Braid-structure in a harper model as an example of phase space tunneling

(1)

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Braid-structure in a harper model as an example of phase space tunneling

Armelle Barelli, Christian Kreft

To cite this version:

Armelle Barelli, Christian Kreft. Braid-structure in a harper model as an example of phase space tunneling. Journal de Physique I, EDP Sciences, 1991, 1 (9), pp.1229-1249. �10.1051/jp1:1991203�.

�jpa-00246408�

(2)

J.

Phys.

I France1

(1991)

1229-1249 SEPTEMBRE 1991, _PAGE 1229

Classification

Physics

Abstracts

03.65 71.15 _75.20

Braid-structure in

_a

harper model

_as _an

example of phase _space tunneling

Annelle Barelli

(*)

and Christian Kreft

Technische Universitfit Berlin, MA 7-2, Strasse des 17. Juni136, D-1000 Berlin12,

Germany

(Received

27

Febmary

1991,

accepted

in

final form

6 May

1991)

Rksnmk. Nous considkrons _un moddle d'dlectron _sun rdseau carrd _en

champ magn6tique

uniforme. A la limite des

champs

foibles, _une interaction _aux seconds voisins suffisamment

grande

entraine _une

dkgdndrescence

des niveaux de Landau les

plus

bas _en quatre sous-niveaux tressds. Un effet tunnel dans

l'espacc

des

phases,

vu sous le

jour

de l'extension de la mdthode BKW initide _par M. Wilkinson,

explique numkriquement

_ce

phdnomdne.

Los effets dus I des

champs magndtiques plus

klevks sont examinds.

Abs~act. We consider _an electron _on _a square lattice in _a uniform

magnetic

field. For small fields _a sufficient

large

second

ne1gllbour coupling splits

the lowest Landau levels into _a braid like structure. Phase _space

tunneling

in the

spirit

of Wilkinson's extension of the WEB

approximation

is verified

numerically

to

explain

the

phenomenon.

Furthermore the

high

fields effects _are also

discussed.

1. InUoducdon.

The

quantum

^behaviour ^of Bloch electrons in _a uniform

magnetic

field has been

investigated

over and _over since the

early

works of Landau

ill

and Peierls

[2]. Many

studies have been

performed during

the fifties in order to

compute

^the ^band

spectrum,

^the

density

of states, the

transport coefficients,

etc.

[3].

In

1976,

Hofstadter

[4]

revealed in _a numerical work the nice fractal structure of the

spectrum

^of ^the

Harper

Model

[5],

_as _a function of the nornlalized

magnetic

flux _per lattice-cell.

The _case of _a two dimensional electron _on _a

periodic

lattice has been of

special

interest in Solid State

Physics during

the last ten years :

superconductor

networks

[6],

normal metal

networks

[7],

^the

Quantum

Han Effiect

[8],

one dimensional chains with

quasiperiodic potentials [9]. Recently

these models have been used also for the

Quantum

Han Effect in

organic

conductors

[10],

in flux

phases

of the Hubbard model

[lla]

and in

Anyon

superconductivitty [12].

(*)

Permanent address: Centre de

Physique Thborique,

C-N-R-S-

Luminy,

Case907, 13288 Marscille Cedex 09, France.

(3)

To describe the Hamiltonian _we

replace

the _momentum

operator P~ by

the

quasi-

momentum operator

K~

=

~

(P~ eA~)

where

A~

^is ^the ^vector

potential (B

= rot

A),

_a is

h

the lattice

spacing,

_e is the electric

charge

of the carriers and h is Planck's constant. For _a uniform

magnetic

field in two dimensions _one has

B

= a

iA2 a2A1 (1.I)

and

#

is the flux per unit-cell and

#o

=

~ the flux

quantum.

e

In

(1.2)

the role of Planck's constant is

replaced by

the

magnetic

flux ratio _y.

Tuning

the

magnetic

field _we

approach

the semiclassical limit _y

- 0. The

corresponding

classical

phase

space at B

=

0 is the

quasi-momentum

_space,

namely

the Brillouin _zone of the

corresponding

lattice.

Topologically

it is 2-torus and the

magnetic

field

simply

transfornls it into _a _non

commutative 2-torus

[13].

Vfhenever _y

=

2

arp/q ~p/q

e l~l

)

the lattice Hamiltonian H _recovers _some

periodicity

and Bloch's

theory applies

_as well. As _a result H _can be

represented by

_a

sclfadjoint

q x q -matrix whose entries _are

periodic

functions of the

quasi-momentum.

Thus if _y is close to a rational

multiple

of 2 _ar, it is

again possible

to compute ^the

spectrum using

semiclassical

methods.

Based _on these facts _many theoretical and mathematical works _were

published during

the last ten years. In this _context the works

by

Wilkinson

[14]

_are of

special importance.

He

applied

WKB method to get a renornlalization _group

analysis

of Hofstadter's spectrum.

Using pseudodifferential

_operator

techniques

Helffer and

Sj6strand [16l

gave a

rigorous proof

of

Wilkinson's ideas.

Another

point

of view _was

developed by

Beflissard and Rammal

using C*-algebraic techniques

to reformulate

[15]

^and extend the semiclassical results

ill, 13].

In _a _recent paper

II?]

the

algebraic approach

has been used to

compute

the Landau

sublevels of _a Hamiltonian _on _a

triangular

lattice. The

comparison

^between ^the semiclassical fornlulae and the exact calculation of the spectrum ^for y ^e 2

drill

_gave _an

amazingly

accurate

agreement

even for

y's relatively large (namely y/2

_ar «10

9b).

In the

present

^work we continue _on this line

by describing

_an

example

of _a

tunneling

effect in

phase

_space.

Recently

the WKB

theory

has been extended

by

^Wilkinson ^and ^Austin

[18]

who

applied

it to a lattice-model with threefold

symmetry.

^We ^have ^used ^this

approach

to

investigate

_a Hamiltonian _on _a square

lattice, already proposed by

M. Wilkinson

[14],

with second nearest

neighbour hopping

ternl of size _e.

Beyond

_a critical value _e

~ e~ the minimum of the classical energy function bifurcates into four

degenerated

minima in

phase

_space.

For _e e~ the

tunneling

between these wells

gets large

and

produces

_an observable

splitting

of each Landau level into four sublevels. Due to the _non

vanishing

real

part

of the

tunneling

action the sublevels form nice braid like _structure _as _one varies the

magnetic

field

(Fig. I).

Following

Wilkinson's

approach

_we

computed

_an

approximate

fornlula to describe this

phenomenon

and

compared

it

numerically

to the exact

spectrum.

^The

agreement

^is

astonishingly good.

We also observed _a nice structure of the

spectrum

near

# Iwo =1/2 (Fig. 9). Using

the

techniques previously developed [11, 16]

we will discuss

qualitatively

the

shape

of the spectrum at such _a

high

field.

(4)

N 9 BRAIDING IN A HARPER-LIKE MODEL 1231

spectrum

of

square

Lattice with

2.Neighbours(E~=1/2)

o-i

, , _~×_

, ; /.

I »'

_,'~"~

i ^"' _i

_," ~i _-~'~"

/" I ,--"~'

,l'

j

^''

I _,"' I

I .' _I.-..'

~;,'

_.,' ^""

"i

» :~ l ;"

)

/ ~

." ,'

j

^/ ~.'

_, ,"

.~ ,'

i ,' '.

I.'" I

[ _~,(

,'

/,'j _/

i~,,

.'i',f7 _",

j"'

_j ;

,

.~

(. ~

" 'h

/~

~, ^_i ^f

'

.,""~' _Ii

^~'

,_

'. ".~

~

~ ,==*~~' _~ ~' '" ',, ~

o

-3 -2.8 -2.6

~

-2.4 -2.2 -2

Fig.

I. Braid structure.

The paper is

organized

_as follows _: in section 2 _we

present

^the ^model ^and some methods useful to reveal its

properties.

Section 3 is devoted to the

application

of semiclassical

expansions.

In section 4 _we

explain

the

tunneling argument

^and ^the ^results ^of our numerical

calculation.

Finally

in section 5 the _case

#/#o =1/2

is studied.

2. Tile model and its

properties.

This section is devoted to the introduction of the model in consideration. In the

following

discussion of its

properties

_we

present

^the ^different

helpful techniques

ⁱⁿ ^this context.

2.1 2D BLOCH _ELECTRONS IN A UNIFORM MAGNETIC FIELD.

Using

the notations defined

in reference

[llb]

_we define the second

neighbour tight-binding

Hamiltonian in _a uniform

magnetic

field B

through

the

magnetic

translations

( by

H=Tj+Tj~+T~+Tj~+e(T)+Tj~+T(+Tj~). _(2.I)

Here _e is the second

neighbour hopping

term and

Ti

and T~

satisfying

the commutation rules

Ii

₉₁

TjT~=e'YT~TI, y=20r " =20r#). (2.2)

#o

Here

#

₌ _{ai a~ B} is the

magnetic

flux

through

the unit cell.

(5)

For _e

=

0 _we _recover the well

analyzed

Hofstadter

Hamiltonian,

with its

interesting nesting

properties

of the

spectrum

first described _in

[4].

Different authors

[4, 16, 17, 20, 21]

worked with this

type

^of models. A _very efficient

approach

for

studying

them consists in

seeing

them _as members of the abstract _non commutative

C*-Algebra generated by Ti

and T~.

Following

this

strategy, emphasized by

Bellissard and Rammal

[I16, llc],

the different

representations

of this

algebra

on Hilbert spaces

give

^rise to different

techniques

used in this context. In this section _we will

constantly

refer to these papers. ^For a mathematical overview _see

[13, 22].

In this article _we will

mainly

_use the matrix

representation

for rational fluxes and the

Weyl representation

in the _y

- 0 limit. Also the

purely algebraic point

of view is

helpful.

For

example

the

symmetries

of the spectrum are

given through explicitly

known

*,homeomorph,

ism. In this way one finds

I) W(8,tZ)#W(E,I-tZ)

it) «(s;a)=«(s,a+n) (neZ)

iii) W(8,

_a

)

# W

(-

_8, _a

).

Here

«(s,

_a

)

denotes the set of

spectral points

of the Hamiltonian with

overlap

_s and flux

a =

y/2

_or

(I.e. iii)

^is proven

by changing (

into _T~

(I

=

1, 2)).

2.2REPRESENTATIONS oF THE ALGEBRA. For rational fluxes

y=2~r~

q

~p A q =

I,

_{p, q} e

Z

_we _get

by

^Bloch's ^theorem a matrix

representation

of

A~.

^on ^the ^direct,

integral

_space

)~ ^d~k

^C ^we ^consider ^the ^matrix ^valued ^functions ^of ^the ^torus

^{T~ (for}

^details

~2

on tiffs way of

writing

Bloch's theorem _see Ref.

[23])

1j(k)

₌

^e"

_~§j

(k

_e

T~)

with

i

~ ~ 2wi~i

Wj=

,

W2=

^~ ^~

Eh#q~q I=0,..,q-I.

°

2«i~(q-I)

e q

This

representation

of H leads _to _a finite difference

equations

with

Bloch-~oundary

condition

gk(I)

=

gk(f

₊

q) e'~

Note that

«

(H)

=

~J

_«

(H(k)

, k_eT2

where

«(H(k))

=

(z/det (z-H(k)) =0)

is

just

the matrix

spectrum.

^This allows to compute the

spectrum numerically using simple algoritllms

for matrix

diagonalization.

Numerically (Fig. 4)

it showed that the band

edges

_occur for k

=

(0,

0 and k

=

(

_or, _or

),

that

is for

points

of

high

symmetry within the Brillouin _zone

(except

when _q

=

2).

(6)

M 9 BRAIDING IN A HARPER-LIKE MODEL 1233

spectrum of

square

Lattice with

2.Neighbour(£=I/4)

I

"'f C"'" "/]~~'Q_%@@ :jj.__ /

,I'

.,

_I' '[._j_).~'z]@ji~~

~'"-""'

____,, ~,:

'"',,"~ ~~~-,[

~~/~

/"'

't

_m+

_~'s-,. ~Wt&Q25#'

^'

fi~ , mq j~~.--

^~w

~f .RW~

y~ ^'K', ""y"' _§~TW

~'~

l @"

:~~

%.~

» o

~~i~ ~~l~i~ _,>~b ~#~~

f

3~~

^lR'

_q- ^p~

^'~x§L

( /

___

~-_,

'

j

,

."'~; _"?, _j, ~#,'~"' ',

~ _,' ___':$.Q£JL' 4i,£.j=___ /

( '%~~ _~/ _l'

lo

~

-. .dk

J~f~~ _.$ ~yP" ^_~MW

..:

-=.$WS.$

)~ff~"

~W'

_,

/

,_/"

~

-3 -2 _-1 0 Enew 1 2 3 4 5

Fig.

2.

Spectrunl

^of ^the Hanfiltonian for _e

=

JR-

spectrum ^of

square

Lattice with

2.Neighbours(£#1/2)

1

-< .,,t ^: ^.,,, ^I,,. ^. ^., ^j/ ^/

.. .;_.._.=. ; ..-.-.;÷ a

'"--' ~'""

~;;""

/

~j~ /

..-j~3~,

~~

WiRKlil _d'~§~" 'i... "~'

q5~T ---'---~-~__'+%~,

j

/. :.. "-:,' ~.-; .-;

~ ~, ,,, .-==-.. _., _.,

~ ;__[ L"Q£

~.,

^=_._'_. ^/

''~

"~'~ q~W~~°""" /"

+

~,

,~"' ,.-~~3§,

.mw

0

-3 -2 -1 0 1 snow 2 3 4 5 6

Fig.

3.

SpectnJm

of the HanfiltoJliaJl for _e

=

1/2.

(7)

p/q-1/2 pfq+if3

1

pfq-i/4 pfq-1/?

3

z

i

a

Fig.

4. BandstnJcture

along

lines of

high

symJnetry ⁱⁿ ^the ^Brillouin zone for _e

=

1/2.

In

figures

1-3 _we

present

^the ^band

spectrum

^for ^different rational

magnetic

fluxes _y and values of _e

=

I/4, 1/2.

The

computation

was done for all rationals

pig

with _q _« 40 in

general

and q «100 in the

region

_y _~ ^~ ^" In

figure

9 _we used denominators _up to q = 307.

10

In order to

perform

_a semiclassical

analysis

of _our Hamiltonian _we _use

Weyl's

_represen-

tation of

Heisenberg's

commutation relations

[q, pi =iy

_on

L~(R)

which

provides

Tj

₌ e~P, T~ ₌ e~~ where p =

(

^~ _is _the _momentum

_operator

_and

q is the

position

_operator.

i

bq

Writing

e~P ₊

e~~P

=

2 _cos

~p)

_we derive H

=

2 _cos

~p)

₊ 2 _cos

(q)

₊ 2 _e _cos

(2

_p

)

₊ 2 _s _cos

(2 q) (2.3)

This formula

represents

^the Hamiltonian in the notion of _a

symbol

of _a

pseudodifferential operator.

^We will refer to this

picture

_as tile « Classical

Representation

_» of the Hamiltonian.

The

analysis

of the classical

symbol

is the aim of the next section.

2.3 CLASSICAL HAMILTONIAN. The

phase

_space

analysis

of the classical Hamiltonian

H~p,

_q

=

f~p)

+

f(q), f (x)

= 2 _cos

(x)

₊ 2 _e _cos

(2 x)

is the basis of the semiclassical

analysis

of _our model and will

give

_a first hint of the character of the

spectrum.

Unlike in the

ordinary Schr6dinger Equation,

the kinetic term is not

quadratic

with

respect

to p. It is not

strictly positive,

but

equal

to the

potential

term. In tiffs

case the

system

^exhibits ^critical

points

_away from p = 0.

Since

H~p,

_q

)

is

periodic

with

period

2 _or in both variables it is sufficient _to restrict the

analysis

_to the unit cell

(Brillouin zone)

T

=

[0,20r]~. _According

_to renormalization

(8)

N 9 BRAIDING IN A HARPER-LIKE MODEL J235

arguments

^this

periodicity produces

tile recursive _structure of the spectrum

[4, 16] by tunneling

between different unit cells.

In table

I,

_we

give

the location and the nature of the critical

points

_as functions of the

parameter

e e

[0, ii-

In

figure

5 the levels set of _H for different values of _e _are shown.

Table I. Location and nature

of

the critical

points.

ki,

e 0

ki

=

k~

₌ 0 maximum maximum

ki

" °

>

k2

" "

saddle

point

_maXlmUE~

k~ = ~r,

k~

=

0

kj

=

0, k~

= ± Arccos

(- I/4 e)

~

=

~j~rc~os )) _k~~ ^~

^~°~~ ^~°~ ^~~~ ^~~~~~~

^~"~~

ki

= ± Arccos

(- I/4 s), k~

= or

k~ =

k~

= ± Ar~~~

(_ 1j4 g)

^does ^not ^exist ^minimum

kj

=

k~

= or minimum maximum

For _s

~ the minimwn _at k~ =

(or,

_or

)

is

regular, namely

the second derivative of H is 4

positive

definite. lvhile for _s

~ we find four

regular

minima with _same _energy located _at 4

k~ ₌

(±Arccos ),

^±Arccos

_),

while at

(or, or)

there is _a local

regular

e e

maximum. Notice that these four minima _are

exactly symmetric through

_a fourfold rotation.

The transition between the two

shapes

_occurs for _e

= e~ = when the Hamiltonian _near the 4

extremals is of order four

H~=i/4(kc+P,kc+q)=-3+ ~P~+q~)+O~P~+q~). _(2.4)

In the semiclassical

analysis

the behaviour of the Hamiltonian _near the critical

points

determines the character of the spectrum ^for ^small fluxes. For _e

~ near the critical

point,

H 4

behaves like _a harmonic oscillator with Landau levels linear in _y. For

e = a Bohr-

4 Sommerfeld

quantization

method

gives

rise to levels

proportional

to

y~.

If

e ~ we

get

a

4

phase

_space

equivalent

to the double well

problem

for the

ordinary Schr6dinger _operator.

Note that in the latter _case _more local maxima embedded in the spectrum appear.

They give

rise to other Landau levels

emerging

from the

point (E

=

2,

_y

=

0

).

(9)

0.

,

i

0.0

'

0.0

0.2 0A 0.6 0.8 1.0

(10)

3. Semiclassical

expausioo

at weak

magnetic

field.

To

specify

these arguments we derive _a semiclassical

expansion

of the _energy

eigenvalues

of

our Hamiltonian _up to order

y~

_near the bottom wells. In order to do _so, _a _more

syrnlnetric representation

of

A~

^is very useful.

3.I ANOTHER REPRESENTATION AND THE HARMONIC OSCILLATOR APPROXIMATION.

According

to reference

[I Ii

_we will

expand

the Hamiltonian

(2.3)

around the critical

points

and will _use the

following representation

~

;(k~+&q>

J ^" ^e J " ~

,

where

Ki

and

K~

are

operators fulfilling Heisenberg's

commutation relation

[Ki, K~]

= I and

k

=

(ki, k~)

e

R~.

Since _we _are interested in the behaviour of H _near the bottom wells _we choose for k _one of the critical

points

I.e, k

=

k~

=

(Arccos ),

^Arccos ^vith e ~

l/4.

A fornlal

4 e 4 _e

expansion

of H

gives

_a harmonic oscillator

H=-8()1+y16j)1(Kj+Kj+o(y3/2~ _(3.i>

with

spectrum

E~=-~+y~~(2n+1)+O(y~) _(nefll). _(3.2)

So the bottom well energy is

given by

the Landau levels

E~.

The _same

analysis

_near other critical

points gives

:

kj=k~=0 E~=4+4e-y(1+4e)(2n+1)+O(y~) (nef4) (3.3)

kj=k~=or E~=-4+4e+y(1-4e)(2n+1)+O(y~) (nefll). (3.4)

(kj, k~>

₌

(0,

_« _or

(kj, _k~>

=

(«,

0

)

E[=4e-y~~~~~~(2n+1)-)+O(y~) (nefll). (3.5)

For _e

=

1/2

_we

get

^there ^three ^families ^of ^Landau ^levels ^with

negative slopes converging

to

6,

2 and ₊ 2

respectively.

3.2 SECOND _ORDER _EXPANSION.

Following

the strategy

explained

in reference

[lib]

_we

get further corrections to this formula if _we consider the

higher

order

expansion

of the Hamiltonian _as _a

perturbation

of the harmonic oscillator. Let _us set :

H=Eo+y(Ho+V).

We

apply

standard

perturbation theory [24]

_on the harmonic oscillator Hamiltonian

Ho

with _a

potential

such that

V

=

/ (K/

+

K/)

y

~~

~

(K)

₊

K()

₊ _O

_(y

~/~)

(3.6)

4 ₈ 48 ₈

Since V is _a

polynomial

in

Ki K~

all terms in the

perturbation expansion

_are _so and the matrix

(11)

Mm-mm

a)

x x

x

Ma~-Hw

x

b)

Fig. 6.- Semiclassical and exact spectnJm for _e =1/2: a) harmonic

approximation,

b) tunneling effect. Solid line _: senficlassical approximation _; crosses exact spectrum.

(12)

elements

in Vi n')

_can be

computed.

We derive

En(Y) "~~~)~

+Y

~~(~f~ (2n+1)-£j(88~+1)(2n+1)~+88~j+o(y~).

(3.7).

Note that _we do not find _terms of odd order in

/

_in _the

energy

expansion,

since all the matrix elements

containing

odd _powers of

Ki K~

vanish.

Formula

(3.7)

for the

asymptotic

behavior of the

spectrum

has been verified

numerically

and found to be

quite

accurate

(see Fig. 6).

Some

exceptional

_cases,

arising

in the model studied

here,

have been

analyzed

in _more details. The results will be

presented

in the next section.

4.

Tuonefiog

Mithio the unit cell.

In this section _we will summarize Wilkinson's extention of WKB

[18]

and

apply

it to _our

problem

of

tunneling

^inside a unit cell. As stated above this leads _to _an

explanation

of the braid structure formed

by

the

splitting

^of ^the lowest Landau Levels in

figure

I.

As _seen in section 2.3 the classical _energy function in _our model shows for _e

~ e~ =

I/4

_a

bifurcation of the

single

minimum in the center of each unit cell into four

degenerate

minima

A, B,

C and D

(Fig. 7).

In section 3 _we

explained

how each of them

gives

rise to Landau levels

E~(y)

^defined

by

formula

(3.7). Equivalently

this result is found

by

Bohr-Sommerfeld

quantization

of the closed classical orbits

dA, dB,

etc. near the bottom wells

A, B,

etc.

SA(E)

= p

dq

d~

defines the classical action related to each orbit. Due to the fourfold symmetry ^the orbits _can be transformed into each other

by

rotations of

or/2

and the action

integrals

are the _same

s~(E)

=

s~(E)

=

sc(E)

=

s~(E>.

In

quantum

mechanics _we associate with each level

E~(y)

four

degenerate

one-well

eigenstates (A), (B), C)

and

(D)

localized _near the bottom wells

A, B,

C and D

respectively. Projecting

the Hamiltonian onto this

approximate

basis leads to _an effective 4 _x 4 matrix operator

HT.

As usual the

splitting

of the level

E~

^is

given by

the

eigenvalues

of

HT.

The matrix elements _are the

tunneling amplitudes

between the related states.

They

_are

computed by

means of the

imaginary _part

of

tunneling

actions Im

(S~~),

^Im

(S~c),

etc.

defined

through

Im

(SA~)

=

Im _p

dql.

CAB

Here

eA~

is _a closed

complex path

in the

complex

_energy surface

H(q,

_p

=

E

~p,

_q

EC ) joining

the classical orbits

d~

and

d~ (see Fig. 7).

Such _a

path

exists since

H~p,

_q

)

is real.

Again by _symmetry

all of them enclose the _same _area

~lll(~i)"~lll(~AB)"'

"~lll

(SDA).

(13)

ciassicataJiowedorbit

rcaJpanofcompiexpath

/ /

A B

R 54)

D c

dassicflollowedorbit imaginuypartofcomplexpath

/ ~

A S~) ^B

Fig.

7. -Paths in

phase

_space.

(14)

N 9 BRAIDING IN A HARPER-LIKE MODEL 124J

4.I EFFECTIVE TUNNELING HAMILTONIAN. Due to the rotational symmetry ^of ^the

problem

the effective Hamiltonian

H~

has the form

t r

I

H~=

^° ^~

(4.I)

r

I

₀ _t

t r

I

0

The matrix element _t=

re'°

=

(A(H(B)

is in

general complex. By symmetry

r =

(A (H( C)

=

(C (H( A), describing

the interaction between the second

neighbours (I.e. (A)

and

C)),

has to be real in order to get a hermitian matrix.

The

eigenvalues

of

H~

are

Ai=r+2rcos0, A~=r-2rcos0 A~=-r+2rsin0, A~=-r-2rsin0.

For

particular

values of _r and _t

= r e~° these

eigenvalues

_can _cross.

According

to Wilkinson _t is in

general

_a

complex

number. At the lowest order in y

(as

_y

-

0)

its

amplitude

_r is

given by

the usual

tunneling

_energy

[25]

w Im

SA~ (E)

r ₌ y exp

2 _" _Y

where _w

= 2 _or

(bSA/@E)~

is the oscillator

frequency.

For E _we take the energy of the

quantized orbit,

that is E

=

E~(y)

of section 3.

The Bohr-Sommerfeld

quantization

_can not determine the

phase

of each state

(A),

(B), C)

and

(D).

However the

phase

of the

cyclic product

^of matrix elements around _a closed

loop

in

phase

_space

gets

a well defined value _:

jAjHj BjjBjHj CjjcjHj DjjDjHjAj ^=r~e~'°

Re

S~

The

phase

4 0

= + gr is defined

through

the action S~

along

_a closed

path joining

all Y

four _contours

A, B,

C and D. The term

(2. or/2)

is _a Maslov index

emerging

from the two

tuming points along

such _a

path (Fig. 7).

To compute ^this ^index we use the rule described for instance in reference

[26] according

to which it is

equal

to the number of stable fixed

points

minus the number of unstable fixed

points

divided

by

two.

The coefficient _r is

analogously given by

_a term

decaying exponentially

in Im

S~c.

A closer

analysis

of the

complex paths (see below) _suggests

that Im

(S~c)

~Im

(S~~)

and the correction due _to _r _can be

neglected.

Inserting

the coefficients into the matrix

gives

the

splitting

of the _energy

E~~>~~

= ± ~~°

e

~~~

cos

~~

+ ^"

"

~

~ _~ ~

(4.2) E~~~>

= =

i~°

e+~

Sin

tl~

+

i

The corrected bottom well energy is

given by E)~~= E~(y)+E~~~

_k

_=1,2,3,4

_where

E~(y)

is

computed

in

equation (3.7).

JOURNAL DE PHYSIQUE I T I, M 9, SEPTEMBRE I991 49

(15)

Thus the

broadening

of levels in the braids is controlled

by exp(- (Im _S~( /y),

while the

frequency

of the oscillations observed in

figure

I is

given by

Re

(S~/y).

To evaluate this formula _we need the action

integrals

Im

S~(E~(y))

and Re

S~(E~(y)).

The

specific

form of the

Hamiltonian, namely H~p,

_q

)

=

f~p

₊

f (q )

=

E allows to

compute

real and

complex paths

in the energy shell

explicitly (see Appendix).

We get ^four ^solutions ^of-

p(q)

for each value _{of q}

corresponding

to the different

paths connecting neighbouring

wells.

The Riemann surface I

=

(q,

_p e

C~

_H

(q,

_p

= E

)

for 3

~ E

~ 2 is

computed through using

variables

f

₌ e'~ and

a~ ⁼ e'P For _e

=

1/2,

1is then

given by

the

polynomial equation

~4-2 ~3- _jE- _f(t)j ~2-2

~ ⁺ i

=

o

(4.3)

with

f(f)

=

f

₊

f (f~

₊

_f ~~).

Since the

degree

of the

polynomial

is four _we

get

^four 2

Riemann sheets. The

computation

of the _cuts is tedious but

elementary

and shows that _we

get

ten cuts.

Using

Hurwitz's theorem

[27]

it follows that the

genius

of this

manyfold

is 7. An

explicit

calculation of this manifold

permits

to check that indeed I admits 7 holes. It also follows from such _an

analysis

that several _non

homotopic paths

_are

contributing

to the action.

We have

neglected

all but the most natural

path

in _our calculation which _we believe is the shortest. The numerical result

suggests

^that ^this _guess ^is correct. We have not

investigated

the

topology

of this manifold further

[28].

4.2 COMPARISON WITH EXACT SPECTRUM. The

possibility

to compute

numerically

the

exact

(Sect. 2.2)

_as well _as the

approximated

_spectrum

(last section)

allows to test the accuracy of the semiclassical method for finite _y.

We _are faced with _two _sources of _error

limiting

the

precision.

The exact

spectrum

is _a

bandspectrum

while the

approximation

is

given by

_a

point spectrum.

So the

knowledge

of the

eigenenergies

is restricted to the

bandlength

which is

exponentially

small in _y. on the other hand the

computation

of the

approximated

spectrum ^involves a numerical

integration limiting

the

precision,

too. As _a result the effect of the _«

diagonal tunneling

_»

(I.e.

between contours A and

c~

_can not be

observed,

since the coefficient _r is of the _same order _as the bandwidth. This is connected to the fact that the

complex path

from A to C is of the _same

length

_as the _one

between

repeated

contours in different unit cells.

The observed

crossings

of braids _are real

crossings

due to the

argument given

in section 4.I

as

long

_as _we _can

neglect tunneling

between various cells. Since tiffs

tunneling

between cells does not

destroy

the rotational fourfold symmetry we suspect ^that braids _cross _anyway. A _test for it will be to compute ^the ^Chem numbers

corresponding

to the gaps from either side of the

crossing.

The

change

in the

sign

of these Chem numbers would be _a

proof

that real

crossings

indeed _occur.

If _one restricts the

comparison

to the

splitting

of

energies considering

(E~~~

E~~~)/2

and (E~~~

E~~~)/2

_one _can check the next

neighbour tunneling

inside _a unit cell.

Using

also the

mean

energies

E~~>^~~

= (E~~~ + E ~~~)/2 and E~~>^~~

= (E~~~ + E ~~~)/2 ^allow ^to ^test ^the harmonic

approximation.

The data of the exact

(no index)

and the

approximated (index w)

spectrum

are

presented

in table II for selected values of _y. In

figure

6a _we show the

comparison

for _a wide range of _y. Note that _r _can be estimated

numerically

from the difference E~~~~~ E~~>~~

and is found to be much smaller than the

splitting.

The harmonic

approximation

works

fairly

well up to _a ₌ ~

=

0.02.

Figure

6b shows that 2 _or

the

exponential damping

and the oscillation observed in the braid structure is well described

by

the above

theory

of

tunneling.

The

qualitative _agreement

with formula

(4.2)

is

astonishing

well _up to _a

=

0.015.

(16)

N 9 BRAIDING IN A HARPER-LIKE MODEL J243

Table II.

CoJnparison of

seJniclassical and _exact spectruJn at selected _y.

p q y

( E(I)

_{E (2)~}

j~ ~(i)

_{~ (2)~}

j~ ~(3)

_{~ (4)~}

j~ ~(3)

_{~ (4)~}

/~

l 100 0.0628 6.04742 _x 10-6 5.97555 _x 10-~' 3.55317 _x 10-~' 3.32270

x 10-6

98 0.0641 7.05507 _x

10-6

_6.75341

x10-6

5.52107 _x

10-~

5.55445 _x

10-6

96 0.0654 10.9706 _x 10-6 10.6403

x10-6

3.25535 _x

10-6

3.44692 _x

10-6

92 0.0682 16.7132 _x

10-6

16.5907 _x

10-6

8.33658 _x

10-6

_7.74342

x

10-6

90 0.0698 17.7426 _x

10-6

16.9970 _x

10-6

15.9590 _x

10-6

_16,1280

x

10-6

85 0.0739 43.1212 _x 10-6 42.7613 _x 10-6 7.90317 _x

10-6

7.69656 _x 10-6

70 0.0897 276.175 _x

10-6

276.030 _x

10-6

326.767 _x

10-6

_425.853

x

10-6

39 0.1611 034.93 _x

10-6

106.16 _x 10-6 6 662.52 _x

10-6

8 309.58 _x 10-6

5. Senddassic _near q = 2.

In this section _we will discuss the observed spectrum near a =

~

=

We will however _not

2 _or 2

give

_a full detailed calculation.

Following

the method described in section

2.2,

the

original

Hamiltonian

H=2cos

(q)+2cos ~p)+2ecos (2q)+2ecos (2p)

is first transformed into :

H

=

e~~

Wj

+ e~P W~ +

e[e~~~ W)

+ e~~P

W(]

₊ h-c-

(5.I)

where

Wi= [~ ~j

= «j,

lI§= [~. j~j

= «~ with _«j, «~ and _«~, Pauli matrices

. i

0

~~ 0 -1

At _a

=

1/2

the

spectrum splits

into two bands

given by

_:

E=(q,p

=

2 _e

ices (2 q)

₊ _cos

(2p)1

_± 2

~/cos2 _(q)

₊

_{cos2 ~p)}

=

2(-

2 _e ₊ 2 Ed ±

Q~) _(5.2)

with A

=

cos~ (q)

₊

cos~ ~p).

E-

(q,

_p

)

is

represented

in

figure

8.

S-I AT A

=

2 _: LANDAU _LEVELS. In this

particular

case the band

edge corresponding

to A

= 2 is

given by E(

=

2(2

_s _±

/)

_reached _at _the _critical

_point _kj

~ ⁼

k~~

₌ ^0.

H is

quantized according

to section 3,I and here _we

simply

have to

apply

the

following

rule _p

-

kj~

+

/~ _Kj

_and

q -

k~

~

+

/~ _K~

_where

_y'

= y or

H

= s

[cos (2 11 _Kj)

+ cos

(2 11 _K2)11

+ 2 _C°S

(/~ _Ki)

"1 ⁺ ~ _~°~

~~ _~2~

"2 ~~"~~

with

1=[~ _~j

0

(17)

6

4

2

o

-4

~

0.8

0.4 0 0.2 0.4 0.6 0.8 1

Fig.

8.

Energy

shell at _a

=

1/2

and

e =

1/2.

Expanding

H _up to the order

y'~

_one _gets

:

_ ~

/ ,48/±1(2n+1)+~(16s±/)x

~~

~~~* _~

/

~~

x

Ii

₊

(2

_n ₊ I

)~l

±

j/ Y'~

₊ o

(y'3) _(5.4)

which describes the Landau levels _near the band

edge E(

=

2(2

_s ±

/).

5.2 A

=

0 _: _A DIRAC OPERATOR. Here the band

edge

differs from the

Et

=

4 _s and this value is obtained _near the critical

point ki

~ ⁼

k~~

₌ ^" ^therefore

2 the

quantization

rule is _p

-

"

+11 _Kj

_and

q -

"

+

llK~.

_The Hamiltonian is then

2 2

expressed

_as follows H

= e

jars (2 W _Kj

+ CDs

(2 W _K2)1

_' _{2 Sin}

(/~ _~i

"i ~ _"~

_~

"(j

~~

(18)

Expanding

H up ^to first order in

y'

one obtains _:

H=-4s-2$(Kj«i+K~«~)+4ey'(K)+K))+O(y'~/~). _(5.6)

This Hamiltonian _can be

easily diagonalized

because of its

complete equivalence

with the Dirac Hamiltonian

[16].

Following

the strategy

previously developed [lib]

let _us _set

Ho

=

Kj

_«j +

K~

_«~

=

~

j

then

Kj

+

iK~

0

H(

=

K)

₊

K(

+

I«~[Kj, K~]

=

K)

₊

K(

_«~

(5.7)

The

eigenvalues

of

H(

_are 2 _n and 2 _n ₊ 2 at this order in

y'.

The

perturbation

calculation

performed

_on

^H~ leads,

_up to order

y'~,

_to _the _final _result

:

Et

=

-4s±

(2ny')~/~ (l ^~'~ )~~~+4sy'(2n+1) ^ey'~[1+ ^(2n+1)~]. ^(5.8)

2

5.3

QUANTIzATION

FAR FROM THE WELLS, A

= ~.

This case

gives

rise to a line of 16 _e

critical

points

defined

by

A

=

cos~ (q)

₊

cos~ ~p)

= ~

(see Fig.

8 for the _case _s

=

1/2).

16 _s

Since _we _are not at a bottom well _we cannot _a

priori

^follow ^the ^results ^of ^section ^3.

However _a look at

figure

9 shows that the

spectrum

^looks very

much

like the

spectrum

^of a

spectrum

of square Lattice with

2.Neighbours(£=1/2)

0.56

,

-

-~i ~ ~~ -~ i-

~~~ _~ - ₌ - ~ ~ _~'

~.~~ ~ , ,'~, ~~ _{%_} '

©Z~

~i

₉

~

i$

£

_-

~i _'~~@~~, ~ "/H~ _. ' _~'.-

' ' > _' ~'' .

" "' ~" ~ '

W _~ k'k '

~ ~' ' @ _$ _~~. '~~ ~ '

~ " ' ' ~ ~

~ l# 'il ( '~~ i / 1-

o ~ '~W

~ . ~o # ~ ' $ .

~ WI ~ ~h / %

) ~ m. ~ "-*" +'

' ~ _'~~ ''~

~~', '

" >

~ . - 0 -

~

^~

^~ ^~ ^~ ^~,

^~$

~ ~ "

. *~ -M' - - ~.-

-

_* _-' _-

~W .

=

'

~

~ ~

~~ ~

"' '~

" "

"-

,

"~,-

o-s

-2.4

-2.2

Fig.

(19)

quantized

version

A'~

_{of A}

(namely

the Hofstadter

spectrum),

folded

through

the relation

(5.2).

So let _us first _use the semiclassical

expansion given

in section 3 for

A'~

near

A'~

=

0, namely

A~

=

y'(K)

₊

K()

^~

^~ (K)

₊

K()

₊ O

(y'~) _(5.9)

The

eigenvalues

_e~ of

A'~

are then

given by

e~ =

y'(2n+1)-~[l

+

(2n+1)~]+O(y'~). _(5.10)

With

that,

_we _are able to compute ^the

corresponding eigenvalue Ep

of the initial Hamiltonian

using equation (5.2),

Ep =-4s+4se~-2/. _(5.ll)

In

figure

10 _we

computed Ep

for different values of _n. These _curves fit

qualitatively

well with the

picture

of the exact spectrum near « =

1/2

in

figure

9.

En

~ -0.1 -0.08 -0.06 -0.04 -0.02

Fig.

10. Senfidassical result

Ej(y)

for _n

= 1, 2,

,

7.

However,

_we _see that the exact

spectrum

^for s

=1/2 gives

values of energy below the minimal value

ES

=

5/2 permitted by

formula

(5,ll).

This is due to the next order correction of the effective Hamiltonian.

Following

the work

by

Helffer and

Sj6strand [16]

_one _can represent the Hamiltonian H

by

_mean of the

quantization

of _a classical function in the form _:

H

=

H~°~

₊

Y'H~~~

₊

y'~

H~~~ ₊ _O

(y'~) _(5.12)

(20)

where

H~°~= -2(-2

s+2 Ed

Q~).

_The _rules

_given

_in

_[16] _permit

_to

_compute

H~~~ which is here

H~~~(q,

_p ₌ 4 _s sin

(q)

sin

~p) (5.13)

It is called the first

subprincipal symbol.

Using perturbation theory,

_we _can in

principle compute

corrections due to H~~~,

H~~~ The order

y'

correction due to H~~~

actually

vanishes. So _we need to

compute

H~~~ _to _compare with the

picture.

The calculation of H~~~ is much harder and _we did not

perform

it. Thus _we cannot conclude. However it is to be noticed that the second order correction to the energy comes from two terms : from the first order

perturbation

correction due to

H~~)

and from the second order correction due to H~~~ which has the form

,~

(n

_(H~~~(

Jn)

_(~

~~~

_~

i

~(o) ~jo)

If

Ej°~

₌ ²

(-

2 _e ₊ 2 _se

~

/)

_is _the _level _such _that

_E)°~

~

E(°~

for any Jn # n at the

given

value of

y',

then

8E~

_~

0, showing

that this term

produces

_a correction with the

right sign.

5.4 SOME _WORDS _ABOUT _THE _MINIWELLS. In fact H~~~ breaks the invariance

along

the

family

of _curves A

= const. This will

give

rise to local minima in the effective operator ; these local minima _are what Helffer and

Sj6strand

called _« miniwells _».

Following

their

th~

^these

miniwells should create their _own

system

^of ^Landau ^levels ^if we

replace y' by y' [29].

Moreover,

because of rotation symmetry ^these ^Landau ^levels ^should

give

rise to minibraids

by tunneling

effect. We _are not able to _see these minibraids _on the

picture

and this is

probably

because the values of

y'

for which it should be observed _are too small to be _seen in _our calculation.

6. Conclusion.

It has been

numerically

verified that Wilkinson's extention of WKB

applies

also to _a

tunneling phenomenon

within _a lattice unit cell. The

resulting

canonical invariant formula

(4.2)

describes the spectrum

quite

well for small

magnetic

fields.

It has also been demonstrated that the second

neighbour

Hofstadter Hamiltonian possesses

an

interesting

behaviour in tiffs kind of models _as the interaction

parameter

_s ^is ^varied.

Seeing

the Hamiltonian _as _a member of _an abstract _non commutative

C*-Algebra

is found

to be very

helpful

in

determining

various

aspects

^of ^the quantum ^behaviour

using

all the different

representations

mentioned above.

Especially

the

comparison

of the classical

picture

and the

algebraic representation

localized

near critical

points

in

phase

_space allows _a

prediction

of the structure of the

spectrum.

The semiclassical

expansion

_near the bottom wells in power of _y has

already

been studied

rigorously

both in the

algebraic

framework

[11, 13]

^and ^with

pseudodifferential operators techniques [16, 30,

3

Ii.

The WKB method has also been the

subject

of many

rigorous works,

even for the _case of

Harper's

model

[16]. However,

the

pseudodifferential

_operators

approaches

used up to now involve

breaking

the

phase

_space symmetry. ^This

example

shows that there is _a need for _a

phase

_space treatment of the

tunneling

effect. To get

general

results

one should

develop

a

rigorous

^version ^for the localization of the Hamiltonian to

phase

_space

regions

_near the considered classical