Semiclassical methods in solid state physics : two examples

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Semiclassical methods in solid state physics : two examples

Jean Bellissard, Armelle Barelli

To cite this version:

Jean Bellissard, Armelle Barelli. Semiclassical methods in solid state physics : two examples. Journal

de Physique I, EDP Sciences, 1993, 3 (2), pp.471-499. �10.1051/jp1:1993145�. �jpa-00246736�

Classification

Physics

Abs _tracts

75.10 71.25.C 03.65F

Semiclassical methods in solid _state physics

_:

two examples

Jean Bellissard and Armelle Barelli

Laboratoire de

Physique Quantique

Universit6 Paul Sabatier 118, route de Narbonne F-31062 Toulouse

Cedex,

France

(Received

22

July

1992,

accepted

in final form 9

August 1992)

Rdsumd. Ce travail est _{une revue} de deux

problbmes

motiv6s _par l'6tude des modbles bidi- mensionnels _pour la

supraconductivitd

I haute

temp6rature

critique. La

premibre partie

_con-

cerne l'6tude du spectre

d'6nergie

_pour des dlectrons de Bloch bidimensionnels soumis £ _un

champ magndtique

uniforme. Une analyse

semi-classique

permet d'en

comprendre

les propr16t6s qualitatives et quantitatives. La deuxibme partie est _un plaidoyer _pour l'utilisation des m6thodes du "Chaos

Quantique"

dans l'6tude des systbmes de fermions fortement corr616s. La distribution des dcarts de niveaux d'un modble t J _en deux dimensions, _en fournit

une illustration.

Abstract. We present ^here a review of two problems motivated

by

2D models for

high

Tc

superconductivity.

The first part _concerns the energy spectrum ^of 2D Bloch electrons in _a

uniform

magnetic

field. A semiclassical

analysis

provides a qualitative as well _as

a

quantitative understanding

of this spectrum. In the second part we make the _case for the

application

of

"Quantum Chaos" to

strongly

correlated fermion systems. ^It is illustrated by the level

spacing

distribution for the t J model in two dimensions.

1met R. Rammal for the first time at the Paris

meeting

on Statistical Mechanics in

January

1983. Pierre Moussa introduced _us to each other. Some

similarity

between Rammal's work _on the

Sierpinski

lattice

[75]

^and some one dimensional almost

periodic

models studied

by Bessis,

Moussa and

myself _[15]

_was the _reason

why

he wanted to make _my

acquaintance.

During

the short discussion _we had

together

at this

meeting,

it became

immediately

clear that _we had two

points

in _{common : we} had both

got

our

degree

in _a French

university,

and both worked hard _to understand the structure of the Hofstadter

spectrum [55].

On the first

point,

_we shared the

hardship

of

being

outsiders in the French research institu- tions

heavily

dominated

by

the smart but

arrogant

"Grandes Ecoles" alumni.

On the

second,

_{we were} both fascinated

by

the

complexity

and the

importance

of

magnetic

field effects in Solid State

Physics.

Bloch electrons in _a

magnetic field, superconductor

or normal metal

networks,

the

quantum

Hall

effect,

electronic

properties

of

quasicrystals, high Tc superconductors

and

quantum

chaos

472 JOURNAL DE

PHYSIQUE

I N°2

have been the focus of _our

frequent

and

continuing

dhcussions _over the _years since then. Not to mention the

subjects

of

daily life, professional matters, politics

^and

private problems

_as well.

Scientific

ambition,

_a

_strong

need for affection and _a wounded

pride secretly

underlied his life. Rammal died _{on a}

Friday

in the middle of _a scientific discussion after _years of heartbreak for his

country falling apart,

^and _years ^of acute

physical pain, despite

the

hope

that medical

technology

_gave him for the better.

Jean Bellksard

Toulouse, July

1992

1. BlocI~ electrons in _a

magnetic

field.

Electronic motion in _a

crystal subject

to _a

magnetic

field has been _one of the main

topics

in Solid State

Physics.

A

magnetic

field breaks the time reversal

symmetry

and reveals details of the

microscopic

structure of _a

crystal.

The Hall

effect,

the de Ham _van

Alphen effect,

electronic

diamagnetism

in

metals,

the Meissner

effect,

flux

quantization

of vortices in

superconductors

and _more

recently

the

magnetoresistance

oscillations in

slightly

^disordered

metals,

and the

quantum

^Hall

effect,

are

examples

of the

complexity

_a

magnetic

field introduces in

homogeneous

materials.

The first

study

of such

problems

_goes back to Landau in 1930

[58]

for the motion of electrons in a

metal,

in the effective _mass

approximation.

It _{was soon} followed

by

the work of Peierls

[72]

^{for Bloch electrons. But it took until the}

fifties,

with _{a paper}

by Luttinger [64]

to start _a

systematic study

of the

problem.

The main remark

coming

out of these works is that

magnetic

translation

operators [94]

are

generating

the effective Hamiltonian

describing

the electronic motion in the

independent

electron

approximation.

The _case of _two dimensional

systems

in

a

perpendicular

uniform

magnetic

field became

increasingly important

after the

development

of electronic devices such _as MOSFETS _or

heterojunctions _[2].

It is

nowadays possible

to

get

metallic thin films 200

1

_thick.

_Many

_{of the afore mentioned effects}

are

actually

enhanced in two dimensions.

More

recently,

the

discovery

of

high Tc superconductors

in

copper-oxides _[12]

led several

experts

to propose that the main mechanism

leading

to

Cooper pairs,

lies

probably

in the

planes

of the copper

atoms, reducing

^the

problem

to two dimensions.

Moreover,

the flux

phases approach

reduces the

problem

to the _case of

independent charged particles

in _a lattice and _a

magnetic

^field. Even

though

^{this last}

approximation

is

questioned nowadays,

it led _many

physicists

to go back to the

question

of the 2D electronic lattice motion in _a uniform

magnetic

field.

If _we denote

by

U and V the two

magnetic

translations

describing

such _a

system, they

_are

unitary operators fulfilling

the

following

commutation rules _:

UV ₌

e~~'"VU (I)

Here,

_{o =}

#/#o

is the ratio between the

magnetic

flux

# through

the unit cell and the flux

quantum lo

₌

hle (h

is Planck's constant and _e the electron

charge).

To describe the electronic motion in _a

crystal,

let _us consider first the _case for which _{a =} 0

corresponding

to the absence of _a

magnetic

^field.

Only

those energy levels _near the Fermi energy

participate

in

conduction,

_so that _{we can} restrict ourselves to a finite set of bands

crossing

^{the Fermi surface. For}

simplicity,

let _us restrict ourselves to the _case of

only

_one such band. Let

E(k) (k

₌

(ki, k2) being

the

quasimomentum)

be the

corresponding

band function.

By

Bloch

theory, E(k)

is

periodic

with

respect

to the

reciprocal

lattice. Peierls then

proposed

to

replace

k

=

(ki, _{k2) by}

the dimensionless

operator

^K =

(Ki, K~),

where

K»

₌

) I"h£ ^A»)

= COnSt

(P» eA»)

,

»

in the

expression

of E to

get

the effective hamiltonian. Here A

=

(Ai,A~)

is the vector

potential,

_a~

(p

₌

1, 2)

are the lattice

spacings

^{in each direction.}

Remembering

that

e~~i

and

e~~? satisfy

the commutation relation

(I),

the Peierls Hamiltc- nian _can be written _{as a}

convergent

_power series in the

operators

U and V.

The

rigorous justification

of this Peierls substitution has been done in

[IS, 85]

^{and leads} to corrections _to the Peierls Hamiltonian. But whatever the

corrections,

the effective Hamiltonian

always

admits _a

representation

in the form

He~

_"

~j hmi,1112(°)U~~

v~~~~~~~~~~~

'

(2)

(~l~i,~1~~)~zz

where

hm~,m~(al's

_are ^{smooth functions} of _a. This series

usually

_converges

absolutely.

The

simplest

_case consists in

looking

at _a square lattice for which

we

ignore

the

higher

order terms in

(2).

It

gives

rise to the sc-called

"Harper

model"

namely _[49]

_:

HHarper

_" ^u +

u~~

_{+ v +}

v~~ (3)

The

analogy

between

(I)

and the canonical commutation relations

(where

h ₌

h/2x), jQ, pi

₌ ;h

,

(4)

between the

position Q

and the momentum P

operators,

is realized if _we

replace

U

by e~~

,

V

by

e~Q and 2xo

by

h. Therefore _a

plays

the role of _an effective Planck constant in

(I).

l'he main difference is that h is _a fixed

parameter

_once and for

all,

whereas _a is

proportional

to the

magnetic

field and _can be varied like _a tunable

parameter.

We thus have realized _a concrete

system

in which _{one can} test the semiclassical methods. In

a

crystal

with lattice

spacing

of order of

11,

_and

a

strong

realistic

magnetic

field of order of 10 T _we

get

_a of the order

of10~~,

which is

quite

small and

justifies

_a semiclassical

approach.

However if _we build

artificially

_a network with lattice

spacing

of the order of

lpm,

in _a

magnetic

field of few

Gauss,

_{we can}

get

o te I. Such networks _were indeed built

by

the Grenoble _group

[68, 69] by

_mean of

superconductors.

The de Gennes

[34]

^{and Alexander}

[I] theory

for such networks

permits

to relate in _a

simple

_way the

groundstate

_energy ^{of the}

Harper

Hamiltonian

(3)

to the normal

metal-superconductor

transition _curve in the

temperature-magnetic

field

phase

_space. In this

problem

however the electron

charge

e must be

replaced by

the

Cooper pair charge 2e,

in

lo.

1.I ALGEBRAIC _APPROACH. Another

approach

to

(I)

consists in

considering

the _case for

which _{o =}

p/q

is _a rational number. Then U~ and V _are

commuting.

In _an irreducible

representation

_{we can}

always represent

U and V

by

mean of _{q x q} matrices

e~~iu

_and

e'~?v

where _u and _{v are} unitaries such that

u~

= v~

= I _uu

=

e~~'P'~vu (5)

Substituting

in the

expression (2)

for the effective

Hamiltonian,

_we

_get

_a

k-dependent

_{q x q}

matrix which _can be

numerically diagonalized

_{on a}

computer, giving

rise to _a

family

of _q subbands.

JOURNAL DE PHYSIQUE I f 3, N'2, FEBRUARY lW3 17

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PHYSIQUE

I N°2

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-3.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Fig-I- Spectrum

of

Harper's

model

(Hofstadter's butterfly).

For

Harper's model,

this

diagonalization

is

especially simple

for the subband

edges

are ob- tained for

k~

= 0 _or

x/q

mod 2x. It gave rise to the famous Hofstadter's

butterfly _[55] (see Fig. I).

As _{one can see} from

figure I,

the

eigenvalue spectrum

_as the

magnetic

flux

varies,

exhibits _a fractal _structure of

high complexity.

The first mathematical

difficulty,

in this numerical

approach,

is to show that whenever the flux

o becomes

irrational,

_{one can}

approximate

the

spectrum by

_mean of the rational _one. This

can be done

using C*-algebra techniques (see [71]).

A

C*-algebra

A is _a

complex

vector space, with _a bilinear

product,

and _an antilinear involution

a E A _-- a* _E A

satisfying (ab)*

₌ b*a*. Moreover A has _{a norm} for which it is _a Banach

space

(namely

each

Cauchy

_sequence

converges)

and satisfies

llall~

=

lla*all (6)

A unit I is _an element of A such that aI

= Ia

= a

,

Va _e A. Therefore from

(6) _))I)

= I. A

C*-algebra

has not

necessarily

_any

unit,

but if _some unit exists it is

unique.

One _can

always enlarge

_a

C*-algebra by adding

_a unit

by

brut force. A

*-homomorphism

_p A _-- B between

two

C*-algebras

^A and

B,

is _a linear _map such that

p(ab)

=

p(a)p(b), p(a*)

₌

p(a)*.

This structure is

actually

_very natural from two

points

of view.

First of

all,

the

topology generated by

this _norm is

purelj~ algebraic,

because

(6)

tells _us that

[[a[)~

^{is the}

spectral

radius of a*a

(a purely algebraic concept).

This is confirmed

by

the fact that if _p is _a

*-isomorphism

from A to B then p is norm

preserving.

Thus the _norm is _an

algebraic

invariant.

Moreover,

the antilinear involution

permits

to define

positive elements, namely

those elements of

A

which _can be written in the form a*a for _{some a E}

A.

This

positivity _property

is essential in

Quantum

Mechanics in that it

permits

to define energy and

groundstates

for instance.

Then,

if _{a E}

A, _))a

II is the smallest _non

negative

number "c" such that 0 < a*a < cI.

It _can be shown

[78]

^{that there is} a

unique largest C*-algebra Au (up

to

isomorphism) generated by

two "abstract" unitaries U and V

satisfying (I). Au

contains _a unit. It is called the "'rotation

algebra"

and had been defined and studied

by

^Rieflel

[78]. Moreover, given

_any

compact

subset I of

R,

one can show that there is _a

C*-algebra AI generated by

elements of the form

(2)

where the

hm~,m~

are continuous functions of _o

on I and vanish all but for _a finite number of them.

AI

has

always

_a unit. In _a certain _sense

AI

_can be _{seen as}

Uo~ IAn. Actually

for _a e I there is _a natural

*-homomorphism

_pa

AI

_--

Ao

which consists in

evaluating

all the

hm~,m~'s

at _o.

Given _an element _a in _a

C*-algebra AI

with _a unit

I,

its

spectrum SpA(a)

is the set of

complex

numbers _{z e} C such that zI

a has _no inverse element in A.

The main result

justifying

the numerical calculation is _:

Theorem i Let H

= H* _a

self adjoint

element

of AI,

where I is _a

compact

^subset

of

^R.

Then the

spectrum E(a)

₌

Sp

A

(pa(H))

is continuous with

respect

to _a

namely

its _gap

edges

are continuous

functions of

_a. ^~

A

C*-algebra

_can be

represented by operators

in _a Hilbert _space.

Namely

_a

representation

x is the data of _a Hilbert _space

7i,,

called the

representation

_space, and of

a

*-homomorphism

x A _--

B(7i,)

into the

C*-algebra

of bounded

operators

_on

7i;

For the rotation

algebra

several

representations

have

physical meaning.

For

instance, Harper

in 1955

[49]

^{used the}

following

one

7i,

₌

£~(Z), namely

the Hilbert _space of

quantum

states

on a discrete lD

chain,

and the

operator x(U)

is the translation

by

_one, whereas

x(V)

is the

multiplication by e~~'(~~°'~)

if N is the

position _operator.

It is not difficult to _see

then,

that the

Schr6dinger equation x(HHarper)~l

_"

^E~I,

^where

HHarper

is

given by (3)

is

nothing

but the

Harper equation

~l(n

+

I)

₊

~l(n 1)

+ 2 _cos

2x(z

_on

)#i(n)

₌

E~I(n) (7)

We _can also represent U and V _on

f~(Z~) namely

in _a 2D

lattice, by

_mean of the discrete

magnetic

translations. At

last,

_{one can} also

represent

them _on

L~(R) by e~w~i

and

e~W~?

where ₇

=

2xa, 1(2

is the

position operator

and

Ill

_"

-I).

Many

other

representations

^{do exist} and _are useful for

practical

_purposes.

1.2 SEMICLASSICAL ANALYSIS. To go forward _on the

knowledge

of the Hofstadter _spec-

trum, we will _use the semiclassical

analysis namely

_we will

develop

_an

asymptotic

calculus _as

a -- 0. This _can be done _as follows

setting

₇

= 2xo

we use the

representation x(U)

₌

^e'~~~+w~~~ x(v)

=

e~~~?+w~?~

,

(8)

and

expand

the effective Hamiltonian

(3)

in _powers of

@.

If k ₌

(ki, k~)

is chosen _{as a} maximum _{or a} minimum k

=

kc

of the band function

E(k),

one

gets

for H _an

expression

of the form

H =

E(kc)

+

I(fif~~ _)~vI(~ltv

₊

_O(7~~~)

,

(9)

47G JOURNAL DE

PIIYSIQUE

I N°2

Energy

-2 / ~

/

/

~~'~~

"

o

' -2.5

-2.75

-3

-3.25

3 5

-3.75

o o.02 o.04 o-fit, @,I»I i,1 ^~~~

Fig.2. Comparison

between _exact spectrum and semiclassical formulae for the Landau levels in the Harper model; points _are extracted from the numerical exact spectrum, ^full curves represent

semiclassical formulae

given by Equation (11).

where

M~v

is the effective _mass matrix _near the extremum and +

(resp. -)

is chosen for

a minimum

(resp. maximum).

The

O(7)

term is

nothing

but Landau's Hamiltonian in

the effective _mass

approximation.

The

corresponding eigenvalues

_{are easy} to

get, namely (n EN)

_:

En

₌

E(kc)

+

j _~~j+j~

₊

_o(72)

,

(lo)

1 2

where _{ml, m2 are} the

eigenvalues

of

M,

and _{n E} N labels the

corresponding

Landau levels.

The

O(7~)

term _can be obtained from

(9) by perturbation theory.

In

particular

if H is

poly-

nomial in U and

V,

_{one can}

_get

_an

expansion

of

En

in

(10)

at every order in _7. In

particular

for

Harper's equation,

the Landau levels _are

given by (see Fig. 2)

Et

= +

4

7(2" +1)

+

)~ ll

⁺

^(2~

⁺

^1)~l fit"~

₊

_("

₊

_l)~l

₊

^(7~)j ₍₁₁₎

Such _an

expression

can

actually

be

generalized

_near each rational value

p/q

e

Q

of _a. For

indeed, using

the matrices _u and

v in

eq.(5),

and the

generators Up

=

pp(U)

and Vp =

pp(V),

the

sub-C*-algebra

of

Mq(C)

©

Ap generated by

U'

= u ©

Up

^{and V'} = v © Vp ^is

isomorphic

to

Ao

if _o

=

p/q

₊

fl.

Therefore in this

representation,

the effective Hamiltonian _pa

(H)

can be

seen as a q x q matrix

kp

with elements in

Ap.

In much the _same way we

get

an

expansion

near

6 =

2xfl

te

0,

if _we

replace _Up

and Vp

by e~(~i+W~i)

_and

e~(~?+W~?)

_and

developing

in _power8 of

li.

_{The difference is that} for 6

= 0 the "band function"

E(k)

is

replaced by

_a matrix valued functions

7i(k). Diagonalizing 7i(k) gives

a sequence of _q subbands

Ei(k),

,

Eq(k),

which

gives

the

spectrum

of

pp/q(H),

whenever _{we vary} k.

Correspondingly

_we

get k-dependent eigenstates )I)k

,

I =

I,

, q.

Such _a "classical" mechanics

corresponds

to motion in _a fiber bundle. Here the

((k, )I)~)

e

T~

_x

C~;

k _e

T~)

is _a line bundle associated _to the i~~ subband.

To describe the

spectrum

_{near a} ^band

edge,

we

pick

_{up a} band

(namely

_we choose _{I e}

[I, q]),

and let

kc

be the value of k for which

E;(kc)

is the band

edge

_we consider. Then

kc

is _a maximum _{or a} minimum of

E;.

In order _to reduce the

problem

to the _{case a --}

0,

_we

replace

the matrix valued Hamiltonian

kp _by

an effective band Hamiltonian

by

_mean of _a standard Schur

complement

formula

tip(z)

=

(<lfip4

₊

(<lfipo;

~~

~ Q;fiplo

,

(12)

where

ii)

₌

(I)k_k~, ^Q;

=

Iq (I) (ii

and

Iq

^{is the unit matrix in}

Mq(C).

The

eigenvalue

E of

kp

near

E; (kc)

is then

given by

the

equation

_:

E =

eigenvalue

^of

jip(E)

Expanding (12)

in _powers of

li _gives

_{rise to}

an

expansion

^{of the} _same ^form as

eq.(9)

where

now M is the effective _mass for the subbands. So _we

get

^{Landau sublevels} at each band

edge.

However the

O(6)

term

splits

into two

pieces

where

fl;(k)

=

(I)kk(I(.

The first term is

proportional

to (b( and

gives

rise to a

discontinuity

of the derivative of the band

edges (namely

for

n =

0)

at _{o =}

p/q (I.e.

6

=

0).

Notice that

@@

is the

density

of _states at the band

edge.

The other term is

proportional

to 6 and _accounts for the curvature of the i~~ bundle introduced

by

^{the matrix} Hamiltonian

~(k).

It

produces

_an

asymmetry

of the derivative of the band

edges

near o =

p/q (see Fig. 3).

Formula

(13)

_was written for the first time

by

Wilkinson

[92]

^{for the}

Harper

model. Rammal

[76]

^{rederived it for}

Harper's

model and related the derivative

fiE;,n(6) /fi6

to the

magnetization

of a

superconducting network, using

Abrikosov's

theory.

The formula

(13)

_was derived in full

generality (namely

for _any model

given by Eq.(7))

in

[13] (see

also

[51, 77])

where it _was called the Wilkinson-Rammal formula.

1.3 BANDTOUCHING. The Wilkinson-Rammal formula is valid

Only

at a band

edge

_sep-

arated from other

bands, namely

whenever

E;(kc)

is _a

simple eigenvalue

^of

~(kc).

On the

contrary,

^if two bands touch at k

=

kc,

the

asymptotic

is different and

depends

_upon the order of the contact between them. Whenever two bands touch each

other,

the contact is

generically

conical

(see Fig. 4). Degenerate perturbation theory

indicates that the effective Hamiltonian

jip(z)

in

(12)

must be _a 2 _x 2 matrix

corresponding

to the

eigenprojection

:

1<1(<1+ 1<'1("1 = fl

,

if

Ei(kc)

₌

E;>(kc),

I and I'

being

the labels of the

touching bands,

and

kc

^the conical

point.

One _can show that the lowest order term in the

jip(z) expansion

is

given (after

_a suitable choice of

coordinates) by

_a Dirac

operator, namely

jip(z)

₌

E; (kc

+

Vi l~,

^~

~~

~~~

~'~~~

⁺

O(d) (14)

'1 2

478 JOURNAL DE

PHYSIQUE

^I N°2

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-1.2 -1.0 -0.8 -0.6 -0.2 0.4 0.6 1.0

Fig-S- Asymmetry

of the central band

edges

for the

Harper

model _{near a}

=

1/3.

Thus Landau sublevels _are

replaced by

Dirac levels

namely

Ez,~z =

E; (kc)

+

24@sgn(n)

₊

_O(6)

,

(15)

where _n e Z is the Dirac label.

For

Harper's model,

it _was shown in

[17]

^{that there is} no gap ^at E

= 0. In

particular

if

o =

p/q,

where _q is _even, the middle gap is closed

producing

_a conical

bandtouching.

This is what _{we can see} in

figure

4 at _{a =}

1/2

where the Dirac levels _are

given by (6

₌

2x(a 1/2))

Ef~

₌

+2(2n(6()~'~ (l *)

+

O((6(~'~)

_n > 0

(16)

' 2

~

Non

generic touching

had been studied in

[9].

Playing

with such _a semiclassical

analysis permits actually

to reveal the

topology

of the classical bands

underlying

the

system.

^A bunch of Landau sublevels will reveal the existence of _a local extremum in _a subband. The

slope

of these levels

permits

to _measure the local

curvature of the subband

near this extremum. The

expansion

of this level in _power of 6

permits

in

principle

_to

compute

the

higher

order derivatives of the subband function. In much the _{same way,} Dirac levels reveal the existence of _a conical

bandtouching.

At last saddle

points

also

produce

_a

special packing

of sublevels. This has been studied

by

Helfler and

Sj6strand [52].

^It

permits

to understand in

particular why

the Hausdorfl dimension of the Hofstadter

spectrum

_may not vanish

[88].

L

o

r

g ~

ki

k2

r

FigA.

Conical contact between bands at half flux in the

Harper

model.

1.4 TUNNELING _EFFECT. The

previous analysis

does not account for

exponentially

small

terms in _a

(or

_a

p/q)

which _may be

produced by

the

tunneling

effect.

Actually considering

E(k)

to be the classical Hamiltonian where

ki (resp. kz) represents

classical momentum

(resp. position)

_{we may}

get tunneling

between wells in the k ₌

(ki, k2) phase

_space. Because

E(k)

is

periodic

in

k,

_any well is

actually infinitely degenerate by

_mean of the translation in

reciprocal

_space. It will

produce

_a

broadening

of the Landau levels and sublevels which

gives

an

exponentially

small correction _{as a or a}

p/q

_-- 0. This

broadening

has been described

by

Wilkinson

[92]

^and

Helfler-Sj6strand _[52].

Before

going

into

it,

let _us restrict ourselves to the

simpler

case for which _we have

classically degenerate

wells in the cell of

periods

of the

reciprocal lattice,

close to each other. Such _a

phenomenon

_appears for the

following

model studied

by

Wilkinson

[92]

and Barelli-Kreft

[10]

Hw=U+U~~+V+V~~+t2(U~+U~~+V~+V~~), ₍₁₇₎

with12

>

1/4. Indeed, if12

_<

1/4, Hw

admits _a

unique

classical minimum at

ki

_"

k2

"

x mod 2x which bifurcates at 12 _"

1/4

into _a local maximum and

gives

rise to four minima

symmetric by

_a fourfold rotation around

ki

"

k2

_{" x.} This

symmetry produces

_an exact

classical

degeneracy.

Thus the

corresponding

Landau sublevels _are

exactly degenerate

modulo

O(tY").

To describe the

tunneling

effect between these four

wells,

_we

replace

the Hamiltonian

by

_a

4 _x 4 effective Hamiltonian for each value of the Landau quantum number

n.

Using

the rotation

480 JOURNAL DE

PHYSIQUE

I N°2

symmetry,

this matrix admits the

following

form _:

0 t r

il

~~~~~~~

~"~7~~

~

' ~~~~

t _r

?

0

where

En (7)

h the n~~ Landau sublevel

computed

_as

before,

_r

= f E R and t e C.

Using

WKB

techniques _[50, 93],

_{one can} write t

(resp. r)

ⁱⁿ terms of

tunneling

action8 _{Sn_n_}

(re8p.

Sopp)

^between

neighbouring

wells

(resp, diagonally opposite wells)

in the form _:

where w

aslov's corrections.

It

_turns out

that

_(Imsopp( > _(Imsnn.(

so

that

r

_becomes

negligible

compare to t.

contrary to the

Schr6dinger

_case, the real

part

of _{n_n} does

terms in

the

^pression

of

_level

splitting when 7 varies.

E~

₌

+~de-lIm(sn_n

^)jH _~~~

(~~(s~

~

/~

_~

i)

~

~~~~-(lm(Sn

n,)(/~ ~i~

(fi~(

s

/

~

#) ^(~°)

The

corresponding

Landau level

splits

into four levels like _a braid _{as 7} varies

(see Fig. 5).

The _same

phenomenon

has been also observed with Dirac sublevels

[9].

^In

figure

6 is repre- sented the

braiding

of Dirac sublevels

produced by

the

touching

of two bands

along

five conical

points

with _{an exact} fourfold

symmetry

around the fifth _one. One _{can see on}

figure

7 that the WKB formulae fit

quite

well with the

diagonalization

method.

The

broadening

of Landau sublevels _can be described in much the _{same way}

by using

the

reciprocal

_group translation

symmetry

ⁱⁿ the

k-phase

_space instead of the fourfold rotation

symmetry

used above.

We

replace

then the Hamiltonian _we started

from, by

_an effective Hamiltonian

Htunnei acting

in the Hilbert _space

generated by

the Landau

eigenstates

of

given quantum

^number n associated to each well. Since each such well is labelled

by

_a

point

in _a 2D

lattice, Ht~nn~j

can be

seen as

acting

_on

f~(Z~) by

_mean of

magnetic

translations.

However,

when _one

computes

the

corresponding flux,

_{one can see} that _now

Htunn~j

e

Ao>

with a'

=

lla

mod I instead

[92, 52].

^This scale invariance of the

spectrum

_was

proposed

for the first time

by

Azbel [6] ^who

emphasized

the role of the continuous fraction

expansion

of _a. The Gauss _map a'

-- I

lo (see

in

[57]) precisely generates

^this

expansion.

If H has the fourfold

symmetry

of the

original lattice,

so does

Htunn~j.

Moreover the _ma- trix element of

Htunnei

between two wells located at sites I and I' is

proportional

to

e~~"'/~

where

Sill

is the

tunneling

action between the wells and I'. So that

as 7 -- 0

only

the

nearest

neighbouring

wells should be accounted for. This leads _to the

following approximate

Hamiltonian _:

Htunnei

_"

7~e~'~~'~'H

(Uo> ⁺

UJ~

+ Va> +

VJ~)

+ O

(e~~'/~)

_,

⁽²¹⁾

with

S'

_> ImS.

This formula shows that the

broadening

of each Landau sublevel is

actually given by

_a

Harper

Hamiltonian, properly renormalized;

thus the

corresponding

spectrum is

again

the Hofstadter

spectrum of

Square

^Lattice with

2.Neighbours(ml/2)

oi

j , ~,~

, j =.w..

I -<'

I ,, ,

~ ,,

~ _,/ _--.~'~

,,' _i ---'

_.,,' _,

, ,,"

, :..

, ., _>_:-'

,.'~.

.~

/ ',

/ ' /

0 '>

/

$' _{l [ ]}

I

o

-3 -2.S -2.6 -2.4 -2.2 -2

Energy

Fig-b-

Braid structure for Landau sublevels in _a

Harper-like

model with second nearest

neighbour

interaction

(from [io]).

one.

Repeating

^this construction

again

and

again gives

rise in

principle

to the fractal _structure of the

spectrum.

However the

previous analysis

works

only

for well

separated

Landau sublevels. Near the band

centers,

^{Helfler and}

Sj6strand

have

developed

the _same kind of ideas and

they

have shown that for

Harper's model,

each

step

of the renormalization

analysis

_can be

locally

described

by

_mean of four standard

forms,

_one of them

being

^the

Harper model,

the others

taking

the

possible

bandtouchings

into account.

The _same

analysis

has been

performed by

Kerdelhud

[56]

^{for the}

Harper

model _{on a}

triangular

or

honeycomb

lattice.

The main result of Helfler and

Sj6strand

is contained in the

following

theorem

[52]

Theorem 2 There is C _>

0,

^{such that}

if

_o admits _a continuous

fraction expansion of

the

form [al,a2,

,

an, with _an > C

Vn,

then the

spectrum E(a) of Harper's equation

has _a

zero

Lebesgve

_measure.

We remark however that if C _>

I,

the set of such a's has

zero

Lebesgue

_measure. A

conjecture (The

Ten Martini

problem

of Kac

[17])

is that

E(a)

should have _{a zero}

Lebesgue

_measure for

any irrational _a.

482 JOURNAL DE

PHYSIQUE

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1.5 FLUX _{PHASES THEORY.} The

discovery

of _copper oxide

superconductors

in 1986 has

probably

been _one of the _most

important

events in Solid State

Physics

ⁱⁿ _years. ^{The main}

reason is that it

provides probably

_{a new} mechanism for

pairing

^{of electrons. The}

corresponding

forces _are much

stronger

^{than the}

phonon coupling

in BCS

theory, producing higher

critical

temperature. Yet,

this mechanism is still not understood.

Nevertheless the _enormous effort since then

by

theoreticians has

permitted

to concentrate the studies _{on a} small number of models. It is

right

_now

accepted by

most of the

experts

that the

key phenomena

_are to be found in _a 2D

model, representing

the Fermi

plectrons

in the

Copper orbitals,

the

Oxygen being

there

only

to create an effective

doping by

holes in these

orbitals. One

possible

model is the sc-called Hubbard model in two

dimensions,

with _a

strong

electron-electron

interaction,

which _can be

approximated by

the sc-called _t J model _{as an} effective Hamiltonian.

A

great

^{deal of work has been done in}

finding

a

good approximate ground

state for such _a model. One of the

approaches proposed

_was the sc-called flux

phase approximation.

The basic idea in such _an

approach

is that in the t J

model, spin

variables _can be considered _as slow whereas

charge

variables _are

rapid.

Therefore _a kind of

Born-Oppenheimer approximation

leads to _a model in which the

spins

_are

frozen, creating

_an effective

magnetic

field

acting

_on

charges

(here holes).

^{In this adiabatic}

approximation,

the

charged particles

become

independent,

and the

Hamiltonian

becomes similar _to the

Harper

model. However the

magnetic

field is _no

longer

/

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Fig.?.

figure crosses

^are from exact _spectrum ^{whereas full} _curves ^are

coming

_from

(from

_{lower figure} : points are ^{extracted from} _{numerical exact} spectrum ^and curves correspond

to

emiclassical expansions

(from