Download Semiclassical methods in solid state physics : two examples

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>. <jpa00246736>
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Submitted on 1 Jan 1993
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j_
pJ1ys. I France
(1993)
3
471-499
1993,
FEBRUARY
471
PAGE
Classification
Physics Abs
tracts
71.25.C
75.10
03.65F
Semiclassical
methods
Bellissard
Jean
Laboratoire
Physique
de
(Received
22
solid
Armelle
and
Cedex,
Toulouse
in
physics
state
examples
two
:
Barelli
Quantique
Universit6
Paul
118,
Sabatier
Narbonne
de
route
F-31062
France
July 1992,
accepted
in
final
form
August 1992)
9
de deux problbmes
motiv6s
l'6tude
des
modbles
bidiest
une
revue
par
supraconductivitd
I haute
temp6rature critique. La premibre partie conpour
l'6tude
du
des
dlectrons
Bloch
de
bidimensionnels
soumis £ un
spectre d'6nergie
cerne
pour
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
illustration.
une
Rdsumd.
Ce
mensionnels
travail
la
Abstract.
We
here
present
a
review
of
problems
two
superconductivity.
The first part
the
concerns
energy
uniform
magnetic field. A semiclassical
analysis provides
understanding of this spectrum.
second
In the
part we
fermion
"Quantum
Chaos"
strongly
correlated
to
systems.
Tc
distribution
for
the t
J
model
in
two
a
by
motivated
spectrum
qualitative
make
It is
the
2D
of 2D
as
case
illustrated
models
Bloch
well
for
as
the
by the
for
electrons
a
high
in
a
quantitative
application of
level spacing
dimensions.
the
Paris
meeting
each
other.
Some
Statistical
Mechanics in January
on
similarity
between
Rammal's
work on
us
dimensional
almost
studied
the Sierpinski lattice [75] and
periodic
models
by
Bessis,
some
one
why
he
wanted
make
Moussa and myself [15] was the
acquaintance.
to
reason
my
During the short discussion we had together at this meeting, it became immediately clear
had both got our degree in a French
that we had two points in
university, and
common
:
we
1met
R.
Rammal
1983.
Pierre
both
worked
for
the
first
Moussa
introduced
hard
understand
time
at
to
of the
Hofstadter
structure
spectrum [55].
hardship
of
being
outsiders
the French
instituin
research
we
dominated
by the smart but arrogant
"Grandes
alumni.
Ecoles"
tions heavily
both
fascinated
by the complexity and the importance of magnetic
On the second, we
were
field effects in Solid State Physics.
metal networks, the quantum
Bloch
electrons in a magnetic field, superconductor or normal
Hall effect,
electronic
properties of quasicrystals, high Tc superconductors and quantum chaos
On
the
first
to
point,
shared
the
the
PHYSIQUE
DE
JOURNAL
472
N°2
I
dhcussions
the years since then. Not
of our frequent and continuing
over
professional
and
private problems as well.
subjects
daily
life,
politics
of
matters,
to
underlied
his
Scientific ambition, a strong need for affection and a wounded pride secretly
after years of
heartbreak
discussion
life. Rammal died on a Friday in the middle of a scientific
medical
for his country falling apart, and years of acute physical pain, despite the hope that
technology gave him for the better.
have
the
been
focus
the
mention
Bellksard
Jean
Toulouse, July 1992
electrons
BlocI~
1.
motion
Electronic
in
in
a
magnetic
a
field.
crystal subject
to
magnetic
a
has
field
been
one
of the
Physics. A magnetic field breaks the time reversal
symmetry and
the microscopic
of a crystal. The Hall effect, the de Ham van Alphen
structure
diamagnetism in metals, the Meissner effect, flux quantization of vortices in
oscillations
disordered
in slightly
and
recently the
magnetoresistance
more
quantum Hall effect, are examples of the complexity a magnetic field introduces
Solid
State
main
reveals
topics
in
details
of
effect, electronic
superconductors
metals, and the
in homogeneous
materials.
first study of such problems goes back to Landau in 1930 [58] for the motion of electrons
approximation. It was soon followed by the work of Peierls
metal, in the effective
mass
The
in
a
Bloch
[72] for
systematic
electrons.
translation
operators
[94]
until
took
it
But
study of the problem.
The
generating
are
fifties, with a
coming
the
main
remark
the
effective
by Luttinger [64]
paper
works
of these
out
Hamiltonian
is
describing
to
start
a
that
magnetic
the
electronic
independent electron approximation.
The case of two
dimensional
systems in
perpendicular
uniform
magnetic field became increasingly important after the development
a
of electronic
devices
such as
MOSFETS
heterojunctions [2]. It is nowadays possible to get
or
metallic
thin films 200 1
thick. Many of the afore
mentioned
effects
actually enhanced in
motion
in
the
are
dimensions.
two
recently,
discovery
high Tc
superconductors
in copper-oxides [12] led several
Cooper pairs, lies probably in the
planes of
dimensions.
Moreover, the flux
two
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
More
experts
to
the
that the
propose
the
atoms,
copper
of
leading
reducing the problem to
main
mechanism
to
field.
If
denote
we
unitary
by
operators
U
V the
and
fulfilling
the
two
magnetic
following
UV
#/#o is the
hle (h is
quantum lo
Here,
o
=
ratio
=
To
describe
corresponding
the
to
participate
energy
crossing the Fermi
band.
Let
By Bloch
electronic
the
between
Planck's
absence
motion
of
the
constant
a
describing
translations
commutation
rules
such
a
they
system,
e~~'"VU
(I)
=
magnetic
and
e
the
are
:
flux
# through the
electron
unit
cell
and
the
flux
charge).
in a crystal, let us
consider
magnetic field. Only those
first
the
energy
case
levels
for
near
which
the
a
=
0
Fermi
conduction, so that we can
restrict
ourselves
to a finite
set of bands
restrict
For simplicity, let us
ourselves
of only one such
to the
case
E(k) (k (ki, k2) being the quasimomentum) be the corresponding band function.
Peierls then proposed
theory, E(k) is periodic with respect to the reciprocal lattice.
in
surface.
=
N°2
SEMICLASSICAL
replace
to
k
(ki, k2) by
=
dimensionless
the
K»
METHODS
operator
) I"h£
=
SOLID
IN
K
STATE
(Ki, K~),
=
A»)
473
where
eA»)
(P»
COnSt
=
PHYSICS
,
»
expression
the
in
nian
to
get
the
are
the
lattice
rigorous justification
The
corrections
always
to the
admits
Peierls
a
hamiltonian.
Here
spacings in each
A
(Ai,A~)
=
of this
Peierls
Hamiltonian.
substitution
representation
in the
He~
~j
been
has
whatever
But
is
the
vector
direction.
e~~i and e~~? satisfy the
commutation
relation (I), the
series
in
the
convergent power
operators U and V.
as
a
that
written
be
can
=
effective
E
1, 2)
of
potential, a~ (p
Remembering
[IS, 85]
in
done
corrections, the
the
Peierls
Hamiltc-
leads
and
to
Hamiltonian
effective
form
(2)
v~~~~~~~~~~~
hmi,1112(°)U~~
"
'
(~l~i,~1~~)~zz
hm~,m~(al's
where
simplest
The
in
terms
(2).
gives
It
smooth
are
case
functions
in
rise
the
to
analogy
(I)
between
and
at
the
This
a.
usually
series
lattice
square
a
u~~
u +
"
jQ, pi
ignore
we
namely [49]
absolutely.
the higher order
:
v~~
+ v +
commutation
canonical
converges
which
for
"Harper model"
sc-called
HHarper
The
of
looking
consists
(3)
(where
relations
h
h/2x),
=
(4)
;h
=
,
main
and
thus
We
crystal
10 T
we
is that
field
and
have
realized
with
lattice
get
plays
a
fixed
a
varied
be
tunable
a
of
11,
is
which
is
operators,
of
once
and for
an
if
realized
effective
role
we
replace
Planck
all, whereas
a
is
by e~~
U
constant
V
,
(I).
in
proportional
l'he
the
to
parameter.
which
system in
order
of
of10~~,
order
the
parameter
like
concrete
a
spacing
of the
a
h is
can
P
momentum
2xo
difference
magnetic
a
position Q and the
by h. Therefore
the
between
by e~Q
and
one
can
quite small
the
test
justifies
and
methods.
semiclassical
magnetic
realistic
strong
a
field
of
of
approach.
semiclassical
a
In
order
artificially a network with lattice spacing of the order of lpm, in a magnetic
indeed built by the Grenoble
get o te I. Such networks
group
were
we
can
by
superconductors.
The
de
Gennes
Alexander
of
[68, 69]
[I] theory for such
[34] and
mean
networks
Hamiltonian
permits to relate in a simple way the groundstate energy of the Harper
in
the
metal-superconductor
transition
temperature-magnetic
field
(3) to the normal
curve
phase space. In this problem however the electron charge e must be replaced by the Cooper
pair charge 2e, in lo.
However
if
field of few
build
we
Gauss,
ALGEBRAIC
1.I
which
o
=
representation
where
u
and
is
unitaries
are
number.
always
can
we
v
rational
a
such
u~
Substituting
matrix
in
which
the
U
and
U~
(I)
V
V
by
mean
in considering
commuting. In an
consists
are
of q
x
q
matrices
the
e~~iu
for
case
irreducible
and
e'~?v
that
=
v~
=
I
uu
expression (2) for the effective
numerically diagonalized on
be
can
Then
represent
to
and
approach
Another
APPROACH.
p/q
e~~'P'~vu
(5)
=
Hamiltonian,
a
computer,
we
get
giving
a
rise
k-dependent q x
to a family of
subbands.
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I
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for k~
I). As
0
=
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or
1.1,:'.
',
-1.0
diagonalization
2x.
0.5
0.0
-0.5
1.0
2.0
1.5
.,
3.0
3.5
edges
are
2.5
(Hofstadter's butterfly).
model
mod
..
.:
-1.5
Harper's
Harper's model, this
For
tained
of
.,,i>11"
,.,.
-2.0
-3.5
,
~~
~,.
'll.,
/
Fig-I-
It
is
especially simple
rise
gave
famous
the
to
for
subband
the
one
can
see
from
A unit
I is
an
element
of A
such
C*-algebra has not necessarily
enlarge a C*-algebra by adding
two C*-algebras A and B, is a
that
aI
=
=
one.
involution
a
that
structure
if p is
a
is
actually
*-isomorphism
very
Ia
such
map
from
natural
from
A to B
Banach
a
=
,
that
(6)
Va e A.
Therefore
from
unit
exists it is unique.
A
p(ab)
*-homomorphism
=
then
(6) ))I)
One
A
p
p(a)p(b), p(a*)
points of view.
purelj~ algebraic,
is
First of all, the topology generated by this
norm
concept). This
(a
of
purely
algebraic
a*a
spectral
radius
the
is
[[a[)~
This
This
lla*all
any unit, but if some
unit by brut force.
a
linear
flux
the
=
llall~
ob-
butterfly [55] (see
the magnetic flux varies,
Hofstadter's
figure I, the eigenvalue spectrum as
of
high
exhibits a fractal
complexity.
structure
difficulty, in this numerical approach, is to show that whenever
The first
mathematical
becomes
irrational, one can approximate the spectrum by mean of the rational
o
be
done
using C*-algebra techniques (see [71]).
can
with a bilinear
product, and an antilinear
A C*-algebra A is a complex vector
space,
A
has
A
A
b*a*.
for which it is
a*
satisfying
(ab)*
Moreover
E
E
a
norm
a
-(namely
converges)
and
satisfies
each
Cauchy
space
sequence
Fig.
/
~
./6Y'
~3"
'NAG.
lk
W=.,_.)""'
=.=W
=
--
can
B
=
I.
A
always
between
p(a)*.
two
p is
norm
because
is
preserving.
(6)
confirmed
Thus
the
tells
that
us
by the fact
norm
is
an
N°2
algebraic
antilinear
of A
be
which
can
Quantum
in
It
is the
be
can
in
generated by
two
algebra"
subset
compact
the
(2)
form
number
a
the
hm~,m~'s
I
e
Given
The
A
the
For
in
of
show
number
continuous
unit.
a
E
a
define
to
In
such
"c"
of
functions
certain
a
C*-algebra AI
a
such
C
H
that
sense
AI
pa
--
o
a*a
0 <
I and
on
AI
Ao
with
has
a
cI.
<
vanish
be
can
seen
which
numerical
inverse
calculation
all
7
Many
other
trum,
--
represent
use
This
of
is
can
is
set
:
where
with
continuous
is
I
respect
compact
a
to
a
namely
subset
its
gap
of R.
edges
~
I)
+
~l(n
+
U
and
At
last,
V
on
one
1) + 2
also
can
be
semiclassical
done
setting
e'~~~+w~~~
x(U)
in
Namely
space.
and of
space,
a
representation
a
*-homomorphism
2D
them
-I).
=
E~I(n)
lattice, by
on
then, that
see
nothing but the
to
is
(7)
mean
of
L~(R) by e~w~i
the
discrete
and
e~W~?
"
useful
7
the
on
will
we
2xo
=
practical
for
x(v)
=
)#i(n)
a
represent
and Ill
analysis namely
follows
as
on
forward
To go
ANALYSIS.
the
2x(z
cos
f~(Z~) namely
2xa, 1(2 is the position operator
representations do exist and are
will
we
0.
the
SpA(a)
=
SEMICLASSICAL
1.2
a
=
all
C*-algebra of bounded operators on 7i;
algebra several representations have physical meaning. For instance, Harper
£~(Z), namely the Hilbert space of quantum states
the following one
7i,
chain, and the operator x(U) is the translation by one, whereas x(V) is the
translations.
where
evaluating
the
~l(n
also
finite
a
Actually
in A.
"
can
in
consists
multiplication by e~~'(~~°'~) if N is the position operator. It is not difficult
Schr6dinger equation x(HHarper)~l
E~I, where HHarper is given by (3)
Harper equation
magnetic
for
a.
the
We
but
Uo~ IAn.
as
spectrum
element
of AI,
element
is
I, its
unit
a
no
represented by operators in a Hilbert
7i,, called the representation
space
be
into
rotation
zI
self adjoint
a
Sp A (pa(H))
=
Hilbert
a
that
the
H*
=
E(a)
can
1955 [49] used
discrete
lD
a
on
been
are
positive elements, namely those elements
A. This positivity property is essential
and groundstates for
instance.
Then,
energy
some
*-homomorphism
functions of
B(7i,)
--
can
justifying
C*-algebra
data
one
in
a
e
z
spectrum
continuous
x
475
unique largest C*-algebra Au (up to isomorphism)
a
U and V satisfying (I). Au contains a unit. It is called
defined and studied by Rieflel [78]. Moreover, given any
that
there is a C*-algebra AI generated by elements of
is
unitaries
had
natural
a
Let
i
is the
PHYSICS
o.
result
the
A
x
is
at
element
Theorem
are
STATE
define
to
a*a for
negative
there
the hm~,m~
AI has always
numbers
main
Then
where
an
complex
permits
it
non
and
R,
I of
there
for
that
"abstract"
them.
of
form
that
[78]
the
"'rotation
in the
smallest
shown
permits
involution
written
Mechanics
A, ))a II
E
a
SOLID
IN
invariant.
Moreover, the
if
METHODS
SEMICLASSICAL
we
purposes.
knowledge of the Hofstadter
specdevelop an asymptotic calculus as
the representation
use
e~~~?+w~?~
(8)
=
,
and
If k
expand
=
the
(ki, k~)
gets for H
an
effective
is
chosen
expression
(3)
Hamiltonian
a
maximum
of the
form
as
H
=
E(kc)
+
or
in
a
powers
of
minimum
@.
k
I(fif~~ )~vI(~ltv
=
+
kc of the
band
function
E(k),
one
(9)
O(7~~~)
,
JOURNAL
47G
PIIYSIQUE
DE
I
N°2
Energy
-2
~
/
/
/
~~'~~
"
o
'
-2.5
-2.75
-3
-3.25
3
5
-3.75
Fig.2.
between
Comparison
exact
spectrum
model; points are
extracted
from
formulae given by Equation (11).
semiclassical
Harper
the
M~v
where
is
a
the
effective
(n EN)
effective
the
(resp.
minimum
matrix
mass
maximum).
near
semiclassical
numerical
the
formulae
and
extremum
for
(resp.
+
O(7) term is nothing but
The
corresponding eigenvalues
:
En
E(kc)
=
+
j ~~j+j~
+
Landau
the
full
spectrum,
exact
The
approximation.
mass
and
the
~~~
i,1
@,I»I
o-fit,
o.04
o.02
o
curves
-)
easy
in
chosen
for
Hamiltonian
in
is
Landau's
are
levels
represent
to
namely
get,
o(72)
(lo)
,
1
where
The
m2
ml,
O(7~)
nomial
can
=
7(2" +1)
+
expression
an
indeed, using the
the
and n E N labels the corresponding
levels.
Landau
(9) by perturbation theory. In particular if H is polyexpansion of En in (10) at every order in 7. In particular
levels are given by (see Fig. 2)
of M,
obtained
from
V, one can get an
equation, the Landau
4
Et
Such
eigenvalues
be
in U and
Harper's
for
the
are
term
can
p/q
of
)~ ll
u
and
Mq(C)
©
+
(2~
+ 1)~l
fit"~
+
("
+
l)~l
(7~)j
(11)
+
generalized near each rational value p/q e Q of a. For
eq.(5), and the generators Up
pp(U) and Vp
pp(V),
v
V'
Ap generated by U'
and
is
isomorphic
©
©
Up
Vp
u
v
actually
matrices
sub-C*-algebra
+
2
be
in
=
=
=
=
the effective
Hamiltonian
pa (H) can be
Ap.
In
much
the
expansion
get
seen
as a q x q
same
way we
an
near
6
2xfl te 0, if we replace Up and Vp by e~(~i+W~i) and e~(~?+W~?) and developing in power8
E(k) is replaced by a matrix valued
"band
function"
difference is that for 6
0 the
of li. The
Ei(k),
subbands
Eq(k), which
of q
functions 7i(k). Diagonalizing 7i(k) gives a
sequence
Correspondingly we get k-dependent
gives the spectrum of pp/q(H), whenever we vary k.
I,
eigenstates )I)k I
q.
Ao if
to
o
=
+
fl.
Therefore
kp
matrix
with
in
this
elements
representation,
in
=
=
,
=
,
,
SEMICLASSICAL
N°2
Such
a
T~
C~; k
x
"classical"
describe
To
maximum
the
the
or
is
the
bundle
near
value
of E;.
The
(I)k_k~, Q;
eigenvalue E of kp
(<lfip4
now
in
fl;(k)
bundle.
((k, )I)~)
the
Here
e
subband.
splits into
The
of the
band
rise
to
subbands.
the
for
two
Q;fiplo
~
in
Mq(C).
equation
:
jip(E)
expansion of
an
(12)
,
matrix
unit
eigenvalue of
=
li gives
of
(I)kk(I(.
(<lfipo;
E; (kc) is then given by the
near
mass
term
+
(I) (ii and Iq is the
Iq
=
powers
effective
O(6)
the
However
fiber
a
the i~~
~~
E
M is the
in
477
formula
=
Expanding (12)
motion
to
PHYSICS
STATE
edge, we pick up a band (namely we choose I e [I, q]),
E;(kc) is the band edge we consider.
Then kc is a
order to reduce the problem to the case a -- 0, we replace
Hamiltonian
by mean of a standard
by an effective band
In
=
ii)
SOLID
which
kp
tip(z)
where
IN
band
a
Hamiltonian
to
associated
for
of k
minimum
a
complement
Schur
a
valued
matrix
line
spectrum
kc be
let
and
corresponds
mechanics
T~)
e
METHODS
So
we
the
Landau
get
form
same
sublevels
at
as
eq.(9)
each
where
band
edge.
pieces
first term is proportional to (b( and gives rise to a discontinuity
p/q (I.e. 6
edges (namely for n
0). Notice that
0) at o
@@ is the density of states at the band edge.
introduced
of the i~~ bundle
The other
for the
term is proportional to 6 and
accounts
curvature
~(k). It produces an asymmetry of the derivative of the band edges
Hamiltonian
by the matrix
p/q (see Fig. 3).
near
o
Rammal
Formula (13) was
written for the first time by Wilkinson [92] for the Harper model.
fiE;,n(6)
/fi6
the
magnetization
derivative
Harper's
model
and
related
the
rederived
it
for
to
[76]
theory. The formula (13) was derived in full
network, using Abrikosov's
of a superconducting
generality (namely for any model given by Eq.(7)) in [13] (see also [51, 77]) where it was called
where
=
derivative
of the
=
=
=
=
the
Wilkinson-Rammal
1.3
BANDTOUCHING.
arated
from
of the
conical
jip(z)
contact
The
bands
touch
at k
them.
between
E;(kc)
whenever
is valid Only at a band edge sepsimple eigenvalue of ~(kc). On the
formula
Wilkinson-Rammal
namely
bands,
other
if two
contrary,
formula.
is
a
kc, the asymptotic is
=
Whenever
bands
two
different
touch
(see Fig. 4). Degenerate perturbation theory
corresponding
in (12) must be a 2 x 2 matrix
1<1(<1+ 1<'1("1
=
each
other,
indicates
the
to
depends
and
that
the
contact
is
eigenprojection
order
generically
Hamiltonian
effective
the
the
upon
:
fl
,
if
Ei(kc)
One
choice
can
of
=
E;>(kc),
show
that
I'
I and
lowest
the
coordinates) by
jip(z)
being
a
=
Dirac
the
order
labels
term
of the
in
Vi
and kc the
is
conical
given (after
a
point.
suitable
namely
operator,
E; (kc +
the
touching bands,
jip(z) expansion
l~,
'1
~
~~~
~~
2
~'~~~
+
O(d)
(14)
JOURNAL
478
PHYSIQUE I
DE
N°2
jliflfl
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....;,I:'...
.,
-1.2
-1.0
Fig-S-
Asymmetry
Thus
Landau
-0.8
-0.6
of the
central
sublevels
are
-0.2
replaced by
Ez,~z
where
n
o
e Z is the
Harper's
For
p/q,
=
what
we
can
Ef~
generic touching
Playing with such
shown
a
1/2
=
had
of
a
local
permits
the
same
in
the
in
way,
Dirac
a
subband
principle
to
the
near
compute
levels
reveal
produce a special packing
understand
[52]. It permits to
may
not
near
a
1/3.
=
O(6)
+
where
vanish
[88].
this
the
the
Dirac
the
*)
(l
is
gap
no
producing
(15)
levels
0. In particular if
bandtouching. This is
2x(a
1/2))
given by (6
E
at
=
conical
a
are
=
~
O((6(~'~)
+
2
n
(16)
> 0
in [9].
analysis permits
system.
subband.
also
spectrum
studied
semiclassical
a
extremum
of
curvature
been
there
in [17] that
gap is closed
middle
+2(2n(6()~'~
=
underlying
bands
classical
model
namely
24@sgn(n)
+
'
Non
Harper
,
was
the
q is even,
in figure 4 at
see
the
levels
Dirac
E; (kc)
=
for
1.0
label.
Dirac
model, it
where
edges
band
0.6
0.4
A
bunch
of
The
slope
of
higher
existence
of
permits
of
a
conical
of
sublevels.
This
has
in
particular
why
the
been
of the
Hausdorfl
reveal
to
this
the
in
At
last
power
of
In
saddle
and
the
of the
existence
function.
Helfler
dimension
the
measure
level
subband
bandtouching.
studied by
topology
the
will
levels
derivatives
order
reveal
to
sublevels
expansion
these
The
extremum.
actually
Landau
local
of 6
much
points
Sj6strand
Hofstadter
METHODS
SEMICLASSICAL
N°2
IN
SOLID
STATE
PHYSICS
479
L
o
r
ki
g
~
k2
r
FigA.
Conical
between
contact
bands
half
in
the
previous analysis
be produced by
the
at
flux
Harper
model.
for exponentially
small
not
account
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
exponentially small correction as a or a p/q -- 0. This broadening has been described by
an
Wilkinson [92] and Helfler-Sj6strand [52].
TUNNELING
1.4
in
terms
a
The
EFFECT.
(or
p/q)
a
which
may
does
=
Before going into it, let us restrict
ourselves
degenerate wells in the cell of periods of the
phenomenon
for the following model
appears
the simpler case for which we have classically
Such a
reciprocal lattice, close to each other.
studied by Wilkinson [92] and
Barelli-Kreft
[10]
to
Hw=U+U~~+V+V~~+t2(U~+U~~+V~+V~~),
with12
>
mod
x
2x
1/4.
Indeed, if12
which
bifurcates
symmetric by a fourfold
classical degeneracy. Thus
<
at
12
rotation
the
(17)
1/4, Hw admits a unique classical
minimum
1/4 into a local maximum and gives rise
"
ki
around
k2
"
corresponding
"
Landau
x.
This
sublevels
to
ki
"
four
produces
symmetry
are
at
k2
"
minima
an
exactly degenerate
exact
modulo
O(tY").
To
4
x
4
describe
effective
the
tunneling
Hamiltonian
effect
for each
between
value
these
of the
four
Landau
we
replace
quantum
number
wells,
the
n.
Hamiltonian
Using
the
by
rotation
a
JOURNAL
480
matrix
this
symmetry,
admits
PHYSIQUE
DE
following form
the
~"~7~~
t
En (7) h the n~~ Landau
techniques [50, 93], one
where
Sopp)
neighbouring
between
where
il
?
r
computed
sublevel
r
~
t
WKB
N°2
:
0
~~~~~~~
I
0
before,
as
(resp. r)
~~~~
'
r
f E
=
wells
t
in
and t e
R
tunneling
(resp, diagonally opposite wells) in the
write
can
action8
of
terms
form
C.
Sn_n_
Using
(re8p.
:
w
corrections.
aslov's
It
that
out
turns
the Schr6dinger
contrary to
terms
the
in
>
(Imsopp(
the
case,
E~
of
level splitting
+~de-lIm(sn_n )jH
~~~~-(lm(Sn n,)(/~
=
~
r
that
part
of
pression
so
(Imsnn.(
real
becomes
7
when
~~~
~i~
to
compare
negligible
varies.
(~~(s~
/~
/
~
(fi~( s
~
i)
~
#)
(~°)
corresponding Landau level splits into four levels like a braid as 7 varies
phenomenon has been also observed with Dirac sublevels [9]. In
same
sented the braiding of Dirac
sublevels
produced by the touching of two bands
points with an exact fourfold symmetry around the fifth one. One can see on
formulae fit quite well with the diagonalization
WKB
method.
The
The
broadening
The
reciprocal
symmetry
replace
the
Hilbert
each
seen
the
the
then
Hamiltonian
Since
each
acting on f~(Z~) by
corresponding flux, one
as
scale
invariance
emphasized the role of the
in [57]) precisely generates
If H
trix
has
fourfold
the
of Htunnei
element
Hamiltonian
S'
>
this
a
effective
of
Ao>
e
proposed
expansion of
by using
way
same
of the
fourfold
the
rotation
Htunnei acting in
Hamiltonian
given quantum
translations.
was
fraction
the
instead
(see Fig. 5).
figure 6 is reprealong five conical
figure 7 that the
number
associated
n
lattice, Ht~nn~j can be
a
However, when one
computes
2D
with
for
the
a.
The
a'
first
lla
=
time
Gauss
mod
by
a'
map
instead
I
Azbel
--
[6] who
I
lo (see
expansion.
action
should
original lattice,
of the
wells
located
between
the
two
be
accounted
so
sites
at
wells
I
and
does
and
I'.
This
leads
+ Va> +
VJ~)
for.
Htunn~j. Moreover the maI' is proportional to e~~"'/~
So that as 7 -- 0 only the
to
the
following
approximate
:
Htunnei
This
continuous
much
point in
Htunn~j
now
spectrum
symmetry
tunneling
neighbouring wells
nearest
of the
between
Sill is the
where
that
see
can
magnetic
of
mean
by
labelled
is
in
space
from, by an
eigenstates
Landau
well
described
k-phase
the
in
started
we
the
such
be
can
symmetry
generated by
space
well.
[92, 52]. This
with
sublevels
group
used above.
We
to
Landau
of
translation
t.
does
n_n
"
7~e~'~~'~'H
(Uo> +
UJ~
+ O
(e~~'/~)
(21)
,
ImS.
formula
Hamiltonian,
shows
that
properly
the
broadening
renormalized;
thus
of each
the
Landau
actually given by
spectrum is again the
sublevel
corresponding
is
a
Harper
Hofstadter
SEMICLASSICAL
N°2
spectrum
METHODS
Square
of
SOLID
IN
Lattice
PHYSICS
,
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j
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[
I
o
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-3
-2.6
-2.4
Energy
Fig-b-
Braid
Repeating this
one.
Landau
sublevels
in
a
Harper-like
model
with
second
-2
nearest
neighbour
fractal
structure
(from [io]).
interaction
of the
for
structure
-2.2
construction
again
again gives
and
rise in
principle
to
the
spectrum.
sublevels.
previous analysis works only for well separated Landau
Near the band
have
ideas
and
they
shown that
and Sj6strand have developed the
kind
of
same
renormalization
analysis can be locally described by mean
for Harper's model, each step of the
standard
forms, one of them being the Harper model, the others taking the possible
of four
However
the
Helfler
centers,
bandtouchings
The
or
into
The
main
Theorem
result
We
(The
remark
Ten
of
Helfler
>
with
an,
performed by
and
0,
an
Kerdelhud
Sj6strand
such
that
> C Vn,
is
if
o
then
[56] for the
contained
admits
the
in the
Harper
following
continuous
a
model
on
theorem
fraction
spectrum E(a) of Harper's
a
triangular
[52]
expansion of the
equation has a
,
measure.
however
Martini
irrational
been
is C
There
2
form [al,a2,
Lebesgve
zero
any
account.
analysis has
same
honeycomb lattice.
a.
that
if C > I, the set of such a's
of Kac [17]) is that E(a)
problem
has
zero
should
Lebesgue
have
a
zero
measure.
Lebesgue
A
conjecture
measure
for
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PHYSIQUE
DE
FLUX
probably
PHASES
been
one
reason
is that it
forces
much
are
temperature.
The discovery of
important events
THEORY.
of the
most
provides probably
a
mechanism
new
superconductors in 1986 has
Physics in years.
The main
pairing of electrons. The corresponding
BCS theory, producing higher critical
copper
in
Solid
for
oxide
State
stronger than the phonon coupling in
mechanism
is still not
understood.
Yet, this
by theoreticians
has permitted to
the
concentrate
by
of
the
that
right
accepted
studies
experts
most
on
a
now
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
Hubbard
model in two dimensions, with a strong
orbitals.
One possible model is the
sc-called
electron-electron
interaction, which can be approximated by the sc-called t
J model as an
the
Nevertheless
effective
effort
enormous
small
number
of
since
models.
then
It is
Hamiltonian.
A great deal of work has been done in finding a good
One of the approaches proposed was the sc-called
model.
idea
in
such
an
approach
is
that
in
the t
J
model, spin
approximate ground state
phase approximation.
for
flux
variables
can
be
considered
such
The
as
a
basic
slow
rapid. Therefore a kind of Born-Oppenheimer approximation leads
effective
magnetic field acting on charges
to a model in which the spins are frozen, creating an
(here holes). In this adiabatic approximation, the charged particles become independent, and
However the magnetic field is no longer
similar to the Harper model.
becomes
Hamiltonian
the
whereas
charge
variables
are
N°2
SEMICLASSICAL
METHODS
SOLID
IN
STATE
PHYSICS
483
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K
Energy
0.6
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1-.
l~~
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Fig.?.
figure
(from
to
crosses
lower
emiclassical
figure
expansions
are
:
(from
points
from
are
exact
extracted
spectrum
from
whereas
numerical
exact
full
spectrum
curves
and
are
curves
coming
from
correspond
JOURNAL
484
PHYSIQUE
DE
I
N°2
necessarily uniform, and it is still a dynamical variable in that it must be chosen to minimize
concentration.
Fermi
energy, given the holes
sea
studies
dealing with the ansatz of a
this problem is already difficult,
Because
were
many
Hofstadter's
uniform minimizing magnetic field. So that we are back to
spectrum.
that the miniOne of the main
conjectures based upon this series of approximation,
was
#/#o should be equal to the
concentration
b of charges (b
mizing dimensionless
flux o
calculations
Hofsnumber Of charges per site). This conjecture was based upon
numerical
On
in the
semiclassical
limit
tadter's
further proved to be
correct
spectrum (see in [59]) and was
[77], namely b -+ 0 for the Hofstadter like models. This is what gave birth to anyons, a kind
They can be reinterpreted in terms of
of quasipartide
with a magnetic flux
attached
to it.
parastatistics, thanks to the fact that the indiscernability of identical particles in 2D leads to
representation of the Braid Group [30].
a
because of its
fundamental
character, it has
Even though such an idea was
very appealing
been
verified in experiments [39] and one
future by now [3]
not
can
say that it has hardly any
of high Tc superconductors.
at least in the
context
One of the main problems with such an approach is that the system should be driven spontaneously toward a state breaking the time reversal symmetry.
mechanism
Even though such a
doubt
that it can be broken in the t
is a priori possible, there is some
J model with spin 1/2
the
=
=
[37].
problem is
boson approach
Another
slave
a
[79].
However,
that
minimizing magnetic flux
the uniform magnetic
the
shows
that
filled
band
is
certainly
field
ansatz
uniform.
not
gives
rise
to
indeed
For
an
unstable
state
Lieb
be
and
One
half
and
the
case
sea.
This
done
was
producing only
the "anyon rule"
a
a
that
bipartite lattice, preliminary rigorous results by
magnetic field ansatz with flux 1/2 mod I should
of holes
small
should produce a slight
amount
a
distribution.
flux
possible way to approach the
periodic flux to use again
a
of
for
uniform
suspect
may
we
magnetic
in the
that
indicate
Nevertheless
correct.
disorder
the
at
Loss [62]
uniform
non
the
in [8] to find out that the local
correction
to the ground
global
&
=
b
was
still
valid
as
magnetic flux problem was to consider the
approach for the stability of the Fermi
disorder
introduced
by the periodicity, was
by
of a form factor, and
state
energy
mean
semiclassical
b
-+
0.
recognized that
probably because of the
the small field limit & -+ 0 and the long
as
period one do not commute,
so-called "magnetic
breakdown"
caused
by tunneling effect in phase space near separatrix [87]. Therefore, the understanding of the
minimizing magnetic flux is still a pending question.
This long discussion
between
at the heart of the exchanges
Rammal
and one of us (JB)
was
during the years 1990-1991.
The very fact that the ground state of the
Hubbard
the t
J model
hard to compute
or
was
so
However
led
them
2.
Quantum
While
one
shown
in [3] it has
think
that
to
of
the
usual
been
approaches
were
not
suitable
for
a
result.
chaos.
us
(JB) kept talking
for
months
of
quantum
chaos
with
Rammal
on
the
basis
phenomena should be important and universal, one day in April 1990, Rammal
called him on the phone to claim that in the Hubbard
model on a finite lattice, the eigenvalues
avoided crossings as the coupling
tend to exhibit
varied [38]. That
constant
many
was
was
an
indication
that
chaotic
behaviour, whatever that means,
should
in these
models.
some
occur
Still, a clear test of such an idea had to be done, and even with it the question of the mechanism
unclear for the next
producing this effect remained
Apparently
Rammal
took this idea
year.
that
such
SEMICLASSICAL
N°2
very seriously because
May 1991 [80].
he
METHODS
produced
several
IN
SOLID
with
works
STATE
his
PHYSICS
students
on
485
chaos
quantum
before
of "Quantum Chaos"
several
topics mainly developed since the
name
by a community of physicists who would probably disagree on a unified
definition of the subject.
however
would say that it
Most of them
the quantum
signature of classical chaos on
concerns
the dynamical properties of a system. As its stands, such a definition
reflects a pragmatic point
of view taking into
the fact that most of the studies have
concerned
account
systems with two
degrees of freedom so far. It is very significant that the title of the Les Houches 1989
summer
school proceeding was
"Chaos and Quantum Physics". As explained by the organizers in the
foreword, "the expression "Quantum Chaos" at first makes us think of a quasi-exponential
proliferation of publications from the last decade hiding more or less legitimately behind a
find
We
end
scientific
the
under
of the
seventies
ill-defined
label".
subject developed around three kinds of technical approaches
(RMT), Semiclassical Theory (ST) and Dynamical Localization (DL).
The
Matrix
Random
Theory
2.I
THE
THEORY.
First of all, the oldest
RANDOM
MATRIX
approach is Random
MatriX
Theory (RMT). It was proposed in the fifties by Wigner [9 II as a convenient way of representing
the Schr6dinger operator for a nucleus.
The idea is that such a
Hamiltonian
is very complicated
and can be approximated in many
dimensional
matrix
HN,
respects by a finite (but large)
where N is the dimension, the elements of which are
random
variables.
The simplest example
consists in taking these
variables to be gaussian, identically
distributed
and independent. This
ensemble of matrices is called a GOE (gaussian orthogonal ensemble) or GUE (gaussian unitary
ensemble) depending upon whether HN is real symmetric or complex hermitian.
The main
features
(a)
(b)
the
the
One
can
Wigner
these
of
ensembles
are
:
density of states of such matrices is given by a semicircle law [91],
level spacing
distribution
exhibits a repulsion of neighbouring levels.
approximate the level spacing distribution by the one for 2 x 2 matrices
surmise
fl
m
Z~~sPe~'b~
the
other
hand
follows
if the
random
P(S)
This
is in
Physics
This
eJlicient
checked
particular
describe
to
has
RMT
tool
in
of
the
the
disordered
1984, Bohigas
systems
by
relevance
The
Porter
the
level
spacing
dis-
[74] and
[26] proposed
(23)
e~'
=
random
giving
developed by Dyson
been
calculation.
sixties
Anderson's
in
case
=
diagonal (or quasidiagonal)
is
matrix
2(GUE)
namely [2 II
law
Poisson
a
I(GOE)
=
(22)
fl
tribution
the
:
p(s)
On
called
rise
to
and
potential
model
Anderson
Mehta
[40]
to
theory for
a
eventually by Bohigas
of such
[70] used
in
Solid
State
localization.
the
point of becoming
Physics started
Nuclear
et al.
very
a
to
be
[48] in 1982.
RMT as a
statistical
tool to investigate
degrees of freedom. The first example was
the Sinii
billiard.
Following a general idea of Berry [21], they established a phenomenological
rule, namely that a "generic classically integrable system" should have a level spacing distribution p(s) given by a Poisson law implying some
kind of level clustering,
whereas a "generic"
classically chaotic system should obey one of the Wigner distributions
GOE or GUE depending
the
Hamiltonian.
In such a statement,
the word
symmetry properties of the quantum
upon
In
spectra
of
quantum
et
al.
systems
with
small
to
use
number
the
of
JOURNAL
486
"generic"
to
say
has
more
not
precise meaning yet,
a
symmetries (such as
negative
of
constant
the
harmonic
[23]
rigorous results of Sinii [83] permit
that special systems having extra
surface
arithmetic
geodesic motion on an
or
follow
rule.
the
GUTZWILLER'S
APPROACH
N°2
recent
fail to
may
I
indicates
it
But
oscillator
[24, 81])
curvature
SEMICLASSICAL
THE
2.2
though
even
integrable systems.
for
least
at
PHYSIQUE
DE
The
FORMULA.
technical
next
tool
approximation and first established
used in
upon a WKB
a
by Gutzwiller [46] in the context of impurity levels in Solid State Physics, and later by Balian
and Bloch [7] in the case of eigenmodes of an Optical cavity.
(q,p) E R~ x R~, classical phase space) be a classical
Hamiltonian
Let 7i(x) (x
corresponding to a quantum one k. We consider the case of N degrees of freedom, and denote by
h/2~ the Planck
that k has a discrete
h and h
We
spectrum and denote
constant.
assume
their multiplicities.
eigenvalues
counted
with
by ED < El <
En
the
corresponding
<
<
"Quantum
Chaos"
formula
is
based
=
=
If
~~~~
(it is a smooth
d(E, e) by :
approximation
of the
Dirac
d(E, e)
(E~
e2)
measure)
we
~j b<(E
=
'
the
define
smoothed
density
of
states
(24)
En
n
Using
a
mately
Feynman path integral, and
written
phase
stationary
a
[47] d(E,e)
method
be
can
approxi-
as
d(E, e)
m
I(E)
+
jqq
~j A;(E)e~~~?(~)'~e
j
~~
(25)
~
J
In
expression
this
d(E)
is
"Weyl
the
d(E)
asymptotics"
[90]
or
~ / d~~xb(E
=
classical
term,
namely
:
7i(x))
(26)
Moreover, in (25) we sum up over the family of periodic orbits indexed by j, corresponding to
E. Then, 2j(E) is the period of this orbit, S;(E) is its
action £$_~ f. p~dq~, and
energy
the
7j a
focal
terms
phase shift (the Maslov index of j) which
points in the phase space. At last A;(E)
of the Floquet matrix of orbit.
formula
large
takes
into
is
counting
a
account
the
factor
(sometimes infinite)
existence
which
oscillating
can
~f
caustics
or
be
written
in
(as
-+ 0)
density of states
from
away
its Weyl classical
limit.
different
properties depending upon whether the
This
has very
sum
classical dynamics defined by 7i(x) on the energy shell 7i(x)
E is quasi-integrable or chaotic.
This is due to the exponential
proliferation of classical periodic orbits in the latter case.
This
must
be
shows
taken
into
that
account
a
to
number
describe
fluctuations
the
of
of
terms
h
the
=
2.3
DYNAMICAL
regime
by the
exhibit
standard
a
Lo
map
Several
CALIzATION.
diffusive
behaviour
[32] given by
in
classical
phase
space.
Hamiltonian
The
prototype
systems in a strongly chaotic
of such systems is provided
:
Pt+i
=
pt + Ii sin qt
qt+i
"
qt +
pt+i
mod
(27)
2~
,
N°2
SEMICLASSICAL
where
a
be
can
1979
via
a
t~o
~
Even
89]
on
a
diffusion
a
the
For
constant.
is
map
a
which
be
can
~~~~
~~~~
mathematical
map",
only rough
"sawtooth
standard
map
has
progress
which
made [18, 31,
be ergodic,
been
happens
estimates
to
finite
at
time
have
[36].
obtained
been
487
~~~
~~~~
though this result is not rigorous, recent
slightly different model, the sc-called
has
and
PHYSICS
for
restricted
described
STATE
SOLID
time, and describing a rotator kicked periodically in time. Such
describing the behaviour of a Hamiltonian system near a
resonance
to two degrees of freedom [63].
Chirikov [32] showed
that for K > 4, the typical
behaviour
report,
diffusive
diffusion
with
(see
Fig.
constant
8)
process
:
model
his
In
IN
discrete
t is the
generic
METHODS
This
system
evolution
is
quantized, and is
unitary Floquet
a
be
can
given by
known
the
as
operator
e~'lie~~+
F
kicked
"
problem" [28]. The
rotor
quantum
given by
F
~"Q
(~g)
#
,
Q and
Actually,
position
where
P
ih.
here Q> P and h
such as the period
parameters,
rotor,
is led
one
the
are
usual
are
operators
momentum
two
quantities.
kicks, the
effective
Planck
between
T
expression
the
to
and
dimensionless
of the
heft
If
on
circle, satisfying [Q, PI
introduce
physical
to
a
=
wants
one
inertia
of
moment
classical
I of the
constant
~)
"
(30)
showing again
that it can be tuned by varying the kicks frequency for
instance.
argued by Fishman, Grempel and Prange [42] that F should have
the basis of an
similar to the one leading to
Anderson
argument
on
nh (n E Z) and similar
momentum
space is quantized according to pn
It has
been
spectrum
the
Then
like in
classical
Anderson
one,
so
that
if
we
the
As h
-+
the
compare
that
shows
localization
the
diffusion
of the
quantum
T
The
discrepancy
interference
between
effects
which
((pt
PO )~)
one
for
time
constant
system
(like
the
insulator
can
theory
diffusion
transition
be
inital
in
state
the
classical
and
T.
increase
to
related
D is
the
to
breaking
time
T
and
to
system
m
(
G3
const
(31)
j~
h
and
behaviour
is due to
existence
of quantum
quantum
quantum state in a finite region of phase space.
classical
diffusion
local12ation
has been
and
extended
quantum
to
viewed
works
an
chain
to
the
It(n)
a
<
t
continues
relationship between
systems with a higher number of degrees of freedom.
standard
map in which It is actually a quasiperiodic
localization
over
lD
to
converges
classical
trap
The
Such
evolution
quantum
Beyond this point the quantum
(see Fig. 8).
Shepelyansky [33] supplemented by numerical
calcu-
agree
classical
length f
the
time
average
functions
two
fixed
0 and
while the
classical
stay bounded
argument by Chirikov, Izrailev and
lations
the
model.
mechanics,
quantum
average
An
localization.
=
the
point
pure
a
if
=
as
a
Itof(win,
v-dimensional
=
to
absolutely
simple example
of
is
then
given by
a
time, namely
,wv_in)
w2n;
2, all states
v
are
increases
exponentially
constant)
is expected leading
A
function
lattice
in
localized
fast
with
continuous
momentum
but
K.
now
If
v
space,
the
> 3
spectrum
for
which
localization
an
for
Anderson
large
1(
the
length
metal-
[29, 82].
JOURNAL
488
PHYSIQUE
DE
I
N°2
E
20
20
Fig.8.
Time
straight
the
values
WHAT
2.4
IS
times
been
has
a
property
the
motion
energy
quantum
other
parameter
is
there
long
suitable
always
behaviour
time
time
do
to
converge
there
However,
as
free
a
stationary
behaviour
states.
of the
complexity
us
Arnold
first
theorem
(Ii,
IN, Pi,
,
that
means
Therefore
spectrum
the
these
of
of
initial
on
in
them
[5]
the
asserts
,@N)
E
-+
all if the
at
consider
as
0.
case
that
R~
x
curves
=
4;
different
under
time
T~,
not
a
-+
0 and t
-+
evolution
that
however
has
that
classical
systems.
While
at
a
finite
the
that
an
or
plays the
effective
The
constant.
Planck
semiclassical
quantities
quantum
h
constant
ratio
flux
a
a
[86].
Planck's
problem
oo,
-+
to
see
the
number
even
not
?
and
classically integrable
is a
canonical
change
such
Quantum
by Berry
"
remarked
conditions
I
do
mechanics
surprised
be
of
notion
as
fixed
at
0.
definitely
quantum
system
there
h
limits
Notice
of
h
as
limits
two
the
all,
namely
parameter,
in
would
we
h
quantum
K
for
deterministic, in the long run,
This paradoxical
situation
appears.
chaos
extra
counterpart
between
quantities appear
They are unchanged
Hamiltonian.
of the
Let
"clash"
a
rotor [20]1 It
do long time
Where
no
is
classical
their
regime
chaotic
system is
when
an
First
proportional to h. We have seen in section
role of h and is no longer a universal
In the kicked
constant.
rotor
which
be
physically
tuned
desired.
constant
occurs
can
as
notation
let us
denote by h this elsective
To simplify
Planck
limit corresponds to h -+ 0. It is known that in most systems the
some
~
loo
the
explicit
more
dependence"
"sensitive
a
in
map
represent
make
us
Hamiltonian
more
Of
terms
Let
the
classical
becomes
standard
points
and
previously presented.
characterizing
a
the
80
(from [16]).
constant
of
mechanics
In
for
energy
classical
technicalities
the
description
explained in
stochastic
kinetic
the
to
60
QUANTUM CHAOS ?
THEN
behind
[20], chaos is
the
of
Planck
effective
lying
Chaos"
short
evolution
corresponds
line
of the
40
Hamiltonian
systems
effects
simple
as
commute.
Precisely when we
they give rise to the
of
classical
deal
with
spectrum
in the
chaos
states, for h > 0, have
degrees of freedom [44].
quantum
of
Hamiltonian
of
for
7i.
The
variables, the action
be expressed as
can
Liouvilleand
a
angle
function
SEMICLASSICAL
N°2
of the
actions
replacing
I
,IN) only.
IL>-
=
by njh
each Ii
METHODS
where
is
nj
The
Efff
smooth
some
(as
cases
into
this
have
seen
the
account
(ni,
nN)
and
n,
=
7i is
now
Still,
expression
as
if two
in
levels
not
the
is
cross
we
consists
then
in
:
nNh)
first
can
say if 7i
7i(1, @)
procedure
(32)
,
of
term
section
exactly integrable,
489
get the eigenvalue spectrum
I) exponentially
of the tunneling effect due to
h -+ 0, each eigenvalue is well
occurence
regions.
classical
If
enough,
we
PHYSICS
STATE
quantization
[41]
to
7i(nib,
=
~~
If 7i is
EBK
integer,
an
SOLID
IN
7io(1)
=
terms
exact
classical
them
written
ef(1,
in
powers
be
@)
added
degeneracies
by the quantum
independently.
be
+
must
labelled
follow
can
expansion
an
small
of h.
to
In
take
between
numbers
as
(33)
,
(I, @), for e small, the KAM theorem (see [43]) asserts
tori, I
const, survive but a theorem by Poincard [73]
that only a discrete set of periodic orbits of 7io survives
the
perturbation.
asserts
Perturbation
theory in quantum mechanics can be used uniformly with respect to Planck's
unstable
periodic orbits of 7io corresponds in quantum meof
constant
[19]. The
occurence
chanics to level crossings producing (as h varies), after
perturbation,
kind of anticrossings
some
(like the Jahn-Teller effect of molecular physics). On the other hand, classical chaos precisely
develops near
unstable
periodic orbits. So we are led to admit that this local "hyperbolicity"
is associated
with avoided crossings of levels [45].
the level splitphenomena occur which
However in the fully chaotic regime,
some
new
cause
eigenstates is
ting to increase. For indeed the tunneling effect between unperturbed
resonant
classical
illustrated
in [27]. In
enhanced by the
chaotic
This has been
beautifully
transport.
this latter
orbits in the complex phase space,
the usual
WKB theory, requiring classical
case
does not provide the
level splitting. Everything looks like if the
transport of the wave
correct
barrier
function through the dynamical
efficient by using the classically chaotic
mcmore
was
results in level splitting are of the
tion than by using the usual tunneling elsect.
These
same
order of magnitude as the
spacing between levels.
mean
elsect on Rydberg Hydrogen
A good example of such a phenomenon is given by the
Zeeman
number splits
quantum
atoms (see Fig. 9). If the magnetic field is small enough, each principal
sublevels roughly linear in B. However, bne can see that rapidly these
into a bunch of Zeeman
they tend to avoid
sublevels will start creating a lot of crossings. Due to higher order
terms
f is
where
that
most
each
other.
becomes
function
unperturbed
number
If the
rapidly as
larger
very
enough
smooth
a
of the
B
of such
increases.
the
if
instead
=
crossings
avoided
Moreover
classically
in
of
invariant
can
one
becomes
that
see
the
regime (namely
chaotic
B
oscillate
too high, each level will
splitting at the avoided crossings
large) than in the regular region
[35].
remark
We
then
h
-+
that
select
0 will
we
h
consider
quantum
0, and
-+
numbers
"n"
-+
a
given
oo,
and
window
we
will
arguments give an insight into what should be called
assign quantum numbers
the quasi-integrable regime, one
can
they exhibit
sometimes
avoided crossings. For indeed, in this
small (typically of order e~"~ by WKB approximation)
very
These
separation
In
the
of
chaotic
fuzzy due to
level spacing.
in the
chaotic
order
For
case
classical
the
energies [E-, E+],
same
situation.
Quantum Chaos. First of all, in
unambiguously to levels, even if
latter case, the level splitting is
compared to the mean spacing
h.
regime however,
level
of
be in
the
quantum
oscillations,
and
this
(and this
regular
than
reason
in the
the
numbers
level
splitting
is
somewhat
one.
meaningless. The spectrum is
of magnitude of the
mean
paradoxical) the spectrum is more rigid
become
is of the
order
JOURNAL
490
EE
PHYSIQUE1
DE
No2
io-~
-2
/f2
E
Fig.9.
magnetic
the
is in
atom
2.5
LEVEL
tics
of level
which
of
define
(I)
three
a
fl <
illustrate
to
scales
the
of
a
variation
that
=
=
=
point, let
depending
discrete
Eo(h)
set
approximately
compute
us
H
upon
Ei(h)
<
<
statis~
the
h,
parameter
some
Ei+i(h)
of
<
of h.
for h
h'- h
if
=
O(b~) perturbation
theory for
individual
levels
convergent;
is
h'-
(it) a short scale b,, such that in the
eigenvalue distribution
does not change,
large
number
of
times;
a
(iii) a macroscopic scale bM along which
We
the
strong
a
(~
Hamiltonian
a
variation
scale b~, such
B/Bc is
magnetic field B. ~
2.3~
T).
B
corresponds
The
i
to
perturbative regime whereas here fl > 60,
in
units
is
i
this
consider
us
being given by
sensitive
microscopic
Ei(h)
Let
atoms
atomic
regime (from [35]).
chaotic
To
REPULSION.
eigenvalues only
We will
a
in
~~/(-2E)~.
=
strongly
repulsion.
spectrum
expressed
field
is fl
here
parameter
natural
namely
Rydberg Hydrogen
for
sublevels
Zeeman
dimensionless
the
the
y
~~-9
2_
h
range
but each
O(b,)
=
shape
macroscopic
the
shape
macroscopic
the
pair (Et Ei+i)
of
levels
individual
spectrum
of the
is
of
the
oscillates
modified.
that
assume
6p < b, < 6M
Now
let
us
consider
[ho> fill of size
from ho to hi,
to
the
the
level
O(b~) along
the
main
separation
which
variation
subspace generated by
H(hi)
Ei(h)
between
Ei(h), Ei+i(h)
of H(h) can be
states
H(ho)
I and
"
+ I.
bH(h)
m
one
fl~
by
a
2
varies
corresponding
matrix
2
x
interval
an
As h
write
can
(ei
Consider
crossing.
avoided
one
described
Thus
Ei+i(h).
and
exhibit
+ J
fi)
(34)
,
PWrS(I,I+1)
where
the
the
sum
representation
all
concerns
of 2
x
2
pairs of eigenvalues (I,I
given by Pauli's
matrices
+
I)
one
which
J
=
anticross,
(ai,
a2>
a3).
and
The
we
have
constant
used
et
METHODS
SEMICLASSICAL
N°2
the
represents
appreciable
variation
mean
only
value
Ei+1)/2
+
which
we
by hypothesis. So
scale bM
PHYSICS
STATE
neglect
can
et
0.
*
491
here
Here
because
is
vi
that
if
symmetric fi
is real
interval
k=o
L
has
et
vi
it
an
in R~.
vector
some
component of
E R~ (namely the second
[h-,
consider
Let us
h+] with size O(b,) and let us decompose
now
an
O (b, /b~)
subintervals
Uf]o~[hk, hk+i] of size O(b~).
variation of H(h) along this interval
Hence the total
then be written
can
as :
Remark
H(h)
the
on
(Et
of
SOLID
IN
vanishes).
into
N
=
N-I
H(h+)
By hypothesis
change,
not
variables,
such
on
that
so
Therefore
the
interval
an
one
whereas
can
fl~
=
«
(35)
v)~~
large, but the macroscopic shape of
v)~~'s (as k varies) as identically
N is
the
the
consider
rapid
the
H(h-)
make
oscillations
spectrum
distributed
does
random
independent.
them
sum
N-1
£,
~(k)
~~
~
i
k=0
becomes
The
Gaussian
a
random
spacing Ei+i
level
2
x
2
with
matrix
zero
mean
N gets
as
large.
Et is then given by
s=Ei+i-Ei=2]Vi)
namely
distribution
its
p(s)
p(s)
have
We
here
CHAOS
2.6
HA
=
of
STRONGLY
describing
#-yj=
Wigner
V E
R~
if
V E
R~
2D
a
strongly
real symmetric
case
complex hermitian
explaining why
argument
an
surmise
if
CORRELATED
£
t
=
the
Z~~se~'b~~
Z~~s~e~'~'~~
=
beginning
a
IN
model
J
t
given by
is
fermion
n~,_a) c$,,cy,a (I
(I
lattice
ny,-a
law
Let
SYSTEMS.
FERMIONS
correlated
Wigner
the
case
should
be
consider
now
us
universal.
the
gas
£
+ J
S~
(36)
Sy
,
j~-yj=1,~,yEA
i,~~yEA
with periodic boundary conditions, c~,a is the fermion
creator
number of electrons of
of
electron
of spin a, n~,, = cl,c~,, is the
operator at the site z E A
an
spin a at z, and S~ is the spin operator for electrons at site I. HA acts on the Hilbert subspace
"Gutzwiller
number n~ = n~,t + n~,i < I at each site (the sc-called
of states
with occupation
A is
finite
a
lattice
in
Z~, taken
projection" ).
The
the
particular
to
its
of
number
total
No is
number
of
that
is
N
which
value
particles £~(n~,t
holes.
avoid
To
Since
even.
maximizes
the
25(~)
0 or I depending upon
thermodynamic limit namely
(no magnetization here).
model
This
speak
t1
of
describes
prton
any
chaos
in
it has
such
-+
electron
an
whether
N
a
any
model?
=
Nt
frustration
total
dimension
oo,
No is
No IN
gas
with
whether
=
unclear
the
+ n~~ i)
classical
we
spin
is
of
the
even
-+
b
+
Ni
will
is fixed
assume
conserved
we
and
equal
to
No where
N
bipartite and in
25(~)
Ni
Ni
that
A is
will
fix
=
corresponding sector.
Since N is even,
odd.
physics
will
The
correspond
to the
or
0
(b is the doping concentration), SIN
-+
large number of quantum particles. It is quite
a
limit, the particles being fermions. So how can we
JOURNAL
492
Even though we do not know yet
eigenvalue spectrum is very sensitive
distribution.
spacing
To do so, we
phenomena liable to hide a possible
the
symmetry,
Poisson
a
dilserent
In
the
One
physical
must
however
question,
that
to
answer
to
N°2
be
we
So
parameters.
one
can
enough,
careful
check
can
not
whether
the
its
level
measure
introduce
to
extra
repulsion. For instance if our
Hamiltonian
admits a
not be coupled together and the eigenvalues of H in two
distribution
then check that the level spacing
will satisfy
level
will
sectors
independent.
will be
sectors
PHYSIQUE I
DE
can
law.
model
J
t
our
it is
important
therefore
restrict
to
H to
of each
sector
a
This
symmetry.
invariance, we
N, No, the total spin S(~) Moreover, due to translation
we
indexed by k. Lastly the J term
admits a spin
choose a given quasi-momentum
must
sector
rotation
invariance
which forces us to choose a spin sector.
numerical
point of view, the choice of the finite lattice A is done by defining a
From the
why
is
choose
must
£ of Z~
sublattice
symmetry,
generated by
will
we
two
where
and p are
identified
q
be
can
integers.
two
with
the
(q>P)
"
£
Thus
this
In
way
figure
we
10.
have
can
square
16
avoid
To
y
Y
(~q>P)
or
matrix
sites
18
size
too
sides
~j
~
(we
big
"
yZ
xZ +
=
whose
~
in
Z~.
and
x
E
order
In
to
the
preserve
square
choose
X
A
vectors
are
A
and
and
x
=
y
Z~/£.
and
the
number
of sites
becomes
~2 ~ ~2
~
choose
for
N
the
to
even
spin frustrations)
choose No
memory,
we
avoid
computer
shown
as
"
I
(one
hole).
Now
the
compute
break
this
difficulty lies in the spin symmetry.
hard
It is actually
too
each spin sector.
calculation
it is better
To avoid such a long
by modifying the J-term as
technical
main
elements
matrix
symmetry
to
in
£
(Ji (Sf)S)~)
S(~)S)~))
+
+ Jjj
S(~)S)~))
to
(37)
,
jz-yj=1,~,YEA
with
At
far
mean
Ji # Jjj. The limit Jjj -+ oo is actually the
last, the level spacing must be normalized
The
normalization
apart in the spectrum.
level separation be one.
This
be done
can
Fig. II) and replacing
approximation of p(E).
The
calculation
In [65] it is also
the
Ising
in
has
been
shown
that
which
allow
to
a
obviously integrable.
comparison between levels
is
usually chosen in such
by calculating the density
is
way
a
of
as
states
the
local
p(E) (see
f~$ P(E)dE where p is some smooth
Ei by the number fi
operation is called "unfolding the spectrum" [25].
performed in [65] (see Fig. 12) and exhibits a GOE
distribution.
the level spacing
distribution
becomes
Poisson.
as Jjj
-+
oo
levels
This
model
order
=
The previous analysis is certainly very preliminary.
investigate
whether
system is integrable. Actually
can
a
and still unpublished
investigations of various models indicate that such a method is
recent
justified. For instance, a Heisenberg spin chain in lD, known to be integrable by mean of a
DISCUSSION
2.7
it
But
Bethe
In
shows
Ansatz
holes
One
the
way
use
such
field, by computing
a
a
shows
should
situation
could
be
exhibits
same
calculation
numerical
SPECULATIONS.
RMT
[22]
the
much
AND
that
a
the
be
to
Poisson
lD t
that
distribution
model
J
it
satisfies
to
[4].
with
a
one
hole
should
distribution
Poisson
also
be
integrable [60].
[84]. Presumably
with
A
two
[61].
dilserent
method
level
used
test
spacing
the
integrability
distribution
on
the
of
a
2D
transfer
Ising
model
matrix.
with
While
magnetic
for
a
zero
N°2
METHODS
SEMICLASSICAL
4
SOLID
.
.
.
.
.
.
.
.
.
.
.
.
m
m
m
m
3
2
j
IN
STATE
PHYSICS
493
_i
3
2
/
4
'
'
/
'
/
'
/
'
/
m',
'
I'
m
'
'
/
'
/
/
m
m
'
m
m
/
'
,
/
b
(
,
m
/
m
m
m
/
,
/
,
/
,
/
,
"
,
"
/
,
/
,
/
,
/
,
,
-3
Fig.10.
Lattice
-2
sites
for
/
/
2
-1
the t
J
model;
upper
figure
:
16
3
sites; lower figure
18
sites.
PHYSIQUE
DE
JOURNAL
494
I
N°2
250
/h.~w~",
q~,;w'
~
,
.
iso
:
°
°
a
°
.
loo
°
6
2
~
-2
-4
0
4
E
Fig-I
Density
I.
of
Jz
of the t
states
0.8
.
model
N
for
=
18
and
Jz
0.25
=
(from [65]).
.
0.6
~
i~
0A
0.2
~
0.5
2
1.5
2.5
3
s
Fig.12.
and
Poisson
Level
spacing
distributions
P(s)
distribution
are
shown
as
full
in
the t
lines
Jz
model
(from [65]).
for
N
=
18
and
Jz
=
0.25.
Ideal
GOE
N°2
SEMICLASSICAL
magnetic field [67] it is known to
every magnetic field. Such a test
The
question
next
IN
could
integrable,
give an
indication.
GOE
character
be
the
concerns
METIIODS
SOLID
it is
STATE
unclear
of the
PHYSICS
integrability
whether
for the t
result
495
All
model.
J
holds
for
matrix
obviously real, so that this result should not be surprising.
are
the approach in terms of slave bosons, flux phases or
However
superconductivity suganyons
should be broken.
gests that, at least for low energy excitations, the time reversal
symmetry
We do not see anything like that here. It is however
sufficient
because this
not a
argument,
breaking of the time reversal symmetry alsects only the ground state. For one hole, the Na~
elements
of the
gaoka
Hamiltonian
[66] requires
theorem
that
the
ground
unique
be
state
that
so
time
symmetry
reversal
Moreover, the path integral approach used in quantum Monte Carlo calculations [54] shows that for spin 1/2, no time
reversal
symmetry breaking is required, namely
there is no need to
introduce
complex
field.
Therefore,
does not expect any of the
a
gauge
one
approximation like flux phases or anyons to work unless in trivial case, namely whenever the
should
be
not
broken.
dimensionless
flux is 0
or
1/2
mod
I.
Another
important question raised by this result
the validity of the quasiparticle
concerns
approximation in terms of a Fermi liquid theory. Even though quasiparticle peaks have been
observed in a t
J model or in the 2D Heisenberg ailtiferromagnet, if the theory
made of
were
well defined
independent quasiparticles we would expect a Poisson
distribution
How
to
occur.
reconcile the two points of view ? Presumably, while level repulsion do occur
to
at high enough
the density of states is large,
whereas
elementary
at low
energy, in the region where
energy,
excitations
produce
level
clustering. However this has not been tested yet. If this is so a
may
liquid theory should be only a low energy approximation, and RMT should give the explanation
for
the
If
such
that
find
to
"scars"
neither
bosons
Lastly,
another
[53]
be
may
functions.
in such
relevant
fermions,
nor
Green's
in
fails, namely if GOE
explanation for the
explanation
an
ought
we
background
incoherent
but
scars
still
holds
appearance
would
We
case.
conspiring
to
groundstate
the
near
quasiparticles
of
then
have
produce
a
new
stable
a
peaks.
kind
then
energy,
We suggest
of
excitations,
excitation.
important question
the conductivity.
that
the
linear
We conjecture
concerns
behaviour
of the resistivity
with
of the
temperature ill] should be a
respect to
consequence
random
matrix
approach. However, again, only low energy
excitations
relevant
for such
are
calculations.
Therefore
if RMT holds at small
and explains the linear law of resistivity,
energy
the quasiparticles
approach becomes questionnable.
an
computation
The
operator.
current
basic
Field
Randomness
two
will
requirements.
conductivity
involves
required
is
structure
Quantum
in
of
This
the
requires the knowledge of the matrix
locality properties of the
Hamiltonian.
It
describe
to
the
Hamiltonian
in
of
terms
random
elements
means
for
that
matrices.
the
some
Like
Theory,
translation
invariance
and locality are
essential
ingredients there.
Hamiltonian
which
constrained
by these
only
for
the
of
the
is
part
not
occur
investigating this question [14].
We are currently
Acknowledgments.
This
with
authors
Bohigas,
D.
is
paper
R.
summary
a
Rammal.
cited
A.
Poilblanc,
in
Their
the
and
a
review
solution
reference
could
list.
We
of
problems
some
not
have
want
Connes, B. Dou~ot, P. Duclos, R.
R. Seiler, C. Sire, J. Sj6strand
been
raised
found
in
particular during
without
the
contribution
discussions
of
many
especially
J.-C. Angles d'Auriac, O.
Fleckinger, B. Hellser, C. Kreft, G. Montambaux,
to
and
thank
M.
Wilkinson.
496
JOURNAL
PHYSIQUE I
DE
N°2
References
[ii
Ando,
[2] T.
Phys.
[3]
Superconductivity of
Alexander,
Phys. Rev. B
S.
(1983)
27
Fowler,
A-B-
(1982)
54
B.
Angles d'Auriac,
T.
J.-C.
[5]
Arnold,
Stern,
F.
Angles d'Auriac,
perturbations, and the
V-I-
percolation approach
A
to
effect of disorder,
the
of
properties
Electronic
two-dimensional
Mod.
Rev.
systems,
437-672.
J.-C.
[4]
networks.
154i-1557.
Dougot,
for
Response of
submitted
anyons,
Flux
the
Phys.
to
Phase
electromagnetic
to
(1992).
Rev.
preparation (1992).
Hsu, in
Mathematical
Stamp,
P-C-E-
search
muon
of
methods
Springer,
mechanics,
classical
(1978).
York
New
JETP 19
conduction
electron
in a magnetic field, Sov. Phys.
[6] M.Ya. Azbel, Energy spectrum of a
(3) (1964) 634-645; M.Ya. Azbel, On the spectrum of difference equations with periodic coejJicients,
Math.
Sov.
[7]
Dokl.
(1964)
5
1549-1552.
Domain :
of Eigenfrequencies for the Wave Equation in a Finite
Smooth
Boundary Surface, Ann. Phys. Go (1970) 401-447;
of Eigenfrequencies for the Wave Equation in a Finite Domain : II. Electromagnetic
Distribution
Field,
Spaces, Ann. Phys. 64 (1971) 271-307;
Distribution
Riemannian
of Eigenfrequencies for
III.
Oscillations,
Ann. Phys.
the
Wave Equation in a Finite
Eigenfrequency
Density
Domain
:
Paths, Ann. Phys.
Solution of the Schrbdinger Equation in Terms of Classical
69 (1972) 76-160;
R.
Balian, C. Bloch,
I.
Three
85
(1974)
Distribution
Dimensional
Problem
with
514-545.
Spectrum of
[8] A. Barelli, J.
Bellissard, R. Rammal,
approach, J. Phys. France
algebraic
Barelli,
[9] A.
Fleckinger,
R.
(1990)
51
2D
Bloch
electrons
analysis of Harper-like
Semiclassical
in
a
periodic
magnetic field:
2167-2185.
models, Phys.
Rev.
B 46
(1992)
11559.
[10] A. Barelli, C. Kreft,
J.
Phys.
[ll]
B.
[12]
J-G-
I
France
1
Batlogg, Physica B
Bednorz,
(1991)
169
Press
Harper
model
(1986)
188.
as
an
example of phase
space
tunneling,
in
Solid
B 64
Physics
State
Application,
Vol. 2, D-E-
& M.
Takesaki
uniform magnetic field in
Eds., Cambridge University
of Almost
periodic
Schr6dinger
2D
:
Evans
electrons
in
a
(1988).
J.
Bellissard,
[15] J.
Bellissard,
Rev.
and
a
7.
Phys.
Z.
C*,Algebras
Bellissard,
Operator Algebms
in
structure
1229-1249.
Miller,
K-A-
[13] J.
[14]
Braid
(1991)
Lett.
under
D.
49
(1992).
progress
Bessis, P. Moussa,
(1982)
Chaotic
states
Phys.
operator,
701-704.
Localization
Mathematical
[16] J. Bellissard, A. Barelli~ Dynamical
:
Quantum Measurement, P. Cvitanovic~ I-C- Percival~ A. Wirzba
Framework, in Quantum Chaos,
Eds., Kluwer Pub]. (1992) 105-
i30.
[17]
J.
Bellissard,
(1982)
[18] J.
Bellissard,
Phys.
[19] J.
B.
Simon,
S.
Vaienti,
Cantor
spectrum
for
the
Mathieu
Almost
equation,
J.
Anal.
Func.
48
408-419.
144
(1992)
Rigorous
Diffusion
Properties
for the
Sawtooth
Map,
Commun.
Math.
521-536.
Bellissard, M. Vittot, Heisenberg's picture and non
commutative
geometry
mechanics, Ann. Inst. H. Poincar6 52 (1990) 175-235.
in quantum
of
the
semiclassical
limit
[20]
quantum-to-classical
Berry, Some
asymptotics,
quantum physics, M.-J. Giannoni, A. Voros, J.
London, New York, Tokyo (1991).
M-Vand
[21]
M-V-
(1977)
Berry, M. Tabor,
Level
clustering
375-394.
[22] H. Bethe, Z. fir Phys.
71
(1931)
205.
in
the
NATO
ASI, Les Houches, session LII, Chaos
Eds., North-Holland,
Amsterdam,
Zinn-Justin
regular
spectrum,
Proc.
Roy.
Soc.
London
A
356
SEMICLASSICAL
N°2
METHODS
STATE
SOLID
IN
PHYSICS
497
Bleher, On the
distribution
of the number of lattice points inside a family of convex
ovals
Preprint (1992); P-M- Bleher, Z. Cheng, F-J- Dyson, J-L- Lebowitz,
Distribution
of the error term
for the number of lattice points inside a shijted circle, Preprint (1992); P-M- Bleher, The energy
level spacing for two
harmonic
oscillators
with golden mean
mtio of frequencies, Preprint (1992).
[23] P-M-
[24]
BogomoIny,
E-B-
[25] O. Bohigas,
B.
Georgeot,
M.-J.
matrix
theories
Random
LII, Chaos
Amsterdam,
and
quantum
London,
New
Giannoni,
and
York
[27]
Tomsovic,
S.
(1990)
65
of fluctuations of chaotic quantum
Quantum Systems, G. Casati Ed., Plenum Press,
Chamcterization
in
three
Chen,
Y.H.
E.
Girvin,
Canright,
Dimensions,
Witten,
Statistics
S.M.
Phys.
and
Chirikov,
B.V.
Levels, Phys.
Chaotic
on
Rev.
Girvin,
Rev.
A.
Lett.
Halperin,
Flux
in
Brass,
A
Universal
a
pendulum
dimensional
one
under
with
system
345-348.
two
(1989)
63
in
quantum
a
Int. J.
Mod.
Phys. B 3 (1989) io01; G.S.
Dimensions, Phys. Rev. Lett. 63 (1989) 2291Superconductive Pairing of Fermions
and
Semions
B.I.
2295-2298.
Statistical
Properties of
[31] N-I- Chemov, Ergodic and
of the 2~ Torus, to appear in J. Stat. Phys. (1992).
(1979)
Effects
Transport
Classical
D.L. Shepelyansky,
Anderson
transition
frequencies, Phys. Rev. Lett. 6211 989)
Wilczek,
F.
S-M-
2294; G-STwo
Ullmo,
D.
B.V.
incommensurate
Canright,
[33]
spec-
New-
5-8.
[29] G. Casati, I. Guarneri,
[32]
Zinn-Justin
Chirikov, F.M. Izrailev, J. Ford, Stochastic
behavior of
perturbation, Lect. Notes in Phys. 93 (1979) 334-351.
[28] G. Casati,
periodic
a
in
Houches, session
Eds., North-Holland,
Les
(1985).
Bohigas,
Lett.
[30]
Voros, J.
A.
ASI,
NATO
Preprint (1992).
Chaos,
Arithmetical
dynamics,
chaotic
physics, M.-J. Giannoni,
York, Tokyo (1991).
[26] O. Bohigas, M.-J. Giannoni, C. Schmit,
Behavior
tm, NATO ASI Ser. Chaotic
O.
Schmit,
C.
Instability of Many
Piecewise
Linear
Dimensional
Hyperbolic
Automorphisms
Phys. Rep.
Systems,
Oscillator
52
263-380.
Chirikov, F.M. Izrailev, D-L- Shepelyansky, Dynamical Stochasticity in Classical and Quan~
Mechanics, Sov. Sci. Rev. C 2 (1981) 209-267; B.V. Chirikov, F.M. Izrailev, D.L. Shepelyansky, Quantum chaos : localization vs ergodicity, Physica D 33 (1988) 77-88.
B-Vtum
[34]
P.G.
Diamagn£tisme
Gennes,
de
Acad.
Sci.
C.R.
Acad.
Atomes de Rydberg
[35] D. Delande,
VI, Paris (1988), unpublished.
[36] J.
supraconducteurs
grains
de
(Paris) B 292 (1981) 9-12; Champ critique
Sci. (Paris) B 292 (1981) 279-282.
Dolbeault,
unpublished
[37] B. Dougot, W.
Krauth,
results
private
Champs
en
Statiques
Dougot,
[40]
F.J.
66
P.C.E.
(1991)
Dyson,
Stamp,
Intenses,
seuil de percolation, C-Rsupraconductrice ramifi£e,
d'Etat,
Thbse
Universit6
Paris
communication.
Dougot, R. Rammal, Instability of ferromagnetism
[38]
model, Phys. Rev. B 41 (1990) 9617.
Lett.
d'un
boucle
(1992).
B.
[39] B.
prds
d'une
Response
Functions
and
at
finite hole density
Dynamics of
the
in
WAH
the U
=
System,
cc
Hubbard
Phys.
Rev.
2503.
M.L.
Mehta, J.
Math.
Phys.
4
(1963)
701-712.
[41] A. Einstein, Zum Quantensatz von Sommerfeld und Epstein, Verh. der D. Physikal Ges. 19 (1917)
Corrected
Bohr-Sommerfeld
82-92; L. Brillouin, J. Phys. Radium 7 (1926) 353-368; J.B. Keller,
Conditions for Non Separable Systems, Ann. Phys. 4 (1958) 180-188.
Quantum
LocalizaRecurrences
and
Anderson
[42] S. Fishman, D.R. Grempel, R.E. Prange, Chaos, Quantum
Localization
tion, Phys. Rev. Lett. 49 (1982) 509-512.; D.R. Grempel, S. Fishman, R.E. Prange,
potential : an exactly solvable model, Phys. Rev. Lett. 49 (1982) 833-836.
in an
incommensumte
[43] G.
Gallavotti,
Elements
of
Classical
Mechanics, Springer
Texts
and
Monographs
in
Physics (1983).
[44] P. Gaspard Dynamical Chaos and Many-Body Quantum Systems in Quantum Chaos, Quantum
Measurement, P. Cvitanovic, I.C. Percival, A. Wirzba Eds., Kluwer Pub]. (1992) 19-42.
JOURNAL
498
Silverstone,
[45] S. Graffi, T. Paul, H.J.
ings, Phys. Rev. Lett.
Gutzwiller, J.
Classical
(1987)
59
PHYSIQUE
DE
N°2
I
Overlapping
Resonance
Quantum
and
Avoided
Cross-
255-258.
Phys.
(1971)
[46]
M.C.
[47]
Hamiltonian
semi-classical
Gutzwiller, The
quantization of chaotic
systems, NATO ASI,
Houches, session LII, Chaos and quantum physics, M.-J. Giannoni, A. Voros, J. Zinn-Justin
Amsterdam, London, New York, Tokyo (1991).
Eds., North-Holland,
Math.
12
343-358.
M.C.
Les
[48]
R.U.
Haq,
[49]
P.G.
Harper,
Phys.
[So]
Soc.
Single
Bohigas, Phys.
O.
Band
A 68
Lond.
Heading,
J.
Pandey,
A.
(1955)
Introduction
An
of
Motion
874-
i18,
Hirsch,
[54]
J-E-
[55]
D.R.
Fye, Phys.
R.M.
Hofstadter, Energy
magnetic field, Phys. Rev.
[56] Ph. Kerdelhud,
hexagonales.
[57]
A.Va.
[58]
L.D.
Khintchine,
[59] P. Lederer, D.
Lieb,
Private
thanks
E.
to be
du
III,
Bulletin
de
la
of
pe-
Harper
de
chaotic
Hamiltonian
of
systems
electrons
Bloch
in
a
rational
p£riodique
Math.
[71] G.
Pastur,
Phys.
this
Press (1964), Chicago.
(1930) 629-637.
63 (1989) 1519.
remark.
published (1992)~
Rev.
Pedersen,
Spectral
75
et
Orsay (1992).
64
Lett.
Rev.
Paris-Sud~
de
thank
we
(1944)
65
the
and
Electrons
on
information
in
(1980)
Properties
to
Potential,
Periodic
a
prior
publication.
Springer Verlag,
Motion,
Quantum
Sire,
C.
for
authors
Stochastic
Berlin,
Phys.
Rev.
spin-fermion
models,
of the (H,T) phase
diagram
chaos
in
11?.
Experimental
France
determination
Lettres
(1983)
44
L-853
L-858.
Pannetier, J. Chaussy, R. Rammal, J.-C. Villegier, Experimental Fine Tuning of
Superconducting
Network
Lett. 53 (1984)
in a Magnetic Field, Phys. Rev.
L.A.
irrational
or
triangulaires
sym£tries
avec
Universit£
2D
[70]
scars
:
Univ.
Metalle, Z. Phys.
Pannetier, J. Chaussy, R. Rammal,
superconducting network, J. Phys.
a
[69] B.
Soc16t6
(1989).
43
Thesis~
Lichtenberg, M-A- Liebermann, Regular
Heidelberg, New York (1983).
J.M. Luttinger,
The Effect of a Magnetic
Field
814-817.
84 (1951)
Montambaux, D. Poilblanc, J. Bellissard,
G.
submitted
Lett. (1992)
to Phys. Rev.
Y. Nagaoka, Phys. Rev. 147 (1966) 392.
[68] B.
of
I'£quation
Mdmoire
magn£tique
spectre,
A-J-
[67] L. Onsager, Phys.
de la
Communication.
Lieb for
[62] E. Lieb, M. Loss,
Bulletin
2239-2249.
Rice, Phys.
E.H.
[66]
wave
Fractions, Chicago
T.M.
(1962).
Methuen
Harper II,
de
2521.
functions
(1976)
hi£rarchique
(1986)
56
Poilblanc,
J-B-
[65]
Lett.
and
der
[61]
[64]
Rev.
Continuous
Proc.
1515-1518.
Schrbdinger
de
Field,
(1990).
40
of classically
(1984)
14
Magnetic
Uniform
a
London,
M6moire
1,
semi-classique
pour
ii?, Fasc. 4,
Diamagnetismus
[60]
[63]
B
io86.
in
l'6quation
pour
Tome
53
levels
Equation
Structure
Landau,
Fasc.
eigenfunctions
Lett.
Methods,
Integml
semi-classique
J. Sj6strand, Analyse
Mathdmatique de France,
Bound-state
[53] E-J- Heller,
riodic orbits, Phys. Rev.
Electrons
892.
[52] B. Helfler,
Socidtd
(1982)
48
Conduction
Phase
to
[51] B. Helfler, J. Sj6strand, Analyse
Mathdmatique de France, Tome
Lett.
Rev.
of
Systems
Disordered
in
One
Body
the
Frustration:
1845-1848.
Approximation,
Commun.
179.
C*-algebras
and
their
automorphism
Academic
groups,
London,
Press,
New
York
(1933)
763-
(1979).
[72]
R.E.
Peierls,
Zur
Poincard,
Les
Diamagnetismus
des
Theorie
von
Leitungelectronen,
Z. fr
Phys.
80
79i.
[73] H.
Nouuelles
M£thodes
(1892-1894-1899), reprinted by Dover,
[74]
C.E.
Porter,
Statistical
Theories
Mdcanique
de la
New
of Spectra
York
:
Cdleste
Tomes
I,II,III,
Gauthiers-Villars
(1957).
Fluctuations,
Academic
Press
(1965).
SEMICLASSICAL
N°2
[75] R. Rammal,
Spectrum of
[77]
France
Rieflel,
M.R.
Mdmoire
[83]
M.
Kelly
approach
semiclassical
France
to
-algebra
dans
Short
(1984)
Weisbuch
Bloch
electrons
191.
Eds., Springer
in
magnetic
a
to
Congress of Math-
the
Europhys.
Lett.
(1990)
11
451.
Preprint
chaos,
Arithmetic
Princeton
de
et
libert£, Mdmoire de D.E.A.
techniques semi~classiques,
(1992).
statistical
stochastic
Shepelyansky, Some
properties of simple classically
quantum systems,
Localization of diffusive
in multi~leuel
Physica D 29 (1983) 208-222; D-L- Shepelyansky,
excitation
systems, Physica D 28 (1987) 103-114.
Ya.G.
Sinai
(1991)
Mathematical
problems
in the
theory of quantum
chaos, Lect.
Notes
Schr6dinger
Equation
[86] S. Smale,
The
Mathematics
of Time, Springer Verlag, Berlin~ Heidelberg,
communication.
[87] P.C.E. Stamp, private
[88] C. Tang, M. Kohmoto, Global Scaling Properties of the Spectrum for
Equation, Phys. Rev. B 34 (1986) 2041.
[89] S. Vaienti,
Map,
[90] H.
Nachr.
Discontinuous
Sawtooth
Ergodic Properties of the
Weyl, fiber die Asymptotische Verteilung der Eigenwerte,
(1911)
E-P-
[92] M.
Sac.
[93] M.
23
in
Math.
1469
,
41-59.
[84] C. Sire, D. Blaison, unpublished results (1992).
Microlocal
Analysis for the
Magnetic
Periodic
[85] J. Sj6strand,
C.I.M.E.
in Lect.
Notes in Math. (1989).
Lectures
questions,
[91]
45
& C.
Communication
Chaos
Sarnak,
D.L.
Phys.
Quantique dans un systdme d plusieurs degr£s
Rayonnement (1991); Ph. Lecheminant, Chaos Quantique
et
Matidre et Rayonnement (1991).
de D-E-A-
Matidre
[82]
of Microstructures,
J.
499
1803-1830.
C*
rotation
Dougot,
B.
Rondeau,
[81]
froctals,
algebraic
An
(1990)
51
Irrational
[79] J. Rodriguez,
P.
on
PHYSICS
STATE
(1978).
ematicians
[80] C.
excitations
Fabrication
Bellissard,
J.
field,
[78]
SOLID
303.
p.
Rammal,
J. Phys.
R.
IN
harmonic
[76] R. Rammal, in Physics and
(1986)
METHODS
Stat.
Konigl.
related
(1980).
York
Quasiperiodic
a
J.
New
and
Schr3dinger
(1992)
Phys.
67
Ges.
Wiss.
251.
G6ttingen
i10-i17.
Wigner,
Wilkinson,
Lond.
A
Wilkinson,
(1990)
Review
SIAM
Critical
391
(1984)
E-J-
9
(1967)
properties of
Austin,
1.
electron
eigenstates
in
incommensurate
systems,
Proc.
Roy.
305-350.
Phase
space
lattices
with
threefold
symmetry~
J.
Phys.
A
:
Math.
Gen.
2529-2554.
Group, Phys. Rev. A 134 (1964) 1602-1607;
Translation
[94] J. Zak, Magnetic
Group II : Irreducible
Representations, Phys. Rev. A 134 (1964) 1607-1611.
Magnetic
Translation