J., Bad Schandau conference on _{localization,} Eds. W. Weller and P. Ziesche, Teubner,

Leipzig (1988).

[6]

DUBROVIN B. A. and Novikov S. P., Sov.

_Phys.

JETP 52

₍₁₉₈₀₎

511 ;

DUBROVIN B. A. and NOVIKOV S. P., Sov. Math. Dokl. 22

₍₁₉₈₀₎

240 ;

NOVIKOV S. P., Sov. Math. Dokl. 23

₍₁₉₈₁₎

298.

[7]

HALDANE F. D. M. and REZAYI E. H.,

Phys.

Rev. B 31

₍₁₉₈₅₎

2529.

[8]

AROVAS D. _P., BHATT R. N., HALDANE F. D. _M., LITTLEWOOD P. B. and RAMMAL R.,

Phys.

Rev. Lett. 60

₍₁₉₈₈₎

619 and

_unpublished

_report.

[9]

ZAK _J.,

_Phys.

Rev. A 134

₍₁₉₆₄₎

1602 and 1607.