Ground state of the Fermi gas on 2D lattices with a magnetic field

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Submitted on 1 Jan 1990

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Ground state of the Fermi gas on 2D lattices with a

magnetic field

R. Rammal, J. Bellissard

To cite this version:

Ground

state

_{of the Fermi gas}

on

2D lattices with

a

magnetic

field

R. Rammal

₍₁₎

and J. Bellissard

₍₂₎

(1)

Centre de Recherches sur les Très Basses

Températures,

CNRS, BP 166 X, 38042 Grenoble

Cedex 05, France

(2)

Centre de

Physique Théorique,

CNRS,

Luminy,

Case 907, 13288 Marseille Cedex 09, France

(Received

on

_February

_{20, 1990,}

_accepted

in

final form

on _{May 29,}

₁₉₉₀₎

Résume. 2014 On 6tudie

l’énergie

des électrons de Bloch sous

_{champ magnétique}

en fonction du

flux

_magnétique

_03A6 et du taux de

_remplissage

03BD. On calcule

_l’énergie

totale

_{E(03A6, 03BD)}

à l’aide d’une nouvelle méthode de

_{quantification semi-classique.}

Le minimum absolu est atteint pour un choix

optimal

d’un _quantum de flux _par

_particule.

Ce

_phenomene

se

_produit

_pour différents réseaux et

sous des conditions assez

_générales.

Pour 03BD

_{fixé, l’énergie}

est une

_ligne

brisée en fonction de 03A6. Toutefois,

l’énergie

du fondamental est une fonction

_régulière

de 03BD. Les

implications

de ces

résultats pour la stabilisation des

Anyons

et les

phases

de flux sont discutées.

Abstract. - The

energy of 2D Bloch electrons in a

_magnetic

field is studied as a function of the

filling

fraction 03BD and the

_magnetic

flux 03A6.

Using

a new semi-classical

_quantization

method the total

energy

E(03A6, 03BD)

is calculated and shown to have an absolute minimum which

_corresponds

to one

flux _quantum per

particle.

This

optimal

flux

phenomenon

is shown to occur under

large

conditions and for different lattices. An

_{explicit cusp-like}

behavior of E 03BDs. 03A6 at fixed 03BD is found both for the absolute and for the relative minima of E. Furthermore, the

_ground

state energy is

shown to be a smooth function of 03BD. The

_implications

of our results for the stabilization of

_Anyons

and the flux states are discussed. Classification

Physics

Abstracts 03.65 - 05.30F - 71.25

1. Introduction.

Very

recently,

the search for a

_possible

failure of the Fermi

_{liquid theory}

in

_high

Tc

superconductors

was at the

_origin

of a new

_piece

of

_physics

for the Landau levels of Bloch electrons

_[1].

One of the consequences of the convoluted structure of the energy

spectrum

is the

possible

stabilization of the Fermi sea

_by

the gaps which appear at _every rational

_magnetic

flux

_0.

_Indeed,

Anderson

_{[1, 2]}

was the first to realize the

_possibility

of an

_optimal

flux

corresponding

to one flux

_quantum

_per

_particle.

More

_precisely,

for a

_{given filling}

fraction v of fermions in a 2D

_lattice,

the actual

_ground

state is achieved

_{for 0}

= _v. It turns out that this relation is at the heart of the

_{Anyon superconductivity}

_[3],

where it is called the

« fundamental relation ».

_Since,

this

_conjecture

has received a rather _strong numerical

support

from various authors and for different lattice structures

_{[4, 5].}

_Furthermore,

the

_plot

of the total energy at fixed v exhibits

_clearly

a

_cusp-like

structure with an absolute minimum at

41

= v

[5].

This numerical _support has been facilitated

_by

the _present

_knowledge

of the

energy

spectrum.

Indeed,

the

_spectrum

of

_{tight-binding}

electrons on a 2D lattice in a

_magnetic

field has an

_extremely

rich structure. The

_generic

features of the

_{single particle}

_spectrum

(self-similarity, nesting

properties,

gap-labelling, etc.)

are

_by

now well studied

_{[6, 7]}

for various 2D lattices.

Actually,

the

_complete

absence of

_analytical

or

_rigorous

results for this _very

_important

conjecture

motivated the work

_presented

in this _paper. A _summary of our results has

_already

been announced elsewhere

_[8].

A detailed

_exposition

_{is the main purpose of this paper.} The method we used is a semi-classical

_approach

for Bloch electrons in a

_magnetic

field. A recent

powerful

formulation of this

_problem

is the

_object

of reference

_[9].

Within the same

_algebraic

framework,

we

investigated

the zero field limit as well the case of an

arbitrary

rational flux.

The

_present

paper is a direct illustration of the results obtained in reference

_[9].

The paper is

organized

as follows. In section

2,

we work out the case of zero

_magnetic

_flux,

where the

_property

of

_optimal

flux

_{at 41}

= v is

explicitely

obtained. Section 3 is devoted to the

case of an

arbitrary

rational flux

_41.

The detailed structure of the energy

E ( 41, v )

as a function

of 41

and v is worked out in section 4 : cusps,

symmetry,

analyticity

of the

_ground

state energy,

etc.

_Concluding

remarks are the

_object

of section

5,

where the

_possible

extensions of our

results are

_briefly

indicated.

2. Fermi sea _{energy : weak field} limit.

In this

_section,

we consider the Fermi sea _energy of N

_spinless

fermions on a 2D lattice

(Ns

sites,

Np

plaquettes).

At _very weak

_magnetic

field,

the lowest Landau levels are

given by

the standard

_{expression :}

E,, = e 0 + 8 H n + 1

to first order in the

_magnetic

flux. Here 2

eo denotes the lower

edge

of the zero field band

_(e.

a. _Eo = - 4 t for a _square

_lattice)

and

_/3

is

defined via

_f3H

= 2 _{t y, y} = 2 77-

0 .

Each level admits a

_degeneracy

_D,

which is

_equal

to the

Oo

ratio of the total flux

(magnetic

field x

_system

area)

to the flux

_{quantum 41 o.}

It turns out that for

_{vNs -}

N

_fermions,

the

appropriate

variable for the

energetics

is

where v is the

lattice

_filling

factor

and 0

is the reduced flux

_through

each of the

Np

plaquettes (.0

o = 1 is the

sequel).

The ratio

N pl N

is a

_geometrical

factor

_equal

to 1

_(resp.

2 and

_1/2)

for a _square

_{(resp. triangular}

and

_honeycomb)

lattice.

_Omitting

the

_{origin of energy}

eo, the total

energy E

is

given by

where

_{o =}

8 HX

Lis the chemical

_potential

at zero field.

As a function

_{of X,}

E exhibits oscillations

(Fig. 1)

and assumes the same value 2

_{E / N (0}

= 1 at X =

1, 2,

3, ...

Each of these

degenerate

states

corresponds

to the

complete

filling

of

_only

the first X =

1, 2, 3, ...

Landau levels. Close to these

degenerate

minima, E

has

a down

cusp-like

behavior. The cusps are

_symmetric

and the oscillation of the

corresponding

Fig.

1. -

Cusp-like

structure of the Fermi sea

_{energy E}

of Bloch electrons, with

_(a)

and without

(b)

the

underlying

lattice

_potential.

Here v is the lattice

_filling

_{fraction, 0 / 0 ()}

the reduced flux per

plaquette

and

_Co

is the chemical

_potential

at zero

_magnetic

field.

The infinite

_degeneracy

of the

_ground

state _energy E is lifted if the

_{crystal potential}

is considered. To see this consider the case of the

_simple

square

lattice,

where to the second order in _{y :}

The same calculation can now be

_repeated,

with the final result

_{(n - X - n + 1 ) :}

In

_particular,

for X = n filled levels :

This

_implies

that X = 1 is selected as the absolute minimum of

E,

the other minima are

reached at X =

2, 3,

... etc. Therefore the infinite

degeneracy

is lifted

by

the

crystal potential.

Furthermore the

_{symmetric cusp-like}

structure is

_preserved

at X = n :

The above result thus

_provides

a

_{perturbative proof}

of the

_{conjecture (k}

= v.

However,

two

questions

are raised

_by

this

_{finding :}

how

general

is the behavior of E found here for the square lattice ? Is the result thus obtained stable when further terms in

_equation

_(2.2)

are

In order to answer the first

_question,

we consider the

_general

case worked out in reference

[9],

and

_{corresponding}

to :

Here

_{£ (k )}

is the zero field

_dispersion

relation,

a -

_{a/ak1 - ia/ak2, L1}

=

a2/akf+ a2/aki.

Using

an obvious

_notation,

we

_get

as before a

_{simple expression}

for E. In

_particular

for x= n:

where The

previous

conclusion holds also in this

_general

case. In

_particular,

for the cases where

E (k) = e (- k)

holds

_(i.e.

lattice with inversion center

_symmetry),

the last term in

_equation

(2.4)

vanishes and one is reduced to the

_previous

situation of the square

lattice,

the

_only

difference

_being

the value of

_(>0)

and

_{A2E (0).}

Therefore,

for usual lattice structures

(square, triangular, honeycomb, generalized

square,

etc),

the absolute minimum of the

Ns

ground

state _energy is reached for X =

1,

i.e.

for cp =

N S

1). It is

_important

to notice that this

Np

conclusion is true

_{providing 41}

0. For

_{inversion-symmetric}

_lattices,

this appears as the

_only

condition for the

_validity

of the obtained result. The

_geometrical

factor

_TVg/Ap

assumes the

values

1,

1/2

and 2 for square,

triangular

and

_honeycomb

lattice

_{respectively,}

and then the numerical

_finding

of reference

_[4]

is recovered.

In order to answer the second

_question,

we have considered the effect of the next order

perturbation

calculation on

_En.

In the case of the

simple

square

lattice,

we have

[9]

valid to third order in y,

where à = 64 t,

à’ = 24 x 64 t 2and

8 H = 2 t Y.

A

_simple

calculation leads to :

This shows that the absolute minimum is realized at n = 1.

Furthermore,

the

cusp-like

behavior is

_preserved.

However,

at this order in y, the

symmetry

of the _cusp is broken.

Indeed,

at X = n the

_slopes

are

_{given by :}

Therefore,

in addition to the

_asymmetry

of the cusp, an

_{explicit dependence}

on n takes

_place

in

_{equation (2.7).}

This

_general

behavior of an

_asymmetric

_cusp is

_expected

to be a

_generic

sufficient condition for a

_symmetric

_cusp of the total energy of N fermions at H =

Hn

corresponding

to the

_filling

of the first n

_levels,

is

This

condition,

which is satisfied to second order in y, is violated at the next

order,

as can be seen in

equation (2.6).

Example

1 : Consider first the case of a

_filling

factor v on the

_simple

_square lattice. To second order in y, one has

If the next order corrections

_{0 ( y 3)}

to Landau levels are taken into account, one obtains :

Clearly,

the minimum of E is

_always

reached at X = 1. It is

interesting

_to notice that the free fermions limit is obtained from

_{equation (2.9) by taking X}

= _{oo .}

Furthermore,

we mention

the very

high

accuracy of our result

_{(Eq. (2.9))}

in

_comparison

with available numerical data

[5].

Example

2: Let us mention the case of a

_generalized

_square lattice with

_long

_range

hopping :

first

_(tl),

second

_(t2)

and third

_(t3)

n.n. interactions. For this

_model,

the Landau

levels are

_{given by}

For _{the total energy, and} a

_filling

factor

This result

_generalizes

the

_previous

one

_{(Eq. (2.10))}

and can be used to check numerical results.

3. Fermi sea _{energy :}

_arbitrary

flux.

The

_previous

result can

_actually

be

_generalized

to an

_arbitrary

flux.

Starting

from the weak field limit y -

0,

a

_{periodic potential splits}

the Landau levels into subbands and this results into

_broadening

and a _very convoluted rich

_spectrum

_[7].

The situation becomes

_simpler

at

rational

_{flux 0}

_{= p}

_/q,

_{where q}

subbands appear in the

spectrum.

Each subband has the same

spectral weight

1 lq.

For

_instance,

if the

_filling

factor is v =

1 lq,

then the lowest subband can

be

_completely

filled. Unless accidental

_degeneracy,

_{the q}

subbands are

separated by

and s are relative

_{integers labelling}

the different gaps in the spectrum

(gap labelling

theorem

[10]).

The

_{couple r}

=

1, s = 0

corresponds

to the so-called main

gap, where the IDS is

exactly

equal

to

_0.

The stabilization of the Fermi sea at the

_{flux 0}

= v is

actually

a _consequence of the

_following

_argument. Let us start with

simple

Landau levels _et, each

_having

the

_degeneracy

D. For a

_system

of N

_fermions,

if the flux assumes the

_{value p n}

such that N =

nD,

then the

ground

state of the

_system

will have n lower levels

_completely

filled and all

_higher

ones

_empty.

For our _purpose, it is

_important

to realize

_{that, when 0}

=

p n’

the

ground

_state is

isolated,

separated

from the lowest excited states

_by

a finite _gap. At

_{0 = qb n,}

when the

_highest

occupied single particle

level is not

_{completely occupied,}

the

_ground

state is

_degenerate.

Assume

_{now 0}

=

0 n

for some _n, and turns on the interaction with the

_{periodic potential}

of the lattice. We claim

that,

to _{any order in}

_{perturbation theory,}

the

ground

state remains

non-degenerate.

The

_{non-degeneracy}

of the

_ground

state cannot be altered

_by

the

interactions,

even if the radius of convergence of the

perturbation theory

is zero.

_Having

established that the

_ground

state is realized at n = 1 for weak

flux,

this result remains valid at on

_arbitrary

finite value of the flux.

Our

_argument

is

_mainly

based on the

_continuity

as a function of the

_strength

of the interaction. The situation is somewhat similar to the

_stability

of the shell structure of atoms

and nuclei. Another

_{important ingredient}

of the

_argument

is the

special

feature of 2D Landau levels in the sense that their

_degeneracy

is a

_topological

data : the total flux

_through

the

lattice. A

_{possible improvement}

of our

_argument,

_{using homotopy}

_concepts,

is under

consideration.

4.

_Analyticity

and

_cusp-like

structure.

As outlined in reference

_[9],

the semi-classical

_approach

allows for the calculation of the Landau level structure near an

arbitrary

rational flux

p /q.

The

expansion

_parameter

is

0 -plq

and

_explicit

results have been obtained. In this

section,

we discuss some

_general

properties

of the Fermi sea _energy

E(o, v )

as a function of the

flux 0

and the

filling

fraction v.

4.1 ANALYTICITY. - The

_following

statement can be

_{proved rigorously}

_[11].

If

Eo ( i, ) =-= E (-0 = v, v )

is the

_ground

state _energy, then

_Eo

is a smooth function of v

except

possibly

at v = 0 and at _every rational value v _{= p}

/q

for which the _gap at the Fermi level closes.

By

smooth we mean differentiable at _every order. This result does not exclude the

possibility

that

_{EO(t,) be analytic}

with

_respect

to v. The

proof

of the

analyticity

_property,

without

_being

difficult,

requires

however an elaborate

_machinary

of non-commutative

geometry.

A detailed account will be

_published

in a

_separate

_paper

_[11].

Let us

_give

a hint on

the

_proof.

The fundamental idea is the

_analytic

form of the Hamiltonian when viewed as an

element of the rotation

_{algebra A (a ),}

where

_{a =- 0}

_{/ 0 0}

in the notation of the

present

paper. As recalled in reference

_[9],

any element of

.ae ( a)

can be

_expanded

as a Fourier series :

In the case of the Hamiltonian

H, a -

am ( a )

is a continuous

_application

_{and am}

_corresponds

to

_{a_ m ( a ).}

_Furthermore,

_{-r (a)}

=

ao ( a )

where r denotes the trace _per unit volume. Take for instance the case of the

_Harper

problem :

H = U + v + U* + V *.

In this _{case, any}

continuous function

_{f (H)}

of H admits Fourier coefficients which are continuous in a.

energy

EF-

If

Ep

belongs

to a _gap, then

_PF

can be written

_(Cauchy

_formula)

as

But z - H is

_analytic

in a. Therefore the Fourier coefficients of

PF

are

_analytic

and the Fermi sea _energy

_ï(PF7:f)

is also

analytic.

4.2 EXISTENCE OF THE CUSPS

_(Figs

1 and

2).

- In

general,

the

_following

_proposition

holds.

For a rational

_{value p /q}

_{of 11,}

the total energy

E (0, v )

exhibits a _cusp

_(as

a function of

_¢)

in the

_vicinity

of a dense

_set

= mq p

of flux values

₍₀

_{- 4>}

- 1

_),

m and n

_being

the gaps

nq

labels.

Furthermore, for

= v the

_slope

difference

(aE+ /ao

(aE- /ao )

is

given by

the

width of the gap labelled

by m

=

0, n

= - l. Here we

_adopted

the notation m -

_no

for the IDS _{with the gap}

_{(m, n ).}

This

proposition

can be

_{proved rigorously [11].}

The

following

argument,

valid at

rb

= _v, illustrates the heart of the

proof (the

extension _to other

cusps is

immediate).

The local behavior of

_{E(0, i, )}

_{at 0}

= v is controlled

by

the first Landau level at the subband

_edge

we

consider.

_{For c/J}

v, the IDS is

bigger

than v above the Ist Landau

level,

whereas it is smaller

below. In order that the IDS be

_equal

to v

_{(at 0}

a

_v),

the first Landau level at the

top

e+ of the last filled subband

_(«

valence subband

_»)

must be

_partially

filled.

_Similarly,

at

0 -5

v, the same

_argument

_implies

that the first Landau level at the bottom e- > e + of the

next subband

_(«

conduction subband

_»)

is

_partially

filled. So the local behaviour of

E (0, v )

is controlled

_by

the formula

_giving

the levels as a function of

c/J -l!...

In

_particular,

q

the

_slopes

_{aE::!:- /04>}

_{at 0}

= v ± 0 are

_{governed by}

the

_edge

of the

_{corresponding partially}

filled subbands. This

_argument

holds for _{every cusp due} to

_{a jump}

of the Fermi level

_{(Fig. 1).}

The calculation of the

_slopes

can be

_performed

as follows. Let us first assume that

rb

= v +

_8,

where v is a fixed fraction and

_f3

« v.

_/3

= 0

corresponds

_to a Fermi level located

Fig.

2. - Picture of the Fermi level

jump (a)

at a = v _{between the filled subbands and the empty} ones.

at e’

_{(0 ),}

i.e. the

_top

of the last filled subband.

_{For Q ±}

_0,

the Fermi level moves to the

_right

and the total _energy becomes :

The first term in

_{equation (4. 1) corresponds}

to the energy if the « valence » subbands are

completely

filled

_{at 0 = v}

+

_8.

The second contribution describes the deficit

(- 6 )

in energy due to the

_partial

_filling

of the first Landau level at the

right

of e+ . The first order terms in

₆

have been retained and then

The

_{physical meaning}

of the coefficient a is that of the

_{magnetization (to}

the

_right)

of the

« valence » subbands

_{at 0 == v}

_{(see below).}

_Therefore,

Similarly,

the behaviour of

_{E (0, v )}

_{at 8}

s 0 is

_{given by :}

and then

This

_implies

in

_particular

that

_{E(.O, v )}

admits a _cusp

_{at 4> ==}

v . The difference between the

two

_slopes

is

_{given by :}

_{aE’laO -}

_{aE-la-0 =}

?_ - e, which is the energy gap between the

« valence » and « conduction » subbands. This result is

_actually

a

_particular

case of a more

general

formula. For a

_{given v,}

_{assume 0}

chosen such that the n lowest subbands are filled

(e.a.

local minima of

_E).

_{Therefore, 0 = zlm}

and :

(A

further

_{generalization}

of

_{Eq. (4.7)}

will be found

_below).

More

_generally,

if the

_ground

state is defined

_{by 0 = t,}

=

p

/q,

then the other

singularities

of

_{E(o, i, )}

are located

_{at cP}

_{given by}

_{p /q}

= m -

_no.

Here m and n are the gap labels.

Therefore,

the cusps also occur at :

The constant

_{0 «-- q5 --}

1 restricts the allowed values of m to :

0 - m - n + E.

for q

n ::> 0 and

_{p lq :> m :::. n}

+ p

/q

for n « 0. This leaves us with a dense set of flux

_values,

where E admits a _cusp behavior.

At this

_point

two

_remarks

are of order.

First,

the

generic

_{cusp is}

asymmetric :

conclusion.

Secondly,

our results have been obtained for a fixed number N of fermions and a

fixed unit cell area. This has to be

compared

with the

_pertinent

discussion of reference

_[12]

relative to the

_analyticity

of the _energy under different conditions and for different

_problems.

4.3 EXPLICIT FORMULA FOR THE CUSPS. - The

following

formula is a

_{generalization}

of

equation (4.7)

(see

proof below)

- -1 , .

where

N(E, 0 )

refers to the IDS at flux

_41.

An immediate consequence of

equation (4.8)

is the

_{following :}

However,

for e’ :5 e :5 e -, i.e. within the gap, the gap

labelling

theorem

_implies

that

2013 =

_{integer independent}

_{of 0}

or E.

So,

a 4>

The

_result,

_{equation (4.8)}

leads to an

_explicit

formula for the

_slopes,

and can be used for

practical

purposes. The

proof

is a

_simple

matter of calculation. For

_this,

_starting

from

equation (4.1) giving

the definition of a :

a

_{partial integration gives :}

and then

The same calculation holds for aE-

_/a4J

and this

_{implies equation (4.8).}

It is

_interesting

to observe that

_{equation (4.10)}

is somewhat the « dual » of the formula

[ 13]

giving

the

_{compressibility}

of

_interacting

fermions,

restricted to the lowest Landau level :

The

only

difference is

that, v

in

_{equation (4.11)}

refers to the

_filling

the lowest Landau

level,

whereas v in

_equation

_(4.10)

is the lattice

_filling

factor.

Remark. The

_{analysis leading}

to the cusp structure can

_actually

be

_repeated

in the cases where the gap closes. A

typical example

is

provided

by

the

_simple

square lattice

at q5 =

1/2.

In this

case the _gap closure occurs at e =

0,

and the

(relativistic)

Landau levels are :

_En

= ±

₍₈

n y )1/2

Thus,

there is no more _cusp and this is consistent with the absence of the _gap.

4.4 MAGNETIZATION OF THE FILLED SUBBANDS. - In the calculation of the

_slopes,

a new

quantity appeared :

a. It turns out that a can be

_{expressed simply, using}

the differential calculus outlined in the

_Appendix.

For the sake of

_simplicity,

we consider the case where the

Hamiltonian does not

_{depend explicitly}

on the

_{flux a = 0}

_/0()

_(e.a.

_Harper

_problem).

In this

case aH = 0. We are interested

_{by a m JÙ}

= ar

(PH) - a

_{T (PH)}

where P is the

eigen-a«

projector

on

_energies

below the Fermi level and T is the trace _per unit volum.

_Applying

equation (A.1 )

of the

_Appendix

with A = P and B =

H,

one obtains

The first term can be calculated as shown in the

_appendix.

_Indeed,

one has

and

This

_{implies :}

Further

_{simplifications}

are obtained from :

and this leads to the final

expression

This

_{original expression}

for A can be

_interpreted

as the difference

between the «

_{magnetization »}

M- of the filled subbands

_(fermions)

and that M’ of holes

occupying

the

_empty

subbands :

with

5. Conclusion.

Let us first summarize our main results. In this _paper, we have

_presented

a detailed

_study

of

behavior of

_{E(o, v )}

at fîxed v has been shown to take

_place.

In

_particular,

a

_cusp-like

structure is exhibited. We have calculated the associated

_slopes

_{at 0 == 1,}

as well as the

relative minima of

_{E(l/J, v).}

In

_general

_{the cusps} occur on a dense set of values of

_0.

The

procedure

we used is based on semi-classical

ideas,

which we believe are the

appropriate

tools

to handle the

_singular

perturbation

problem

at hand : selection of the lowest Landau level at

weak field in the

_{infinitely degenerate}

case of free fermions.

The

_{compressibility}

_property

of the Fermi gas in the

ground

state

_(i.e.

_at

_v)

results from

_general

considerations

_leading

to the

analyticity of E(O = v, v )

as a function of v.

Finally,

the

_asymmetry

of the _cusp at absolute or local minima of

_{E (0, v )}

_{vs. 0}

is due to the finite

_{magnetization}

of Landau subbands at rational flux. An

_explicit

and

_{original expression}

for this

_quantity

has been derived here.

We _{conclude this paper with three remarks :}

i)

the

_{possible analogy}

between the stabilization

_{at v}

_by

_{the gaps and Peierls}

instability

has been noticed in reference

_[14].

In both cases, one is reduced to a 1 D

_problem

with a one

particle

_spectrum

having

an infinite number of _gaps. The interactions between

fermions mediated

_by

the lattice are at the

_origin

of the stabilization

phenomenon

in both

cases. Numerical

_support

[15]

for this

picture

in the 1 D Peierls

problem

is well known.

However,

we believe that the Fermi _gas

problem

in a

_magnetic

field is more

_transparent

because of the absence of mean-field

_{approximation}

for instance.

_Furthermore,

beside a _very

simplified

model

_[16],

there is no

_{general proof for}

the existence of Peierls

_instability.

For this

reason, we believe that some

_deep

relation should be found between these two

_{problems ;}

ii)

a more

_{rigorous justification}

of our

_argument

in favor

_{of 0 = 1,}

as absolute

_minimum,

is

called for. In some sense the situation is reminiscent of the

_proof

of the

_{labelling-gap}

theorem

[10].

Therefore,

we believe that

_homotopy

considerations are

_probably

behind this

_general

result ;

iii) through

this

_paper

we have considered

only

the case of a uniform

_magnetic

field. A natural

question

arises : what

happens

if the flux distribution is not so

_{regular ?}

Partial answers in the case of a

_periodic

flux are now available and will be

_published

elsewhere.

Appendix.

In reference

[17]

a new

_type

of differential calculus has been used. The purpose of this

appendix

is to recall the derivation

rules,

for the elements of the

_family

_{{A ( a )}}

of rotation

algebras.

This allows us to

_{manipulate algebraic quantities}

as a function

of a ( == l/J / l/J 0 in

the

notation of the

_present

_paper).

Two

_principal

rules are of

importance

for us :

(1)

For A and B in the universal rotation

_{algebra :}

where a =

v /va

and

ai£ =

alak,,.

The last term in

_{equation (Al)}

stems from the

non-cummutative

_aspect

_{of A(a).}

(2)

If

A -

1

exists,

then

the Wilkinson-Rammal

_formula,

etc.

_Here,

we illustrate

(Al)

in the case : A = B =

P,

where P is an

_{eigenprojector. Using}

p 2

=

P,

, one deduces from

(Al) :

Multiplying by

P and

taking

the trace _per unit volume leads to :

where C -

_{[dlP, d2P]}

is the commutator of

_dlP

and

_d2P.

A similar result holds for

Then,

using T (C) =

0,

one deduces :

Stated

otherwise,

where Ch

_{(P )}

is the Chern class of the

_projection

P.

_{Equation (A4)}

is the so-called Streda formula

_[13].

References

[1]

ANDERSON P. _W., Kathmandu Summer School Lectures

_{(1989) unpublished}

notes.

[2]

ANDERSON P. W.,

Phys.

Scr. T 27

₍₁₉₈₉₎

660 ;

WIEGMANN P. B., Ibid. p. 160 ;

LEDERER _P., POILBLANC D. and RICE T. _M.,

_Phys.

Rev. Lett. 63

₍₁₉₈₉₎

1519.

[3]

CHEN Y.-H., WILCZEK F., WITTEN E. and HALPERIN B. I., Int. J. Mod.

_Phys.

B 3

₍₁₉₈₉₎

1001 ;

CANRIGHT G. S., GIRVIN S. M. and BRASS A.,

Phys.

Rev. Lett. 63

₍₁₉₈₉₎

2291 and 2295.

[4]

HASEGAWA Y., LEDERER P., RICE T. M. and WIEGMANN P. B.,

Phys.

Rev. Lett. 63

₍₁₉₈₉₎

907 ;

NORI F. et _al.,

_{preprint (1989) ;}

LANTWIN C.J.,

preprint (1989) ;

HASEGAWA H. _{et al.,}

_{preprint (1989).}

[5]

MONTAMBAUX G.,

Phys.

Rev. Lett. 63

₍₁₉₈₉₎

1657.

[6]

AZBEL M. Ya., Zh.

_Eksp.

Teor. Fiz. 46

₍₁₉₆₄₎

929

_{(Sov. Phys.}

JETP 19

_{(1964) 634) ;}

HOFSTADTER D.,

Phys.

Rev. B 14

₍₁₉₇₆₎

2239 ;

CLARO F. H. and WANNIER _G.,

_Phys.

Rev. B 19

₍₁₉₇₉₎

6008 ;

RAMMAL R., J.

_Phys.

France 46

₍₁₉₈₅₎

1345 ;

CLARO F. H.,

Phys.

Status Solidi B 104

₍₁₉₈₁₎

K31.

[7]

GUILLEMENT J. P., HELFFER B. and TRETON P., J.

_Phys.

France 50

₍₁₉₈₉₎

2019 and references therein.

[8]

RAMMAL R. and BELLISSARD J., Eur.

_Phys.

Lett.

₍₁₉₉₀₎

to be

_published.

[10]

For a

_{general exposition,}

see BELLISSARD J., LIMA R. and TESTARD _D., in « Mathematics +

Physics,

Lecture on recent results », Vol. 1, Ed. L. Streit

(World

Sc. Pub.,

Singapore,

Philadelphia)

1985, p. 1-64.

[11]

BELLISSARD J. and RAMMAL R., in

_preparation.

[12]

TESANOVI0106 Z., AXEL F. and HALPERIN B. I.,

Phys.

Rev. B 39

₍₁₉₈₉₎

8525.

[13]

See for instance Eds. R. E.

_Prange

and S. M. Girvin, The _Quantum Hall Effect

_{(Springer-Verlag,}

New

_York)

1987.

[14]

KOHMOTO M. and HATSUGAI Y.,

preprint (1989).

[15]

MACHIDA K. and NAKANO M.,

Phys.

Rev. B 34

₍₁₉₈₆₎

5073.

[16]

KENNEDY T. and LIEB E. H.,

Phys.

Rev. Lett. 59

₍₁₉₈₇₎

1309. For the case of

finite-gap

spectra,

see BELOKOLOS E. _D., Theor. Mat. Fiz. 45

₍₁₉₈₀₎

268 and 48

₍₁₉₈₂₎

604.