1

 Adiabatic
theory and topological indices
in molecular physics and solid state physics 

or

Geometric and topological aspects

of slow and fast coupled dynamical systems
in quantum and classical dynamics.

29 April 2002,

Lectures notes for lectures given in :

Service Phys. Théorique de Saclay, march-april 2002,

http ://www-spht.cea.fr/fr/

and M.A.S.I.E. Spring School, Warwick, march 2002.

http://www.maths.warwick.ac.uk/~mark/symposium

Frédéric Faure,
Laboratoire de Physique et Modélisation des Milieux Condensés (LPM2C)
(Maison des Magistères Jean Perrin), CNRS
BP 166 38042 Grenoble Cedex 9 France
email : frederic.faure@ujf-grenoble.fr

web page : http ://lpm2c.polycnrs-gre.fr/faure

2
Contents

1 Introduction and Overview 7

1.1 Fast and slow coupled systems in physics . . . . . . . . . . . . . . . . . . . 8

1.1.1 Examples in the macroscopic world: . . . . . . . . . . . . . . . . . . 8

1.1.2 Examples in the microscopic (quantum) world . . . . . . . . . . . . 10

1.2 Geometrical and topological aspects . . . . . . . . . . . . . . . . . . . . . . 12

1.3 Classical versus Quantum models . . . . . . . . . . . . . . . . . . . . . . . 13

1.3.1 Expression of the dynamics for a single system . . . . . . . . . . . . 13

1.3.2 Hamiltonians for a slow-fast system . . . . . . . . . . . . . . . . . . 13

1.3.3 Eective dynamics in ber bundle . . . . . . . . . . . . . . . . . . . 14

1.4 In the next chapters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Berry's Connection and Berry's phase in quantum mechanics 17

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2
2.2 The Foucault pendulum and parallel transport in T S . . . . . . . . . . . . 17

2.2.1 The Levi-Civita Connection . . . . . . . . . . . . . . . . . . . . . . 18

2.2.2 Holonomy and curvature . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.3 Fiber bundle description . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 The canonical quantum bundle over (Hilbert space) . . . . . . . . . . . . . 24

2.3.1 The projective space (H) . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.2 Berry's connection between the bers of H → (H) . . . . . . . . . . 26

3 The Semi-classical limit. Example of the Angular momentum dynamics 31

3.1 The free rigid body motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1.1 In classical mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1.2 In quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 The Classical limit of the Angular momentum dynamics . . . . . . . . . . 34

3.2.1 The su(2) algebra and the coherent states . . . . . . . . . . . . . . 35

3.2.2 Expectation values of operators . . . . . . . . . . . . . . . . . . . . 36

3.2.3 Schrödinger equation and classical limit . . . . . . . . . . . . . . . . 37

2
3.2.4 The quantizated line bundle over S . . . . . . . . . . . . . . . . . 39

3
4 CONTENTS

4 Topological aspects in the Semi-quantum model of slow-fast coupled sys-

tems 43
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2 Model for coupling between: slow rotation and n quantum vibrational levels: 44
4.2.1 The semi-quantal model . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2.2 Modications of bands by an external parameter λ: . . . . . . . . . 46
4.2.3 Quantum model: (rotation is quantized). . . . . . . . . . . . . . . 48
2
4.3 Model with more interesting topological phenomena: Slow motion on P ,
dimension 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.3.1 Classical mechanics: . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.3.2 Quantum mechanics: . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.3.3 Slow Vibrations coupled with 3 electronic states: . . . . . . . . . . . 52
4.3.4 Band spectrum in B.O approximation: . . . . . . . . . . . . . . . . 52
2
4.3.5 Topology of a vector ber bundle F over P : . . . . . . . . . . . . . 53
4.3.6 Quantization of vibrations: . . . . . . . . . . . . . . . . . . . . . . 54
4.3.7 Summary: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3.8 Remark on Index formula for the sphere (angular momentum phase
space) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3.9 Remark on the surface of degeneracy S in the model between bands
2-3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.3.10 Remark on Semi-classical expansion for ~ → 0; Weyl formula with
correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3.11 Remark on Naturality of index formula . . . . . . . . . . . . . . . 58
4.3.12 Index theorem and group theory: . . . . . . . . . . . . . . . . . . . 59
4.4 Main Born-Oppenheimer theorem of adiabaticity . . . . . . . . . . . . . . 60
4.5 Berry's connection, Chern Class and Characteristic Classes . . . . . . . . . 63
4.5.1 Berry's Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.5.2 Characteristic Class . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.5.3 Important physical remarks: . . . . . . . . . . . . . . . . . . . . . . 66
4.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5 Topological aspects in the Classical model of slow-fast coupled systems 69

5.1 A simple class of models. Topology of the tori bundle. . . . . . . . . . . . . 69
5.2 Semi-quantum model; Energy Bands and their topology by semi-classical
calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.3 Relation with classical and quantum monodromy: . . . . . . . . . . . . . . 72
5.4 Classical Hannay connection; Semi-classical correspondence with Berry's
connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.4.1 Classical Hannay connection . . . . . . . . . . . . . . . . . . . . . . 74
5.4.2 Semi-classical correspondence between Hannay and Berry's connection. 77
5.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
CONTENTS 5

6 Topological Chern indices and the Integer Quantum Hall eect 81

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.2 Band Spectrum and topology for the Quantum Hamiltonian . . . . . . . . 82
6.2.1 A dimensionless model . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.2.2 Band Spectrum and topological Chern indices . . . . . . . . . . . . 84
2
6.2.3 Decomposition of L (slow ) in eigenspaces of T̂x and T̂p : . . . . . . . 85
6.2.4 Topological indices of the bands . . . . . . . . . . . . . . . . . . . . 86
6.3 Interpretation of the Topological Chern indices: the Quantized Hall conduc-
tivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.3.1 Delocalized Bloch waves and Localized Wannier waves . . . . . . . 86
6.3.2 Physical consequence on temporal evolution
. . . . . . . . . . . . . 88
6.3.3
∂U
Addition of a weak external electric eld ~ =
= Ey ~uy . . . . . .
E 88
∂~
r
6.4 Born-Oppenheimer Approximation with Strong Magnetic eld; Eective dy-
namics in a Landau Level . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.4.1 Formula for Chern indices Cn : . . . . . . . . . . . . . . . . . . . . 90
6.4.2 Sum of Chern indices in a Landau Level . . . . . . . . . . . . . . . 91
6.5 Semi-classical calculation of Chern indices in a Landau Level . . . . . . . . 91
6.5.1 Classical Hamiltonian and trajectories of H(q, p) . . . . . . . . . . . 92
6.5.2 Quasi-modes : Quasi-mode on a contractible trajectory of type (0, 0) : 93
6.5.3 Semi-classical spectrum and tunnelling eect: . . . . . . . . . . . . 94
6.5.4 General result: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.5.5 Total Chern index: . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.5.6 The Chern indices for a chaotic dynamics : . . . . . . . . . . . . . . 97
6.6 References : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6 CONTENTS
Chapter 1
Introduction and Overview
The objective of these lectures is mainly to present a general framework for the description
of dynamical systems containing a slow sub-dynamical system coupled with a faster sub-
dynamical system, shortly called slow-fast coupled systems.
In the examples, we will mainly be concerned with nice examples in physics of such a
situation, namely small molecules.
We will describe the total dynamics in term of ber space, where the slow motion takes
place in the base space, whereas the fast motion takes place within the bers.

This problematics can be considered in classical mechanics, or in (mixed description)

Classical-Quantum mechanics, or in Quantum mechanics.

Using semi-classical rules, we will present the relations between the topological and
geometrical properties respectively in these three possible descriptions.

In particular we will see the strong similarity between the semi-classical limit and the
adiabatic limit.

The usefulness of using dierent descriptions will be clear for example, when we will
see that topological characterization of ber bundles within the mixed Classical-Quantum
description (i.e. the Born-Oppenheimer description) gives a nice insight in the full exact
quantum spectrum: the precise numbers of energy levels in each bands.

To summarize:

The subject of these lectures is:

Geometric and topological aspects of slow and fast coupled

dynamical systems in quantum and classical dynamics.

In this introduction, we give some preliminary explanations to these three adjectives.

The author aknowledges the Service of physique théorique of Saclay for its kind hospi-
tality during two months, and the M.A.S.I.E. group (Mechanics and Symmetry in Europe)
for good scientic discussions and collaborations.

7
8 CHAPTER 1. INTRODUCTION AND OVERVIEW

1.1 Fast and slow coupled systems in physics

We give here few examples of slow-fast mechanical coupled systems in physics.

1.1.1 Examples in the macroscopic world:

The ordinary Pendulum

with length l(t) varying slowly. See gure.

The Classical adiabatic theorem says: The enclosed area in phase space I =
E(t)/ω(t) is approximatively conserved.
One says that I is an adiabatic invariant.
(More precisely, if dl/dt ∼ ε, then |I(t) − I(0)| < Cε for t ∈ [0, T /ε]).
pθ Fast motion follows constant action I

θ
l(t) ω 2 =g/l

Slow parameter l(t)

We have say nothing about the angular position.

The Foucault pendulum:

The fast oscillations direction of the Foucault Pendulum follow parallel transport on the
earth.

θΗ

N
S

ωΤ

θΗ

S
1.1. FAST AND SLOW COUPLED SYSTEMS IN PHYSICS 9

After one revolution, appearance of a geometrical phase (holonomy): the Hannay

angle
ZZ
ϕHannay = dΩ = 2π (1 − sin (latitude))
hemisphere

For a rotating pendulum, the phase shift is a dynamical phase + geometrical

phase:

∆ϕ = ωT + ϕHannay

(This is a general relation).

Revolution and Precession of planets

This is the rst historical application of the averaging method by Lagrange, Laplace, Gauss.
They observed the absence of secularity in the variations of the major axes a of orbits.

In other terms, a is a adiabatic invariant.

Slow precession

Fast Revolution

Statistical systems.

Ergodicity on energy surface, gives conservation of phase space volume:

V(E) = V ol {(q, p), H(q, p) ≤ E},

and then gives entropy conservation S(E) = k log V(E).
(This gives important results in statistical physics).
10 CHAPTER 1. INTRODUCTION AND OVERVIEW

l(t)

Biological systems
Fast dynamics: the reproduction cycle of a predator-prey system.
Slow dynamics: climate evolution, modied by the vegetable food consumption
by the prey.

1.1.2 Examples in the microscopic (quantum) world

In these lectures we will consider mainly the following examples.

Small molecules,

are a very rich place to observe slow and fast coupled systems:
−15
with fast electrons (typical period τe ' 10 → 10−16 s.),
−14
slower vibrations of the nuclei (τv ' 10 → 10−15 s., )
slower rotation of the molecule (τrot ' 10−10 → 10−12 s.).

Spin precession

of a neutron in a slowly varying external magnetic eld ~ (t).

B
The Quantum Adiabatic theorem tells: The spin state follows approximately the
instantaneous eigenstate |spin >' |±B~ >, up to a quantum phase.
~ 1
RR
After one period T of B (t), the nal phase is θDynamical = ωT plus θBerry = dΩ.
2 hemisphere
1.1. FAST AND SLOW COUPLED SYSTEMS IN PHYSICS 11

|s 0> |s f>=|s 0> exp− i( θDyn+ θBerry )

B

Quantum Hall eect:

Bi-dimensional Electrons a strong Magnetic eld , in a periodic potential , with a weak

external electric eld .

• Fast Cyclotron motion on circles.

 Slower precession of the circles.

 Slower motion of quasi-momentum over the Brillouin zone.

Bz
y
Ey

jx
X x

In these lectures, we will consider only conservative (i.e. Hamil-

tonian) systems having a nite number of degree of freedom.

Remarks

From the above examples,

• One can distinguish two class of problems:

 (A)  Slow-Fast systems : Fast system coupled with a slower system (recipro-
cal inuences)
12 CHAPTER 1. INTRODUCTION AND OVERVIEW

 (B)  Time adiabatic systems : Fast system with slower external time-dependent
parameters (uni-directional inuences).

Property: (B) is a special case of (A).

• The coupling strength between the fast and slow dynamics is not necessary
weak.
( non perturbative description).
Interesting topological phenomena can then appear...

1.2 Geometrical and topological aspects

They concern ber bundles with connections which naturally occur in the previous
situations of slow and fast coupled systems.
A ber bundle is a continuous collection of isomorphic space.
Here the base space is the dynamical space of the slow sub-system, or the parameter
space for Time-adiabatic systems.
The Fiber space is the dynamical space of the fast sub-system.

Total space

Fiber =
Fast Phase space
or Fast Hilbert Space

Base space =
Slow Phase space (A)
or Parameter space (B)

Note that from a global point of view, one can consider the Topology of the bundle
related to possible twists of the bers:

Fibers

Base space
Trivial bundle SW=0 Moebius strip SW=1 Trivial bundle SW=0
1.3. CLASSICAL VERSUS QUANTUM MODELS 13

1.3 Classical versus Quantum models

In the above examples, the fast or slow motions can sometimes be consider in a quantum
or classical description, independently. We recall some basic facts.

1.3.1 Expression of the dynamics for a single system

• The classical dynamics is express by Hamilton equation
in Phase space X = (q, p) ∈ P (dimension 2n):

dq ∂H(q, p)
=
dt ∂p
dp ∂H(q, p)
=−
dt ∂q

• The quantum dynamics is expressed by Schrödinger equation in Hilbert space

H:
d|ψ(t)i
i~ = Ĥ|ψ(t)i
dt
• Semi-classical rules relate the Classical and Quantum dynamics,
in the limit  ~/action → 0:

Phase space P ↔ Hilbert spaceH

Hamiltonian H ↔ Hamiltonian Ĥ

1.3.2 Hamiltonians for a slow-fast system

For a Slow-fast system (class A), there are then 4 possibilities of description:

Slow \ Fast Classical in PF ast Quantum in HF ast

Classical in PSlow Function Htot (Xslow , Xf ast ) Matrix Symbol Xslow → Ĥf ast (Xslow )
(Classical ) (Semi-Quantum )

Quantum in HSlow No meaning (?) Ĥtot in Hslow ⊗ Hf ast

(Quantum )

Remarks

• Superposition property in quantum dynamics:

→ One can not treat the inuence of a quantum system on a classical
system .
Then the Quantum-Classical description has no real dynamical meaning.

• The Semi-quantum description is the Born-Oppenheimer approximation.

14 CHAPTER 1. INTRODUCTION AND OVERVIEW

1.3.3 Eective dynamics in ber bundle

• For a Classical dynamics, with integrable fast motion:

 The fast motion in bers follows tori with constant action, and follows
the (geometric) Hannay connection plus a dynamical phase.

 The slow motion on base space is described (rst order) by averaging

method: Hef f ective (Xslow ) = hHtot (Xslow , Xf ast )if ast motion

• For a Semi-Quantum dynamics, with discrete spectrum of Ĥf ast (Xslow ):

 The fast state in bers is an eigenstate |ψn (Xslow )i of Ĥf ast (Xslow ), with
energy En (Xslow ). Its follow the (geometric) Berry connection plus a dy-
namical phase.

 The slow motion on base space is described (rst order) by : Hef f ective (Xslow ) =
D E
ψn (Xslow )|Ĥf ast (Xslow )|ψn (Xslow ) = En (Xslow ) .

Energy
Fine structure: slow motion

Large structure: fast motion

∆Ε ∼ h ω
• Quantum dynamics: group of levels in the spectrum:

1.4 In the next chapters

• Rigorous denitions of the dierent descriptions:

 Classical Htot (Xf ast , Xslow ) on Pslow × Pf ast ,

 
 Semi-Quantum Xslow → Ĥf ast (Xslow ) on Pslow × Hf ast ,

 Quantum Ĥtot on Hslow ⊗ Hf ast

• Adiabatic theorems and Semi-classical Correspondences between the three

descriptions, at a topological and a geometrical level.

Slow \ Fast Classical, Integrable in PF ast Quantum in HF ast

Classical in PSlow (Classical ) (Semi-Quantum )
Topology of tori bundle Topology of eigenstates bundle
Hannay's connection Berry's connection
Quantum in HSlow No meaning (?) (Quantum )
Number of states in group of levels
Density of states
1.4. IN THE NEXT CHAPTERS 15

• Geometry of vector bundles, Connections, curvature, (Semi-classical) Computa-

tion of topological Chern indices.

• Remarks:
the Hannay connection needs integrability;
the Berry connection is very general in Quantum mechanics.

Fast Integrable Motion

Slow Motion
16 CHAPTER 1. INTRODUCTION AND OVERVIEW
Chapter 2
Berry's Connection and Berry's phase
in quantum mechanics

2.1 Introduction

The purpose of this chapter is to present the quantum dynamics in Hilbert space from
a geometrical point of view, initiated in physics by Berry in 1984: The Hilbert space
is seen as a line bundle over its Projective space, with a natural connection, the Berry's
connection. (In geometry, this is common since a longer time, and the Berry's connection
is sometimes called the Chern's connection).
Here we will present this connection as a Levi-Civita connection which is the con-
nection which denes parallel transport of tangent vectors on a surface embedded in R3 .
This chapter contains more geometric informations than physics, and is to prepare the
subsequent chapters.
Other important tools related to ber bundles in physics will be presented in next
chapters.

• The rst section introduces the Levi-Civita connection on the sphere. This is an
introduction to vector bundles from an intuitive point of view,

• The second section denes the Berry's connection and shows that any quantum evo-
lution follows this natural connection plus a dynamical phase.

2.2 The Foucault pendulum and parallel transport in

T S2
References: [13][5].
Objective of this section: introduction to a vector bundle, the tangent bundle T S 2,
with its connection.

17
18CHAPTER 2. BERRY'S CONNECTION AND BERRY'S PHASE IN QUANTUM MECHANICS

We consider the sphere S 2 which is the earth surface, xed in an inertial frame (actually
the earth rotates in this frame, with period T = 24 hours). The attached point of the
Foucault pendulum is forced to follows a path γ(t) on the sphere, which is parallel to the
equator.
In this section, we will also imagine any possible path γ(t) on the sphere S 2.

2.2.1 The Levi-Civita Connection

We suppose small oscillations of the pendulum, and simplify the physical discussion by ar-
guing that (from conservation of angular momentum along the local z axis) the oscillations
3
try to maintain their direction in space R , but are constrained to the tangent plane of the
sphere. The result is that the oscillations follows the so-called parallel transport on the
sphere, or Levi-Civita connection which we now dene.

v2
v1
x1 x2
δx

Figure 2.1: Foucault Pendulum

Tx2 v1

dv P

v2

Tx
1

Figure 2.2: Levi-Civita Connection

For a given point x ∈ S 2 , we note Tx S 2 the tangent plane. Over each point x(t) = γ(t)
2
of the path, we denote vx ∈ T S the tangent vector (which could be the velocity of the
pendulum measured after each pendulum period). At a given point x ∈ T S 2 , and for a
small displacement δx on the path, there corresponds a small variation dv = vx+δx − vx .
3
Using the scalar product in R , one denes the orthogonal projection Px of vectors of
R3 onto the tangent plane Tx S 2 .
2.2. THE FOUCAULT PENDULUM AND PARALLEL TRANSPORT IN T S2 19

Then the continuous family of tangent vectors vx is said to follow the Levi-Civita
connection or parallel transport over the path γ if:

Dδx v = P dv = 0
↑
Covariant Derivative

Property: The Levi-Civita connection conserves the norm of the transported

vector.
Indeed: d |v|2 = d hv, vi = 2 hv, dvi = 2 hP v, dvi = 2 hv, P dvi = 0.

Remarks

• We have said that for physical reasons the Foucault pendulum follows this Levi-Civita
connection. This is an exact result if γ is parallel to the equator, and can be shown
using Coriolis forces in the pendulum frame [2].

• For any slowly varying path γ(εt) ∈ S 2 with ε→0 (slow compared to the pendulum
oscillations), the Foucault pendulum would follow approximatively the Levi-Civita
connection, with an error less than ε on the time interval t ∈ [0, 1/ε]. This results from
the Classical Adiabatic theorem [3][15], but non directly because this theorem
needs a fast rotational motion of the pendulum around the z axis (see discussion in
[11]).

• This connection is geometric in the sense that in that limit, the direction of the
pendulum depends only on the geometry of the path γ on S 2.

2.2.2 Holonomy and curvature

Consider now any closed path γ S 2 . After one loop, one can compare the
on the sphere
2
initial and nal directions of the pendulum vi ,vf ∈ T0 S . These two vectors have the same
length, but they dier by an angle h (γ), which depends only on the geometry of the path
γ , and is called the holonomy of the Connection on the path γ .
From the second picture, one easily guess that h(γ) is related to the surface enclosed
by the so called curvature integral:

d2 s
ZZ ZZ
h (γ) = θ = ΩS = curvature = : holonomy ∈ [0, 2π]
S S R2

For the adiabatic pendulum the Holonomy is the Hannay angle. (See below for a
more precise denition of the Hannay angle).
20CHAPTER 2. BERRY'S CONNECTION AND BERRY'S PHASE IN QUANTUM MECHANICS

N
θ

θΗ γ

S S

Figure 2.3: Holonomy

2.2.3 Fiber bundle description

Over each point x ∈ S2

is attached a tangent plane Tx S 2 . Each tangent plane Tx S 2 denes
2
the ber over the point x in base space S .
The collection of these bers is called the tangent bundle T S 2. More precisely, T S2
2
is a real vector bundle of rank 2 over S . In general, the rank of a vector bundle is
the dimension of each ber.
Two dierent tangent plane Tx and Tx0 are not identical, and there is no unique and
0
natural way to identify them. If γ is a path which connect x and x , we have seen that
Tx and Tx0 can be identied with the parallel transport (or Levi-Civita connection) in the
bers over the path γ. We insist that this connection has been dened from the scalar
3
product in R .
The whole set of bers can have a global twist over the base space as in the well S2
1
known example of the Moebius strip which is a real vector bundle of rank 1 over S .

Topology of a real vector bundle of rank 1 over S1

The topology of such a bundle is characterized by an integer which represents the number
of twists: the Stiefel-Whitney index SW ∈ Z2 = {0, 1}. (see [5][8]).
(This characterization is an intrinsic one: two bundles are isomorphic if there is a
bijective mapping between them which respects the bers. So the third example on the
3
gure is trivial even if it can not be continuously deformed in R to the rst example which
is also trivial).

Topology of a complex vector bundle of rank 1 over S2

Because each tangent plane Tx S 2 is oriented, one can identify Tx S 2 with the complex plane
C, and consider the bundle T S 2 as a complex vector bundle of rank 1 over S 2 .
2.2. THE FOUCAULT PENDULUM AND PARALLEL TRANSPORT IN T S2 21

Fiber T x1
T x2
h(γ )

N N

x1
δx x2 γ

Base space Base space

S S

Figure 2.4: Vector bundle description

Fibers

Base space
Trivial bundle SW=0 Moebius strip SW=1 Trivial bundle SW=0

Figure 2.5: Real vector bundle of rank 1 over S 1.

22CHAPTER 2. BERRY'S CONNECTION AND BERRY'S PHASE IN QUANTUM MECHANICS

As explained in the gure, the topology can be characterized by an integer C ∈ Z called

the Chern index. (or rst Chern class).

F(J)

J Clutching function

θ e i ϕ(θ)

ϕ(2π)=ϕ(0) + C2π

C: Chern Index

Figure 2.6: Complex line Bundle over S 2. The topology is characterized by the homotopy
type of the clutching function, C ∈ Z.

Property:
Chern(T S 2 ) = +2

(See other examples below.)

proof: see gure.

North Hemisphere

Glue ϕ= 0 π 2π 3π 4π
Equator

Chern = 2

South Hemisphere

Figure 2.7: Topology of the tangent bundle T S2

2.2. THE FOUCAULT PENDULUM AND PARALLEL TRANSPORT IN T S2 23

Remarks:

• The connection (curvature, holonomy,.. ) is a geometric structure on the ber

bundle (sensitive to continuous deformations of the metric).

• The Chern index C ∈Z is topological property, robust (i.e. not sensitive) under
continuous deformations.

• Geometrical structures can be used to compute topological properties. Example:

Gauss-Bonnet curvature formula:
Z Z
2

1 4π
Chern T S = Curvature = =2
2π S2 2π

• The sign denition of the Chern index we use is from algebraic geometry conventions
[7] and is opposite to algebraic topology conventions [8].

Topology from the zeros of a global section

A Global section of the bundle is a continuous choice of vectors in each ber.

For a real vector bundle of rank 1 over S1 (as the Moebius strip), the Stiefel-Whitney
index SW is the parity of the number of zeros of a section (0 = even number of zeros or
1 =odd number zeros), see gure.

Fibers

Base space
Trivial bundle SW=0 Moebius strip SW=1

Figure 2.8: Moebius strip, Zero of sections.

For a complex vector bundle of rank 1 over S 2, the Chern index C is obtained by the
opposite of the sum of the orientations (±1) of isolated zeros of a generic section.

Remark: We saw two possibilities to compute the Chern index. This is very general.
First from an integral curvature, which more generally use tools of cohomology (inte-
grals). The second possibility is from zeros of global sections and use tools of homology
(intersections) (historically the rst denition of Chern classes). The two approaches are
related by Poincaré duality [7].
24CHAPTER 2. BERRY'S CONNECTION AND BERRY'S PHASE IN QUANTUM MECHANICS

+1

C=1+1=2
+1

Figure 2.9: A section of T S2 is a vector eld on S 2. It gives the topology of the bundle
T S 2.

2.3 The canonical quantum bundle over P (Hilbert space)

2.3.1 The projective space P (H)

For a quantum system described by the state |ψ(t)i ∈ H in Hilbert space H , the
Schrödinger equation is linear:

d|ψ(t)i
i~ = Ĥ(t) |ψ(t)i
dt
so if |ϕ(0)i = λ|ψ(0)i then |ϕ(t)i = λ|ψ(t)i.
It is therefore natural to identify vectors |ψi ∼ λ|ψi, for any λ ∈ C. They are
represented by an equivalence class [ψ], which a dimensional one complex vector space in
H.
The set of equivalence class is the projective space:

P (H) = (H \ {0}) / ∼

So H → P (H) is a complex ber bundle of rank 1 over P (H).

ϕ λϕ ψ λψ

:H
λϕ
ψ
ϕ λψ

[ψ]
[ϕ]

Hilbert Space H
Projective space P(H)

Figure 2.10: Projective Space P (H)

2.3. THE CANONICAL QUANTUM BUNDLE OVER P (HILBERT SPACE) 25

λ|ψ(t )>

λ|ϕ(0)> |ψ( t)>

|ψ(0)> :H

[ψ(t )]
[ψ(0)]

Projective space P(H)

Figure 2.11: Quantum evolution in P (H)

Because it is linear, the quantum dynamics in H projects onto P (H) (i.e. gives well
dened trajectories on P (H)).
Remark that if dimC H = n then dimC P (H) = n − 1.

Example of H = C2
P (C2 ) = P1 ∼
= S2 is the Riemann Sphere, or Bloch Sphere, and H = C2 is a complex
ber bundle over P (C ) ∼
2
=S 2
: (so called  tautological bundle) , with

Chern C2 → P1 = −1

Proof:
If (|+i, |−i) is a orthonormal basis of C2 ,
a
: if b 6= 0


|ψi = a|+i + b|−i =b b |+i + |−i
∼ z|+i + |−i = |ψz i : (z = a/b ∈ C) unique representative in P C2

but if b = 0, i.e. |z| → ∞,

|ψ∞ i ∼ |+i : unique point at innity

So we have shown that P C2 is the complex plane z ∈ C with the innity identied to a

unique point. This is a sphere S 2 .

The stereographic (or inhomogeneous) coordinate on S 2 ∼ = P C2 are z = cotg (θ/2) e−iϕ ∈

C.
Consider now the orthogonal projection of |−i onto the ber F iber[ψ] . This gives a global
section of the bundle |s[ψ] i = P[ψ] |−i = |ψihψ|−i which is zero only if [ψ] = [+]. One can show
that this zero has orientation −1.
Remark: the normalized states hψ|ψi = 1, form the Hopf bundle S 3 → S 2 .

26CHAPTER 2. BERRY'S CONNECTION AND BERRY'S PHASE IN QUANTUM MECHANICS

[+] Projective Space P 1

[ψ]
θ

Im(z)
[−] z

Re(z)
Complexe plane z=a/b

Figure 2.12: Stereographic coordinates on P (C2 )

Remarks

• If|ψi = a|+i + b|−i is a spin 1/2 state, then [ψ] ∈ P (C2 ) is identical with the mean
~
s = hψ|Ŝ|ψi ∈ S 2 in R3 .
spin direction ~

2.3.2 Berry's connection between the bers of H → P (H)

Similarly with theT S 2 case, the Scalar product in H denes a Levi-Civita connection on
the bundle H → P (H).
A section |ψ (X)i is parallel transported over δX if

D|ψi = P d|ψi = 0

|ψihψ|
but P = hψ|ψi
so

|ψihψ|dψi
D|ψi = =0 ⇔ hψ|dψi = 0
hψ|ψi

Connection
|ψ>
|ψ+δψ>

|ψ+δψ>
|δψ>

|ψ> P=|ψ><ψ|

δX

Hilbert space Projective Space

Figure 2.13: Connection on H → P (H)

2.3. THE CANONICAL QUANTUM BUNDLE OVER P (HILBERT SPACE) 27

0
If an other section has an additional phase |ψ (X)i = eif (X) |ψ(X)i (and therefore is
0 0 if
not parallel transported), then |dψ i = idf |ψ i + e |dψi,

hψ 0 |dψ 0 i = i df hψ 0 |ψ 0 i

This connection is called the Berry's connection in physics (but is known since a
long time in mathematics, see [7]).

Quantum evolution and the Berry's Connection

Consider any quantum dynamics dened by the Schrödinger equation (with possibly time
dependent operator Ĥ(t))

d|ψi
i~ = Ĥ(t) |ψi
dt
This gives

hψ|Ĥ(t)|ψi
hψ|dψi = −i dt = i df hψ|ψi
~
with
df hψ|Ĥ(t)|ψi Eψ(t)
=− =−
dt ~hψ|ψi ~
so the quantum evolution follows the Berry's connection plus an additional dynamical
phase (related to the mean energy Eψ(t) ).

|ψ(dt)>
|ψ(0)> df

Con. :H

[ψ(0)] [ψ(dt)]

Projective space P(H)

Figure 2.14: Quantum evolution in H → P (H) with the dynamical phase.

28CHAPTER 2. BERRY'S CONNECTION AND BERRY'S PHASE IN QUANTUM MECHANICS

Suppose now that [ψ(t)]

is a closed trajectory in P (H). After a period T, the state
iϕ
comes back to the original ber, so |ψ(T )i = e |ψ(0)i. The total phase is

ϕ = ϕBerry + ϕDyn (2.1)

0
ϕBerry = h (γ) : Holonomy or Berry s phase (2.2)
Z T
Eψ(t)
ϕDyn = − dt : Dynamical phase (2.3)
0 ~

h(γ) |ψ(0)>

ϕD
Evolution
|ψ(T )>
Con.nection

[ψ(0)]=[ψ(T)]

Figure 2.15: Berry's phase

As we have shown this Berry's phase is quite general in quantum mechanics and occurs
in dierent contexts when there is a closed (or almost closed) trajectory [ψ(t)] in P (H):

1. Adiabatic motion: a closed loop of in parameter space X(t) gives (from quantum
adiabatic theorem) an approximated closed loop of [ψ(X(t))]: this is the original
adiabatic Berry's phase. See below.

2. Semi-classics: motion of a wave packet on a closed trajectory. See more details

below. Then
I
1 Action
ϕBerry = pdq = ,
~ ~
and gives quantization rules: ϕBerry ≡ 0 [2π] ⇔ Action = n h, n ∈ N.
(This bundle description with wave packets is the basis of geometrical quantization
theory with complex polarization, see [21][6]).

Proof: from [q̂, p̂] = i~, and T̂Q = e−iQ(p̂−p)/~ , T̂P = eiP (q̂−q)/~ , one deduces T̂P−1 T̂Q−1 T̂P T̂Q =
eiAction/~ . On the other hand, as explained above, these unitary operators generate parallel
tranport of wave packets at mean position (q, p). 
2.3. THE CANONICAL QUANTUM BUNDLE OVER P (HILBERT SPACE) 29

Parallel
[q(T)p(T)]= [q(0)p(0)] transport

Figure 2.16: Quantization rule

Other properties (*) Natural metric (Fubiny-Study) on P (H), ds2 = d2 ([ψ] , [ψ + dψ]) =
1 − |n hψ|ψ + dψin |2
ds ∆E
The velocity of [ψ(t)] measured with this metric is v = dt
= ~
, with ∆E 2 =
hψ|Ĥ 2 |ψi − hψ|Ĥ|ψi2 the energy uncertainty.
Reference: Anandan et al [1].
30CHAPTER 2. BERRY'S CONNECTION AND BERRY'S PHASE IN QUANTUM MECHANICS
Chapter 3
The Semi-classical limit. Example of
the Angular momentum dynamics
This chapter is an introduction to quantization of reduced phase space, coherent states, and
some semi-classical limit concepts, in the simple example of angular momentum dynamics.

We will begin with the simple and concrete example of the free rigid body dynamics.

We explain at the end of the chapter, why and how the semi-classical limit is
related to the adiabatic limit .

3.1 The free rigid body motion

For a precise theory of the free rigid body motion, see Arnold [2], or Ratiu [9] for symplectic
reduction theory.

Here, we only give a short description of some (known) results, inspired from [12].

See also [10].

3.1.1 In classical mechanics

Consider a free rigid body, that is a rigid body with no external forces (for example a
falling rigid object).

Consider a xed inertial frame Re with origin at the body's center of mass, and a frame
Rb xed with respect to the body.

A conguration of the rigid is specied by its orientation with respect to an inertial

frame, that is by the rotation matrix R ∈ SO(3) which relates frames Re and Rb . The
conguration space is then the group SO(3) of rotation matrices which is 3 dimensional
(2 angles for the direction of rotation, and one angle for the magnitude of rotation).

The phase space P = T ∗ SO(3) includes the momenta denoted by J~ and is therefore
6 dimensional.

31
32CHAPTER 3. THE SEMI-CLASSICAL LIMIT. EXAMPLE OF THE ANGULAR MOMENTUM DYNA

The reduced phase space: SJ2~

Because of symmetry by rotation of the problem, the total angular momentum is conserved.
This means that, with respect to the inertial frame Re , the total angular momentum
J~e ∈ R3 is a xed vector.
Euler showed how to simplify the equations of motion by going to the frame Rb attached
to the body. In this frame ~
J(t) moves. But J~ = RJ~e so J~ = J~e is ~ (t)
constant, so J

moves on the surface of a 2 dimensional sphere SJ2~ .

e
This sphere is called a reduced phase space of the problem (it is shown in [9] that
SJ2~ has indeed a canonical reduced symplectic two form).
e
The kinetic energy is

Jy2 Jz2
  1  2 
1 J
H J~ = JI
~ J~ =
−1 x
+ + (3.1)
2 2 Ix Iy Iz
where I is the moment of inertia tensor, a symmetric positive denite matrix, we suppose
diagonal in the frame Rb , with eigenvalues (Ix , Iy , Iz ).
The energy levels (which coincide with the trajectories) are depicted on gure 3.1.
There are 6 xed points, 2 minima of energy, 2 saddle points, and 2 maxima.

Jz
Max

Je

Min
Saddle
Jx
Jy

Iz < Ix <I y

Figure 3.1: Rigid body trajectories on the reduced phase space SJ2~ .
e

The equation of motion of J~ can be written as:

~
dJ(t) ∂H(J)~
= ∧ J~
dt ∂ J~
Each xed point on the sphere corresponds to special cases when J~ coincide with one
of the principal axis of the body. The motion of the body in Re is then a rotation around
3.1. THE FREE RIGID BODY MOTION 33

J~e . Try to launch a book, and observe these stationnary points, called relative equilibria
(because the motion is still on a circle).

The ber bundle over SJ2~ ; A Berry's phase from the XVIII century
e

Let us denote by MJ~e the points of phase space P which corresponds to a given J~e . Of
course MJ~e ∼
= SO(3) is 3 dimensional. Because J~e is constant of motion, the dynamics stay
inside MJ~e .
We have seen that the reduced phase space SJ2~ dened above, miss to describe the
e

rotation of the body around the vector J~e . This means that SJ2~ is obtained from MJ~e by
e
identication of points related by such a rotation:

SJ2~e = MJ~e / ∼≡ SO(3)/SO(2)

So MJ~e is actually seen as a ber bundle over the sphere SJ2~ . The bers are circles, and
e
the dynamics of the body takes place in the bers, whereas the reduced dynamics is on
SJ2~ . See gure 3.2. With our conventions, the topology of this ber bundle is Chern = 2,
e
2
(because it is isomorphic to the unit subbundle of T S ).

∆θ

Jz

Je
Je

∆θ

Jx
Jy

Iz < Ix <I y Re

Figure 3.2: True trajectories: in a Circle bundle over the reduced phase space SJ2~ .
e

Consider now any closed reduced trajectoryγ on the reduced phase space SJ2~ , with
e
period T and energy E . The actual trajectory in MJ~ is not closed and after the period T ,
e
it comes back in the initial ber, shifted by an angle ∆θ , measured in inertial frame Re .
In [12], R.Montgomery showed that this angle can be express as:

2ET
∆θ = − Ω,
J~
34CHAPTER 3. THE SEMI-CLASSICAL LIMIT. EXAMPLE OF THE ANGULAR MOMENTUM DYNA

where Ω is the solid angle enclosed by the closed trajectory γ on the reduced phase space
SJ2~ . The angle Ω is the holonomy of the path γ with respect to a natural connection in the
e
ber bundle MJ~ . This connection is induced from the natural symplectic form in phase
e
∗
space T SO(3).
The angle 2ET /J is a dynamical phase, and Ω is a geometrical phase, very similar with
(2.1).

3.1.2 In quantum mechanics

The Hilbert space of the quantum free rigid body is

H = L2 (SO(3))

Angular momentum operators are acting in H with the usual commutation relations of
the so(3) Lie algebra:
h i h i h i
Jˆx , Jˆy = iJˆz , Jˆy , Jˆz = iJˆx , Jˆz , Jˆx = iJˆy .

Two rotations act in this problem. A rst rotation which rotates together the rigid
body and the vecteur J~e , is a symmetry of the problem.
A second rotation which rotates the rigid body alone (or equivalently the vector J~ in
the body frame), is not a symmetry of the problem.
With respect to these two rotations the Hilbert space decomposed as:
M (1) (2)
H = L2 (SO(3)) = Hj ⊗ Hj
j∈N

where Hj is an irreducible representation space of the so(3) algebra, with dimension

2j + 1.
In group theory, this decomposition is called the Peter-Weyl formula, see [17].
(2)
The quantum operator Ĥ obtained from (3.1) acts in Hj alone: this express the
quantum reduced dynamics of the rigid body.
(1)
(Note that because Ĥ has no action in Hj , every eigenvalue has multiplicity (2j + 1)).

3.2 The Classical limit of the Angular momentum dy-

namics

In this section we recall some known results about the angular momentum coherent states,
and their role to dene the classical limit, see [14] and [20]. We would like to stress the
correspondence between the quantum dynamics of an angular momentum with a xed
modulus j (integer or half integer: 2j ∈ N), and the classical dynamics of an angular
momentum vector J~ of length 1. J~ belongs to a sphere noted S 2 with radius 1, which is
the classical phase space of the angular momentum.
3.2. THE CLASSICAL LIMIT OF THE ANGULAR MOMENTUM DYNAMICS 35

3.2.1 The su(2) algebra and the coherent states

The quantum hermitian operators of the spin Jˆx , Jˆy , Jˆz form an irreducible representation
of the su(2) algebra, in a Hilbert space Hj with dimension 2j + 1 :
h i h i h i
Jˆx , Jˆy = iJˆz , Jˆy , Jˆz = iJˆx , Jˆz , Jˆx = iJˆy .
In order to have a nice correspondance with the classical limit, we now rescale these
operators by:
ˆ 1 ˆ
J~new = J~
2j
ˆ ˆ
We also write J~ insteed of J~new , so:

h i 1 h i 1 h i 1
Jˆx , Jˆy = i Jˆz , Jˆy , Jˆz = i Jˆx , Jˆz , Jˆx = i Jˆy .
2j 2j 2j
So dene

1
~ef f =
2j
1
(remark that ~ef f = 2j
plays the role of an eective Planck constant, ~ef f → 0 is the

classical limit).
A basis are the vectors
  |m >, m = −j, −j + 1, . . . , +j , eigen-vectors of the Jˆz operator:
Jˆz |m >= m
j
|m >. An element of the group g ∈ SU (2) is represented by the unitary

operator
     
α) = exp −iα3 Jˆz /~ef f exp −iα2 (2j)Jˆy /~ef f exp −iα1 Jˆz /~ef f ,
R̂(~ (3.2)

acting in Hj , α
~ = (α1 , α2 , α3 ) are the Euler angles.
where

The state |m = −j > corresponds to the classical vector J ~ = (0, 0, −1). In order
to obtain a quantum state |J ~ > associated to the classical vector J~ with any spherical
coordinates (θ, ϕ) we only need to apply the rotation operator (3.2) on |m = −j >, with
~ = (0, θ − π, ϕ), see gure 3.3 (a). Such a state |J~ >= R̂(~
α α)| − j > is called a coherent
state. One can show that ([20]):

jΘ2
 
2 Θ
< J |J~ >
~0 = cos 4j
'1− + o(Θ2 ), (3.3)
2 2

where Θ is the angle between J~0 and J~ on the sphere.

More generally, for every state |ψ >∈ Hj , one can dene its Husimi distribution
[20],[18, 19]:
2
~ = < J|ψ
Husψ (J) ~ > ,

which is a positive function on the sphere.

36CHAPTER 3. THE SEMI-CLASSICAL LIMIT. EXAMPLE OF THE ANGULAR MOMENTUM DYNA

From (3.3), we see that in the classical limit j → ∞, the Husimi distribution of a
coherent state |J~ >becomes localized on the phase space, at point J~ with a width
√
' 1/ j . Let us also remark that if J~0 and J~ are opposite, then the function < J~0 |J~ > is

zero with order 2j .

As a second example, consider the state |m >. One obtains that ~ =
Hus|m> (J)
2  
< ~
J|m > is maximum on the line J~ = m/j as expected, with a width of order
z
√ ~ > has a zero
' 1/ j , and < J|m of order (j − m) at point J~ = (0, 0, 1), and a zero of

order (j + m) in J~ = (0, 0, −1).

Jz
0
1
0
1
1
0
0
1
0
1111 (j−m) zeros
000
0
1
0
1
0
1
0
1
0
1
θ 0
1
0
1
Jcl 0
1
0
1
1
0
2j zeros 1
0
0
1 11
00
ϕ Jy Classical support of state |m>
= classical trajectory.
Jx
1
0
(j+m) zeros
(a) (b)

2
Figure 3.3: (a) Husimi distribution HusJ~(J~0 ) = < J~0 |J~ > of a coherent state |J~ > , with
its zeros.
2
(b) Husimi distribution Hus|m> (J~0 ) = < J~0 |m > of the state |m > with its zeros.

3.2.2 Expectation values of operators

One can compute [20]:

~ Jˆz |J~ >= Jz ,

< J|
~ Jˆx |J~ >= Jx ,
< J|
~ Jˆ2 |J~ >= (Jz )2 + ~ef f 1 − J 2 ,


< J| z z

and more generally, the expectation value of an operator Ô over coherent states gives a
function on the sphere, noted
~ =< J|
O(J) ~ Ô|J~ >
called the Berezin symbol of the operator (or Normal symbol). The operators con-
~
structed from the elementary operators Jˆ as above, have a symbol which admits a formal
series in power of ~ef f :
~ =
O(J) ~
O0 (J) ~ + ...
+~ef f O1 (J)
principal symbol
3.2. THE CLASSICAL LIMIT OF THE ANGULAR MOMENTUM DYNAMICS 37

The map Ô → O is injective [4], so the symbol characterizes the operator. The rst
~ is the principal symbol, or classical observable. In the limit ~ef f → 0 the
term O0 (J)
∞
symbols are dense in the space of C functions on the sphere. See [4][16] for the more
general deformation quantization framework.
Thanks to the injectivity of the symbol, one can dene the star product of two symbol
a ∗ b, to be the product of the two operators (see gure):

∗ b = â b̂
ad

a
^a
b
b^
a*b ^ab^

Operators
Function on phase space

Figure 3.4: Symbols and star product.

Then one has the property (common to all star products, and choice of quantization):

a ∗ b = ab + i~ef f {a, b} + o (~ef f )

where the second term involves the Poisson brackets of the symbols. The star product
allows to work (as possible) with symbols on phase space insteed of operators.

3.2.3 Schrödinger equation and classical limit

The quantum dynamics is dened by Hamiltonian Ĥ , a self-adjoint operator in Hj . A

quantum state |ψ(t)i evolves under the Schrödinger equation:

∂|ψ(t) >
i~ef f = Ĥ|ψ(t) > (3.4)
∂t
∂
Remark: one put the factor 1/j in front of
∂t
in order to have a nice classical limit.

The symbol of Ĥ is written

~ Ĥ|J~ >= H0 + ~ef f H1 + . . .

H =< J|
Consider a coherent state ~
|ψ(0) >= |J(0) >. One can show that this state is approxima-
tively a coherent state for not too long time evolution:

~ > = O(t)
|ψ(t) > −eiα(t) |J(t)
38CHAPTER 3. THE SEMI-CLASSICAL LIMIT. EXAMPLE OF THE ANGULAR MOMENTUM DYNA

where the dynamics of point ~ on classical phase space Sj2

J(t) is given by classical Hamil-
ton equations of motion:
~
dJ(t) ∂H0 (J)~
= ∧ J~
dt ∂ J~

Jz

Jcl (t)

Jcl (0)

Jy

Jx

Figure 3.5: Evolution of Husimi distribution of a coherent state: it is approximativelly

coherent, and its mean position ~
J(t) evolves with the Classical Hamilton equations of
motion.

Remarks:

• For example for Ĥ = Jˆz , the motion is a rotation around the z axis, with angular
velocity ω = 1.
• From a geometrical point of view, the symplectic two form on S2 is (in spherical
coordinates):
1
sin θ dθ ∧ dϕ
Ω=
2
∂ ∂
and the equation of motion can be written as dH(.) = Ω (XH , .), with XH = θ̇ +ϕ̇ ∂ϕ
∂θ
the Hamiltonian vector eld.

• Important remark: In order to have a well dened classical limit, we have put a
factor ~ef f in the Schrödinger equation (3.4). This is possible by a redenition of
time units.
In the sequel, we will consider Schrödinger equations without this 1/j factor:

∂|ψ(t̃) >
i = Ĥ|ψ(t̃) >
∂ t̃
In other words we will use a time variable t̃ = t/~ef f . (Correspondingly, the two
form is Ω̃ = Ω/~ef f ).
The main eect of this new denition, is that in the classical limit, the classical
motion viewed in time scale t̃ becomes slower and slower, because typical
velocities on the sphere are
dθ dθ
ṽθ = = ~ef f
dt̃ dt
3.2. THE CLASSICAL LIMIT OF THE ANGULAR MOMENTUM DYNAMICS 39

In this sense, the classical limit ~ef f → 0 will be equivalent to the adiabatic
limit of slow motion.

3.2.4 The quantizated line bundle over S2

For each point J~ ∈ S 2 we have dened a coherent state ~ ∈ Hj .
|Ji The choice of the phase

in the ~ is
denition of |Ji quite arbitrary, so the correct geometrical object to consider is
~.
the one dimensional vector space of vectors proportionnal to |Ji
2
This denes a complex line bundle over S , whose topology is characterized by an
integer Chern index.
The Chern index of this line bundle is

C = −(2j) (3.5)

The easiest way is to see this is to construct a global section of the bundle and compute
its zeros: Consider a reference coherent state |J~0 i, its projection ~ J|
|Jih ~ J~0 i gives a global
section. This section is zero if ~ J~0 i = 0.
hJ| We said above that this happens for J~ = −J~0
with order (2j). A correct inspection of the orientation gives C = −(2j).

Remarks:
h i
• In mathematical terms, the coherent states family denes a map J~ ∈ S 2 → |Ji
~ ∈
P (Hj ), and the line bundle in consideration is the pull back of the canonical bundle
Hj → P (Hj ).)

• This line bundle, and its topology will be important for the application of the index
formula below. (it plays also key role in geometric quantization, see [21, 16]).
40CHAPTER 3. THE SEMI-CLASSICAL LIMIT. EXAMPLE OF THE ANGULAR MOMENTUM DYNA
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65:1697, 1990.

[2] V.I. Arnold. Les méthodes mathématiques de la mécanique classique. Ed. Mir. Moscou,
1976.

[3] V.I. Arnold. Geometrical methods in the theory of ordinary dierential equations.
Springer Verlag, 1988.

[4] Martin Bordemann, Eckhard Meinrenken, and Martin Schlichenmaier. Toeplitz quan-
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[5] T. Eguchi, P.B. Gilkey, and A.J. Hanson. Gravitation, gauge theories and dierential
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[6] F. Faure. Exposé sur la quantication géométrique. http://www-fourier.ujf-

grenoble.fr/~faure, pages 112, 2000.

[7] Griths, Phillip and Harris, Joseph. Principles of algebraic geometry. A Wiley-
Interscience Publication. New York, 1978.

[8] A. Hatcher. Vector Bundles and K-Theory. http://www.math.cornell.edu/~hatcher/,

1998.

[9] T.S.Ratiu J.E. Marsden. Introduction to Mechanics and Symmetry. 1998.

[10] J.E. Marsden, R. Montgomery, and T. Ratiu. reduction, symmetry, and phases in
mechanics, volume 88. Memoirs of the American Mathematical Society, 1990.

[11] R. Montgomery. The connection whose holonomy is the classical adiabatic angles of
hannay and berry and its generalization to the non integrable case. Comm. in Math.
Phys., 120:269294, 1988.

[12] R. Montgomery. How much does the rigid body rotate? a berry's phase from the 18th
century. Am. J. Phys., 1991.

41
42 BIBLIOGRAPHY

[13] M. Nakahara. Geometry, topology and physics. Institute of Physics Publishing, 2003.

[14] A. Perelomov. Generalized coherent states and their applications. Springer-Verlag.,

1986.

[15] P.Lochak and C.Meunier. Multiphase Averaging for Classical Systems. Springer Ver-
lag, 1988.

[16] Schlichenmaier, Martin. . e-prints: QA/9902066 QA/9910137 QA/9903105, 1999.

[17] C. Segal. Lectures on Lie groups and Lie Algebras. 1995.

[18] K. Takahashi. Wigner and husimi functions in quantum mechanics. Journ. of the
Phys. Society of Japan, 55:762779, 1986.

[19] K. Takahashi. Distribution functions in classical and quantum mechanics. Progress of

Theor. Phys., 98:109, 1989.

[20] W. M. Zhang, D. H. Feng, and R. Gilmore. Coherent states: theory and some appli-
cations. Rev. Mod. Phys., 62:867, 1990.

[21] N.M.J. Woodhouse. Geometric quantization. Clarendon Press, Oxford, 1992.

Chapter 4
Topological aspects in the
Semi-quantum model of slow-fast
coupled systems

4.1 Introduction

• A small molecule:

group of interacting (quantum) nuclei and electrons.

with fast electrons (τe' 10−15 → 10−16 s.),
slower vibrations of the nuclei (τv ' 10−14 → 10−15 s., )
slower rotation of the molecule (τrot ' 10−10 → 10−12 s.).
Characteristics:

• Fast-Slow coupled, quantum, Hamiltonian system with -

nite number of degree of freedom.

• We will be concerned with Topological properties of the spec-

trum, (crude properties but robust against perturbations).
Works of D. Sadovskii, B. Zhilinskii, ....

First example: Slow rotation coupled with fast vibrations of the nuclei

E
(cm −1) N3 N’3 = N3 + ∆N3
Molecule CD 4
Large Vibrationnal structure
∆Ε∼ h ω (3 levels)
Slow rotation
D J
2 N’2
C
S N1
Fine rotationnal structure
N’1
Fast vibrations
J

43
44CHAPTER 4. TOPOLOGICAL ASPECTS IN THE SEMI-QUANTUM MODEL OF SLOW-FAST COU

Observations:

• Group of rotational levels

• Restructuration of groups and exchange of levels with external parameter.

Objectives: Understand these qualitative phenomena

These qualitative phenomena are very common in molecular spectra.

E−BJ(J+1) Molecule CF4

−1
(cm )

1300

ν3

2ν 4

1250

0 10 30 50 J

4.2 Model for coupling between: slow rotation and n

quantum vibrational levels:

4.2.1 The semi-quantal model

Suppose Ĥtot acts in

Htot = Hslow ⊗ Hf ast = Hj ⊗ Cn
Its symbol admits a formal power series in ~ef f = 1/(2j):

~ = hJ|
Ĥf ast (J) ~ Ĥtot |Ji
~ = Ĥ0 (J)
~ + ~ef f Ĥ1 (J)
~ + ....

• This is a operator valued symbol (the Born-Oppenheimer Approximation):

4.2. MODEL FOR COUPLING BETWEEN: SLOW ROTATION AND N QUANTUM VIBRATIONAL LE

  

 J~ → Ĥf ast J~
2 n
 SJ → Herm (Hf ast = C )
Slow F ast


 
• Eigenvalues of Ĥ0 J~ :

     
are ~ ~
E1 J , E2 J , E3 J~ , . . ., form n bands, provided there is no degeneracy.

E
E(J)
E3

E2

E1
Energy bands
2
J in S

We discuss below the important role of degeneracies.

 
• The eigenspaces of Ĥ0 J~ :
     
J~ → Fi J~ = Ker Ĥ0 (J)
~ − Ei J~ Id ⊂ Cn

F(J)

J Clutching function

θ e i ϕ(θ)

ϕ(2π)=ϕ(0) + C2π

C: Chern Index

dene n Complex Vector Bundle of rank 1: F1 , F2 , F3 .

46CHAPTER 4. TOPOLOGICAL ASPECTS IN THE SEMI-QUANTUM MODEL OF SLOW-FAST COU

Their topology is characterized by Chern indices:

C1 , C2 , C3 , . . . ∈ Z

Additivity of the indices:

Because of the triviality of the total bundle Hf ast → Pslow = Sj2 ,
X X
Ci = c1 (Li ) = c1 (⊕i Li ) = c1 (Hf ast ) = 0.
i i

4.2.2 Modications of bands by an external parameter λ:

Property about degeneracies:
If λ ∈ Rn → Ĥ (λ) is generic family of hermitian operators,then degeneracies be-
tween two eigenvalues occur with codimension 3. And more generally degeneracies with
multiplicity k = 2, 3, . . . occur with codimension k 2 − 1 = 3, 8, . . .
2
Because the space of k × k Hermitian matrices is k dimensional.
Matrices with multi-
plicity k are: λId with λ ∈ R.
Or because for a 2 × 2 matrix:
  q
λ1 λ3 + iλ4
Ĥ (λ) = , ∆E = (λ1 − λ2 )2 + 4λ23 + 4λ24 : splitting
λ3 − iλ4 λ2

Remarks:

• for the same reason, in the case of real symmetric matrices, the codimension for
1
a multiplicity k = 2, 3, . . . event, is
2
k (k + 1) − 1 = 2, 5, . . ..

• In many physical problems, constrains by particular symmetries.

 
For model Ĥf ast J~ ,
the external parameters are:

 
~
J ~
J~ =  ∈ SJ2 × R+ = R3


, J
J~

Modication of Chern index at a generic degeneracy:

∆C2 = C20 − C2 = ±1

Proof: in J~ ∈ R3 space,
4.2. MODEL FOR COUPLING BETWEEN: SLOW ROTATION AND N QUANTUM VIBRATIONAL LE

Degeneracy points ∆ C=+1

−
Energy
(cm−1) C3 λ
1000

C2

950

C1 (semi−classical limit)
10 20
Angular momentum J J

C2 C2’ C1’
C1

C1 C1’
J
λ∗
λ =|J|
λ∗
(b) λ =|J|
(a)

C’

C’
C C C
C
(q,p)
∆ C = C’−C

|J| λ
|J|

Local model near an isolated degeneracy:

 
±λ q + ip p
Ĥλ (q, p) = , ∆E = 2 λ2 + q 2 + p2
q − ip ∓λ

giving (Berry 84)

∆C = ∓1

Because topology can be computed from zeroes of a global section :

48CHAPTER 4. TOPOLOGICAL ASPECTS IN THE SEMI-QUANTUM MODEL OF SLOW-FAST COU

Fibers

Base space
Trivial bundle SW=0 Moebius strip SW=1

Here
Ĥλ ≡ B. ˆ,
~ ~σ ˆ : Pauli Matrices
~ = (λ, q, p) , ~σ
B
 
has eigenstates |ψ± B ~ i dene two complex vector bundles F+ , F− of rank 1 over S 2 .
A global section of the upper band is F+ is obtained by projection of the xed state
|+i onto F+ :    
|ψ+ B~ ihψ+ B ~ |+i ∈ F+
 
and has one zero for |ψ+ B~ i = |−i. Careful inspect of orientation gives C+ = −1.


4.2.3 Quantum model: (rotation is quantized)

Opérateur ~ˆ
Ĥtot = Ĥf ast (J) sur Hj ⊗ Cn

Theorem :(C.Emmrich-A.Weinstein, CMP 176, p.701, 1998),

Construct hprojectors
i in Htot , P̂1 , P̂2 , P̂3 , . . . associated with bands,
such that Ĥtot , P̂i = O(~∞
ef f ).

(P̂i is given by its symbol ~ + ~ef f P̂i,1 (J)

P̂i,0 (J) ~ + . . .,
   
with P̂i,0 J~ : spectral projector of Ĥ0 J ~ .)
 
So dene: Ni = Rank P̂i : number of levels in band i

Remarks:
•This result is no so obvious if bands overlap in energy; (gure above).
•Corrections are due to possible tunnelling eect between bands.
•General theorem and Proof below.
Summary:
4.2. MODEL FOR COUPLING BETWEEN: SLOW ROTATION AND N QUANTUM VIBRATIONAL LE

1/2 Quantum (B.O.) Quantum

 
J~ → Hf ast J~ ~ˆ
Operator Ĥ = Hf ast (J) sur Hj ⊗ Cn
Sj2 → Herm (Cn )
Chern indices Ci for bands Number of levels in bands: Ni

Degeneracy points ∆ C=+1

−
Energy E
N3
(cm−1) C3

1000

N2
C2

950

C1 N1
10 20
Angular momentum J J

Question: relation between band topology Ci and Ni ?

Property: (F.Faure, B.Zhilinskii, Phys.Rev.Lett. 85, p.960, 2000)

Ni = (2j + 1) − Ci
i.e. ∆Ni = −∆Ci

Proof (simple):
For a generic contact, the local model gives ∆C = ∓1,
Its quantization:

 
±λ q̂ + ip̂
Ĥλ = , gives ∆N = ±1
q̂ − ip̂ ∓λ
So ∆Ni = −∆Ci .  
Consider a generic deformation of the given symbol Ĥ J~ to the trivial (uncoupled)
situation Ĥ0 = Ŝz , where Ni = (2j + 1), Ci = 0.
.
Simple example: Spin-orbit coupling
A two state (fast) spin ~ (s = 1/2)
S ,
coupled with a (slow) angular momentum J~ with 2j + 1 states, with j1 :
50CHAPTER 4. TOPOLOGICAL ASPECTS IN THE SEMI-QUANTUM MODEL OF SLOW-FAST COU

...

...
~
E :Band +
|1>|+>

...
∆ C=−1 |0>|+> |1>|−>
|0>|−>

λ 0 1 ~
λ

∆ C=+1 |0>|−> |0>|+>

...
|1>|−> |1>|+> : Band −

...

...
 
H J~slow , S
~f ast = (1 − λ) Sz + λ J.
~ S,
~ λ ∈ [0, 1] : parameter

• For λ = 0, H = Sz : Two trivial bands, C± = 0 ; with N± = (2j + 1) levels.

• For ~S
λ = 1, H = J. ~ : Two non trivial bands, C± = ∓1, with N± = (2j + 1) ± 1 levels.

• At λ = 1/2, J~ = (0, 0, −1), gives H = 0; an isolated degeneracy between the two

bands.

j=4
E
N+=9 N+=10
C+=0 C+=−1
0.5
E

1/2 C+=0 C+=−1

0 −0.5
N−=9
C−=0 C−=+1 N−=8
C −=0 C−=+1
−1/2 0 0.5 λ 1
0 0.5 1 λ

What is particular here:

•Dim( phase space S 2 )=2 < Codim(degeneracies )=3.
So Vector bundles have rank 1.
With external parameter λ ∈ R, isolated degeneracies; local model.
2
•Rank 1 vector bundles over S are characterized by

2
S 2, Z ≡ Z


Chern Index : C ∈ H
4.3. MODEL WITH MORE INTERESTING TOPOLOGICAL PHENOMENA: SLOW MOTION ON CP 2

Question: what happends with 4-dimensional compact slow

phase space?

4.3 Model with more interesting topological phenom-

ena: Slow motion on CP 2, dimension 4.

4.3.1 Classical mechanics:

Three vibrations in 1:1:1 resonance on T ∗ R3 = R6 :

q3
3 3
X 1 X
p2i + qi =
2
|Zi |2 = hZ|Zi,


Hvib =
i=1
2 i=1
1
with Zi = √ (qi + ipi ) ∈ C, Z = (Z1 , Z2 , Z3 ) ∈ C3
2 q2

Classical trajectory [Z]

so Z(t) = Z(0)e−it . q1

For a xed energy E = hZ|Zi, a trajectory is associated

to a point [Z] in reduced phase space (dim 4)

CP 2 = C3 \ {0} / ∼,


with Z ∼ λZ, λ∈C

4.3.2 Quantum mechanics:

on L2 (R3 ),
operators q̂i : ψ(~q) → qi ψ(~q) p̂i : ψ(~q) → −i ∂ψ(~
q)
∂qi
,

3
X 1
p̂2i + q̂i2 ,


Ĥ =
i=1
2

• Spectrum:
52CHAPTER 4. TOPOLOGICAL ASPECTS IN THE SEMI-QUANTUM MODEL OF SLOW-FAST COU

E Multiplicity

Polyad: N (N+1)(N+2)/2 3  
X 1 3 3
E= ni + = n1 + n2 + n3 + = N +
3 i=1
2 2 2
N=1
N=0 1 1
multiplicity (N + 2)(N + 1)
:
2
Phase space CP 2 ⇔ Hilbert space HP olyad N

• Semi-classical limit: N →∞

4.3.3 Slow Vibrations coupled with 3 elec-

tronic states:

Matrix symbol: with parameter λ ∈ [0, 1] :magnetic eld,


 [Z] → Ĥf ast,λ (Z) = (1 − λ) Ĥf ast,0
+ λ Ĥf ast,1 (Z)
CP 2 → Herm C3f ast

Slow Fast

•For λ = 0,  
E
−1
+1 T3
No dependence on [Z] ∈ CP 2 : Ĥf ast,0 =  0 , giving
0 T2

−1 T1
1
three trivial bers bundles, rank 1, on CP 2 : T1 , T2 , T3 .
0 λ
E •For λ = 1,
1


Vline +1 Ĥf ast,1 (Z) ≡ |Z ih Z| = <Z|Z>
Zi Zj i,j
: Projector onto line [Z] ⊂
Vorth 0
C3f ast
λ Eigenvalue (E3 = 1): rank 1 ber bundle Vline :the canonical
1
bundle
Eigenvalue (E1 = 0, E2 = 0): rank 2 ber
bundle Vorth .

4.3.4 Band spectrum in B.O approximation:

One compute E1 (λ, [Z]) ≤ E2 (λ, [Z]) ≤ E3 (λ, [Z]), λ ∈ R, [Z] ∈ CP 2 .

4.3. MODEL WITH MORE INTERESTING TOPOLOGICAL PHENOMENA: SLOW MOTION ON CP 2

E Degeneracy surfaces
T3
+1
Vline : fibré en droite
canonique

T2
0
Vorth : fibré orthogonal
rang 2
T1

−1
1/2 2/3 1 λ

represents the decomposition of the trivial bundle CP 2 × C3 :

T1 ⊕ T2 ⊕ T3 = C3 = Vline ⊕ Vorth
Rank 1, trivial Rank 3, trivial Rank 1 ⊕ Rank 2

4.3.5 Topology of a vector ber bundle F over CP 2 :

Characterized by its Chern Class C(F ) ∈ H ∗ (CP 2 , Z)

C(F ) = 1 + Ax + Bx2 , A, B ∈ Z
and its rank: r ∈ N∗ , (B = 0 if r = 1).
2
(x is symplectic two form on CP ).

• Composition property:

C(F ⊕ F 0 ) = C(F ) ∧ C(F 0 ) = 1 + (A + A0 ) x + (AA0 + B + B 0 ) x2

• In the model,

T3 Vline (r=1, A=−1, B=0)

1 = C(C3 ) = C(VLine ) ∧ C(VOrth ) T2
T1 Vorth (r=2, A=+1, B=−1)

C(VLine ) = 1 − x, C(VOrth ) = 1 + x + x2
 54CHAPTER 4. TOPOLOGICAL ASPECTS IN THE SEMI-QUANTUM MODEL OF SLOW-FAST COU

but

C(VOrth ) 6= (1 + A x)∧(1 + A0 x) = 1+(A + A0 ) x+(AA0 ) x2

no solution with integers A, A0 .

So VOrth is a rank 2 undecomposable
bundle.

Physical interpretation:

A spectral gap can not appear inside

the band Vorth , under any perturbation.

• Remark: One needs at least three bands

because:

(1 + A x) ∧ (1 + A0 x) = 1 + (A + A0 ) x + (AA0 ) x2 = 1
⇒ A = A0 = 0 : 2 trivial bands

4.3.6 Quantization of vibrations:

CP 2 → Hilbert space HP olyad N , Z =

√1 (q + ip) → Ẑ = √1 (q̂ + ip̂) ,
2 2
Ĥf ast (Z) → Ĥtotal
Total Hilbert space:

Htot = HP olyad N ⊗ C3Electronics

Numerical results for N = 4:

E N0 =15
1
Nline=21

N0 =15

N0 =15 N =24
orth

−1
1 λ
4.3. MODEL WITH MORE INTERESTING TOPOLOGICAL PHENOMENA: SLOW MOTION ON CP 2

One observes the exchange of elemen-

tary group of ∆N = N + 2 = (N + 1) +
1=6 levels.

Question: relation between N and band topology (r, A, B)?

Atiyah-Singer Index formula (1965),

Fedosov (1990)
(A twist version of the Hirzebruch-Riemann-
Roch Formula )
relating Analysis (number of levels) and
topology of bundles:

N (F ) = Ch(F ∗ ) ∧ Ch(P olyadN ) ∧ T odd(T CP 2 ) /coef de x2

 

with

1 2
Ch(F ∗ ) = r − Ax + A + 2B x2


: Band topology of the dual bundle
2
Ch(P olyadN ) = exp (N x) : geometric quantization of CP 2
3
T odd(T CP 2 ) = 1 + x + x2 : Base space
2

In the above model:

" ! #
x2 (N x)2
 
3 1
N (VLine ) = 1+x+ ∧ 1 + Nx + ∧ 1 + x + x2 = (N + 3) (N + 2)
2 2 2 2
/x2

" ! #
x2 (N x)2
 
3
N (VOrth ) = 2−x− ∧ 1 + Nx + ∧ 1 + x + x2 = N (N + 2)
2 2 2
/x2
56CHAPTER 4. TOPOLOGICAL ASPECTS IN THE SEMI-QUANTUM MODEL OF SLOW-FAST COU

4.3.7 Summary:

•Semi-classical correspondence between

the Topological aspects of Semi Quantum
and the Qualitative aspects of the Quantum
problem :

The semi-quantum Born-Oppenheimer

approximation for rotation-vibration-electronic
coupling in molecules, shows bands with
non trivial topology (vector bundles of
any ranks).

This topology is related to the num-

ber of energy levels in each group of the
quantum problem.

•Bifurcations : A change of band

topology gives an exchange of levels
between groups of levels.

•in Q.C.D., Instantons are solitons of

the gluon eld, with a non trivial topology.

The index formula gives them an axial charge.

The consequences is a breaking of the chiral
symmetry and explains the important mass of the mesons η, η 0 .
The index theorem is one of the deepest
and hardest results of mathematics which
is probably enmeshed more widely with topol-
ogy and analysis than any other single re-
sult (Hirzebruch-Zagier 1974).

4.3.8 Remark on Index formula

for the sphere (angular momen-
tum phase space)

Chern Class of a line bundle is C(F ∗ ) =

∗ 2
1 − Cx ∈ H (S , Z), with C ∈ Z.

N (F ) = Ch(F ∗ ) ∧ Ch(Quantj ) ∧ T odd(T S 2 ) /coef de x2

 
4.3. MODEL WITH MORE INTERESTING TOPOLOGICAL PHENOMENA: SLOW MOTION ON CP 2

with

Ch(F ∗ ) = 1 − Cx : band
2
Ch(Quantj ) = exp ((2j)x) : geometric quantization of S

T odd(T S 2 ) = 1 + (1 − g)x : Base space, genre g = 0

gives

N (F ) = [(1 − Cx) ∧ (1 + (2j)x) ∧ (1 + (1 − g)x)]/x = (2j) + (1 − g) − C

= (2j + 1) − C

4.3.9 Remark on the surface of

degeneracy S in the model be-
tween bands 2-3

In Parameter space (λ, [Z]) ∈ R × CP 2 ,

2
This surface S ⊂ R × CP is homotopic
1 2
to CP ⊂ CP (Sphere: Z1 = 0)

Locally, one has a rank 2 bundle over

1
Normal(CP ):

Rank 2, bundle

Chern =−1

: (Slow) Base Space: Normal(CP1)

Chern = 1
S2

This gives transfert of states:

∆N = (N + 1) + 1
58CHAPTER 4. TOPOLOGICAL ASPECTS IN THE SEMI-QUANTUM MODEL OF SLOW-FAST COU

4.3.10 Remark on Semi-classical

expansion for ~ → 0; Weyl for-
mula with correction

For a line bundle over a Riemann sur-

face,
2
hef f = 1/(2j), Vol(S )=1.

V ol
N (F ) = + (1 − g) − C
hef f

The rst term is Usual Weyl number

of quanta in total phase space
(Below, this will give the local density
of states.)

For a line (r = 1) bundle F over CP 2 ,

number of levels N (F ) is a polynomial in
N:
" !  #
(N x)2

1 2 2 3
N (F ) = 1 − Ax + A x ∧ 1 + Nx + ∧ 1 + x + x2
2 2 2
/x2
   
1 2 3 1 2 3
= N + N −A + + A − A+1
2 2 2 2

Interpretation: hef f = 1/N , Vol(CP 2 ) =

1/2.
So
 
V ol 1 3
N (F ) = 2 + − A + ...
hef f hef f 2

4.3.11 Remark on Naturality

of index formula

The Chern Class is a map:

C : F ∈ V ect(M ) → C(F ) ∈ H ∗ (M, Z)

The main interest of Chern class C(F ) is that coecients are integers.
0 0
But C(F ⊕ F ) = C(F ) ∧ C(F ).
4.3. MODEL WITH MORE INTERESTING TOPOLOGICAL PHENOMENA: SLOW MOTION ON CP 2

•For two bundles over (dim 2n) phases spaces,

F1 → M1 , F2 → M2
one expects:

N ((F1 ⊗ F2 ) → (M1 × M2 )) = N (F1 → M1 ) N (F2 → M2 ) : product of Hilbert spaces

N ((F1 ⊕ F2 ) → M ) = N (F1 → M ) + N (F2 → M ) : Sum of bands
This comes from

Ch(F1 ⊗ F2 ) = Ch(F1 ) ∧ Ch(F2 )

Ch(F1 ⊕ F2 ) = Ch(F1 ) + Ch(F2 )
T odd (T (M1 × M2 )) = T odd (T M1 ) ∧ T odd (T M2 )

So the index formula is an expected for-

mula:

N (Fi ) = [Ch(Fi ⊗ LineN ) ∧ T odd(T Mi )]/coef de xn

But Ch, T odd ∈ H ∗ (M, Q) (not integer

classes).

4.3.12 Index theorem and group

theory:

In the model, for λ = 1, Ĥ1 is constructed

from equivariance by SU(3) :

=
Htot = H polyad H elec = Band "Line" Band "Orth"

15 * 3 = 21 + 24

Weyl formula of group theory gives

correct dimensions NLine , North .

Remark on relations with vector

coherent states, and weight diagramm,
 60CHAPTER 4. TOPOLOGICAL ASPECTS IN THE SEMI-QUANTUM MODEL OF SLOW-FAST COU

induced representations, equivariant

vector bundles:

Irrep: E3 Irrep:
H2 E2

E1

H1

Orbit of : Orbit of :
2
Line bundle over: SU(3)/U(2)=CP Line bundle over: SU(3)/U(1)*U(1)
(Perelomov coherent states) Orbit of :
Rank 2 bundle over SU(3)/U(2)=CP 2

4.4 Main Born-Oppenheimer

theorem of adiabaticity

(C.Emmrich-A.Weinstein, CMP 176, p.701,

1998)

Consider:

• a Phase space Pslow (a symplectic mani-

fold for slow motion),

• an Hilbert space Hf ast (for fast motion)

• a Matrix symbol p ∈ Pslow → Ĥ(p) ∈

Herm(Hf ast ) which can be written

Ĥ(p) = Ĥ0 (p) + ~Ĥ1 (p) + ~2 Ĥ2 (p) . . . ,

 4.4. MAIN BORN-OPPENHEIMER THEOREM OF ADIABATICITY 61

• Hypothesis: ∀p ∈ Pslow , eigenvalues (λi )i=1,...m

of Ĥ0 (p) are separated from the other part
of the spectrum (µj )j=... :

∀i, j, p λi (p) − µj (p) 6= 0

• So eigenvalues (λi (p))i=1,...m dene a sub-

space E(p) ⊂ Hf ast , with orthogonal pro-
jector π̂0 (p).

• E → Pslow is a rank m complex vector

bundle over Pslow .

• Then for any k ∈ N, there is a unique

matrix valued symbol:

π̂(p) = π̂0 (p) + ~π̂1 (p) + . . . ~k π̂k (p)

which denes a self-adjoint operator π̂tot in Htot , such that:

2
π̂tot = π̂tot + O(~k+1 ) : quasi − projector,
h i
Ĥtot , π̂tot = O(~k+1 ) : almost commute.

Remarks

• One can thus modify π̂tot (move slightly the

eigenvalues towards 1 or 0, without moving
the eigen-spaces) to obtain a true projector
0 02 0
π̂tot (i.e. π̂tot = π̂tot ). Let:
0
N = Rank(π̂tot )
N is the number of eigenvalues close to 1 of the principal symbol ~.
π̂0 (J)
0
• The index formula above gives N = Rank(π̂tot )
in terms of topology of the bundle E.
• Generic case: each eigenvalue Ei and eigen-
vector |φi > of Ĥtot , i ∈ [1, . . . dim Htot ],
can be associated with the vector bundle
E or its complement E ⊥ ; i.e. |φi i ∈ Im(π̂(p)) or K̂er (π̂(p))
.

• Consequence: a quantum state which ini-

tially belongs to the space Im(π̂(p)), will
stay in this space forever during its evolu-
tion, with a good approximation (if k high).
 62CHAPTER 4. TOPOLOGICAL ASPECTS IN THE SEMI-QUANTUM MODEL OF SLOW-FAST COU

• Non generic case: by resonances between

two eigenvalues the associated states can
be equidistributed onIm(π̂(p)) and K̂er (π̂(p)).,
as it occurs usually in the tunneling ef-
fect.

Indications for the proof :

By induction on k ∈ N. One works only
with symbols.
Hypothesis for a given k:
π ∗ π − π = ~k+1 A + O ~k+2

[π, H]∗ = ~k+1 F + O ~k+2

Check that the hypothesis is true for

k = 0.
Because [π0 , H0 ] = 0, one can nd a ba-
sis (for a given p ∈ P ) such that:
     
1 0 (λi )i 0 H00 0
π0 = , H0 = ≡ ,
0 0 0 (µj )j 0 H11
and write in this basis:
 
A00 A01
A= , idem f or F.
A10 A11
 
A00 0
Lemme 1: [A, π0 ] = 0, so A = .
0 A11
Lemme 2: F00 = [A00 , H00 ] , F11 = [A11 , H11 ].
Write:

π̃ = π + ~k+1 K
 
K00 K01
with unknown K = such
K10 K11
that:

π̃ ∗ π̃ − π̃ = O ~k+2

[π̃, H]∗ = O ~k+2

Lemme 3: K00 = −A00 , K11 = A11 .

Lemme 4: H00 K01 − K01 H11 = F01 and
H11 K10 − K10 H00 = F10 , i.e.:
(K01 )ij = (λi − µj )−1 (F01 )ij , idem f or K10 .
So Matrix K(p) is determined, giving π̃ .
Lemma 1,2,3,4 are not dicult to prove.

4.5. BERRY'S CONNECTION, CHERN CLASS AND CHARACTERISTIC CLASSES63

4.5 Berry's connection, Chern

Class and Characteristic Classes

We gives here the denitions of the Charac-

teristic classes used in Index formula above.
Consider

H : xed Hilbert space

x∈M : Parameter space (P hase space M anif old)

x-dependent decomposition of H:
A

M  
H= Hi,x , Hi,x = Im P̂x,i , P̂x,i : projector H
2
i
U
•Think that Ĥx x-dependent
is a Hermitian op- H1
erator, with eigen-spaces Hi,x .

•If P̂x,i has a smooth dependence with respect to H

x ∈ M , then Fi : Hi,x → M is a well dened vector
bundle over M of rank mi = dimC (Hi,x ). v
•Consider a parametrized path x(t) ⊂ M . Lift x x
in H × M with respect to the parallel t1
  transport t2 x(t)
(Levi-Civita) in each subspace Im P̂i . M
This denes a unitary family of
operators
Berry
Û (t1 , t2 ) in H, with Hermitian generator K̂ (t2 ):
d
i~ Û (t1 , t2 ) = K̂ Berry (t2 ) Û (t1 , t2 )
dt

Remark:
•Operator K̂ Berry express the Berry's connection.

dx
•K̂ Berry depends only on the tangent vector v= dt
∈ Tx M so K̂ Berry is a 1-form on
M with values in Herm (H).

Property:
X dP̂x,i
Berry
K̂x,v = i~ P̂x,i
i
dt

proof:
64CHAPTER 4. TOPOLOGICAL ASPECTS IN THE SEMI-QUANTUM MODEL OF SLOW-FAST COU

We saw that the Levi-Civita connection on space Hi,x can be expressed by:

|ψ + dψii = Pi (x + dx) |ψii , where |ψi i ∈ Hi,x

Connection
|ψ>
|ψ+δψ>

|ψ+δψ>
|δψ>

|ψ> P=|ψ><ψ|

δX

Hilbert space
P P P
so |ψ + dψi = i |ψ + dψii = i (Pi + dPi ) |ψii = |ψi + i dPi Pi |ψi. Identify with
|dψi
i~ dt = K̂ Berry |ψi..

4.5.1 Berry's Curvature

Consider two tangent vectors v1 , v2 ∈ Tx M , dening

an innitesimal loop in M , with sides vi dt.
The Curvature is the Holonomy on this loop.
It is expressed by an innitesimal unitary opera-
tor
H i
Berry Û = 1 − ĤvBerry dt1 dt2 + o(dt2 )
~ 1 ,v2
A standard calculation gives the Hermitian
generator of this Berry's curvature:

v2 Berry −i h Berry Berry i

Ĥx,v 1 ,v2
= K̂x,v1 , K̂x,v2
~
Remarks:
v1 •Ĥ Berry is a 2-form on M with values in
Herm (H).
Berry
•By construction Ĥx,v1 ,v2
leaves Hi,x invariant, ∀i. The restriction is:

 
Ω̂Berry
i,v1 ,v2
Berry
= P̂i Ĥx,v 1 ,v2
P̂i ∈ Herm (Hi,x )

•If dimHi,x = 1, then ΩBerry

i,v1 ,v2 ∈ R is called the (Scalar) Berry's curvature for level
i.
4.5. BERRY'S CONNECTION, CHERN CLASS AND CHARACTERISTIC CLASSES65

•One can check the formula:

X
Berry
Ĥx,v1 ,v2
= Pi [dv1 Pi , dv2 Pi ] Pi
i

•We will see below that in the semi-classical limit, if the dynamics is integrable,
thenĤ Berry is the quantization of a classical Hamiltonian H Hannay
corresponding to the Hannay connection between Tori.

4.5.2 Characteristic Class

Consider a complex vector bundle F : Hx → M , over compact manifold M.

Dene

 
1 Berry
C(F ) = det 1 + Ω̂ ; Total Chern Class
2πi
Remarks and Properties:

• Ω̂Berry M so C(F ) is a dierential

is a 2-form on form with even degree only.
One writes also: C(Fi ) = 1 + C1 + C2 + . . .,
2k
with Ck ∈ H (M, R): kst Chern Class.

• If RankC (F ) = r, then Ck = 0 for k > r.

If n = dimR M , then Ck = 0 for 2k > n.

• C(F ) is an integral C(F ) ∈ H even (M, Z); which means that Ck gives integers
class
after integration over any closed submanifold S ∈ H2k (M, Z).
m+n
(One check this in the Universal Classifying Grassmanian bundle C → Gm (M ),
and use invariance of dierential forms by pull-back; see Hatcher's Book, or Eguchi).

• if r = RankC (F ) = 1, then

1 Berry
C1 = Ω̂ ∈ H 2 (M, Z)
2πi
characterizes the topology of the bundle F. (See Well's book).
This is not true in general for higher ranks.

• On CP 2 , we used the fact that

H ∗ CP 2 = Z + Zx + Zx2

where x is the (normalized) symplectic 2-form.

66CHAPTER 4. TOPOLOGICAL ASPECTS IN THE SEMI-QUANTUM MODEL OF SLOW-FAST COU

• More generally, for any polynomial P (or formal serie) on R, then

  
1 Berry
Tr P Ω̂ ∈ H even (M, R)
2πi

is a topological invariant (does not depend on the connection).

To have nice relations with ⊕, ⊗, (important for the index formula) dene:

 1 Berry 
Ch(F ) = Tr e 2πi Ω̂ ; Chern Character

!
1
2πi
Ω̂Berry
T odd(F ) = det 1 ; Todd Class
Ω̂Berry
e 2πi −1

Then:

Ch(F1 ⊗ F2 ) = Ch(F1 ) ∧ Ch(F2 )

Ch(F1 ⊕ F2 ) = Ch(F1 ) + Ch(F2 )
T odd (F1 ⊕ F2 ) = T odd (F1 ) ∧ T odd (F2 )

4.5.3 Important physical remarks:

We show here that the Index formula is more precise than just giving the total number of
states.

• The Index formula can be written:

Z
N (F ) = µ
M
µ = Ch(F ∗ ) ∧ Ch(P olyadN ) ∧ T odd(T CP 2 ) /Vol
 

The Volume form µ is interpreted as the local density of states in phase space M.

• µ is still well dened if M is not compact.

• By the Semi-classical Symbol of the Hamiltonian p ∈ M → H(p) ∈ R, one obtains

then the Energy density of states.

• For hef f = 1/N → 0, the expansion of µ is the Weyl formula. (Averaged part of
the Gutwiller Trace-Formula), and involves no dynamics.
4.6. REFERENCES 67

4.6 References

Ro-Vibrationnal coupling:

• B. Zhilinskii, L. Michel , Symmetry, Invariants, and Topology , Phys. report, 2001.

• R.G. Littlejohn, and W.G. Flynn "semi-classical theory of spin-orbit coupling", Phys.
Rev. A ,1992.

Introduction to topology of vector bundles, and Index formula:

• Nakahara Geometry, topology and physics , graduate students series in physics,

Adam Hilger, Bristol and N.Y.

• Allen Hatcher, Introduction to Algebraic Topology , book online: http://www.math.cornell.edu/~hatc

(1998).

• Allen Hatcher, Vector Bundles and K-Theory , book online: http://www.math.cornell.edu/~hatcher/

(1998).

• T. Eguchi, P.B. Gilkey, A.J. Hanson Gravitation, Gauge theories and dierential
geometry Phys. Rep., 1980.

Semi-classical Theory of Slow-Fast systems:

• R.G.Littlejohn, W.G.Flynn Geometric phases in the asymptotic theory of coupled

wave equations Phys.Rev. A,44 (1991).

• C. Emmrich, A. Weinstein Geometry of the transport equation in multicomponents

WKB approximations Comm.Math.Phys., 209, p691 (1996).

• G.Panati,H. Spohn,S;Teufel Space-adiabatic Perturbation theory preprint mp-arc

25/01/2002.

Index formula and geometric quantization:

• B. Fedosov, The Atiyah-Bott-Patodi Method in deformation quantization", Commun.

Math. Phys.,209, p691,2000.

• E. Hawkins, Geometric quantization of vector bundles and the correspondence with

deformation quantization  Commun. Math. Phys., 215, p409, (2000).

Index formula and topology in molecular spectra:

• F.Faure, B.Zhilinskii, Topological Chern indices in molecular spectra. Phys.Rev.Lett.

85, p.960, 2000.

• F. Faure, B. Zhilinskii, Topological properties of the Born-Oppenheimer approxima-

tion and implications for the exact spectrum", Lett. in Math. Phys., 2001.

• F. Faure, B. Zhilinskii, Topologically coupled energy bands in molecules eprint quant-

ph/0204100.
68CHAPTER 4. TOPOLOGICAL ASPECTS IN THE SEMI-QUANTUM MODEL OF SLOW-FAST COU
Chapter 5
Topological aspects in the Classical
model of slow-fast coupled systems

5.1 A simple class of models. Topology of the tori bun-

dle.
~
•Model: A slow angular momentum J(t) coupled with fast Angular momentum ~ .
S(t)
•Total classical phase space :

Ptot = Pslow × Pf ast = Sj2 × Ss2

•Total quantum Hilbert space :

Htot = Hslow ⊗ Hf ast = Hj ⊗ Hs , dim = (2j + 1) (2s + 1)

•with the adiabatic assumption:

js

and the (optional) semi-classical limit for fast variable:

s1

•The classical model is specied by a total symbol:

 
~ S
H J, ~

•Total Dynamics is nearly integrable (well identied tori: Sf1ast × Sslow

1
).

69
70CHAPTER 5. TOPOLOGICAL ASPECTS IN THE CLASSICAL MODEL OF SLOW-FAST COUPLED

•Simple example Spin-orbit coupling :

 
H J, S = (1 − λ)Sz + λJ~S,
~ ~ ~ λ ∈ [0, 1]

•Summary:
~\
Slow J ~
FastS PF ast = Ss2
Classical in Quantum in HF ast = Hs
Classical in PSlow = Sj2 ~ S)
Function Htot (J, ~ Operator ~ → Ĥf ast (J)
Symbol J ~
2 2
(Classical , phase space Sj × Ss ) (Semi-Quantum )

Quantum in HSlow = Hj No meaning Ĥtot

(Quantum in Htot = Hslow ⊗ Hf ast )
Restricted Hypothesis:
 
~
•For every J xed, HJ~ S ~ is a function on Ss2 , with only a minimum min and a

Maximum M ax.
C : this class of models.

Max
S2s Max

: Reeb graph

min
min

•If M ax > min, Topology of the fast trajectories, characterized by degree d∈Z of:

degree of : J~ ∈ Sj2 → Max ∈ Ss2

So Topological subclass of models

C = (∪d Cd ) ∪ Singulars

d=0
d=−1 d=1
d=2
... ...

path

Class C
5.2. SEMI-QUANTUM MODEL; ENERGY BANDS AND THEIR TOPOLOGY BY SEMI-CLASSICAL C

1 2
Topology of tori bundle (T → Sslow ):

ChernHannay = 2 d
Examples:

~ J).
H = B( ~ S~ ∈ Cd
with J~ (θ, ϕ) → B
~ (θ0 , ϕ0 ) of degree d:
•d = 1, ~ = J,
B ~ ~S
H = J. ~ CHannay = 2
•d = 0, ~ = (0, 0, 1)
B H = Sz , CHannay = 0
•d 6= 0, ~
B(θ 0
= θ, ϕ = dϕ) 0 ~ J)
H = B( ~ S,
~ CHannay = 2d

B(J) B(J)

0 0 0

d=0 d=1 d=2

5.2 Semi-quantum model; Energy Bands and their topol-

ogy by semi-classical calculation

For H ∈ Cd , there are dimHf ast = 2s + 1 isolated bands,

with Chern index CBerry,m , m = −s → +s.
Property:

CBerry,m = − (2m) d,
∂CBerry,m
CHannay = − = 2d
∂m
Proof: Count the zeros of a global section
of bandFm : J~ → ĤJ~|ψJ,m
~ i = EJ,m
~ |ψJ,m~ i.

Consider a xed coherent state |S ~ 0 i. A global section is |ψJ,m

~ ihψJ,m
~
~ |S0 i.

Same zeroes as the Husimi distribution at point ~0 :

S
  2
~ = hS
HusΨ S ~0 |ψ ~ i
J,m
72CHAPTER 5. TOPOLOGICAL ASPECTS IN THE CLASSICAL MODEL OF SLOW-FAST COUPLED

111
000
000
111
B axis
111
000
111
000
11111
0
1110000
000
000
1110000(s−m) zeros
1111
111
000
111
000
000
111
000
111
000
111
111
000
111
000
111
000
111
000
1
0 11
00
S 00
11
00
11
0 00 Classical support of state |ψm(J) >
11
(= classical trajectory)
(s+m) zeros
1
0

CBerry,m = ((s − m) − (s + m)) d = − (2m) d

Example: s = 2, m = −2 → +2 so 5 bands,

d = 1, C−2 = +4, C−1 = +2, C0 = 0, C1 = −2, C2 = −4,

d = 2, C−2 = +8, C−1 = +4, C0 = 0, C1 = −4, C2 = −8,

Remember in the Quantum model : Nm = (2j + 1) − CBerry,m

So transition d→d+1 gives a redistribution of levels ∆Nm = 2m.

∆N−2 = −4, ∆N−1 = −2, ∆N0 = 0, ∆N1 = +2, ∆N2 = +4,

:∆Ν=+4

:∆Ν=+2

:∆Ν=0

:∆Ν=−2

:∆Ν=−4

5.3 Relation with classical and quantum monodromy:

 
Local model at a transition between Cd and Cd+1 Transition occurs if ~ J~ ∼ 0,
B
for J~ ' J~∗ .
5.3. RELATION WITH CLASSICAL AND QUANTUM MONODROMY: 73

d d+1
λ
0

B(J)
(q,p)
0

d=1 d=2

0 λ

(q, p) : local coordinates for J~ ∈ Sj2 .

 
Generic local model in ~ ∈ R2 × Ss2 :
q, p, S
 
~ = qSy + pSx − λSz
Hloc q, p, S

Parameter space (q, p, λ) ∈ R3 .

Singularity at (0, 0, 0) gives:
∆CHannay = 2, ∆CBerry,m = −2m, ∆Nm = 2m
•For s = 1/2, already considered:
 
−λ p + iq
Ĥloc =
p − iq λ
•This local model is integrable:

1 2
p + q2 ,


N = Sz + {Hloc , N } = 0
2

This local integrable model has a generic (classical and quantum)

monodromy defect
Observed by D.A. Sadovskii,B.I. Zhilinskii, Monodromy, diabolic points, and angular
momentum coupling  Physics Letter A, 256, p235 (1999).
74CHAPTER 5. TOPOLOGICAL ASPECTS IN THE CLASSICAL MODEL OF SLOW-FAST COUPLED

H λ =-6 s=3 H H λ =6
λ =0

N N
N

See Movie on web-page.

Remark: generic only with the special assumption on Reeb graphs.

Monodromy matrix:  
1 0
M= ∈ SL (2, Z)
1 1
Remark: Monodromy is a generic event in integrable systems.

5.4 Classical Hannay connection; Semi-classical corre-

spondence with Berry's connection

5.4.1 Classical Hannay connection

General SetUp:

• A xed Phase space p ∈ Pf ast .

• A Parameter space x∈M

5.4. CLASSICAL HANNAY CONNECTION; SEMI-CLASSICAL CORRESPONDENCE WITH BERRY'S

• A familly of Hamiltonian Hx on Pf ast , Integrable for any x ∈ M.

We write (Ix , θx ) local (x-dependent) action-angle coordinates on Pf ast .

• Adiabatic limit: x (εt) ∈ M varies slowly, ε  1.

The Classical Adiabatic theorem: the trajectories follows tori with (approxima-
tively) constant action.

Objective: explain the precise motion in the tori, in terms of a geometric Hannay con-
nection.
Spin precession example:
Hamiltonian on Pf ast = Ss2 :

 
~ = B.
Hx S ~ S~

with imposed Slow varying Parameter:

~
x(εt) = B(εt) ∈ M = S 2, with ε1
Tori trajectories are Circles around ~
B on Ss2 .

B(t)

Parameter space M Phase space P fast

Adiabatic Averaging method:

The evolution vector eld on M × Ss2
!
~
dx ∂Hx (S) ~ ∈ T M × Ss2


V = (Vx , VS ) = , ∧S
dt ~
∂S
is approximated by its averaged over the motion of ~:
S
hV if ast = h(Vx , VS )if ast = h(0, VS )if ast + h(Vx , 0)if ast
= (0, VS ) +V Hannay
Dynamical Geometric
decomposed
76CHAPTER 5. TOPOLOGICAL ASPECTS IN THE CLASSICAL MODEL OF SLOW-FAST COUPLED

dI
• in the trivial Dynamical fast motion (In Action-Angle coordinates, VS : dt
=
0, dθ
dt
= ω(I) = ∂H
∂I
).

• and the Hannay Hamiltonian Vector eld V Hannay .

More explicitly:
On Ss2 , dene Rx,α : (Ix , θx ) → (Ix , θx + α) the rotation by the same angle α on each
Tori.

Z 2π  
Hannay dα dRx,α −1
V = R Vx
0 2π dx x,α

Pfast

α α
α
α
Ix
M

x x+dx
Vx Ix+dx
Pfast
 
• V Hannay
= h(Vx , 0)if ast = dx
dt
, VSHannay has component in Pf ast = Ss2

• Vector eld VSHannay Hannay

is generated by a Hamiltonian Kx,V
x
.
Hannay
Hannay Hamiltonian function K is a one-form on M valued in Hamil-
tonian function in Pf ast .

It denes the Hannay Connection between tori, which connects tori with same actions
R
I = pdq .
Example of spin precession:

 
KVHannay ~ ~ ~
= B ∧ Vx .S,
x
5.4. CLASSICAL HANNAY CONNECTION; SEMI-CLASSICAL CORRESPONDENCE WITH BERRY'S

Vx

B(t+dt)
B(t)

B(t) Vhannay
B^V x

Parameter space M Phase space P fast

Hannay's curvature:
is the Hamiltonian vector eld (valued two- form):

ΩHannay = VvHannay Hannay

 
v1 ,v2 1
, Vv 2

or Hamiltonian function (valued two- form):

HvHannay
 Hannay Hannay
,v
1 2
= Kv1 , Kv2

By construction, they preserve the Tori. The Hannay curvature is then just a shift
Hannay
angle Ωθ in each Tori:

∂H Hannay
ΩHannay
θ =
∂I

Example of spin precession:

     
HvHannay ~ ~ ~ ~ ~
= (~v1 ∧ ~v2 ) .B B.S = (~v1 ∧ ~v2 ) .B Hx S
1 ,v2

ΩHannay
R
Remark: the Tori bundle has global topology: C Hannay = θ =2

5.4.2 Semi-classical correspondence between Hannay and Berry's

connection.

•Now Ĥx is the quantization of Hx , on xed Hilbert space H.

•The eigenspaces of Ĥx dene a x-dependent decomposition of H:
M  
H= Hm,x , Hm,x = Im P̂x,m , P̂x,m : projector
m

Where m is the quantum number of quantized Tori, with Action: Im = m h.

•The quantization of Rx,α is the unitary operator:

X Im (α/2π) X
R̂x,α = e−i ~ P̂m = e−imα P̂m
m m
78CHAPTER 5. TOPOLOGICAL ASPECTS IN THE CLASSICAL MODEL OF SLOW-FAST COUPLED

Hannay
•The quantization of Kx,v is then

Z
Hannay dα dR̂x,α −1
K̂x,v = i~ R̂ v
2π dx x,α
Property:

Hannay Berry
K̂x,v = K̂x,v : Hamiltonian connection

Hannay Berry
Ĥx,v = Ĥx,v : Hamiltonian curvature

proof:

Z X Z dα
1 Hannay dα dR̂x,α −1 0 dP̂m0
K̂ = R̂x,α v = ei(m−m )α P̂m
i~ x,v 2π dx m,m 0
2π dx
X dP̂m 1 Berry
= P̂m = K̂
m
dx i~ x,v

5.5 References

Semi-classical computation of Chern indices

• F.Faure "Topological properties of quantum periodic Hamiltonians" Jour. of phys.

A: math and general, 33 , 531-555 (2000)

• F. Faure, B. Zhilinskii, Topological properties of the Born-Oppenheimer approxima-

tion and implications for the exact spectrum", Lett. in Math. Phys., 2001.

Classical and Quantum Monodromy in integrable systems

• J.J. Duistermaat, On Global action-angle coordinates ,Pure Appl. Math., 33,p.687-
706, (1980).

• R.H. Cushman and L.M. Bates, Global Aspects of classical integrable systems ,
Birkauser,Basel, (1997).

• Vu Ngoc San, Quantum Monodromy and Bohr-Sommerfeld Rules , Letters in Math-

ematical Physics 55, p. 205-217, (2001)

• D.A. Sadovskii,B.I. Zhilinskii, Monodromy, diabolic points, and angular momentum

coupling  Physics Letter A, 256, p235 (1999).
5.5. REFERENCES 79

Adiabatic and Averaged method

• P.Lochak and C.Meunier,Multiphase Averaging for Classical Systems ,Springer Ver-

lag, (1988).

• V.I. Arnold Mathematical Methods in Classical Mechanics Springer 1978.

Classical Hannay's Connection

• J.H. Hannay, Angle variable holonomy in adiabatic excursion of an integrable Hamil-

tonian J. Phys.A, 18,p. 221-23, (1985).

• Richard Montgomery  The connection whose holonomy is the classical adiabatic an-
gles of Hannay and Berry and its generalization to the non integrable case , Comm.
in Math. Phys., 120, p.269-294, (1988).

Semi-classical correspondence between Hannay and Berry's Connection

• M.V. Berry, Classical Adiabatic angles and quantal adiabatic phase J. Phys. A, 18,
p.15-27, (1985).

• J.Asch, On the Classical limit of Berry's phase integrable systems , Commun. Math.
phys.,127,p.637-651., (1990).

• A. Weinstein,Connections of Berry and Hannay Type for moving Lagrangian Sub-

manifolds , Advances in Mathematics, 82, p.133-159, (1990)
80CHAPTER 5. TOPOLOGICAL ASPECTS IN THE CLASSICAL MODEL OF SLOW-FAST COUPLED
Chapter 6
Topological Chern indices and the
Integer Quantum Hall eect

6.1 Introduction

Non interacting bi-dimensional electrons in a Magnetic eld B , in a periodic potential

V,

 
1  e ~ 2 ~= 1 1
H(x, px , y, py ) = p~ − A + V (x, y), A − B y, B x
2m c 2 2

with a weak external electric eld E .

Bz
y
Ey

jx
X x

•Motion of electrons;
Slower motion of quasi-momentum over the Brillouin zone.

•If B is strong (not assumed for now): Fast Cyclotron motion on circles; Slower
precession of the circles;

81
82CHAPTER 6. TOPOLOGICAL CHERN INDICES AND THE INTEGER QUANTUM HALL EFFECT

σxy (e 2/h)
E Band structures
Bz 2
0 Quantum
y Classical
Ey n=1 2
−1
0 1
Y
n=0 1
jx
C=0
X x
V E
Fermi
Landau Levels

Prop: (Thouless et al. 82)

jx e2 X
σxy = = Cn , Cn ∈ Z
Ey h lled bands
h/e2 = 25812.807 Ω

Observed by V.Klitzing et al. (1980).

Remarks:

• Recent experiments of Albrecht et al. (PRL 86,147 2001) to observe Hofstader

Spectrum.

• Plateaux are explains by intermediates localized states, due to disorder.

• Strong B is important: gives Landau gaps and plateaux.

• The Fractionnal Quantum Hall eect is related to interactions between elec-

trons.

6.2 Band Spectrum and topology for the Quantum Hamil-

tonian

6.2.1 A dimensionless model

• We consider non-interacting electrons.

• Electron of mass m in plane (x, y) subject to a bi-periodic potentiel V (x, y) (period

X)

• a transverse constant magnetic eld ~ = B ~ez .

B

• The classical phase space is T ∗ R2 ≡ R4 , with coordinates (x, px , y, py ).

6.2. BAND SPECTRUM AND TOPOLOGY FOR THE QUANTUM HAMILTONIAN83

• The Hamiltonian function is:

1  e ~ 2
H(x, px , y, py ) = p~ − A + V (x, y)
2m c
 
with ~ = rot A
B ~ = (∂x Ay − ∂y Ax ) ~ez .

~ = − 1 B y, 1 B x = − 1 ~r ∧ B~,


• The symetric Gauge is the choice: A 2 2 2
and gives:

 2  2 !
1 eB eB
H= px + y + py − x + V (x, y)
2m 2c 2c

• For the quantum dynamics, x̂, p̂x , ŷ, p̂y are operators ([x̂, p̂x ] = i~, . . .),

• The Hilbert space is

Htot = L2 R2 = Hx ⊗ Hy

and the Schrodinger equation reads:

d|ψi
i~ = Ĥ|ψi
dt

with Hamiltonian operator Ĥ obtained by quantization of H.

• Consider canonical linear transformation, with new dimensionless fast and slow
variables (and idem for quantum operators):

( √
S
xf = px + 2√1 S y
Xf ast = √~
S
pf = ~ y
p − 2√1 S x
S
px − 2Y1 y

xs = ~X
Xslow = S 1
ps = − ~X py − 2X x
with the quanta of surface:
hc
S=
eB
One can check that indeed:

[x̂f , p̂f ] = i, [x̂s , p̂s ] = i hef f

(and other commutators are zero).

with the dimensionless parameter eective Planck constant :

S
hef f = : inverse of number of quanta ux per cell
X2
84CHAPTER 6. TOPOLOGICAL CHERN INDICES AND THE INTEGER QUANTUM HALL EFFECT

hef f → 0 is the adiabatic limit (not assumed yet)

Dene the dimensionless time

t̃ = ωt
with the cyclotron frequency
eB
ω= .
mc
and the dimensionless Hamiltonian and potential

1 1
H̃ = H, Ṽ (a, b) = V (−Xa, −Xb)
~ω ~ω
Then the Schrödinger equation reads:

d|ψi ˆ
i = H̃|ψi, in Htot = L2 (Rslow ) ⊗ L2 (Rf ast ) = Hslow ⊗ Hf ast
dt̃
and
1 2  p p 
pf + x2f + Ṽ ps + hef f pf , xs + hef f xf


H̃ (Xs , Xf ) =
2

Remark that H̃ (Xs , Xf ) is bi-periodic w.r.t. Xs = (xs , ps ).

We write now:
h = hef f

6.2.2 Band Spectrum and topological Chern indices

Translation operators in L2 (Rslow )

(T̂x ψ)(xs ) = ψ(xs − 1)

(T̂p ψ̃)(ps ) = ψ̃(ps − 1) : (Fourier)

T̂x = exp(−ip̂s /~), T̂p = exp(ix̂s /~).

From periodicity of Ĥ :
[Ĥ, T̂x ] = [Ĥ, T̂p ] = 0.

but:
T̂x T̂p = e−i/~ T̂p T̂x ,
So if
1
N= ∈ N∗ .
2π~
6.2. BAND SPECTRUM AND TOPOLOGY FOR THE QUANTUM HAMILTONIAN85

then
[T̂x , T̂p ] = 0 : Hypothesis

Remark:

1
N= →∞ : Semi − classical limit (and adiabatic limit)
2π~
1 A
( if
2π~
= B
∈ Q, consider T̂x = exp(−iB p̂s /~ef f ), then [T̂x , T̂p ] = 0).

6.2.3 Decomposition of L2 (Rslow ) in eigenspaces of T̂x and T̂p :

  
T̂x |ψ >= exp(iθ1 )|ψ > RR dθ1 dθ2
HT ore (θ1 , θ2 ) = |ψ > such that , L2 (Rslow ) = HT ore (θ1 , θ2 ) (2π)2
T̂p |ψ >= exp(iθ2 )|ψ >
θ~ = (θ1 , θ2 ) ∈ Tθ are Bloch Parameters
ψ̃(ps ) is 1-périodique so ψ(xs ) is (h = 1/N )-discrete, and also 1-periodic.

θ 2 /(2πΝ)
N=5

−1 0 1 q
1/N

A basis of HT ore (θ1 , θ2 ) is then:

1 X
|j, θ~ >≡ ψj,θ~ (x) = √ exp (−in1 θ1 ) δ(x − qj − n1 ), j = 1, . . . , N
N n1 ∈Z
 
1 θ2
with qj = j+ .
N 2π
So
dimC HT ore (θ1 , θ2 ) = N.

Band spectrum of Ĥ in HT ore (θ1 , θ2 ) ⊗ L2 (Rf ast ):

Ĥ|ϕn (θ1 , θ2 ) >= En (θ1 , θ2 )|ϕn (θ1 , θ2 ) >, n ∈ N.

86CHAPTER 6. TOPOLOGICAL CHERN INDICES AND THE INTEGER QUANTUM HALL EFFECT

|ψ(θ)>
n

fibre

θ2

θ1

6.2.4 Topological indices of the bands

Suppose that En (θ1 , θ2 ) is isolated eigenvalue ∀θ. (true for generic Ĥ ).

The eigenspaces dene Complex line bundles:

Fn → Tθ

With topology characterized by Chern index:

Cn ∈ Z : Chern index of band n

(Formula to compute Cn below).

6.3 Interpretation of the Topological Chern indices: the

Quantized Hall conductivity

The usual proof of

jx e2 X
σxy = = Cn
Ey h lled bands
uses Kubo formula.
We present an equivalent but more dynamical proof, which relates Cn with the
adiabatic transport of a wave packet.

6.3.1 Delocalized Bloch waves and Localized Wannier waves

In a xed band n, consider a C∞ section |ψi of the vector bundle Fn → Tθ :

Z Z
|ψ >= dθ1 dθ2 |ψ(θ1 , θ2 ) >, |ψ(θ1 , θ2 ) >∈ Htore (θ)

Each state |ψ (θ)i ∈ Htore (θ) is a delocalized Bloch wave,

but |ψ >∈ L2 (R2 ) is a localized Wannier wave.
6.3. INTERPRETATION OF THE TOPOLOGICAL CHERN INDICES: THE QUANTIZED HALL COND

Wave packet motion

y j
x

Ey

Consider the translated localized Wannier state

Z Z
|ψn1 ,n2 >= Txn1 Tpn2 |ψ >= dθ1 dθ2 ein1 θ1 +in2 θ2 |ψ(θ1 , θ2 ) > .

Conversely:
X
|ψ(θ1 , θ2 ) >= e−in1 θ1 −in2 θ2 |ψn1 ,n2 > .
n1 ,n2 ∈Z2

Generalization: if
Z Z
|φ >= dθ1 dθ2 eif (θ1 ,θ2 ) |ψ(θ1 , θ2 ) >,

with any function f such that:

f (θ1 + 2π, θ2 ) = f (θ1 , θ2 ) + N1 2π

f (θ1 , θ2 + 2π) = f (θ1 , θ2 ) + N2 2π,

(Integers N1 , N2 ∈ Z characterize the homotopy type of f)

Dene the mean position of |φi on the plane by:

X
< n1 >= n1 . |< ψn1 ,n2 |φ >|2
n1 ,n2
X
< n2 >= n2 . |< ψn1 ,n2 |φ >|2 ,
n1 ,n2

Then:
< n1 >= N1 , < n2 >= N2 ,
So the mean position of |φ > is quantized.
g(θ) = n cn einθ = eif (θ) with f (2π) =
P
(This is a simple property of Fourier Series: if
R 2π
f (0) + N 2π then n n|cn |2 =< g|p̂θ |g >= 2π 1
f 0 dθ = N , with courant operator p̂θ =
P
0
1
i
d/dθ.).
88CHAPTER 6. TOPOLOGICAL CHERN INDICES AND THE INTEGER QUANTUM HALL EFFECT

6.3.2 Physical consequence on temporal evolution

Z Z
−iĤt/~
|ψ(t) >= e |ψ0,0 >= dθ1 dθ2 e−iE(θ1 ,θ2 )t/~ |ψ(θ1 , θ2 ) > .

has a dynamical phase f (θ1 , θ2 ) = −E(θ1 , θ2 )t/~ with homotopy type N1 = N2 = 0 .

So |ψ(t) > spreads on the plane, but its mean position < n1 >,< n2 > is zero.

6.3.3 Addition of a weak external electric eld ~ =

E ∂U
= Ey ~uy
∂~r

Potential energy E = −eU (~r), gives a slow motion of the quasi-impulsion ~k(t) =
~
~θ(t)/X :

d~k ∂E ~ = eEy ~uy

=− = eE
dt ∂~r
gives

θ1 (t) = θ1 (0)
θ2 (t) = −ωt with ω = eEy X/~

Then each Bloch state ~

|ψ(θ(t))i follows Berry's connection plus Dynamical phase
in the bers.
After one period T = 2π/ω ,

e−iĤT /~ |ψ(θ1 , θ2 ) >= exp (iφD (θ1 , θ2 ) + iφB (θ2 )) |ψ(θ1 , θ2 ) >,

with dynamical phase φD (θ1 , θ2 ) with homotopy N1 = N2 = 0

and Berry's phase φB (θ1 , θ2 ) of the path θ2 (t), which is homotopic to φB (θ1 ) ≡ 2πCθ1
, so N1 = C , N2 = 0.
(Exponentially small Landau-Zener corrections).
Consequence:
after one period T, the mean position has shifted by integer number of cells

δ < n1 >= C, δ < n2 >= 0

The mean velocity of the electron is then:

δ < n1 > X CX
Vx = =
T T
If the band is lled by electrons, the density is one electron per cell: ρ = 1/X 2 .
The current density is then

e2
jx = ρeVx = C Ey
h
6.4. BORN-OPPENHEIMER APPROXIMATION WITH STRONG MAGNETIC FIELD; EFFECTIVE D

So
jx e2
σxy = = C
Ey h

6.4 Born-Oppenheimer Approximation with Strong Mag-

netic eld; Eective dynamics in a Landau Level

• With no hypothesis on B, the dynamics of electrons has two degrees of freedom:

not integrable.

• Suppose now the adiabatic limit:

hc
hef f = 1
eBX 2
Then the classical electron has a fast cyclotron rotation, and a slower precession
of these circles.

Bz
y
Ey

jx
X x

Born-Oppenheimer description:
We treat Xslow = (xs , ps ) as xed classical parameters,
X̂f ast = (x̂f , p̂f ) as quantum operators, and consider the spectrum of:

 
Xs → ĤXs = H̃ Xs , X̂f

it gives a discrete spectrum (fast motion):

E1 (Xs ) , . . . , Em (Xs ) , . . . : Landau Levels

The slow dynamics (precession of circles) in Landau Level m is described by the

eective bi-periodic Hamiltonian:

2
Hm,ef f (xs , ps ) = Em (xs , ps ) , Xslow = (xs , ps ) ∈ Tslow
90CHAPTER 6. TOPOLOGICAL CHERN INDICES AND THE INTEGER QUANTUM HALL EFFECT

Below, we consider a xed Landau Level, and write:

H(q, p) = Hm,ef f (q, p) , m xed

which is an eective Bi-periodic Hamiltonian on R2 :

H (q, p) = H (q + 1, p) = H (q, p + 1) ' V (q, p)

Remark: no degeneracies between lnadau Bands: Em (Xs ) < Em+1 (Xs ).

So the quantized operatorĤ = H(q̂, p̂) acts in a space HN (θ) with dimension N .

The Landau Level m has N subbands, n = 1 → N , which Chern indices Cn .

σxy (e 2/h)
E Band structures
Bz 2
0 Quantum
y Classical
Ey n=1 2
−1
0 1
Y
n=0 1
jx
C=0
X x
V E
Fermi
Landau Levels

6.4.1 Formula for Chern indices Cn :

1) Integral curvature:

 h i 
i ∂Pn ∂Pn
R
Cn = 2π Tθ
Tr Pn ,
∂θ1 ∂θ2
Pn dθ2 dθ2 , Pn = |ϕn (θ)ihϕn (θ) |
i
R
= 2π Tθ
(< ∂θ1 ϕn |∂θ2 ϕn > − < ∂θ2 ϕn |∂θ1 ϕn >) dθ2 dθ2

2) From the zeros of a global section Pn |z0 i = |ϕn (θ)ihϕn (θ) |z0 i, with a xed
coherent state |z0 i, z0 ∈ Tqp .

So from the zeros of Bargmann or Husimi functions bϕ(θ) (z0 ) = hz0 |ϕn (θ)i:
X
Cn = (±1),
θ tq bϕ(θ) (z0 )=0

with sign ±1 depending on orientation of the zero.

6.5. SEMI-CLASSICAL CALCULATION OF CHERN INDICES IN A LANDAU LEVEL91

6.4.2 Sum of Chern indices in a Landau Level

N
X
Cn = c1 (F1 ⊕ . . . ⊕ FN ) = c1 (Htore ) = +1
n=1

Which means that the rank N vector bundle Htore → Tθ is non trivial.
This formula gives the Classical Hall conductivity in the semi-classical limit.
Five Dierent Proofs:
Proof 1: We saw a basis of Htore (θ), with states |j, θi, j = 1 → N .
This gives a trivialization of the bundle Htore → Tθ over θ1 ∈ [0, 2π[, θ2 ∈]0, 2π[, with
transition function at θ2 ≡ 0:
 
0 1 0
.. ..
. .
 
T (θ1 ) = 
 
..

 . 1 
iθ1
e 0

Remark that c1 (Htore ) = c1 (det (Htore )) (from c1 (L1 ⊗ . . . ⊗ LN ) = c1 (L1 ⊕ . . . ⊕ LN )),

and the line bundle det (Htore ) has transition function det (T (θ1 )) = eiθ1 .
We deduce that c1 (Htore ) = c1 (det (Htore )) = +1. 
Proof 2: Consider E = V ect(|0 >, . . . , |N >) ⊂ Hplan a xed space of dimension N +1,
spanned by N + 1 rst states of Harmonic oscillator. Consider the orthogonal projection:
Pθ : Htore (θ) → E .
One shows that Pθ is into, so the rank N vector bundle Htore (θ) → Tθ is realized as a
subbundle of the rank N + 1 trivial bundle: E → Tθ .
Its orthogonal is a rank 1 bundle L → Tθ (i.e. Lθ ⊕ Htore (θ) = E ), and we calculate
its Chern index with the zeros of a global section. One nd c1 (L) = −1. So c1 (Htore ) =
c1 (E) − c1 (L) = 0 + 1 = +1.
Proof 3: There is a more standard presentation of Htore (θ) as the space of Holomorphic
P
sections of a line bundle over Tqp . The space Tθ is the Jacobi variety and n Cn = 1 results
from the Abel inversion theorem.
Proof 4: By computation, from integral curvature formula.
Proof 5: By semi-classical analysis, from the topology of the classical Reeb Graph.
see below.

6.5 Semi-classical calculation of Chern indices in a Lan-

dau Level

(Generalization of the T.K.N.N. calculation)

2
H(q, p) is bi-periodic on (q, p) ∈ R :
92CHAPTER 6. TOPOLOGICAL CHERN INDICES AND THE INTEGER QUANTUM HALL EFFECT

X
H(q, p) = cn1 ,n2 exp(i2πn1 q) exp(i2πn2 p).
n1 ,n2 ∈Z2

cn1 ,n2 = c̄−n1 ,−n2 ∈ C, (n1 , n2 ) ∈ Z : Fourier coecients

Hilbert space of the plane:

HP lan = L2 (Rslow ),
Quantization of H(q, p):
X 1
Ĥ = cn ,n exp (i2πn1 q̂) exp (i2πn2 p̂) + hermitian conjugate.
2 1 2
n1 ,n2 ∈Z 2

6.5.1 Classical Hamiltonian and trajectories of H(q, p)

Example :

H(q, p) = H0 (q, p) + H1 (q, p),

H (q, p) = cos(2πq) + 0.1 cos(2πp),
 0
H1 (q, p) = P exp −100(q − q0 )2 − 10(p − p0 )2 , q0 = 0.45 p0 = 0.5



(operator P makes periodic on the plane).

p E
A
(0,0)

B D

(0,0)
D
B C E
C (0,-1)

(0,+1)
A E
F
(0,0) F

q 0 1 q
6.5. SEMI-CLASSICAL CALCULATION OF CHERN INDICES IN A LANDAU LEVEL93

Question : Understand the values of Chern indices Cn from the classical trajectories, or Reeb Graph ?

Solution :
the classical dynamics is integrable, so stationnary states are approximated by
quasi-modes (WKB approach) :

|ψ̃n (θ1 , θ2 ) >' |ϕn (θ1 , θ2 ) >

We just have to study the dependence of the quasi-modes with (θ1 , θ2 ).

6.5.2 Quasi-modes : Quasi-mode on a contractible trajectory of

type (0, 0) :

1
Γ

0 1 q

Energy Ẽ :

S(Ẽ) = (k + 1/2)h + o(h), k∈Z

with S(Ẽ) : enclosed surface

Remark : Ẽ does not depend on (θ1 , θ2 ).

Quasi-mode on a non-contractible trajectory of type (0, ±1) :
Energy Ẽ(θ2 ) :

S(Ẽ) = (k − θ2 /2π)h + o(h), k ∈ Z, with S(Ẽ) : right side surface

proof :
R
after one period on trajectory q(t), p(t) the phase is ϕ=− Γ
qdp/~ = S/~, and perio-
dicity condition is ϕ = θ2 [2π]..
Remark : Ẽ(θ2 ) and the support Γ(θ2 ) depend on θ2 .
94CHAPTER 6. TOPOLOGICAL CHERN INDICES AND THE INTEGER QUANTUM HALL EFFECT

p
Γ
1

0 1 q

6.5.3 Semi-classical spectrum and tunnelling eect :

p E (0,+1)
(0,+1) (0,0) (0,−1) non contractible trajectories
1
(0,−1)

critical trajectory

contractible trajectories (0,0)

0 θ* 2π θ
q 2 2

p
Γ2 Γ1 Γ2 Γ2

|ψ2 > |ψ1 > |ψ2 > |ψ2 > q

n1 −1 n1 n1 n1 +1

Numerical spectrum :
Above example, with N = 11, band n=6 :

C1→4 C5 C6 C7 C8→11
0 +1 −1 +1 0
6.5. SEMI-CLASSICAL CALCULATION OF CHERN INDICES IN A LANDAU LEVEL95

A B C

0 2π θ

(B) (B) (C) (C)

p
(A) (A)

6.5.4 General result :

Consider the support of the quasi-mode of band n :

Sn : θ2 ∈ R → Supp (|ψn (θ2 ) >) .

With homotopy In = I(Sn ) ∈ Z :

TQIn [Supp (|ψn (0) >)] = [Supp (|ψn (2π) >)] .

Theorem :

Cn = I(Sn )

proof : uses zeros of Husimi function.

Practical computation from the Reeb graph :
96CHAPTER 6. TOPOLOGICAL CHERN INDICES AND THE INTEGER QUANTUM HALL EFFECT

(c)
C =0
8
Ε R
2
C B
A C =−1
B 6

Ε C =+1
1 5

0 1 q

6.5.5 Total Chern index :

In order to recover :
N
X
Cn = +1
n=1

First dene (Sn + Sn+1 ) by removing the jumps (does not change the homotopy) :

E E

θ2 θ2

Then
N N N
!
X X X
Cn = I(Sn ) = I Sn = +1.
n=1 n=1 n=1
6.5. SEMI-CLASSICAL CALCULATION OF CHERN INDICES IN A LANDAU LEVEL97

6.5.6 The Chern indices for a chaotic dynamics :

There are no more nice WKB quasi-modes.

Numerical results, in the Kicked Harper model : (time-dependent model, parameter γ)
Poincaré sections :

p 1
z

1
z
γ=0.2 γ=0.5 γ=0.7 γ=1.2

−π

C=
98CHAPTER 6. TOPOLOGICAL CHERN INDICES AND THE INTEGER QUANTUM HALL EFFECT

6.6 References :

Integer Hall Eect :

• K.V. Klitzing, G. Dorda, M. Pepper, "New method for high-Accuracy Determination

of the ne structure constant based on quantized Hall resistance", Phys. Rev. Lett.
45 , p.494 (1980).

Chern indices in Integer Hall Eect :

• D. J. Thouless and M. Khomoto and M. P. Nightingale and M. den Nijs Quantized

Hall conductance in a two-dimensional periodic potential  Phys. Rev. Lett. 49, p.
405, (1982).

• J. E. Avron and R. Seiler and B. Simon Homotopy and quantization in condensed

matter physics  Phys. Rev. Lett. 51 p.51 (1983).

• M. Khomoto, Topological invariant and the quantization of the Hall conductance 

Ann. Phys. B, 160, p. 343,(1985).

Semi-classical Computation of Chern indices :

• P. Leboeuf and J. Kurchan and M. Feingold and D. P. Arovas, Phase-space locali-

zation : topological aspects of quantum chaos  , Phys. Rev. Lett. 65, p.3076, (1990).

• F. Faure , Approche géométrique de la limite semi-classique par les états cohérents et

mécanique quantique sur le tore , Thesis of the university Joseph Fourier , Grenoble,
93 -112, http ://lpm2c.polycnrs-gre.fr/faure/, (1993).

• Y. Colin de Verdière, Fibrés en droites et valeurs propes multiples  , Séminaire de

théorie spectrale, p 9-18, (1993).

• F. Faure, Topological properties of quantum periodic Hamiltonians " J. Phys. A,

Math. Gen. 33, p. 531-555, ( 2000).