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Introduction to the Quantum Hall Effects
Lecture notes, 2006
Pascal LEDERER
Mark Oliver GOERBIG
Laboratoire de Physique des Solides, CNRS-UMR 8502
Université de Paris Sud, Bât. 510
F-91405 Orsay cedex
2
Contents
1 Introduction
7
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 History of the Quantum Hall Effect . . . . . . . . . . . . . . . 9
1.3 Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Charged particles in a magnetic field
2.1 Classical treatment . . . . . . . . . . . . . . . . .
2.1.1 Lagrangian approach . . . . . . . . . . . .
2.1.2 Hamiltonian formalism . . . . . . . . . .
2.2 Quantum treatment . . . . . . . . . . . . . . . . .
2.2.1 Wave functions in the symmetric gauge . .
2.2.2 Coherent states and semi-classical motion
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3 Transport properties– Integer Quantum Hall Effect (IQHE)
3.1 Resistance and resistivity in 2D . . . . . . . . . . . . . . . . .
3.2 Conductance of a completely filled Landau Level . . . . . . . .
3.3 Localisation in a strong magnetic field . . . . . . . . . . . . .
3.4 Transitions between plateaus – The percolation picture . . . .
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The Fractional Quantum Hall Effect (FQHE)– From Laughlin’s theory to Composite Fermions.
4.1 Model for electron dynamics restricted to a single LL . . . . .
4.1.1 Matrix elements . . . . . . . . . . . . . . . . . . . . . .
4.1.2 Projected densities algebra . . . . . . . . . . . . . . . .
4.2 The Laughlin wave function . . . . . . . . . . . . . . . . . . .
4.2.1 The many-body wave function for ν = 1 . . . . . . . .
4.2.2 The many-body function for ν = 1/(2s + 1) . . . . . .
4.2.3 Incompressible fluid . . . . . . . . . . . . . . . . . . . .
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CONTENTS
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6 Hamiltonian theory of the Fractional Quantum Hall Effect
6.1 Miscroscopic theory . . . . . . . . . . . . . . . . . . . . . . . .
6.1.1 Fluctuations of ACS (r) . . . . . . . . . . . . . . . . . .
6.1.2 Decoupling transformation at small wave vector . . . .
6.2 Effective theory at all wave vectors . . . . . . . . . . . . . . .
6.2.1 Approximate treatment of the constraint . . . . . . . .
6.2.2 Energy gaps computation . . . . . . . . . . . . . . . .
6.2.3 Self similarity in the effective model . . . . . . . . . . .
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7 Spin and Quantum Hall Effect– Ferromagnetism
7.1 Interactions are relevant at ν = 1 . . . . . . . . .
7.1.1 Wave functions . . . . . . . . . . . . . . .
7.2 Algebraic structure of the model with spin . . . .
7.3 Effective model . . . . . . . . . . . . . . . . . . .
7.3.1 Spin waves . . . . . . . . . . . . . . . . . .
7.3.2 Skyrmions . . . . . . . . . . . . . . . . . .
7.3.3 Spin-charge entanglement . . . . . . . . .
7.3.4 Effective model for the energy . . . . . . .
7.4 Berry phase and adiabatic transport . . . . . . .
7.5 Applications to quantum Hall magnetism . . . . .
7.5.1 Spin dynamics in a magnetic field . . . . .
7.6 Application to spin textures . . . . . . . . . . . .
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4.3
5
4.2.4
4.2.5
4.2.6
Jain’s
4.3.1
Fractional charge quasi-particles .
Ground state energy . . . . . . . .
Neutral Collective Modes . . . . . .
generalisation – Composite Fermions
The effective potential . . . . . . .
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Chern-Simons Theories and Anyon Physics
5.1 Chern-Simons transformations . . . . . . . . . . . . .
5.2 Statistical Transmutation – Anyons in 2D . . . . . . .
5.2.1 Anyons and Chern-Simons theories . . . . . . .
5.2.2 Fractional charge and fractional statistics . . .
at ν
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8 Quantum Hall Effect in bi-layers
131
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
8.2 Pseudo-spin analogy . . . . . . . . . . . . . . . . . . . . . . . 133
CONTENTS
8.3
8.4
8.5
5
Differences with the ferromagnetic monolayer case . . . . . . 134
Experimental facts . . . . . . . . . . . . . . . . . . . . . . . . 137
8.4.1 Phase Diagram . . . . . . . . . . . . . . . . . . . . . 137
8.4.2 Excitation gap . . . . . . . . . . . . . . . . . . . . . . 139
8.4.3 Effect of a parallel magnetic field . . . . . . . . . . . . 139
8.4.4 The quasi-Josephson effect . . . . . . . . . . . . . . . . 141
8.4.5 Antiparallel currents experiment . . . . . . . . . . . . . 142
Excitonic superfluidity . . . . . . . . . . . . . . . . . . . . . . 145
8.5.1 Collective modes – Excitonic condensate dynamics . . 148
8.5.2 Charged topological excitations . . . . . . . . . . . . . 150
8.5.3 Kosterlitz-Thouless transition . . . . . . . . . . . . . . 153
8.5.4 Effect of the inter layer tunneling term . . . . . . . . . 154
8.5.5 Combined effects of a tunnel term and a parallel field Bk 155
8.5.6 Effect of an inter-layer voltage bias . . . . . . . . . . . 158
6
CONTENTS
Chapter 1
Introduction
1.1
Motivation
Almost thirty years after the discovery of the Integer Quantum Hall Effect
(IQHE, 1980)[1], and the fractional one (FQHE, 1983)[2], two-dimensional
electron systems submitted to a perpendicular magnetic field remain a very
active field of research, be it experimentally or on the theory level [3]. New
surprises arise year after year, exotic states of electronic matter, new materials with fascinating quantum Hall properties keep triggering an intense
activity in the field: see for example the number of papers on graphene which
appear on cond-mat since the discovery of the QHE in this material in 2005
[4, 5] or the appearance of excitonic superfluidity in quantum Hall bilayers
[6]. The continuous increase in sample quality over the years is a key factor in the discovery of new electronic states of two dimensional matter. In
heterostructure interfaces such as the GaAs/AlGaAs system, one of the prototypical experimental systems, the electronic mobility µ has now reached
values up to µ ≃ 30 × 106 cm2 /Vs, more than two orders of magnitude larger
than the values obtained at the time of the first QHE discoveries in the 80’s.
The discovery of the quantum Hall effects, in particular that of the FQHE,
has taken a large part in a qualitative advance of condensed matter physics
regarding electronic fluids in conducting materials. In large band metallic
systems, the role of interactions was successfully taken into account until the
sixties by the Landau liquid theory [7], which is a perturbation approach:
interactions between electrons alter adiabatically the properties of the free
electron model, so that the Drude-Sommerfeld model keeps its validity with
7
8
Introduction
renormalized coefficients. The theoretical tools corresponding to this physics
have involved sophisticated diagramatic techniques such as Feynman diagrams, which are all based on the existence of a well controlled limit of zero
interaction Green’s function. It was realized in the fifties, with the theory
of BCS superconductivity [8] that attractive interactions cause a breakdown
of the non interacting model, and a spontaneous symmetry breaking (in the
superconductivity case a breakdown of gauge invariance), but even in that
case Fermi liquid theory seemed an unescapable starting point for conducting
systems. The various other hints of the breakdown of perturbation theory,
such as the local spin fluctuation problem, the Kondo problem, the Mott insulator problem, the physics of solid or superfluid 3 He in the sixties/seventies
or even the Luttinger liquid problem in the eighties did little to suggest the
intellectual revolutions which were in store with the discoveries of the fractional quantum Hall effects and of superconducting high Tc cuprates [9]. (The
heavy fermion physics is somewhat of a hybrid between the former and the
latter electronic systems.)
The fundamental novelty of both those phenomena, which involve electron systems in two dimensional geometries, is that the largest term in the
Hamiltonian is the interaction term, so that perturbation theory is basically
useless. If one tries to do perturbation theory from a limiting case, such
as the incompletely filled Landau level in the Hall case, or the zero kinetic
energy in the High Tc case, one is faced with the macroscopic degeneracy of
the starting ground state. There is no way to evolve adiabatically from this
degenerate ground state to the physical one. In both cases, interactions do
not alter the quantitative properties of a pre-existing ground state. They are
essential at determining the symmetry and properties of the ground state
which results from the lifting of a formidable degeneracy. Thus the whole
aparatus of perturbation theory turned ou to be inadequate for a theoretical
understanding of the QHE, as well as of High Tc superconductivity. New
methods have had to be devised, and new concepts emerged to account for
unexpected exotic phenomena: new particles, new statistics, new ground
states,[11] etc.. The most intuitive method turned out to be very useful and
fruitful. It amounts to guessing the many-body wave function for ≈ 1011
particles for the ground state. This is the incredibly original path chosen in
1983 by Laughlin [12]. More or less the same method led P. W. Anderson to
propose the RVB wave function as the basic new object to describe High Tc
superconductivity in 1987. That wave function is a resonating superposition
of singlet pair products involving all electrons. It looks like a BCS wave
1.2. HISTORY OF THE QUANTUM HALL EFFECT
9
function, where strong correlations prevent the simultaneous occupation of
any site by two electrons. This proposal has been at the center of active
discussions over the last twenty years.
In the context of Quantum Hall Effects, new ideas such as Chern-Simons
theories, which investigate the formation of composite particles when electrons bind to flux tubes, have been very fruitful [15, 16, 17]. These theories ,
based on their topological character, have given flesh to the notion of strange
phenomena such as charge fractionalization, fractional statistics, and so forth,
at work among the elementary excitations of the 2D electronic liquid under
magnetic field.
A new concept, also emerging over the last twenty years, is that of quantum phase transition and of quantum critical points [18]. Quantum phase
transitions occur at zero temperature. They are not controlled by temperature, but by parameters such as pressure, magnetic field, or chemical doping
[11, 18]. In heavy fermions for instance, the quantum critical point separates
a metallic paramagnetic phase from an insulating antiferromagnetic one. At
finite temperature, the “quantum critical regime” involves a broader array
of parameter values. Quantum phase transitions, as we shall see, are present
in a number of Quantum Hall Effects, as a function of magnetic field or of
electron density.
1.2
History of the Quantum Hall Effect
The classical Hall effect
The classical Hall effect was discovered by Edwin Hall in 1879, as a minor correction to Maxwell’s “Treatise on Electromagnetism ”, which was supposed
to be a final and complete account of the physical properties of Nature. Hall
noted that, contrary to Maxwell’s opinion, if a current I is driven through
a thin metallic slab in a perpendicular magnetic field B = Bez , an electronic density gradient develops in the slab, in the direction orthogonal to
the current. This gradient is equivalent to a transverse voltage V , so that
the resulting transverse resistance (the Hall resistance) is proportional to the
field, and inversely proportional to the electronic density: RH = −B/enel .
Here nel is the density per unit surface, and −e is the electron charge.
Things are rather simple to understand with the Drude model, with the
electron equation of motion:
10
Introduction
I
−I
gaz d’électrons 2D
résistance
longitudinale
résistance
de Hall
Figure 1.1: Two dimensional electron system under perpendicular magnetic field. The
current I is driven through the two black contacts. The longitudinal resistance is measured
between two contacts on the same edge, while the Hall resistance is measured across the
sample on the two opposite edges.
dp
p
p
= −e E +
×B − ,
dt
m
τ
where E is the electric field, m is the electron (band) mass , p its momentum
τ the mean diffusion time due to impurities. The stationary solution for this
equation, i.e. that for dp/dt = 0, is
py
B −
m px
0 = −e Ey − B −
m
0 = −e Ex +
px
,
τ
py
.
τ
With the cyclotron frequency ωC ≡ eB/m and the Drude conductivity σ0 =
nel e2 τ /m, one gets
py
px
σ0 Ex = −nel e − nel e (ωC τ ),
m
m
py
px
σ0 Ey = nel e (ωC τ ) − nel e ,
m
m
In terms of the current density j = −nel ep/m, in matrix form
E = ρ j,
History of the Quantum Hall Effect
11
with the resistivity tensor
ρ=
σ0−1 enBel
− enBel σ0−1
!
1
=
σ0
1
ωC τ
−ωC τ
1
!
.
(1.1)
The conductivity follows by matrix inversion,
σ=ρ
−1
=
σL −σH
σH σL
!
,
(1.2)
with σL = σ0 /(1 + ωC2 τ 2 ) et σH = σ0 ωC τ /(1 + ωC2 τ 2 ). In the limit of a pure
metal with infinite τ , ωC τ → ∞, one has
ρ=
0
− enBel
B
enel
0
!
,
σ=
0
enel
B
− enBel
0
!
.
(1.3)
Note that the diagonal (longitudinal ) conductivity is zero together with the
longitudinal resistivity.
The classical Hall effect, deemed by Hall of purely academic interest, and
with no foreseeable application whatsoever is nowadays of current industrial
use, and is still useful in condensed matter physics to measure the carrier
density in conducting materials, as well as to determine their sign.
Landau quantization
Landau was the first to apply quantum mechanics, in 1930, in the study of
metallic systems, to the quantum treatment of electronic motion in a static
uniform magnetic field. He found that problem to be quite analogous in
2D to that of a harmonic oscillator, with an energy structure of equidistant
discrete levels, with a distance h̄ωC . Each level is highly degenerate. The
surface density of states per Landau level, nB , is nb = B/φ0 per unit area,
where φ0 = h/e is the flux quantum , so that nB is the density of flux quanta
threading the surface in a perpendicular field B. Because of their fermionic
character, electrons added to the plane fill in successive Landau Levels (LL),
so that it is natural and useful to define a filling factor
ν=
nel
.
nB
This quantum treatment will be reviewed in chapter 2.
(1.4)
12
Introduction
The Quantum Hall Effect : a macroscopic quantum phenomenon
The IQHE, discovered by von Klitzing in 1980 [1] is, at first sight, a direct
consequence of Landau quantization, and disorder. In fact, as we shall see,
impurity disorder is also a necessary feature: in a tanslationaly invariant
system, the Hall resistivity would have the classical value. In fact Hall quantization appears because of the sample impurity potential, not in spite of it.
The IQHE appears at low temperature, when kB T ≪ h̄ωC , and is defined by
the formation of plateaus in the Hall resistance, which become quantized, for
certain ranges of values of B, as RH = (h/e2 )1/n, where n is an integer, the
integer part of the filling factor: n = [ν]. Each plateau in the Hall resistance
coincides with a zero (exponentially small value in fact) of the longitudinal
resistance (Fig. 1.2). A remarkable fact about the resistance quantization is
that its value is independent of the sample geometry, of its quality (density
and/or distribution of impurities, etc.). The Hall resistance is given entirely
in terms of fundamental constants, e and h. The accuracy of the determination of the n = 1 plateau value reaches 1 part in 109 , so that it is now
used in metrology as a universal resistance standard, the v. Klitzing constant
RK−90 = 25812, 807Ω.
Another surprise followed shortly after the discovery of the IQHE. In
1983, D. Tsui, H. Störmer and A. Gossard found the Fractional Quantum
Hall Effect (FQHE) [2]. This occurs for ”magical” values of the filling factor,
especially within the lowest LL. The first observed fractional plateaus were
at ν = 1/3 and ν = 2/3. Since then, a whole series of plateaux have been
detected. The remarkable aspect is that for fractional ν values, there is a
huge degeneracy of the N body states. Since, apart from impurities, the
only relevant energy is the Coulomb repulsion between particles, one is facing a strongly correlated electron system. Our understanding of the FQHE
is still to-day essentially based on a revolutionnary theory put forward by
Laughlin in 1983: he proposed, by a series of educated guesses, a wave function for N ≈ 1011 particles, written in the first quantization language, which
describes an incompressible electronic liquid state, i.e. one such that elementary and collective excitations are separated from the ground state by a
gap[12]. Following the discovery of other families of fractional QH plateaus
which are not described by the initial Laughlin wave functions, various generalizations have been proposed. B. Halperin generalised in 1983 the Laughlin
wave function to the case of an additional discrete degree of freedom, such as
History of the Quantum Hall Effect
13
3.0
2.0
Ix
Vy
2.5
Vx
1.5
ρxx (kΩ)
2
ρxy (h/e )
2.0
6 54 3
1.5
1
2
1.0
2/3
3/ 2
3/ 4
1/ 2
3/5
3/7
5/9
4/3
5/3
8/5
7/5
0.5
2/5
4/9
4/7
1.0
5/11
6/11
6/13
0.5
5/7
0.0
8/15
4/5
0
0
4
8
Magnetic Field B (T)
7/13
7/15
12
16
champ magnétique B[T]
Figure 1.2: Experimental signature of the quantum Hall effect. Each plateau coincides
with a zero longitudinal resistance. The classical Hall resistance curve is the dotted line.
Numbers label the filling factor ν = n for the IQHE, and ν = p/q (p and q integers for the
FQHE.
14
Introduction
spin [19]. In 1989, Jain generalised the theory to account for observed fractional states with ν = p/(2sp + 1) with s and p integers. He introduced the
notion of “Composite Fermions” (CF). The CF theory allows to understand
the FQHE as an IQHE of CF. This will be dealt with in chapter 4.
1.3
Samples
The discovery of the IQHE and of the FQHE is intimately connected to
the evolution of semiconducting sample preparation to produce 2D electron
gases. The order of magnitude of electron densities in thin metallic films
was not appropriate for the QHE discovery. The electronic surface density of
metallic thin films is of order nel = 1018 m−2 = 1014 cm−2 . As we shall see, the
QHE become observable when the electronic surface density is of the order
of the magnetic flux density, i. e. nel ∼ nB = eB/h. This would amount
to a magnetic field of order ≈ 1000 T, quite out of reach in the laboratory
nowadays, when the largest available fields in a dc regime amount to less
than 50 T, and less than 80 T for pulsed magnetic fields. More intense
fields are available in destructive experiments or nuclear blasts. A useful
quantity
q which sets a length
q scale for the QH physics is the magnetic length,
lB = h̄/eB = 25, 7nm/ B[T]. The magnetic length is such that the flux
2
which threads a surface equal to 2πlB
is the flux quantum φ0 = h/e
Lower electronic densities, typically nel ∼ 1011 cm−2 are reached in semiconducting structures. The samples used at the time of the IQHE discovery
were MOSFETs, shown schematically in the figure 1.3. In such a device, a
metallic film is separated from a semiconductor, which is doped with acceptors, by an oxyde insulating layer. The metal chemical potential is controlled
with a voltage bias VG . When VG = 0, the Fermi level EF lies in the gap
between the valence band and the conduction band, below the acceptor levels
[Fig. 1.3(a)]. Upon lowering the chemical potential in the metal with VG > 0,
one introduces holes, which attract electrons from the semiconductor toward
the interface with the insulating layer. This results in a downward bending of the semiconductor band close to the interface. Electrons attracted
to the interface first fill in acceptor levels, which are below the Fermi level
[Fig. 1.3(b)]. By further lowering of the metal chemical potential, the semi
conductor conduction band can be bent below the Fermi level close to the
insulating layer, so that electrons which occupy states in that part of the
Samples
15
(a)
métal
oxyde
(isolant)
semiconducteur
I
bande de
conduction
niveaux
d’accepteurs
EF
z
métal
oxyde
semiconducteur
V
G
bande de
valence
II
z
(b)
E1
E0
(c)
métal
oxyde
(isolant)
semiconducteur
métal
bande de
conduction
EF
VG
niveaux
d’accepteurs
E
oxyde
(isolant)
z
électrons 2D
bande de
conduction
niveaux
d’accepteurs
EF
VG
bande de
valence
bande de
valence
z
z
Figure 1.3: Metal-Oxyde Field Effect Transistor (MOSFET). The inset I is a schematic
view of a MOSFET. (a) Energy level structure. In the metallic part, the band states are
occupied up to the Fermi level EF . The oxyde is an insulating film. The Fermi level in
the semiconductor falls in the gap between the valence band and the conduction band.
There are acceptor states doped close to the valence band, but above the Fermi level EF
(b)The chemical potential in the metal is controlled by a gate bias VG . The introduction
of holes results in a band bending in the semiconducting part and (c) when the gate bias
exceeds a certain value, the conduction band is filled close to the insulating interface, and
a 2D electron gas is formed. The confining potential has a triangular profile with electric
subbands which are represented in the inset II.
16
(a)
Introduction
AlGaAs
(b)
GaAs
AlGaAs
GaAs
EF
EF
dopants
(récepteurs)
dopants
(récepteurs)
z
électrons 2D
z
Figure 1.4: Semiconducting (GaAs/AlGaAs) heterostructure. (a) A layer of (receptor)
dopants lies on the AlGaAs side, at a certain distance from the interface. The Fermi
energy is locked to the dopant levels. The bottom of the GaAs conduction band lies lower
than those levels so electrons close to the interface migrate to the GaAs conduction band.
(b) This polarisation leads to a band bending close to the interface, and a 2D electron gas
forms, on the GaAs side.
conduction band form a 2D gas. Electron motion , in spite of a finite extent
of the wave function in the z direction is purely 2D if confinement is such
that the energy separation between electronic sub-bands E0 (partially filled)
and E1 (empty) is significantly larger than kB T (inset II in Fig. 1.3).
The problem with MOSFETS is the small distance between the 2D electron gas and the dopants. The latter also act as scattering centers, so that
the mean free path is relatively small, and the electron mobility relatively
low. This problem is dealt with by forming a 2D electron gas at the interface of a semiconducting heterostructure, such as for example in the III-V
heterostructure GaAs/AlGaAs. The two semi-conductors have different gaps
between their valence bands and their conduction bands. When the side with
the largest gap, Alx Ga1−x As, is doped, the receptor dopant levels are occupied by electrons, and the Fermi level is tied to receptor levels, which may
have a higher energy than the bottom of the conduction band in GaAs. The
electrons close to the interface migrate in this conduction band [Fig. 1.4(a)].
This polarisation produces a band bending, now on the GaAs side, which
is not disordered by the dopants. This spatial separation between the 2D
electron gas and the impurities allows to reach larger mobility values than
in MOSFETS. Technological progress in the fabrication of semi-conducting
heterostructures along the last twenty years has allowed to increase mobilities
Samples
17
Density of states
Graphene IQHE:
R H = h/e2ν
at ν = 2(2n+1)
Vg =15V
T=30mK
∼ 1/ν
Usual IQHE:
B=9T
T=1.6K
at ν = 2n
(no Zeeman)
∼ν
Figure 1.5: Quantum Hall Effect as observed in graphene by Zhang et al (Nature 438,
197 (2005)), and Novoselov et al. (Nature 438, 201 (2005))
by two orders of magnitude: The FQHE was discovered in 1983 in a sample
with mobility µ ≃ 0, 1 × 106 cm2 /Vs [2] while samples of the same type reach
nowadays a mobility of µ ≃ 30 × 106 cm2 /Vs.
The discovery of the QHE in graphene in 2005 opens up a new avenue
to experiments and theory in the QHE, because graphene is a qualitatively
new 2D material, with original electronic structure. See figure 1.5
18
Introduction
Chapter 2
Charged particles in a magnetic
field
Our understanding of integer or fractional quantum Hall effects relies mostly
on the quantum mechanics of electrons in a 2D plane, or thin slab, when
submitted to a perpendicular magnetic field. There is a notable exception,
that of the Integer Quantum Hall Effect (IQHE) observed in anisotropic 3D
organic salts such as Bechgaard salts. The IQHE may arise in 3D systems
under magnetic field provided the electronic structure of the material under
magnetic field exhibits the suitable gap structure. However, in the present
lectures, I will adress the main stream of quantum Hall effects physics, that
of electrons the dynamics of which is restricted to a plane. The topic of this
chapter is the single electron quantum mechanics in a plane under magnetic
field. I start with a discussion of the classical mechanics, as a limiting case
of the quantum mechanical case.
2.1
Classical treatment
The equation of motion of a particle (with charge −e and mass m in a
magnetic field B = Bez is as follows:
ẍ = −ωC ẏ,
ÿ = ωC ẋ,
(2.1)
This follows from the Lorentz force F = −eṙ×B – By definition, the cyclotron
frequency is ωC = eB/m. The equation is solved as:
ẋ = −ωC (y − Y ),
ẏ = ωC (x − X),
19
(2.2)
20
Charged particle in a static uniform magnetic field
B
η
R
r
Figure 2.1:
Cyclotron motion of an electron in a magnetic field, around the guiding
center R.
where R = (X, Y ) is a constant of motion. With η = (ηx , ηy ) = r − R, one
has
η¨x = −ωC2 ηx ,
η¨y = −ωC2 ηy ,
(2.3)
and the solution is
x(t) = X + r sin(ωC t + φ),
y(t) = Y + r cos(ωC t + φ),
(2.4)
where r is the cyclotron motion radius, and φ is an arbitrary angle (constant
of motion). The physical meaning of the constant of motion R is transparent:
it is the “guiding center”, around which the electron moves on a circle of
radius r (Fig 2.1).
2.1.1
Lagrangian approach
Lagrangian mechanics starts from the energy function L and the minimum
action principle, which reproduce the equations of motion of the classical
system. This function is defined in configuration space (positions qµ and
velocities q̇µ ). The minimum action principle results in the Euler-Lagrange
equations
d ∂L
∂L
−
= 0,
(2.5)
dt ∂ q̇µ ∂qµ
Classical approach
21
valid for any index µ. The appropriate function in our case is
1 L(x, y; ẋ, ẏ) = m ẋ2 + ẏ 2 − e [Ax (x, y)ẋ + Ay (x, y)ẏ] ,
2
(2.6)
where A = (Ax , Ay ) is a vector potential which is independent of time.
This represents the minimal coupling theory for a charged particle and an
electromagnetic field, written in a covariant form, with Einstein’s convention,
1
Lrel = mẋµ ẋµ − eẋµ Aµ .
2
The conjugate (or ”canonical”) momenta, which will be needed in the Hamiltonian formulation of clasical or quantum mechanics are
px ≡
∂L
= mẋ − eAx ,
∂ ẋ
py ≡
∂L
= mẏ − eAy .
∂ ẏ
(2.7)
The Euler-Lagrange equations yield the equations of motion [Eq. (2.1)]
mẍ = −eẏ(∂x Ay − ∂y Ax ),
mÿ = eẋ(∂x Ay − ∂y Ax ),
(2.8)
where ∂x ≡ ∂/∂x, ∂y ≡ ∂/∂y, and (∂x Ay − ∂y Ax ) = (∇ × A)z = B is the z
component of the magnetic field.
Gauge invariance
A gauge transformation of the vector potential is defined as A′ = A + ∇χ,
where χ is an arbitrary function. The magnetic field is independent of the
gauge (it is “gauge invariant”) since ∇ × ∇χ = 0. A usual gauge in non
relativistic physics is the Coulomb gauge, ∇ · A = 0.1 The gauge (the gauge
function) is not completely determined by the Coulomb gauge condition,
which demands only ∆χ = 0, where ∆ = ∇2 is the Laplacian. Gauge
transformations in 2D are thus defined as harmonic functions. Two gauge
choices are especially useful in the quantum treatment of our problem: the
Landau gauge (e.g. for problems defined on a rectangular sample)
AL = B(−y, 0, 0)
1
Relativistic mechanics use rather the Lorentz gauge, ∂ µ Aµ = 0.
(2.9)
22
Charged particle in a static uniform magnetic field
and the symmetric gauge(e.g. for problems defined on a disc)
AS =
B
(−y, x, 0),
2
(2.10)
the function which transforms from one of these two gauges to the other is
χ = (B/2)xy.
Since velocities ẋ and ẏ, are also gauge invariant, it is clear that conjugate
(or ”canonical”) momenta in equation (2.7) are not. The gauge invariant
momenta (or ”mechanical momenta”) are
Πx = mẋ = px +eAx = −mωC ηy ,
Πy = mẏ = py +eAy = mωC ηx , (2.11)
where we used Eq. (2.2).
2.1.2
Hamiltonian formalism
For the quantum treatment of a one particle system, it is often preferred to
use the Hamiltonian formalism of classical mechanics. The Hamiltonian is
derived from the Lagrangian by a Legendre transformation,
H(x, y; px , py ) = ẋpx + ẏpy − L.
It is an energy function defined in phase space (positions/conjugate momenta). One must express velocities in terms of conjugate momenta, using
equations (2.7), and one finds for the Hamiltonian
H=
i
1 h
(px + eAx )2 + (py + eAy )2 .
2m
(2.12)
The Hamiltonian may also be written in a concise fashion, using the ”relative”
variables, (ηx , ηy ) (using 2.11),
1
H = mωC2 (ηx2 + ηy2 ),
2
(2.13)
where the ”new” variables are nevertheless defined by the variables in phase
space, i.e. (x, y, px , py ).
2.2. QUANTUM TREATMENT
2.2
23
Quantum treatment
The Hamiltonian formalism allows to introduce the canonical quantization,
where one imposes the non commutativity of position with its conjugate
momenta, in terms of Planck’s constant h̄,
[x, px ] = [y, py ] = ih̄,
[x, y] = [px , py ] = [x, py ] = [y, px ] = 0.
Since [x, y] = 0, one sees immediately that [ηx , ηy ] = −[X, Y ]. The fact that
the guiding center components are constants of motion is expressed by [see
also Eq. (2.13)]
[X, H] = [Y, H] = 0.
(2.14)
To compute the commutator between components ηx and ηy , one may use
the formula
∂f
[A, B].
(2.15)
∂B
That formula is valid for two arbitrary operators which commute with their
commutator, [A, [A, B]] = [B, [A, B]] = 0. One gets
[A, f (B)] =
[ηx , ηy ] =
e
m2 ωC2
([px , Ay ] − [py , Ax ])
1
(∂x Ay [px , x] − ∂y Ax [py , y])
eB 2
−ih̄
=
eB
=
or, in terms of magnetic length lB ≡
2
[ηx , ηy ] = −ilB
,
q
h̄/eB,
2
[X, Y ] = ilB
.
(2.16)
The result is of course gauge invariant. A remarkable point is that the dynamics of a charged particle in a magnetic field is perhaps the simplest example
of a non commutative geometry. Notice that, without any knowledge on the
energy level structure, the latter has to be degenerate. In any level chosen at
random, each state must occupy a minimal surface given by the Heisenberg
uncertainty principle,
2
σ = ∆X∆Y = 2πlB
.
24
Charged particle in a magnetic field
In that sense, the real 2D space looks like the phase space of a 1D particle,
where each state occupies a ”surface” 2πh̄. The level degeneracy may thus
be written directly in terms of this minimal surface: the number of states
per level and per unit surface being nB = 1/σ = B/φ0 – i.e. the flux density
in units of the flux quantum φ0 = h/e. Since electrons follow fermionic
statistics, each quantum state is occupied at most by one particle, because of
the Pauli principle. When there are many electrons in the system, the filling
ν of energy levels is thus described by the ratio between the electron surface
density nel and the flux density nB , ν = nel /nB . This ratio is also called the
filling factor.
The Hamiltonian form (2.13), along with the commutation relations (2.16),
is that of a harmonic oscillator – ηx and ηy may be interpreted as conjugate
variables. To exhibit explicitly the harmonic oscillator structure, we introduce two sets of ladder operators, (a, a† ) with
a = √
1
(ηx − iηy ),
2lB
a† = √
1
(ηx + iηy )
2lB
lB
ηy = √ (a† − a),
2i
lB
ηx = √ (a† + a),
2
(2.17)
and (b, b† ) with
b = √
1
b† = √ (X − iY )
2lB
ilB
Y = √ (b† − b),
2
1
(X + iY ),
2lB
lB
X = √ (b† + b),
2
(2.18)
with [a, a† ] = [b, b† ] = 1 et [a, b(†) ] = 0. In terms of ladder operators, the
Hamiltonian writes
1
.
(2.19)
2
The energy spectrum is thus given by En = h̄ωC (n + 1/2), where n is the
eigenvalue of operator a† a. In the context of an electron in a magnetic field
the equidistant levels of the oscillator are called ”Landau Levels” (LL, see
Fig. 2.2). Formally, in fact, the system may be viewed as a system of two
harmonic oscillators,
H = h̄ωC a† a +
H = h̄ωC a† a +
1
1
+ h̄ω ′ b† b +
,
2
2
25
niveaux de Landau
Quantum treatment
4
3
2
1
n=0
m
Figure 2.2: Landau Levels. The quantum number n labels the levels, and m , which is
associated to the guiding center, describes the level degeneracy.
where the frequency of the second oscillator vanishes, ω ′ = 0. The second
quantum number m is the eigenvalue b† b.
The eigenstates are thus determined by the two integer quantum numbers,
n and m, associated with the two species of ladder operators,
√
√
a† |n, mi = n + 1|n + 1, mi,
a|n, mi = n|n − 1, mi (pour n >0);
√
√
b† |n, mi = m + 1|n, m + 1i,
b|n, mi = m|n, m − 1i (pour m >0).
When n = 0 ou m = 0, one finds
a|0, mi = 0,
b|n, 0i = 0,
(2.20)
and negative numbers are prohibited. An arbitrary state may thus be constructed with the help of ladder operators, starting from the state |0, 0i,
(a† )n (b† )m
|n, mi = √ √ |0, 0i.
n! m!
(2.21)
The wave functions, which are the state representation in real space, depend
on the gauge chosen for the vector potential.
2.2.1
Wave functions in the symmetric gauge
To find the real space representation of eigenstates, φn,m (x, y) = hx, y|n, mi,we
must choose a gauge. Here we discuss the symmetric gauge [Eq. (2.10)],
A = (B/2)(−y, x, 0); we must translate equations (2.20) and (2.21) in differential equations, using px = −ih̄∂x and py = −ih̄∂y . With the help of
26
Charged particle in a magnetic field
equations (2.11) and (2.17), one finds the representation of ladder operators
in the symmetric gauge
√ z
¯
a= 2
+ lB ∂ ,
4lB
!
√
z∗
+ lB ∂ ,
b= 2
2
4lB
√
z∗
a† = 2
− lB ∂
2
4lB
√ z
− lB ∂¯
b† = 2
4lB
!
(2.22)
where z = x − iy is the electron position in the complex plane2 , z ∗ = x + iy
its complex conjugate, ∂¯ = (∂x − i∂y )/2 et ∂ = (∂x + i∂y )/2. A state in the
Lowest LL (LLL) is thus determined by the differential equation
2 ¯
z + 4lB
∂ φn=0 (z, z ∗ ) = 0.
(2.23)
The solution of equation (2.23) is a gaussian multiplied by an arbitrary an¯ (z) = 0,
alytic function f (z), with ∂f
2 /4l2
B
φn=0 (z, z ∗ ) = f (z)e−|z|
,
(2.24)
Similarly one finds for the state with m = 0
2
z ∗ + 4lB
∂ φm=0 (z, z ∗ ) = 0,
(2.25)
the solution of which is
2 /4l2
B
φm=0 (z, z ∗ ) = g(z ∗ )e−|z|
,
(2.26)
where the function g(z ∗ ) is anti-analytic, ∂g(z ∗ ) = 0. The state |n = 0, m =
0i must thus be represented by a function which is both analytic and antianalytic. The only function which satisfies both requirements is a constant.
With the normalisation, one gets
φn=0,m=0 (z, z ∗ ) = hz, z ∗ |n = 0, m = 0i = q
2
1
2
2πlB
2 /4l2
B
e−|z|
,
(2.27)
The sign we chose for the imaginary part is unusual, but is convenient for electrons.
For positively charged particles, we would chose the opposite sign, corresponding to the
opposite chirality.
Quantum treatment
27
A state corresponding to the quantum number m in the LLL is found from
equations (2.20) and (2.22),
√ m m
2
z
2
2
∗
¯
φn=0,m (z, z ) = q
− lB ∂
e−|z| /4lB
2
2πlB
m! 4lB
= q
and
1
2
m!
2πlB
∗
√
n
2
φn,m=0 (z, z ) = q
2
2πlB
n!
= q
2
2πlB
n!
1
z
√
2lB
!m
2 /4l2
B
e−|z|
z∗
− lB ∂
2
4lB
z∗
√
2lB
!n
!n
,
(2.28)
2 /4l2
B
e−|z|
2 /4l2
B
e−|z|
,
(2.29)
for a state centered at the origin m = 0 in LL n. An arbitrary state writes
√ m
!n
m
∗
2
z
z
2
2
∗
√
e−|z| /4lB
(2.30)
φn,m (z, z ) = q
− lB ∂¯
2
2lB
2πlB m!n! 4lB
which generates the associated Laguerre polynomials [21].
It is remarkable that, even if functions (2.28) and (2.29) have the same
probability density ,3
∗
2
∗
2
|φn=0,m=j (z, z )| = |φn=j,m=0 (z, z )| ∼
|z|2
2
!j
2
2
e−|z| /2lB
,
j!
√
with a probability maximum at radius r0 = 2jlB (Fig. 2.3), they do not
represent equal energy states.
To conclude the discussion of states |n = 0, mi represented in the symmetric gauge, we compute the average value of the guiding center operator.
With the help of equations (2.18), one finds that
hRi ≡ hn = 0, m|R|n = 0, mi = 0,
but
h|R|i =
3
D√
E
X 2 + Y 2 = lB
It is a Poissonian distribution.
q
√
2b† b + 1 = lB 2m + 1.
(2.31)
28
Charged particle in a magnetic field
0.4
n=1
n=3
n=5
(a)
0.35
|φn,m=0(z,z*)|2
0.3
0.25
0.2
0.15
0.1
0.05
0
0
1
2
(b)
3
r/lB=|z|/lB
n=0
4
5
n=1
2
y/l B
0
0
-2
-2
-4
-4
-4
-2
0
2
4
-4
-2
x/l B
0
2
4
x/l B
n=3
4
n=5
4
2
y/l B
6
4
2
y/l B
4
2
y/l B
0
-2
0
-2
-4
-4
-4
-2
0
x/l B
2
4
-4
-2
0
2
4
x/l B
Figure 2.3: Probability density for a state |n, m = 0i for various values
√ of n. (a) The
density depends only on the radius |z| = r and is maximum at r0 = 2jlB . (b) When
plotted on the plane, the wave function for n ≥ 1 have a ring shape.
Quantum treatment
(a)
29
p
p0
p
x0 x
x
(b)
y
y0
y
<x,y|x0 ,y0 >
<x,y|n=0,m=0>
x
x0 x
Figure 2.4: Coherent states .
This means that both√the particle and its guiding center are located on
the circle of radius lB 2m + 1, but the phase in undetermined. We may
use this to count states, as was done previously, for a disc geometry with
2
radius Rmax
√ and surface A = πRmax : as the state with maximum radius has
Rmax = lB 2M + 1, this yields the number of states in the thermodynamic
2
limit, M = A/2πlB
= AnB , with nB = eB/h, in agreement with the previous
argument about the state minimal surface. Similarly one sees that for the
state |n, m = 0i in level n the relative variable η is localized on a circle with
radius
√
RC ≡ h|η|i = lB 2n + 1
(2.32)
which is also called the cyclotron radius.
2.2.2
Coherent states and semi-classical motion
To retrieve the classical trajectory, (2.4), one must construct semi-classical
states, also called coherent states because they play an important role in
quantum optics. For a 1D harmonic oscillator, a coherent state is the eigenstate of the annihilation operator and it is the state with the minimum value
30
Charged particle in a magnetic field
of the product ∆px ∆x. Such as state can be built from the displacement operator in phase space, which displaces the ground state from hxi = 0, hpx i = 0
to (x0 , p0 ) [Fig. 2.4(a)]. The displacement is done with the operator
D(x0 , p0 ) = e−i(x0 p̂−p0 x̂) ,
(2.33)
where the symbols with hats are operators, not to be confused with variables
x0 and y0 . This operator displaces variables, as we can check using formula
(2.15)
D† (x0 , p0 )x̂D(x0 , p0 ) = eix0 p̂ x̂e−ix0 p̂ = x̂ + x0
and
D† (x0 , p0 )p̂D(x0 , p0 ) = e−ip0 x̂ p̂eip0 x̂ = p̂ + p0 .
The coherent state writes
|x0 , p0 i = D(x0 , p0 )|n = 0i,
(2.34)
where |n = 0i is the 1D harmonic oscillator ground state. Since [D(x0 , y0 ), H] 6=
0 the coherent state is not an eigenstate of the Hamiltonian. Indeed the state
changes with time, and that is how we retrieve the trajectory in phase space
[Fig. 2.4(a)]. x0 and p0 are not bona fide quantum numbers – this would
contradict the fundamental postulates of quantum mechanics, because the
associate operators do not commute. The basis |x0 , p0 i is said to be ”overcomplete” [22].
In general, a displacement operator may be constructed from two conjugate operators, which therefore do not commute. In the case of an electron
in a magnetic field in a 2D plane, we have two pairs of non commuting con2
2
jugate operators at our disposal, [X, Y ] = ilB
et [ηx , ηy ] = −ilB
. With the
first choice, the displacement operator which acts now in real space, writes
−
D(X0 , Y0 ) = e
i
l2
B
(X0 Ŷ −Y0 X̂)
,
(2.35)
and the coherent state (in the LLL) is
|X0 , Y0 ; n = 0i = D(X0 , Y0 )|0, 0i,
(2.36)
where |0, 0i ≡ |n = 0, m = 0i. Since the guiding center is a constant of motion, the displacement operator D(X0 , Y0 ) commutes with the Hamiltonian.
The state (2.36) remains an eigenstate of the Hamiltonian, which is why the
quantum number n is unchanged.
Quantum treatment
31
The dynamics enters with the second pair of operators, with the displacement operator
i
2
D̃(η0x , η0y ) = e lB
(η0x η̂y −η0y η̂x )
,
(2.37)
which generates a displacement to position η 0 = (η0x , η0y ), so that a general
semi-classical state may be written as
|X0 , Y0 ; η0x , η0y i = D̃(η0x , η0y )D(X0 , Y0 )|0, 0i.
(2.38)
The guiding center is thus centered at R0 = (X0 , Y0 ), and the electron turns
around that position on a circle of radius r = |η 0 |. One retrieves thus the
motion represented on figure 2.1, in terms of a gaussian wave packet. To prove
those dynamic properties, remember that a coherent state is an eigenstate of
the ladder operator a, and in our case also of b, with
η0x − iη0y
√
|X0 , Y0 ; η0x , η0y i,
2lB
X
+
iY0
0
√
|X0 , Y0 ; η0x , η0y i.
b |X0 , Y0 ; η0x , η0y i =
2lB
a |X0 , Y0 ; η0x , η0y i =
(2.39)
This can be checked, for example, when expressing the displacement operators in terms of ladder operators (2.17) and (2.18),
D(X0 , Y0 ) = eβb
† −β ∗ b
D̃(η0x , η0y ) = eαa
† −α∗ a
2 /2
= e−|β|
2 /2
= e−|α|
†
∗
eβb e−β b ,
†
∗
eαa e−α a ,
(2.40)
où l’on a défini
η0x − iη0y
X0 + iY0
,
α≡ √
β≡ √
2lB
2lB
where we used the Baker-Hausdorff formula
eA+B = eA eB e−[A,B]/2 ,
(2.41)
which is valid when [A, [A, B]] = [B, [A, B]] = 0. The coherent state writes
thus
2
2
†
†
|X0 , Y0 ; η0x , η0y i = e−(|α| +|β| )/2 eαa eβb |0, 0i
and we find with formula (2.15)
2 +|β|2 )/2
a |X0 , Y0 ; η0x , η0y i = e−(|α|
2 +|β|2 )/2
b |X0 , Y0 ; η0x , η0y i = e−(|α|
h
†
i
†
a, eαa eβb |0, 0i = α |X0 , Y0 ; η0x , η0y i,
†
h
†
i
eαa b, eβb |0, 0i = β |X0 , Y0 ; η0x , η0y i,
32
Charged particle in a magnetic field
which is nothing but equation (2.39).
In order to get the time evolution of the coherent state |α, βi = |X0 , Y0 ; η0x , η0y i,
one uses the time evolution operator on state
i
|α, βi(t) = e− h̄ Ht |α, βi(t = 0)
2 )/2
= e−(|α|
i
e− h̄ Ht
∞
X
(αa† )n
|n = 0, βi(t = 0)
n!
i
(α)n
2
√ |n, βi(t = 0)
= e−(|α| )/2 e− h̄ Ht
n!
n=0
n
∞
X
(αe−iωC t )
−(|α|2 )/2 −iωC t/2
√
= e
e
|n, βi(t = 0)
n!
n=0
n=0
∞
X
= e−iωC t/2 |α(t = 0)e−iωC t , βi,
(2.42)
which yields for the eigenvalue time evolution
α(t) = α(t = 0)e−iωC t ,
β(t) = β(t = 0).
(2.43)
√
√
√
x
2l
Re[β],
Y
=
2l
Im[β],
η
2lB Re[α(t)] et η0y (t) =
(t)
=
Since
X
=
B
0
B
0
0
√
− 2lB Im[α(t)], we retrieve
η0x (t) = η0x (t = 0) cos(ωC t),
η0y (t) = η0y (t = 0) sin(ωC t)
and thus the trajectory given in equation (2.4), identifying
r = |η 0 | and R = (X0 , Y0 ), as mentionned earlier.
Chapter 3
Transport properties– Integer
Quantum Hall Effect (IQHE)
This chapter deals with some aspects of the Integer Quantum Hall Effect
(IQHE) physics, using the quantum mechanics of an electron in a constant
uniform magnetic field, described in the previous chapter. Two main features
allow to understand the IQHE :
• each completely filled LL (for ν = n) contributes a conductance quantum e2 /h to the electronic conductivity,
• when additional electrons start populating the next LL at ν 6= n, they
get localized by the impurities disorder potential in the sample, and
they do not contribute to transport. In the absence of impurities, or,
more precisely, if translation invariance is not broken in the sample,
no plateau can be formed in the Hall resistance, and the classical Hall
result is preserved.
This last feature seems analogous at first sight to Anderson localization in
2D in the absence of a magnetic field [23]. It happens that localisation is
even more relevant in a magnetic field.
3.1
Resistance and resistivity in 2D
Theorists calculate resistivity. Experiments measure resistance. For a classical sytem with the shape of a hypercube of edge length L in d dimensions,
33
34
Transport properties– IQHE
the resistance R and the resistivity ρ are related by the well known equation
R = ρL2−d
(3.1)
Thus, in two dimensions, the sample resistance is scale invariant. The
product R(e2 /h) is dimensionless. It is an easy exercise to show that in the
case of a Hall bar geometry, such as shown in Fig. (1.1), the transverse resistance and the transverse resistivity are equal in 2D, independent of the Hall
bar dimensions. This is a basic ingredient to understand the universality of
the quantum Hall experimental results. In particular it means that one does
not have to measure the physical dimensions of a sample to one part in 1010 in
order to obtain the resistivity to that accuracy. The technological progress
in semiconductor physics which allowed to manufature 2D Electron Gases
(2DEG) with electrical contacts was, in this respect, a decisive one. Even
the shape of the sample, or the accurate determination of the Hall voltage
probe locations are almost completely irrelevant, in particular, because the
dissipation is nearly absent in the QH states.
3.2
Conductance of a completely filled Landau Level
We first discuss the effect of a constant uniform electric field on the Landau
level energy structure. We take the electric field along the y direction.
It is convenient to deal with a sample with rectangular shape, and to assume in a first step (to be relaxed subsequently) that the system is translation
invariant in the x direction.
An appropriate gauge in this rectangular geometry is the Landau gauge
AL = B(−y, 0, 0), so that the Hamiltonian now writes
p2y
(px − eBy)2
H=
+
− eV (y),
2m
2m
where the potential is V (y) = −Ey (electric field pointing in the ŷ direction). The system is translation invariant in the x̂ direction, so that px , or
equivalently h̄k, is a constant of motion, which corresponds to the quantum
number m in the symmetric gauge, i.e. to the guiding center eigenvalue. The
latter, in a state |n, ki, is delocalized along a straight line in the x direction,
contact
L
35
énergie
Conductance of a filled LL
contact
R
µL
µR
NL n
k min
k max y=kl B2
y’
Figure 3.1: LL in the Landau gauge, with a voltage bias between the L and the R
contacts, where chemical potentials are respectively µL and µR . Position yk in the direction
2
y is proportionnal to the wave vector k in direction x : yk = klB
.
2
with coordinate klB
on the y axis. The Hamiltonian is:
H=
p2y
1
+ mωC2 (y + klb2 )2 + eEy,
2m 2
which can be re-written, completing the square, as:
p2y
1
E
+ mωC2 (y − yk )2 + h̄k + C,
H=
2m 2
B
2
where we have set px = h̄k, and yk = −klB
− eE/mωC2 and C is a constant:
2
E
C = − 21 m B
. The energy of the state |n, ki is
εn,k = h̄ωC n +
1
1
+ eEyk + mv̄ 2
2
2
(3.2)
E
~ ∧ B/B
~ 2 and is parallel to the x axis.
where v̄ ≡ − B
is the drift velocity E
This can be derived by deriving explicitly the current:
−e
J~ =
hn, k|(p + eA)|n, ki .
m
D E
Thus there is a net current hJx i along the x axis.
36
Transport properties– IQHE
We conclude that the energy levels follow the electric potential, which
adds to the energy in zero electric field.
Let us now go one step further by considering a slowly varying electric
potential V (y), which we still assume to be translation invariant along x.
We can linearize this potential locally, and repeat the previous analysis: the
energy eigenvalues will not be linear in k any more, but they will roughly
reflect the sum of the LL energy plus the local potential energy. To discuss
electrons in a Hall bar, we take into account the sample edges in the y
direction, which create a confinement potential. The latter results in an
upward bending of Landau levels in the vicinity of the edges, where contacts
allow to measure voltage biases as in figure 3.1.
This justifies the sketch of the LL in figure 3.1, where the LL energy
profile follows the confining potential at the sample edges. The eigenvalues
ǫk are not linear in k, but can be linearized locally: it will still reflect the
kinetic energy, with the local potential energy added to the LL energy.
In order to compute the level contribution to the conductance (along x),
we use formula
eX
In = −
hn, k|vx |n, ki,
(3.3)
L k
where L is the system length along x, and the velocity average value is derived
from the energy dispersion relation
1 ∂εn,k
1 ∆εn,k
≃
.
h̄ ∂k
h̄ ∆k
In the last line, we assume that ∆k = 2π/L is very small, which is certainly
valid if L is very large. Using this, we have
L
L
∆εn,k =
(εn,k+1 − εn,k ).
2πh̄
2πh̄
Thus vk has opposite signs on the two edges of the sample. This means
that in the Hall bar geometry, there are edge currents flowing in opposite
directions. This is not surprising, if we remember the semi-classical picture
of skipping orbits along an edge.
When we sum over vk in equation (3.3, the result depends only on the
edge energies, at kmin and kmax , the edge coordinates (see figure ( 3.1)).
Thus, provided the electric potential has a slow enough variation in direction
y, we may sum over k to get
e
In = − (εn,kmax − εn,kmin ) .
h
vk =
3.3. LOCALISATION IN A STRONG MAGNETIC FIELD
37
The energies at kmin and kmax are given by the chemical potential at the contact points, εn,kmin = µL and εn,kmax = µR . Since the difference in chemical
potentials is controlled by a voltage bias ∆µ = (µR − µL ) = −eV , we see
that the LL conductance is e2 /h, since
In =
e2
V.
h
(3.4)
When n LL are completely filled, we get a conductance
G=n
e2
.
h
Since this is a transverse conductance (the current is in direction x, is zero
along y, and the difference in chemical potentials is in direction y), the resistance tensor we get is
R̂ = Ĝ
−1
=
0 −RH
RH
0
!
,
(3.5)
with the Hall resistance RH = h/e2 n. It is important to realize that this
result, although satisfactory –the Hall resistance only depends on universal
constants e and h, and an integer n–is not sufficient to explain the occurrence of plateaux. In fact, it is fairly easy to show that the result we have
coincide exactly with the classical Hall value at discrete points in the RH
curve, corresponding to n filled Landau levels; it is enough to remember that
ν = hnel /eB = n and to use equation (3.5)to recover the classical value
RH = B/enel . In order for quantized Hall plateaux to be formed, additionnal electrons or holes injected in the system around a density such that n
LL are completely filled must be localized, so as to have no contribution to
transport properties. This localization phenomenon is described in the next
section.
3.3
Localisation in a strong magnetic field
The electric potential Vext (r = R + η) due to impurities is described as
a slowly varying function in the xy plane, so that Landau quantization is
preserved. We now do not assume any more that the impurity potential preserves translation invariance along the x̂ direction. The potential landscape
38
Localisation in a strong magnetique field
Figure 3.2: Semi-classical motion of an electron in a magnetic field in the presence
of an impurity potential. The guiding center follows the landscape equipotentials. The
Hall drift of the guiding center, shown by the arrow is a slow motion compared to the
fast electronic cyclotron motion. Electronic transport is possible when an equipotential
connects the sample edges. If an electronic state is localized within a potential well, it
does not contribute to transport.
has hills and valleys and fluctuates in space around an average value which
is taken to be zero, with no loss of generality. This potential lifts the LL
degeneracy, because the guiding center is not a constant of motion any more.
We see this with the Heisenberg equations of motion
∂Vext
,
∂X
(3.6)
where we used formula (2.15). We see that the guiding center follows the
equipotential lines of the impurity potential (Fig. 3.2). In the case we discussed in the previous section, this led to a Hall current in the direction
orthogonal to the electric field. Equation (3.6) is a generalization of this
result. The guiding center motion is perpendicular both to the external field
and to the local electric field. Quantum states of the LL are thus localized
on equipotential lines corresponding to their energies. The wave functions,
(in the shape of rings in zero potential as in Fig. 2.3) are deformed to tune
to their equipotential lines.
Similarly, we get for the ηx et ηy Heisenberg equations of motion
2
ih̄Ẋ = [X, H] = [X, Vext (X, Y )] = ilB
ih̄η̇x
⇔
η˙x
∂Vext
∂Y
and
2
ih̄Ẏ = −ilB
1
= ηx , mωC (ηx2 + ηy2 ) + V (r + η)
2
l2 ∂V
= −ωC ηy − B
h̄ ∂ηy
Conductance of a filled LL
et
39
η˙y = ωC ηx +
2
lB
∂V
.
h̄ ∂ηx
This provides us with a stability criterion for Landau levels in the presence
of a disorder impurity potenial, since the first terms on the right in the equations above must remain large compare with the terms due to the impurity
potential. The condition reads:
*
∂V
∂η
+
≪
h̄ωC
.
lB
(3.7)
This condition is satisfied provided the variation of the potential over a cyclotron radius is small compared to h̄ωC .
We now have the main ingredients to understand some of the IQHE basic
features. The reasoning below is represented in a schematic fashion on figure
3.3. We have seen in the previous section that the Hall resistance for integer
ν = n is exactly RH = h/e2 n, while the longitudinal resistance is zero. If the
magnetic field intensity is slightly decreased, keeping the electronic density
constant, since the number of states per LL decreases, some electrons have to
promoted to the LL with n + 1. They occupy preferentially the lowest energy
states available, the bottom of basins in the impurity potential landscape.
This is a peculiar form of localisation which is induced by the magnetic field.
Localized electrons do not contribute to the transport. Both the transverse
and longitudinal resistances stay locked at their value for the completely
filled level case with ν = n. The fact that the longitudinal resistance is zero
shows that transport is ballistic. Indeed, we have seen that contributions
to the current from the bulk of the sample compensate, so that transport
is due to n edge channels (edge states), one per completely filled LL. The
current direction on the edge is determined by the potential gradient, which
rises near the sample edge. Edge currents are thus chiral, forward scattering
at one edge is dissipation less, and dissipation can only occur if an electron
circulating along one edge can be scattered backwards by tunneling to the
other edge. This can occur only when edge states trajectories of opposites
chiralities happen to be close to each other. When electronic puddles grow
because more electrons get promoted to level n + 1, they eventually merge
into one another, until equipotentials connect the two edges and eventually
an electronic sea extends over the whole sample. When edges get connected
by equipotentials, dissipation occurs, the longitudinal resistance is finite,
and the transverse resistance varies rapidly as the magnetic field continues
40
Localisation in a strong magnetique field
(a)
ε
(b)
ε
(c)
EF
n
EF
n
ε
EF
états localisés
n
états étendus
densité d’états
densité d’états
densité d’états
NL
(n+1)
Rxx R xy
Rxx R xy
Rxx R xy
h/e2 n
h/e2 n
h/e2 (n+1)
ν =n
B
B
B
Figure 3.3: Quantum Hall effect. In the upper parts of the figure, LL are broadened by
the impurity potential. Their filling is controlled by the Fermi level(EF ). In the middle
part, samples are seen from above, showing equipotential lines, and the gradual filling of
the n-th level (from left to right). The lowest part of the figure is a sketch of the resistance
curves, as the LL filling factor varies. This figure is to be read column by column, the filling
factor increasing from the first column to the last one. In the first column (a), we have a
situation with completely filled LL, ν = n, where the Fermi level sits exactly between LL
n and n + 1, the upper level being empty. The Hall resistance is then exactly RH = h/ne2 ,
and the longitudinal resistance is exponentially small (zero at zero temperature). The
second column describes a situation where the LL n + 1 has a low filling factor. Electrons
occupy potential wells in the sample and do not contribute to electronic transport. This
situation occurs when, at fixed electron density, the magnetic field intensity is slightly
decreased from its value for the complete filling of the n-th LL. The resistance values are
locked at their value for ν = n. In the last column on the right, the n + 1 LL is half
filled: equipotential lines connect the two edges, so that dissipation is allowed through
back scattering processes from one edge to the other. The system changes from ballistic
regime to a diffusive one, and the Hall resistance varies rapidly towards the next plateau,
while the longitudinal resistance reaches it peak value, before decreasing to exponentially
small values. Localized states are on the left and on the right of the extended states in
the center of the broadened LL. When the highest occupied LL has a filling larger than
1/2, the same reasoning applies in terms of holes.
Conductance of a filled LL
41
R L ~ µ 3− µ 2= 0
µ2 = µL 2
3
µ3 = µL
I
I
4
1
6
µ6 = µ5 = µR 5
R ~ µ5− µ3= µR− µL
H
Figure 3.4: IQHE measurements at ν = n. The current I is injected through contact
1, and extracted at contact 4. Between those two contacts, the chemical potential µL is
constant since (a) there is no backscattering and (b) there are no electrons injected or
extracted at contacts 2 and 3 which are used to measure the voltage drop. The chemical
potential µR stays also constant along the lower edge between contacts 6 and 5. The
longitudinal voltage drop thus vanishes, so that the longitudinal resistance is zero, RL =
(µ3 − µ2 )/I = 0. The Hall resistance is determined by the voltage bias between the two
edges µ5 − µ3 = µR − µL .
to decrease until a new conducting channel is formed all along the edges.This
is reached at half filling of level n + 1. Above that filling ratio, the evolution
described so far is reproduced in terms of holes.
To understand the IQHE in terms of edge currents, consider the experimental set up with six contacts, as shown on figure (3.4).
Electrons are injected through electrode 1 and are extracted through electrode 4. The other contacts 2, 3, 5 and 6 are used for voltage bias measurements, with no electron injected or extracted. Because back-scattering is
suppressed when the filling factor is around ν = n, the chemical potentials
µR and µL are constant along each edge. The chemical potential varies only
along input and output electrode 1 and 4. The longitudinal resistance is
measured for instance between contacts 2 and 3, and is found to vanish:
RL = −(µ3 − µ2 )/eI = 0. The Hall resistance is determined by the voltage bias between contacts 3 and 5, RH = −(µ5 − µ3 )/eI = −(µR − µL )/eI.
This situation is precisely that which was described in the previous section,
42
Transitions between plateaus – percolation
where we computed the resistance for n completely filled LL. We thus find
RH = h/ne2 .
3.4
Transitions between plateaus – The percolation picture
The previous section describes a scenario of transitions between Hall plateaus
which reminds us closely of a percolation mechanism: the resistance jumps
from one plateau to the next one when the electron puddles become macroscopic ones and percolate so as to form an infinite electronic sea which extends
to both edges. Percolation transitions are second order transitions, which exhibit critical phenomena, and specific scaling laws for the relevant physical
quantities around the critical point. Those quantities do not depend on microscopic details of the system, they are characterized by critical exponents
which define a universality class.
The transition is controlled by a ”control parameter” K, which could be
the temperature, or, in the case of quantum phase transitions, at zero temperature, by another parameter such as pressure or electronic density [18]. In
our case, the control parameter for transitions between plateaus is the magnetic field intensity. At the critical field Bc , the correlation length diverges,
with a critical exponent ν (not to be confused with the filling parameter)
ξ ∼ |δ|−ν ,
(3.8)
where δ ≡ (B − Bc )/Bc . Dynamic fluctuations may be similarly described
by a correlation ”time”
ξτ ∼ ξ z ∼ |δ|−zν ,
(3.9)
where z is called the critical dynamic exponent. In a path integral formulation, the characteristic time τ is connected to the temperature T through
h̄/τ = kB T . A finite temperature may be considered as a finite size in the
time direction. [25, 24].
At the critical point, physical quantities follow scaling laws which depend
on ratios of dimensionless quantities. One finds for the longitudinal and Hall
resistivity
ρL/H = fL/H
h̄ω τ
,
kB T ξτ
!
Conductance of a filled LL
43
100.0
dxy
max
dB
dxy
N=0
N=1
N=1
max
10.0
(∆B)
−1
dB
−1
(∆B)
N=1
N=1
1.0
0.10
1.00
T(K)
Figure 3.5: Experiments by Wei et al. [26]. The transition width δB and that of
the Hall resistivity derivative ∂ρxy /∂B, measured as a funcition of temperature exhibit a
scaling law with exponent 1/zν = 0, 42 ± 0, 04, for transitions between filling factors 1 → 2
(N = 0 ↓), 2 → 3 (N = 1 ↑) and 3 → 4 (N = 1 ↓).
44
Transitions between plateaus – percolation
= fL/H
h̄ω δ zν
,
kB T T
!
,
(3.10)
where ω is a characteristic measurement frequency, for example in an ac measurement, and fL/H (x) are universal functions. In the following, we deal only
with dc measurements, and ω = 0 properties. For a second order transition,
one expects the characteristic width ∆B of the transition as a function of
the magnetic field intensity, to vary with temperature as
∆B ∼ T 1/zν .
(3.11)
This scaling law was actually found in the measurements by Wei et al. [26],
who found an exponent 1/zν = 0, 42 ± 0, 04 over a temperature interval
varying with more than one order of magnitude between 0, 1 and 1, 3K (figure
3.5).
The two exponents ν and z may be separetly determined if one takes into
account the scaling laws for current fluctuations under applied electric field.
One finds
h̄
h̄
eEℓE ∼
∼ z,
τE
ℓE
where τE ∼ ℓzE is the characteristic fluctuation time, which is connected
to a characteristic length ℓE through equation (3.9). One finds thus ℓE ∼
E −1/(1+z) , and, for the zero frequency resistivity scaling law
ρL/H = gL/H
δ
δ
!
,
,
T 1/zν E 1/ν(1+z)
(3.12)
in terms of universal functions gL,H (x). Other measurements by Wei et al.,
dealing with the current scaling laws, find that z ≃ 1, which leads to ν ≃
2, 3 ≃ 7/3 [27].
The critical exponent for classical 2D percolation is νp = 4/3, smaller
than the experimental value, close to 7/3. The disagreement is probably
due to quantum tunneling effects between trajectories: such processes allow
for back-scattering before classical trajectories actually touch each other.
Chalker and Coddington take into account quantum tunneling in a transfer
matrix approach and find a critical exponentν = 2, 5 ± 0, 5 [29]. Numerical
simulations have reproduced the ν = 7/3 exponent [28].
Chapter 4
The Fractional Quantum Hall
Effect (FQHE)– From
Laughlin’s theory to Composite
Fermions.
In the previous chapter, we have seen that the IQHE with ν = n is understood
on the basis of two main ingredients:
(i) Because of LL quantization, there is an excitation gap between the
ground state with a number of completely filled levels and the next empty
level (ii) Elementary excitations, obtained by promoting an electron to the
next LL are localized and do not contribute to electronic transport. Filled
Landau levels only contribute each a conductance quantum e2 /h.
As emphasized in the Introduction, the observation of the FQHE, first
for a fractional filling of the LLL, with ν = p/(2sp + 1), with integer n and
p, was a sign of the complete breakdown of perturbation theory, such as
diagrammatic analysis based on the knowledge of an unperturbed ground
state. For the fractionally filled LLL, the non interacting ground state has
a huge degeneracy, which prohibits using theoretical techniques used so far
to take electron-electron interactions into account. We know that certain
superpositions of ground state configurations must minimize the Coulomb
interactions. We know from experiments–the observed activated behaviour
of the longitudinal resistance– that there is a gap between the actual ground
state and the first excited states.
The approach described in the previous chapter allows us to deal with
45
46
FQHE – from Laughlin to Composite Fermions
the FQHE, once we have a mechanism which allows to lift the ground state
degeneracy, and to have a gap to the first excited states, be they single
quasi-particles or collective excitations. In that case, we may reproduce the
piece of reasoning of the previous chapter: excited charged quasi-particles are
localized, so that varying the magnetic field intensity around a given exact
fraction of the filling factor leads to plateau formation in the Hall resistivity,
for the same reason as in the IQHE. The difficult part is to identify the non
degenerate ground states, and to characterize their properties, the nature
of excited states, etc.. Before actually introducing the Laughlin and Jain
trial wave functions which solve this problem, we discuss in the next section
the structure of the effective model which describes the electron dynamics
when we make the approximation that it is restricted to the states of a single
partially filled LL. This model will be the basis for the Hamiltonian theory
of the FQHE, which will be developped in the following chapters.
4.1
Model for electron dynamics restricted to
a single LL
Since we are interested at first in describing low temperature properties, only
the lowest excitation energies are of interest. Because we are dealing with
a partially filled LL,(ν 6= n), the relevant excitations are restricted to intralevel dynamics. Furthermore, we consider at first that the spins are fully
polarized, and we do not consider spin flip excitations here. Excitations involving intra-LL transitions are forbidden by the Pauli principle when the
level is completely filled (Fig. 4.1). They are allowed for a partially filled LL.
In that case, the kinetic energy plays no rôle, since all single electron states
are degenerate. We omit this constant in the following. Virtual inter-level
excitations may be considered in a perturbative approach, and give rise to
a modified dielectric function ǫ(q), which alters the interaction potential between electrons within the same level [30]. In contrast with screening effects
in metals, which suppress the long distance part of the Coulomb interaction,
electronic interactions screening in the presence of a magnetic field alters the
potential only for finite wave vectors: for q → 0 and q → ∞, it vanishes, and
ǫ(q) → ǫ, where ǫ is the dielectric constant of the underlying semiconductor.
Since the electron spin is not flipped during such processes within a spin
branch, it plays no rôle. Therefore, we deal in the following with spinless
electrons restricted to a single LL
(a)
ν= N
47
(b)
ν= N
∆Z
h ωC
Figure 4.1: Lowest excitations energies. Each LL is separated in two spin branches
because of the Zeeman effect. (a) For filling ν = n, excitations which couple states within
the same LL are forbidden because of the Pauli principle. Only inter level excitations are
allowed. (b) For fractional filling of the highest occupied LL, (ν 6= n) excitations within
the same LL are allowed and provide the lowest excitation energies. Inter LL excitations,
with energy h̄ωC or ∆z are neglected in the model.
electrons. A more detailed discussion of spin phenomena in the quantum
Hall physics–such as the Quantum Hall ferromagnetism– will be given in a
later part of these lectures.
In second quantized notation, the Hamiltonian restricted to intra LL excitations is
1Z 2 2 ′ †
d r d r ψn (r)ψn (r)V (r − r′ )ψn† (r′ )ψn (r′ ).
Ĥ =
2
(4.1)
It involves states in the n-th level only, ψn (r) = m hr|n, mien,m et ψn† (r) =
P
†
†
m hn, m|rien,m . Operators en,m and en,m are respectively the annihilation
and creation operators for an electron in the state |n, mi. They obey the
fermionic anti-commutation rules,
P
n
o
en,m , e†n′ ,m′ = δn,n′ δm,m′ ,
{en,m , en′ ,m′ } = 0.
(4.2)
Note that the restricted electron fields ψn (r) are not completely localised.
Because the sum over states is restricted to m, we have, with rules (4.2)
n
o
ψn (r), ψn† (r′ ) =
X
m
′ 2 /2l2
B
hr|n, mihn, m|r′ i ∝ e−|r−r |
6= δ(r − r′ ).
(4.3)
The field ψn† (r) creates an electron in the vicinity (4.2) of position r, which
is hardly surprising: this is just another manifestation of the position uncertainty when we restrict the dynamics to a single LL. To have a perfect field
48
FQHE – from Laughlin to Composite Fermions
localisation, one would have to sum over n, i.e. to superpose a number of
LL.
In reciprocal space, the Hamiltonian writes
Ĥ =
1 X
v(q)ρn (−q)ρn (q),
2A q
(4.4)
with the measure q = A d2 q/(2π)2 and the Coulomb interaction potential
v(q) = 2πe2 /ǫq. The operators ρn (q) are the Fourier components of the
electronic density operator in the n-th level ρn (r) = ψn† (r)ψn (r), and one has
R
P
ρn (q) =
=
Z
d2 r
X
m,m′
X
m,m′
hn, m|rie−iq·r hr|n, m′ ie†n,m en,m′
hn, m|e−iq·r |n, m′ ie†n,m en,m′
= hn|e−iq·η |ni
= Fn (q)ρ̄(q),
X
m,m′
hm|e−iq·R |m′ ie†n,m en,m′
(4.5)
where Fn (q) ≡ hn| exp(−iq · η)|ni is the form factor, and we took advantage
of the decomposition r = R + η, which allows to factorize matrix elements
hn, m|e−iq·r |n′ , m′ i = hn| exp(−iq · η)|n′ i ⊗ hm| exp(−iq · R)|m′ i.
(4.6)
In the last line of equation (4.5), we have defined the projected density operator,
X
ρ̄(q) ≡
hm|e−iq·R |m′ ie†n,m en,m′ .
(4.7)
m,m′
4.1.1
Matrix elements
To proceed in practice to actual computations, one needs to compute the
matrix elements which enter expression (4.5) for the density operator. The
simplest way is to use expressions (2.17) and (2.18) for the operators R and
η, in terms of a, a† , b and b† . From now on, we take lB ≡ 1 for simplicity.
Using complex notation, with q = qx − iqy and q ∗ = qx + iqy , we have
1 q · η = √ qa + q ∗ a† ,
2
1 q · R = √ q ∗ b + qb† ,
2
electrons restricted to a single LL
49
so that we get for the first matrix element, with n ≥ n′ , using the BakerHausdorff formula(2.41),
− √i (q ∗ a† +qa)
hn|e−iq·η |n′ i = hn|e
2
2 /4
= e−|q|
2 /4
= e−|q|
|n′ i
− √i q ∗ a† − √i qa
hn|e
X
j
e
2
− √i q ∗ a†
hn|e
2
−|q|2 /4
s
n′ !
n!
−iq ∗
√
2
−|q|2 /4
s
n′ !
n!
−iq ∗
√
2
= e
= e
2
|n′ i
− √i qa
|jihj|e
2
|n′ i
n−n′ X
n′
n!
|q|2
−
(n − j)!(n′ − j)!j!
2
j=0
n−n′
n−n′
n′
L
|q|2
2
!
,
!n′ −j
(4.8)
where we have used
− √i q ∗ a†
hn|e
2


0
|ji =  q n!
pour j > n
1
j! (n−j)!
− √i2 q ∗
n−j
pour j ≤ n
in the third line and the definition of Laguerre polynomials [21],
′
Ln−n
(x)
n′
′
=
n
X
(−x)m
n!
.
′
′
m!
m=0 (n − m)!(n − n + m)!
Similarly we find for m ≥ m′
− √i (qb† +q ∗ b)
hm|e−iq·R |m′ i = hm|e
−|q|2 /4
= e
2
s
m′ !
m!
|m′ i
−iq
√
2
!m−m′
m−m′
Lm
′
|q|2
.
2
!
(4.9)
Defining functions
Gn,n′ (q) ≡
s
n′ !
n!
−iq
√
2
!n−n′
′
Ln−n
n′
|q|2
,
2
!
one may also write without the conditions n ≥ n′ et m ≥ m′ ,
2 /4
hn|e−iq·η |n′ i = [Θ(n − n′ )Gn,n′ (q ∗ ) + Θ(n′ − n − 1)Gn′ ,n (−q)] e−|q|
(4.10)
50
FQHE – from Laughlin to Composite Fermions
and
2
hm|e−iq·R |m′ i = [Θ(m − m′ )Gm,m′ (q) + Θ(m′ − m − 1)Gm′ ,m (−q ∗ )] e−|q| /4 .
(4.11)
For the case n = n′ , we find in equation (4.10) the n-th LL form factor:
−iq·η
Fn (|q|) ≡ hn|e
4.1.2
|q|2 −|q|2 /4
e
.
2
!
|ni = Ln
(4.12)
Projected densities algebra
At first sight, the model defined by (4.4) looks simple. The Hamiltonian
is quadratic in density operators. Such models often have exact solutions.
It happens that the projection in a single LL generates a non commutative
algebra for operators with different wave vectors, which leads to non trivial
quantum dynamics.
Let us compute the commutator [ρ̄(q), ρ̄(k)]. For a one particle operator
P
†
A
A
in second quantized notation, F A (q) = λ,λ′ fλ,λ
′ (q)eλ eλ′ , where fλ,λ′ (q) =
A
′
hλ|f (q)|λ i, the commutation rules in second quantized form follow from
those in first quantization:
h
i
F A (q), F B (q′ ) =
Xh
i
f A (q), f B (q′ )
λ,λ′
λ,λ′
e†λ eλ′ .
(4.13)
The λ index may comprise a number of different quantum indices. This
equation follows from the repeated application of
[AB, C] = A[B, C]± − [C, A]± B
(4.14)
on electronic operators. Equation (4.14) is valid for commutators as well as
anti-commutators.
Using equation (2.16), one finds
[q · R, q′ · R] = qx qy′ [X, Y ] + qy qx′ [Y, X]
= i(qx qy′ − qy qx′ ) = −i(q ∧ q′ ),
where we have defined q ∧ q′ ≡ −(q × q′ )z , and one gets, with the help of
the Baker-Hausdorff formula (2.41)
h
′
e−iq·R , e−iq ·R
i
′
i
′
i
= e−i(q+q )·R e 2 q∧q − e− 2 q∧q
q ∧ q′ −i(q+q′ )·R
= 2i sin
e
.
2
!
′
(4.15)
4.2. THE LAUGHLIN WAVE FUNCTION
51
This yields, with equation (4.13),
!
q∧k
[ρ̄(q), ρ̄(k)] = 2i sin
ρ̄(q + k),
2
(4.16)
for the algebra of projected density operators. This is isomorphous to the
magnetic translation algebra . Indeed, operators which describe electronic
displacements in the presence of a magnetic field have the same commutation
rules. This algebra is closed, and does not depend on the LL n index.
With algebra (4.16), the model is completely defined by the Hamiltonian
(4.4), which writes, in terms of projected density operators
Ĥ =
1 X
vn (q)ρ̄(−q)ρ̄(q),
2A q
(4.17)
where the form factor has been absorbed in the effective interaction potential
in the n-th LL ,
2πe2
2πe2
|q|2
vn (q) =
[Fn (q)]2 =
Ln
ǫ|q|
ǫ|q|
2
"
!#2
2 /2
e−|q|
.
(4.18)
The model has the same structure for all LL. The information about the
level is encoded in the effective potential, which will be discused in the last
section of this chapter. The LLL physics, which will be the main topic in
the remaining parts of this chapter (except the last section), is thus easily
generalized to a LL with higher index: one simply has to take into account
the relevant effective potential, and to replace the filling factor ν by the
partial filling factor of the n-th level, ν̄ = ν − n.
4.2
The Laughlin wave function
In this section, we discuss the arguments used by Laughlin in 1983 to derive the almost exact ground state for the fractionally filled LLL (Lowest
Landau Level), to prove that there is a gap between the ground state and
all excited states, and that there exist factionally charged excitations around
the fractional filling corresponding to the plateaus observed by Tsui, Störmer
and Gossard. Then we will describe Jain’s generalization of Laughlin’s wave
functions.
52
FQHE – From Laughlin to Composite Fermions
It is a good training to examine first the many-body wave function for the
completely filled LLL. In that case there is a gap to excited state which is,
at first sight, a single particle effect, the Zeeman splitting g ∗ µb B (see figure
4.1).1
4.2.1
The many-body wave function for ν = 1
Laughlin exploited a useful property of the single particle Landau Hamiltonian eigenfunctions in the symmetric gauge (see equation 2.28) :
∗
φn=0,m (z, z ) ∝
z
√
2lB
!m
2 /4l2
B
e−|z|
,
so that any analytic function f1 (z)(defined by ∂∂z∗ f1 (z, z∗) = 0 ) in the prefactor of the gaussian belongs to the LLL. All physical results are of course
independent of this gauge choice.
Turning now to the many-body wave function for the full LLL (i.e. ν = 1),
this means that the most general wave function we are looking for has to be
of the form
X |zi |2
ψν=1 ({zi }) ∝ fN ({zi }) exp
− 2
(4.19)
4lB
j
where {zi } means (z1 , z2 , ...., zN ), and fN is analytic in all variables. N is
the total number of electrons, and is equal, since ν = 1 to the total number
of states in the LLL. Since we are dealing with a state where all electron
spins are identical, the spin wave function is symmetrical under exchange of
particles. Since we are dealing with a fermion wave function, the prefactor fN
of the orbital part must be totally antisymmetric under exchange of particles.
It can only be a single Slater determinant with all LLL single particle states
occupied.
This determinant reads:



fN = det 

1
z10 z11
z20 z21
... ...
0
1
zN
zN
... z1N −1
... z2N −1
... ...
N −1
... zN





(4.20)
We will show later on that in fact the gap above the ν = 1 ground state is dominated
by exchange effects, and is much larger than the Zeeman gap.
The Laughlin wave function
53
This determinant, called a Vandermonde determinant, is a polynomial in N
variables, with N zeros. It has a simple expansion as
fN ({zi }) = Πi<j (zi − zj )
(4.21)
Since the highest power of any particle space coordinate zi is N-1, and this
corresponds to a guiding center eigenvalue mmax = N − 1, fN corresponds indeed to a fully occupied LLL, with all states occupied once (as the expression
of equation 4.20 shows).
A striking remark is that this state being the only LLL eigenstate with
P
ν = 1(with fixed center on mass i zi ), it is an eigenstate of the N particle
Hamiltonian for any interaction potential.
In order to analyze properties of ψν=1 , Laughlin resorted to a very original
detour: the so called ”plasma analogy”. The latter amounts to regard the
probability distribution function of particles in the LLL,(putting lB = 1),
2
|Ψν=1 |2 ∝ ΠN
i<j |zi − zj | exp −(1/2)
N
X
l
|zl |2
(4.22)
as the Boltzmann weight of a classical statistical mechanics problem, the
partition function Z of which is given by the norm of the wave function. In
other words,
Z
Z = Πi d2 zi |ψν=1 ({zi })|2
and
|ψν=1 |2 = exp −βUclass .
(4.23)
Since this is a formal analogy, the inverse temperature β which appears here
is arbitrary. For reasons which will appear later, we choose here β = 2/q,
where q will be non trivial later on, but is equal here to 1, so that eventually
Uclass ≡ q 2
X
i<j
(− ln |zi − zj |) +
qX 2
|zl |
4 l
(4.24)
Laughlin remarked that Uclass is the internal energy of a 2D classical one
component gas of interacting particles with charge q in a uniform neutralizing background. Remember that this is an analogy, which has the advantage
of representing an unknown problem, i.e. the probability distribution function of the real integer quantum Hall problem in terms of a different, known,
problem, that of the classical statistical properties of a gas of charged 2D
54
FQHE – From Laughlin to Composite Fermions
particles. In the equivalent classical problem, particles have logarithmic interactions, which are 2D Coulomb interactions, while the real problem has
the same formulation, as we saw above, for any interaction potential.2
To see that the interactions between the classical particles of the equivalent classical problem are 2D Coulomb interactions, remember that in 2D
the flux of the electric field through a circle of radius R (the sphere S1 of the
2D space) is related to the enclosed charge Q by
Z
dx.E = 2πQ
. For a point charge q at the origin, E(r) = qr/r2 so that the electric potential
is Vc = −q ln r/a,3 and Poisson equation in 2D reads:
divE = −∇2 Vc (r) = 2πqδ 2 (r).
(4.25)
Thus the first term on the right of equation 4.24 is interpreted as the
Coulomb interaction energy among N 2D charge q particles. Because the
2
LLL is filled, N = B/φ0 = 1/(2πlB
) = 1/(2π) particles per unit surface
(remember that here lB = 1).
The second term on the right of equation 4.24 represents the potential energy of N particles of charge q interacting with a uniform charged background
2
with charge density ρB = −1/(2πlB
). Indeed
−∇
|z|2
2
= −1/(lB
) = 2πρB
2
4lB
(4.26)
In other words, the uniform background has a charge density which is precisely equal to the density of flux quanta threading the surface. We know
from electrostatics that charge neutrality is the condition for thermodynamic
equilibrium, which corresponds to the most probable states in the partition
function. The overall charge neutrality condition is
nq + ρB = 0
2
(4.27)
The actual interaction between particles in the real problem is in fact a 3D Coulomb
interaction ∝ 1/r, because the electrons in the 2D potential well are immersed in a 3D
space. But the analysis of the filled LLL wave function is entirely independent of any
interaction potential form.
3
a is an arbitrary integration constant,which only changes Vc by a constant, and which
we may later take as a = lB .
The Laughlin wave function
55
Which is satisfied in the ν = 1 LLL ground state, since q = 1. The plasma
analogy tells us more than the overall neutrality condition: it tells us that the
largest values of the probability distribution function |ψν=1 ({zi })|2 is when
charge neutrality is realized locally, otherwise huge costs in Coulomb energy
reduce drastically the contribution of local density fluctuations to Z.
The conclusion for the filled LLL is that it is a strongly correlated liquid,
with random particle positions, but negligible fluctuations on length scales
greater than lB . This statement holds for any interaction potential, and is
true for non interacting particles, because the ν = 1 Vandermonde determinant is the only ground state wave function.
4.2.2
The many-body function for ν = 1/(2s + 1)
Before the discovery of the FQHE in 1983, the ground state of electrons in
the partially filled LLL had been predicted to be a Wigner crystal: electrons
would organise in a triangular cristalline array to reduce their Coulomb interactions. There is indeed some experimental evidence that such is the situation
at low enough filling of the LLL. It is clear however that the Wigner Crystal
cannot produce a FQHE, i. e. a state with a gap above the ground state.
The reason is that a crystal is a state with continuous broken translation and
rotation invariance, so that it has a Goldstone mode, i.e. a collective excitation the energy of which goes continuously to zero with the wave vector. Such
a state does not have a gap above the ground state, in contradiction with the
FQHE phenomenology. Moreover, the Wigner crystal scenario would have
no particular way of selecting “magic” fractional values of the filling factor
observed to correspond to FQHE plateaus.
The explanation of the FQHE at ν = 1/(2s + 1) was proposed by Laughlin the very year it was discovered [12]. He looked for a trial many body
wave function which would respect the constraints and the symmetries of
the problem. Here we sketch the essential steps in the construction of the
Laughlin wave function, starting with the wave function for two particles.
ψ (2) (z, z ′ ).
• The analyticity condition (2.24), for the symmetric gauge imposes that
P
2
ψ (2) (z, z ′ ) = m,M αm,M (z+z ′ )M (z−z ′ )m exp[−(|z|2 +|z ′ |2 )/4lB
], where
m and M are integers.
• Electrons are fermions, with spin polarised electrons, so, as for the
56
FQHE – From Laughlin to Composite Fermions
ν = 1 case, the orbital part of the wave function must be antisymmetric
with respect to permutation of the particles. This limits the choice
to odd m integers. The general two particle wave functions is thus
restricted to be a superposition of functions ψ (2) (z, z ′ ) ∝ (z + z ′ )M (z −
2
z ′ )2s+1 exp[−(|z|2 + |z ′ |2 )/4lB
].
• If we take into account the two body problem for electrons with center
of mass angular momentum M and relative angular momentum m the
following wave function
(2)
2
]
ψM m (z, z ′ ) = (z + z ′ )M (z − z ′ )m exp[−(|z|2 + |z ′ |2 )/4lB
(4.28)
is unique (aside from normalization factors). It is remarkable that,
neglecting LL mixing, this is the exact two body wave function for any
central potential V (|z−z ′ |). The powerful restrictions due to analyticity
(2)
allow to write ψM m (z, z ′ ) without solving any radial equation! There
is only one state in the LLL Hilbert space with center of mass angular
momentum M and relative angular momentum m.
• The corresponding energy eigenvalue Vm0 for the two electron problem
in the LL is independent of M and given by the only matrix element:
Vm0 =
hm, M |V |m, M i
.
hm, M |m, M i
(4.29)
The coefficients Vm0 are called the Haldane pseudo-potentials (generalized to any LL in a later section in this chapter). The discrete energy
eigenstates represent bound states of the (repulsive!) potential. This is
unusual: a repulsive potential has no bound states, only a continuous
spectrum in the absence of a magnetic field. In the presence of a magnetic field, the Lorentz force results in quenching the kinetic energy, so
we may have bound states. In zero magnetic field, two electrons convert their potential energy in kinetic energy and move away from one
another. In a magnetic field, the electrons have fixed kinetic energy, so
they are constrained to orbit around one another.
The discrete spectrum for a pair of particles in a repulsive potential is
a basic feature in the understanding of the FQHE, as it generates a gap
above the ground state energy for all excitations.
The Laughlin wave function
57
Although the exact solution for the two particle problem cannot be generalized to N > 2 in any straightforward fashion, the N particles wave function
proposed by Laughlin obeys the conditions described above. Laughlin generalized the ν = 1 many body wave function, writing
L
ψ ({zj }) =
Y zi − zj 2s+1
lB
i<j
e−
P
j
2
|zj |2 /4lB
.
(4.30)
In the ν = 1 case, s = 0. Note that this is a one variational parameter (the
integer s) trial wave function. The prefactor in Laughlin’s wave function
(4.30) is also called the Jastrow factor. Similar wave functions had been
proposed to describe liquid Helium.
The plasma analogy is very useful in the fractional filling case [12]. Following this picture, we identify the space integral of the wave function square
modulus with the partition function of a classical statistical system, described
by a “free energy” Ucl . The partition function is then
Z=
X
e−βUcl =
ˆ
C
Z
where C represents configurations, so that
−βUcl = 2q
X
i<j
2
d2 z1 ...d2 zN ψ L ({zj }) ,
(4.31)
zi − zj X |zj |2
+
,
ln 2
lB
j
2lB
where q = 2s + 1. The “ temperature” is, as above, β ≡ 2/q in order to get
Ucl = −q
2
X
i<j
X |zj |2
zi − zj −q
ln .
2
lB
j
4lB
(4.32)
What is different for ν 6= 1 as compared to the previous ν = 1 case? We have
to determine the optimal q knowing that the electronic density is nel = ν φB0 =
ν
2 . The neutral background charge density is given by the same expression:
2πlB
2
ρB = −1/(2πlB
), but the charge neutrality condition of the plasma is now:
ρB + qnel = 0 which can be re-written
qν = 1.
(4.33)
The only variational parameter of the Laughlin wave function (4.30) is determined. The exponent of the fractionally filled LLL ground state wave
function with ν = 1/(2s + 1) is 1/ν = 2s + 1.
58
FQHE – From Laughlin to Composite Fermions
4.2.3
Incompressible fluid
In this section, it is shown that the Laughlin wave function is an almost
exact ground state of the many-body problem, and that there is a gap to
any excited state above this ground state. Then the Laughlin wave function
describes an incompressible fluid. Indeed any attempt at altering the volume
(here a 2D surface) of the Laughlin liquid by applying an infinitesimal (2D)
pressure, thereby effecting an infinitesimal work on the system, should fail,
because the excitations needed to describe the change of the system have a
lower bound, cannot be infinitesimally small. The ν = 1 quantum liquid is
also an incompressible fluid, where the gap to excited states is presumably4
due to the Zeeman effect, or possibly the orbital energy h̄ωC . In the fractional
case, the gap is obviously determined by the Coulomb energy ∝ e2 /(ǫr).
In order to evaluate the ground state energy, we need not take into account
the kinetic term, since we are restricted to a single LL. This is certainly
true in the large field limit, since h̄ωc ∝ B, while the Coulomb energy is
e2 /ǫlB ∝ (B 1/2 ). So we need only take into account the latter term.
Suppose that we write the potential energy, quite generally, in terms of
Haldane pseudo potentials
V =
∞ X
X
vm′ Pm′ (ij)
(4.34)
m′ =0 i<j
where Pm (ij) is the projection operator which selects out states in which i
and j have relative angular momentum m.(Note that Pm1 (ij) and Pm2 (jk) do
not commute). Suppose we have a potential defined by vm′ = 0 for m′ ≥ m.
This is a ”hard core potential”. The Laughlin state with exponent m is an
exact energy eigenstate
V ψm ({zi })
(4.35)
Indeed it is clear that Pm′ (ij)ψm = 0 for any m′ < m since every pair has
relative angular momentum larger than, or equal to m.
Suppose m = 3 (Laughlin state at 1/3 filling ). This model obviously has
a minimum excitation energy v1 , which corresponds to allowing at least one
pair to have relative angular momentum 1.
The proof that the Laughlin wave function has a gap to excited states
for the actual Coulomb interaction, follows from the fact that, compared
4
It turns out that the gap in the ν = 1 case is also due to Coulomb interactions, as will
be discussed later on in the section of Quantum Hall ferromagnetism.
The Laughlin wave function
59
to the model hard core potential, the additional Haldane pseudo potentials
of the Coulomb potential (i.e. m′ ≥ 3) can be treated perturbatively, because they are all smaller than v1 . This proof is valid specifically for the
Coulomb potential. Since all Coulomb corrections to the hard core potential
are perturbations, the gap between the ground state and the first excited
state persists. Thus the Laughlin state, almost the exact ground state of the
Coulomb potential Hamiltonian, is that of an incompressible fluid. The excitation gap is a necessary condition for zero longitudinal conductivity, and
zero resistivity, σxx = 0 = ρxx .
Numerical data show that the overlap between the true ground state for
the Coulomb potential and the Laughlin wave function is extremely good.
4.2.4
Fractional charge quasi-particles
A remarkable property of the Laughlin liquid is that its elementary excitations have fractional charge.
Consider the wave function
ψqh (z0 , {zj }) =
N Y
zi − z0
i=1
lB
ψ L ({zj }),
(4.36)
where an additional zero sits at position z0 . The charge density vanishes at
z0 . Expanding formally Laughlin’s wave function as
ψ L ({zj }) =
X
mN −
αm1 ,...,mN z1m1 ...zN
e
{mi }
P
j
2
|zj |2 /4lB
,
and comparing with the expansion
ψqh (z0 = 0, {zj }) =
X
{mi }
mN +1 −
αm1 ,...,mN z1m1 +1 ...zN
e
P
j
2
|zj |2 /4lB
,
we see that, compared to Laughlin’s wave function, all particles are displaced
from one state to the next, mj → mj + 1. In the symmetric gauge where a
√
particle is found on a ring of radius lB 2mj + 1, this means that a hole has
been created at the origin z0 = 0 (“ quasi-hole”). Furthermore, this quasi hole
P
has vorticity. If we examine the phase of ψqh (z0 = 0, {zj }) ∝ j exp(−iθj ),
where θj = tan−1 (yj /xj ), we see that a particle circulating on a closed path
around z0 acumulates a phase 2π.
60
FQHE – From Laughlin to Composite Fermions
In principle, one can describe a wave function with a “quasi-particle”
excitation (with opposite vorticity) in a similar fashion ,
ψqp (z0 , {zj }) = PLLL
N ∗
Y
zi − z0∗
i=1
lB
ψ L ({zj }),
(4.37)
There is a complication here since we are not allowed to use zj∗ in a wave
function which should be analytic in order not to mix in higher LL states. In
order to remain within the LLL manyfold of states, one should use a projector
PLLL on the LLL. A way of doing this is to divide the polynomial part of the
wave function by zj instead of multiplying by zj∗ . By partial integration, this
is equivalent to applying ∂zj to the gaussian factor, which generates zj∗ , up to
a multiplying factor. Given the complication in handling quasi-particle wave
functions, we will deal only with quasi-holes in the following, without loss of
physical generality [32].
In order to check that such excitations have fractional charge, let us use
again the 2D plasma analogy introduced above. The prefactor in the wave
function (4.33) gives rise to a new term in expression (4.32), Ucl → Ucl + V ,
where
N
X
zj − z0 .
ln V = −q
l
B
j
This is interpreted as the interaction potential between the plasma and a
charge 1 ”‘impurity”’ located at z0 . This impurity is screened so as to maintain charge neutrality in the plasma. Since the plasma particles have charge
q, 1/q particles are needed to screen the impurity charge. The quasi-particle
of the Laughlin liquid is thus shown to have fractional charge 5
e∗ =
e
e
=
.
q
2s + 1
(4.38)
Another more direct way to see this charge fractionalisation is to introduce
q quasi-particles at the same point 6
q
[ψqh (z0 , {zj })] =
5
N Y
zj − z0 q
j=1
lB
ψ L ({zj }) = ψ L ({zj }, z0 ),
From now on, we use the generic term ”quasi-particle” for quasi-holes and quasiparticles, except when distinction is necessary.
6
The expression on the left is symbolic .
The Laughlin wave function
61
where we find the Laughlin wave function for N +1 electrons, with the added
one at position z0 . One needs therefore q quasi-particles to add one electron
in a Laughlin liquid, leading to the same conclusion as the plasma analogy
(4.38).
It is interesting to give yet another proof that Laughlin quasi-particles
carry fractional charge, in order to show the essential connection between the
2
fractional quantum Hall plateau, with fractional Hall conductivity σxy = ν eh ,
and the fractionalisation of the quasi-particle charge. Imagine piercing the
sample at the origin with an infinitely thin magnetic solenoid and increasing
adiabatically the magnetic flux φ from 0 to φ0 = h/e. The time variation of
the flux inside the solenoid induces an azimuthal electric field, as Faraday’s
law tells us. This field is such that
I
C
dr.E = −
∂φ
.
∂t
(4.39)
C is a contour surrounding the flux line. If the process is sufficiently slow, the
electric field has low frequency Fourier components only, such that h̄ω ≪ ∆,
where ∆ is the energy gap. There is no dissipation. Because the system is
in a quantum Hall state, the electric field drives a current density which is
radial:
E = ρxy J~ ∧ ẑ.
(4.40)
So we have
~ ∧ dr) = − dφ
(4.41)
J.(ẑ
dt
C
The integral on the LHS represents the total current flowing into the region
enclosed by the contour. Thus the charge inside this region obeys
ρxy
I
ρxy dQ/dt = −dφ/dt
(4.42)
At the end of the process, the total charge is
Q = σxy φ0 = σxy (h/e) = νe
(4.43)
The final step in the argument is that an infinitesimal flux tube containing a
flux quantum is invisible to the particles, and can be removed by a (singular)
gauge transformation which has no physical effect.
This derivation underlines the importance of the fact that σxx = 0 and
σxy is quantized. The existence of fractionally charged elementary excitations
is a direct consequence of the FQHE.
62
FQHE – From Laughlin to Composite Fermions
Numerical data show that there is a finite energy cost to create such
quasi-particles which means that there is a gap between the ground state
described by the Laughlin wave function and its lowest elementary excited
states. This is a necessary condition for the FQHE.
4.2.5
Ground state energy
Beside his ground state wave function proposal, Laughlin showed that it
has lower energy than the Wigner crystal. The latter had been argued to
minimize the Coulomb energy. The Laughlin liquid energy is given by


Z
N
2
Y
hψ L |Ĥ|ψ L i
e2
1 XZ 2 2

d2 zk  ψ L (zi , zj ; {zk })
d zi d zj
=
Z
2Z i6=j
ǫ|zi − zj |
k6=i,j
=
n2el A Z 2 e2
g(r),
dr
2
ǫ|r|
(4.44)
where
2
N (N − 1) Z 2
L
2
ψ
(z
=
0,
z
=
r;
z
,
...,
z
)
d
z
...d
z
g(r) ≡
1
2
3
N
3
N
2
nel Z
(4.45)
is the pair correlation function. This expression takes advantage of the translation and rotation invariance of the wave function 7 and of the fact that there
are N (N − 1) ways of chosing the zi = z1 and zj = z2 pairs in the first line
of equation (4.44). This expression is usually divided by the total particle
number N = Rnel A , and the energy of the homogeneous uncorrelated liquid
E0 = (nel /2) d2 re2 /ǫr is chosen as energy reference. The Laughlin liquid
energy per particle is written in terms of the pair correlation function.
EL =
nel Z 2 e2
[g(r) − 1],
dr
2
ǫ|r|
(4.46)
1X
v(q)[s(q) − 1],
2 q
(4.47)
1
hρ(−q)ρ(q)i,
N
(4.48)
or, in Fourier space
EL =
in terms of the static structure factor
s(q) =
7
rotation invariance results in r = z = |z|.
The Laughlin wave function
63
which is connected to the pair correlation function by Fourier transformation
[25]
Z
d2 r eiq·r [g(r) − 1].
[s(q) − 1] = nel
(4.49)
The pair correlation function (or the structure factor ) thus determines the
liquid structure and describes possibly a short range order. It can be computed from Laughlin’s wave function by Monte Carlo integration [34, 35].
Instead of computing the pair correlation function numerically, Girvin
analysed it in 1984 using symmetries and properties of the 2D one component
plasma [91]. Expanding the Laughlin wave function in terms of z = (z1 −
z2 )/lB and z+ = (z1 + z2 )/lB ,
ψ L ({zj }) =
∼
XX
2 +|z|2 )/8
M m −(|z+ |
aM,m (z3 , ..., zN )z+
z e
,
(4.50)
M m=1
where the tilde on the second sum indicates that the sum is on odd integers,
one finds for the pair correlation function
g(z) =
∼
X
′
2 /4
Am,m′ (z+ )z ∗m z m e−|z|
,
m,m′
where functions Am,m′ (z+ ) depend only on z+ because the other variables
z3 , ...zN have been integrated on. ∂Am,m′ (z+ )/∂z+ = 0 follows from the liquid
translation invariance, and rotation invariance imposes Am,m′ = δm,m′ bm ,
which results in
∼
g(z) =
X
2 /4
m=1
bm |z|2m e−|z|
.
As lim|z|→∞ g(|z|) = 1, and thus lim|z|→∞ m=1 bm |z|2m = exp(|z|2 /4), it is
convenient to rewrite expansion parameters as
P̃
m
2 1
m! 4
bm =
(1 + cm ),
where limits impose limm→∞ cm = 0. The pair correlation function is thus
given as a sinh plus corrections described by the cm parameters.
∼
|z|2 −|z|2 /4 X
2cm
g(z) = 2 sinh
e
+
4
m=1 m!
!
=
−|z|2 /2
1−e
+
∼
X
2cm
m=1
m!
|z|2
4
!m
|z|2
4
!m
2 /4
e−|z|
2 /4
e−|z|
.
(4.51)
64
FQHE – From Laughlin to Composite Fermions
The Fourier transform yields the static structure factor
[s(q) − 1] = −νe−q
2 l2 /2
B
+ 4ν
∼
X
2
cm Lm (q 2 lB
)e−q
2 l2
B
,
(4.52)
m=1
in terms of Laguerre polynomials. The energy (4.47) is finally written as
EL =
∼
νX
ν X
cm Vm0 −
v0 (q),
π m=1
2 q
(4.53)
where v0 (q) is the effective potential in the LLL (4.18), and we have defined
the Haldane pseudo-potentials
Vm0 ≡ 2π
X
2
v0 (q)Lm (q 2 lB
)e−q
2 l2 /2
B
.
(4.54)
q
The advantage of the energy expression (4.53) in terms of Haldane pseudopotentials, defined in terms of the effective potential, allows to describe directly Laughlin liquids in higher index LL (n 6= 0) : pseudo-potentials are
generalised to LL n, using the appropriate effective potential (4.18),
Vmn ≡ 2π
X
2
)e−q
vn (q)Lm (q 2 lB
2 l2 /2
B
.
(4.55)
q
Returning to (4.51), we notice that due to the Laughlin wave function
behaviour when two particles described by z1 and z2 get close to one another,
one has g(z) ∼ |z|2(2s+1) at short distance. This shows that correlations in
Laughlin’s wave function are effective in minimizing Coulomb interactions,
more so than in any state where fermionic correlations would impose only
g(z) ∼ |z|2 . This short distance behaviour ensures that expansion parameters
obey
cm = −1,
pour m < s.
(4.56)
It is also useful to define the ”moments”
Mn = nel
Z
|z|2
dz
4
!n
"
n+2
2
= 2πnel −n! + 2
[g(z) − 1]
∼
X
(n + m)!
m=1
m!
#
cm ,
(4.57)
The Laughlin wave function
65
where the second line is computed with the help of equation (4.51). Following
the plasma analogy,8 charge neutrality imposes M0 = −1 and thus
∼
X
m=1
cm =
1
s
1 − ν −1 = − .
4
2
(4.58)
Perfect plasma screening is expressed by M1 = −1, i.e.
∼
X
m=1
(m + 1)cm =
1
s
1 − ν −1 = − .
8
4
(4.59)
Compressibility properties yield a third sum rule
∼
X
s2
1
−1 2
1−ν
= .
(m + 2)(m + 1)cm =
8
2
m=1
(4.60)
Those sum rules (4.58-4.60) and (4.56) can be used as constraints on the
pair correlations functions (4.51) in connection with Monte Carlo numerical
work. One may use them instead for an approximate determination of the
function: sum rules form a system of coupled linear equations which can be
solved if one sets cm = 0 for m > s + 3, which is a reasonable approximation,
since limm→∞ cm = 0. In this manner, one finds
s=1
s=2
s=3
s=4
cs1
-1
-1
-1
-1
cs3
cs5
cs7
cs9
cs11
cs13
17/32 1/16 -3/32
0
0
0
-1
7/16 11/8 -13/16
0
0
-1
-1
-25/32 79/16 -85/32
0
-1
-1
-1
-29/8
47/4 -49/8
for s = 1, .., 4.
The results for the energy deviate by less than one per cent from numerical results by Levesque et al. [34], as shown on figure 4.2(a). The pair
correlation function apart the correlation hole at small distance, exhibits a
maximum at finite distance, where it is most probable to find a second particle.. This maximum is displaced further away from the origin and becomes
more pronounced if the electronic density is lowered [ν = 1/(2s + 1)]. This
means an enhanced short range order at low densities where a Wigner crystal
66
FQHE – From Laughlin to Composite Fermions
Facteur de remplissage
5
0.1
15
0.3
10
0.2
20
0.4
25
0.5
12
14 15
Energie
-0.1
−0.1
-0.2
−0.2
-0.3
−0.3
-0.4
−0.4
Fonction de corrélation de paires
(b)
1.4 g s(r)
1.2
s=3
s=2
s=1
1.01
0.8
0.6
0.5
0.4
0.2
2
4 5 6
r8
10
10
Figure 4.2: (a) Comparison of our results for the energy (black segments) of Laughlin
states to numerical results by Levesque et al. (gray line) in√units of e2 /ǫlB . The line is the
result of an an interpolation formula U (ν) = −0, 782133 ν 1 − 0, 211ν 0,74 + 0, 012ν 1,7
for Levesque et al.’s results [34]. (b) Pair correlation function for different s. The distance
r is mesured in units of the magnetic length lB . The straight dotted line corresponds to
uncorrelated electrons.
The Laughlin wave function
67
is expected to become more stable. More accurate numerical results confirm
this tendancy (see figure 4.2(b))[39].
4.2.6
Neutral Collective Modes
We have discussed the ground state energy, and analysed elementary excitations (fractionally charged quasi-particles) the energy of which is separated
from the ground state one by a gap. In order to understand FQHE, we now
have to show that collective excitations have a dispersion relation with a
finite gap above the ground state at all wave vectors. Such collective excitations, with wave function |ψq i are likely to be well described within the
”Single Mode Approximation” (SMA)[35]. 9 . In the quantum Hall case,
|ψq i = ρ̄(q)|ψ L i,
(4.61)
where ρ̄(q) is the projected density operator (4.7). Since
ρ̄(q) =
X
m,m′
hm|e−iq·R |m′ ie†n=0,m en=0,m′ ,
the excited state may be interpreted as a superposition of particle-hole excitations (particles in the state |n = 0, mi and hole in |n = 0, m′ i), an average
2
distance qlB
apart. Because of the projection, |ψq i has no component in LL
n 6= 0. The excitation energy with respect to the ground state is
∆(q) =
≃
hψ L |ρ̄(−q)Ĥ ρ̄(q)|ψ L i
− EL
hψ L |ρ̄(−q)ρ̄(q)|ψ L i
f¯(q)
1 hψ L |[ρ̄(−q), [Ĥ, ρ̄(q)]|ψ L i
≡
,
2 hψ L |ρ̄(−q)ρ̄(q)|ψ L i
s̄(q)
(4.62)
where we assumed that the Laughlin state is an eigenstate of the Hamiltonian
, Ĥ|ψ L i ≃ E L |ψ L i, which is an excellent approximation. Moreover the liquid
state rotation invariance has been used to get the second line, as well as
ρ̄† (q) = ρ̄(−q). The projected structure factor is connected to the structure
factor (4.52) through
s̄(q) = s(q) − 1 − e−q
8
2 l2 /2
B
.
Interested readers may find details in the following references [34, 35, 91] (and references in those papers in particular [38])
9
The SMA was used by Feynman in his theory of superfluid He collective modes [40]
68
FQHE – From Laughlin to Composite Fermions
Figure 4.3: Dispersion relation for collective excitations [41]. The continuous curves are
the results in the Single Mode Approximation for ν = 1/3, 1/5 et 1/7; the various symbols
are values obtained by exact diagonalisation [42]. Arrows are the expected reciprocal
lattice parameter moduli for the Wigner crystal at the corresponding filling factors.
Equation (4.62) is precisely the Feynman-Bijl formula, proposed for the description of collective excitations in superfluid He [40]. Using commutation
rules for projected density operators (4.16), one finds
∆(q) = 2
X
k
[v0 (|k − q|) − v0 (k)] sin
2
2
q ∧ klB
2
!
s̄(k)
.
s̄(q)
(4.63)
The dispersion relations are shown on figure 4.3, as well as numerical
results of exact diagonalisations of systems with a small number of particles
[42]. As expected for an incompressible liquid, dispersion relations have
a finite energy gap above the ground state for all wave vectors. They all
exhibit a minimum at a finite wave vector. The latter corresponds to the
reciprocal lattice parameter modulus for the Wigner crystal at the same filling
factor. The minimum, called the magneto-roton minimum, in analogy with
the superfluid He case [40], is a sign of short range (crystalline) order. The
collective mode softening at this wave vector signals a tendancy to Wigner
crystal stabilisation when the filling factor is decreased.
The SMA becomes less reliable at large wave vector, where one expects
4.3. JAIN’S GENERALISATION – COMPOSITE FERMIONS
69
the asymptotic behaviour
2πe∗2
,
2
ǫqlB
i.e. the energy to create a pair made with a quasi-particle of energy ∆qp and
a quasi-hole with energy ∆qh , with well separated components submitted to
Coulomb attraction because of their opposite charges, e∗ and −e∗ .
∆(q ≫ 1/lB ) ≃ ∆qp + ∆qh −
4.3
Jain’s generalisation – Composite Fermions
The Laughlin wave function describes well the FQHE at ν = 1/(2s + 1), but
it fails to apply to the other fractional states which were discovered subsequently, such as ν = 2/5, which is one term in the set ν = p/(2sp + 1). In
order to account for those new states, Haldane [37] and Halperin [43] proposed a hierarchy picture. Following the latter, Laughlin quasi-particles with
sufficient density condense in an incompressible liquid in order to minimize
their Coulomb interaction energy due to their charge e∗ . The state ν = 2/5
would then be a ”daughter” of the Laughlin state at ν = 1/3.
In 1989, Jain proposed an alternative route, the Composite Fermion picture. He first re-interpreted the Laughlin wave function
Y zi − zj 2s Y zi − zj − P |z |2 /4l2
B
j j
e
,
(4.64)
ψ L ({zj }) =
lB
lB
i<j
i<j
as a product of two factors : the first one, i<j [(zi − zj )/lB ]2s , attaches 2s
zeros (vortices with 2s flux quanta ) to particles positions, and the second
one,
Q
χν ∗ =1 ({zj }) =
Y zi − zj i<j
lB
,
(4.65)
can be interpreted as the wave function of a virtual completely filled LL, with
a new (virtual) filling factor ν ∗ = 1 [20]. Indeed, it coincides with equations
4.20 and 4.21.
Jain’s proposal amounts to generalize equation 4.65 by replacing χν ∗ =1 ({zj })
by a Slater determinant for p virtual completely filled LL, χν ∗ =p ({zj }),
J
ψ ({zj }) = PLLL
Y zi − zj 2s
i<j
lB
χν ∗ =p ({zj }),
(4.66)
70
FQHE – from Laughlin to Composite Fermions
Projection to the LLL is taken care of by the projector PLLL , since the function χν ∗ =p ({zj }) contains, if unprojected, high energy components belonging
to LL with p > 1.
What is achieved by this manipulation? The effective number of states
per LL in the virtual levels has been decreased, M → M ∗ = M − 2sN ,
since the first vortex attachment factor has taken 2sN zeros from the system
with N electrons. This amounts to renormalize the magnetic field which is
nothing but the flux density in terms of flux quanta φ0 = h/e, and the filling
factor as well, in the following way
B → B ∗ = B − 2sφ0 nel
et
ν ∗−1 = ν −1 − 2s.
(4.67)
With this picture, we may now re-interpret the FQHE at ν = 1/3 as a
completely filled Composite Fermion level, with ν ∗ = 1, where a Composite
Fermion is an electron with two attached flux quanta. The state at ν = 2/5
is re-interpreted as a state with ν ∗ = 2 (figure 4.4). The CF picture allows to
understand the FQHE of electrons at ν = p/(2sp + 1) in terms of an IQHE
for CF at filling factor ν ∗ = p, since the CF filling factor ν ∗ = hnel /eB ∗ is
connected to the electronic filling factor through
ν=
ν∗
,
2sν ∗ + 1
(4.68)
which is equivalent to expression (4.67).
In the following chapters, we shall elaborate on the physical meaning of
this picture, which is basically a flux counting device, based on the notion
of flux attachment to electrons. It is not an obviously physical approach
to renormalize the magnetic field, which is an external object imposed on
the system. Note however that the magnetic field only enters the theory
with the electronic charge e as a multiplying coupling constant. It is thus
permissible to renormalize the charge, which seems especially relevant given
the fractionalisation of excitation charges discussed above.
4.3.1
The effective potential
Pour mieux comprendre le modèle, on discutera dans cette section quelques
propriétés du potentiel d’interaction effectif (4.17). En raison des zéros des
polynômes de Laguerre Ln (x), la répulsion coulombienne
√ disparaı̂t à certaines
valeurs du vecteur d’onde, notamment à q0 (n) ≃ 2, 4/ 2n + 1, ce qui correspond au premier zéro x ≃ 1, 2/(2n + 1) [31]. Cela mène à des instabilités
Jains’s generalisation
71
ν = 1/3 :
ν∗ = 1
Théorie
de FCs
électron
ν = 2/5 :
ν∗ = 2
quantum de flux libre
vortex
portant 2s quanta de flux (liés)
fermion composite
Figure 4.4: Composite Fermions. The electronic state at ν = 1/3 may be understood
as a CF state with integer filling ν ∗ = 1. CF are electrons carrying each 2s flux quanta.
Similarly, a CF filling factor ν ∗ = 2 describes an electronic filling factor = 2/5.
du système pour la formation des phases de densité inhomogène avec une
périodicité caractéristique Λ ≃ 2π/q0 (n), car il est énergétiquement favorable
pour la densité moyenne hρ̄(q)i d’avoir un maximum à √
q0 . La périodicité Λ
varie proportionnellement avec le rayon cyclotron RC = 2n + 1. Les phases
de densité inhomogène seront discutées plus en détail dans le chapitre 7. Une
P
transformation dans l’espace réel du potentiel effectif, vn (r) = q exp(ir ·
q)vn (q) confirme la apparition d’une échelle de longueur caractéristique. Pour
des petites valeurs de n, cette transformation peut être effectuée de façon exacte, et l’on trouve une somme finie sur des fonctions de Bessel. Dans des
NL plus élevés, n ≫ 1, on peut déduire une loi d’échelle pour le potentiel à
l’aide de Fn (q) ≃ J0 (qRC ), ce qui devient exact dans la limite n → ∞,
ṽ(r/RC )
vn (r) ≃ √
,
2n + 1
2
avec ṽ(x) =


4e
Re K 
πǫx
1−
q
1 − 4/x2
2
2
 ,
(4.69)
où J0 (x) est la fonction de Bessel d’ordre zéro, et K(x) est l’intégrale elliptique complète de première espèce [21].
La figure 4.5(a) montre les résultats pour le potentiel effectif dans les
niveaux n = 1, ..., 5. On remarque la formation d’un palier – à part des petites
oscillations – pour des distances moyennes, superposé au potentiel coulombien habituel, e2 /ǫr, qui est retrouvé à grande distance. Ce palier devient plus
large dans des NL élevés cependant que sa hauteur est diminuée. La forme
72
FQHE – from Laughlin to Composite Fermions
2.02
1.01
(a)
1/r
1.5
1.5
n=1
0.6
n=2
0.5
1/r
n=1
~
v(r)
vn (r)
0.8
(b)
1.75
0.4
n=3
n=4
n=5
1.25
n=2
1.01
n=3
0.75
n=4
0.5
0.5
n=5
0.25
0.2
22
4
4
r
6
6
88
10
10
10
1.0
20
2.0
30
3.0
40
4.0
50
5.0
r/R C
Figure 4.5: (a) Potentiel effectif dans l’espace réel pour les NL n = 1, ..., 5, en unités
de e2 /ǫ. Le potentiel de Coulomb en 1/r est montré pour comparaison (tirets). (b) Les
résultats pour le potentiel (points) sont tracés après la transformation d’échelle (4.69). La
ligne noire représente l’expression approchée ṽ(x) et la ligne grise le potentiel de Coulomb.
(a)
(b)
(c)
r
Figure 4.6: Les fonctions d’onde des électrons dans un NL n ≥ 1 peuvent être
représentées par des anneaux [voir Fig. 2.3(a)]. (a) Si r > 2RC , les anneaux ne se
< 2RC , les anneaux commencent à avoir un recouvrement,
recouvrent par. (b) Pour r ∼
représenté par la surface grise foncée. (c) Le recouvrement n’augmente pas de façon significative lorsque les anneaux sont rapprochés davantage.
d’échelle ṽ(x) est mise en évidence après la transformation des résultats selon
l’équation (4.69) : les points, qui représentent les résultats exacts, tombent
approximativement sur la même courbe (noire). L’approximation (4.69), qui
devient exacte dans la limite n → ∞, décrit la forme du potentiel de façon
suffisamment appropriée aussi pour de plus bas NL à condition que n > 0.
Le point anguleux à r = 2RC dans la forme approchée du potentiel est un
artéfact mathématique – pour x ≥ 2, l’argument de l’intégrale elliptique est
réel tandis qu’il devient complexe pour x < 2, ce qui donne lieu à cette
discontinuité. Cet effet pourrait engendrer des divergences artificielles dans
d’éventuelles dérivées, mais on peut se servir de cette forme du potentiel
uniquement comme support dans des intégrations, ce qui rend la discontinuité inoffensive.
La forme du potentiel peut être illustrée dans une image quasi-classique.
Jains’s generalisation
73
Avec la restriction des champs électroniques au n-ième NL, on a fait une
moyenne sur le mouvement rapide de l’électron, déterminé par la variable η
qui, sans cette restriction, couplerait des états de différents n. Les degrés
de liberté du mouvement des électrons sont donc uniquement leurs centres
de guidage. Comme on l’a vu dans la section 2.2.1, la fonction d’onde d’un
électron dans un niveau de Landau n ≥ 1 tient compte de cette moyenne sur
le mouvement cyclotron et a par conséquent une forme d’anneau de rayon
RC , représentant une densité électronique moyenne [Figs.2.3(b) et 4.6]. Si
la distance r entre les centres d’anneaux, qui sont précisément les centres
de guidage du mouvement cyclotron de chaque particule, est suffisamment
grande (r > 2RC ), la forme des fonctions d’onde de deux particules n’a pour
effet qu’une faible correction du potentiel coulombien. A r ∼ RC , les anneaux commencent à se recouvrir et la répulsion devient donc plus forte. En
revanche, si l’on rapproche les centres de guidage, ce recouvrement ne devient
pas plus grand et la répulsion n’augmente donc pas de façon significative, ce
qui explique la formation du palier dans le potentiel effectif. La répulsion
devient à nouveau plus importante quand le recouvrement est complet à très
petite distance. Or les centres de guidage étant étalés sur la surface minimale
2
2πlB
ne peuvent pas être approchés à des distances plus petites que lB . Pour
n = 0, cette image quasi-classique devient plus problématique parce que les
fonctions d’onde sont de forme gaussienne avec une extension spatiale de
l’ordre de la longueur magnétique. Les électrons devraient donc plutôt être
représentés par un disque de rayon lB , qui constitue également la longueur
minimale, comme il a été décrit dans le chapitre 2 [voir Fig. 2.3(b)].
74
FQHE – from Laughlin to Composite Fermions
Chapter 5
Chern-Simons Theories and
Anyon Physics
Following the CF theoretical proposal, a field theory was constructed to
describe flux attachment to electrons. Such theories are known as ”ChernSimons ” theories in the framework of the generalisation of the Maxwell
theory of electromagnetic fields. Lopez and Fradkin were first to point out
in 1991 [15] their relevance for the FQHE, followed in 1993 by Halperin, Lee
and Read [16], who studied the compressible state at ν = 1/2. The latter
filling factor is the limiting point of p/(2sp + 1) when p → ∞ and s = 1.
This chapter aims at introducing some basic notions about Chern-Simons
transformations, but does not pretend to offer a detailed field theoretical
description. We describe their connections with anyons, i.e. particles in 2D
which obey fractional statistics, and which have a transparent description in
the framework of Chern-Simons theories, the basic notions will be useful in
the follwing chapter.
5.1
Chern-Simons transformations
The Hamiltonian of electrons in a magnetic field writes, in second quantized
form
Ĥ = Ĥ0 + Ĥint ,
75
76
Chern-Simons theories and anyon physics
where the kinetic term is
Ĥ0 =
Z
d2 rψ † (r)
[−ih̄∇ + eA(r)]2
ψ(r),
2m
(5.1)
and Ĥint accounts for interactions between electrons. A Chern-Simons transformation is a singular unitary transformation,
ψ(r) = e−iφ̃
R
d2 r ′ θ(r−r′ )ρ(r′ )
ψCS (r),
(5.2)
where θ(r) = tan−1 (y/x) is the angle formed by vector r and the x axis.
This transformation is clearly singular since the angle θ(r) is not defined for
r = 0. The density is invariant under this transformation
†
ρ(r) = ψ † (r)ψ(r) = ψCS
(r)ψCS (r).
Notice that d2 r′ θ(r−r′ )ρ(r′ ) (see equation 5.2) is an operator which depends
on all electron coordinates. The gradient in expression (5.1) also operates on
the phase factor of the transformation, and one finds
R
−ih̄∇ψ(r) = e−iφ̃
R
d2 r ′ θ(r−r′ )ρ(r′ )
−ih̄∇ − φ̃h̄∇
Z
d2 r′ θ(r − r′ )ρ(r′ ) ψCS (r).
We can thus define a new gauge field, the Chern-Simons vector potential,
Z
h̄
ACS (r) = − φ̃∇ d2 r′ θ(r − r′ )ρ(r′ ).
e
(5.3)
If this potential obeys the Coulomb gauge, as will be shown later on, ∇ ·
ACS (r) = 0, the kinetic Hamiltonian can be re-written as
Ĥ0 =
Z
2
d
[−ih̄∇
†
rψCS
(r)
+ eA(r) + eACS (r)]2
ψCS (r).
2m
(5.4)
The interaction Hamiltonian is invariant, since it depends only on density
operators which are invariant.
In order to analyse this new gauge field and its associated magnetic field,
BCS (r) = ∇ × ACS (r), it is useful to recall some properties of analytic
functions. We take here z = x + iy, unlike our definition in chapter 2. Each
complex function may be written as a sum of a real part and an imaginary
part,
f (x, y) = u(x, y) + iv(x, y).
Chern-Simons transformations
77
The analyticity condition ∂z∗ f (z) = 0 is expressed, in terms of x and y, by
equations known as Cauchy-Riemann differential equations
∂x u(x, y) = ∂y v(x, y),
et
∂y u(x, y) = −∂x v(x, y),
(5.5)
Instead of using the cartesian notation, one may chose the polar coordinate
representation:
f (x, y) = w(x, y)eiχ(x,y) ,
where w(x, y) and χ(x, y) are real functions. The analyticity condition, (∂x +
i∂y )f (x, y) = 0, is now written as
∂x w(x, y) − w(x, y)∂y χ(x, y) + i [∂y χ(x, y) + w(x, y)∂x χ(x, y)]
or, after separation in real parts and imaginary parts, and dividing by w(x, y),
by the Cauchy-Riemann equations in the polar representation:
∂x ln w(x, y) = ∂y χ(x, y)
et
∂y ln w(x, y) = −∂x χ(x, y).
(5.6)
In the simplest case, which is of interest here, f (z) = z = r exp(iθ), this
yields
∂x ln r(x, y) = ∂y θ(x, y)
et
∂y ln r(x, y) = −∂x θ(x, y).
With these equations, we compute easily
[∇ × ∇θ(r)]z = (∂x ∂y − ∂y ∂x )θ(r) = ∆ ln r = 2πδ (2) (r),
(5.7)
where the last step is Poisson equation for a 2D potential. The curl of a
gradient is usually zero, but this is not the case here, because θ(r) is singular
at r = 0, as mentionned above. Similarly, we find
∆θ(r) = −∂x ∂y ln(r) + ∂y ∂x ln(r) = 0.
(5.8)
Together with definition (5.3), this last equation shows that the ChernSimons field satsifies the Coulomb gauge. Equation (5.7) gives for the corresponding magnetic field
BCS
h̄ Z 2 ′
h
= − φ̃ d r ∇ × ∇θ(r − r′ )ρ(r′ ) = − φ̃ρ(r)ez .
e
e
(5.9)
78
Chern-Simons Theories and Anyon Physics
We notice that this magnetic field is 1)intimately connected to the electronic
density, and 2) it is a quantum operator, contrary to the usual B field.
In the mean field approximation, the density operator in equation (5.9) is
replaced by the average density hρ(r)i = nel , so that the field is renormalized
h
B → B ∗ = B + hBCS i = B − φ̃nel
e
(5.10)
where, in terms of filling factor,
B → B ∗ = B(1 − φ̃ν).
(5.11)
If we chose φ̃ = 2s, this field renormalisation is precisely that described by
the CF theory c(4.67).
Let us connect with the trial wave function approach. In the first quantization language, we can rewrite this field transformation as
iφ̃
ψ({zj }) = e
P
θ(zi −zj )
i<j
ψCS ({zj }) =
Y
i<j
zi − zj
|zi − zj |
!φ̃
ψCS ({zj }).
(5.12)
We see that the Chern-Simons transformation ties φ̃ flux quanta (singularity
of order φ̃ in the phase), leaving off the transformation the vortex modulus,
contrary to Jain’s function for φ̃ = 2s (4.66).
5.2
Statistical Transmutation – Anyons in 2D
Chern-Simons theories are especially well fitted to the discussion of anyon
physics. Anyons are particles which live in 2D+1 space, and which obey
fractional statistics, i. e. neither bosonic nor fermionic. All particles known
in the 3D world are either bosons or fermions depending on the behaviour
of their wave function upon interchange of two identical particles.This is
basically because in 3D (and higher dimensions), the rotation group is nonabelian. The components of the angular momentum do not commute. Quantization of angular momenta is in terms of units of h̄/2, as can be seen from
the properties of the Lie algebra for infinitesimal rotations. The classification
in bosons and fermions is not true anymore in 2D, because the rotation group
is a trivial abelian group. Therefore no angular momentum quantization rules
follow from manipulations of the infinitesimal rotations.
Statistical Transmutation – Anyons in 2D
(b)
(a)
z
79
C
A
B
A
B
+
B
A
Figure 5.1: (a) Process for a particle A to follow a path C around a second particle.
In 3D, the path can be lifted off from the plane and thus can be reduced to a point(gray
curves)). (b) Equivalent processus consisting in two successive exchanges of particles A
and B.
Let us look at the interchange of two identical particles. A process T
through which a particle A is adiabatically displaced around another particle
B, is equivalent, modulo a translation, to two exchange processes E (Fig. 5.1).
We assume that particles are localized enough so we can neglect their wave
function overlap. From an algebraic point of view, which takes into account
the homotopy of the processes, one may write:
E2 = T
modulo a translation. Let us first discuss the 3D case where the path C lies
in the x − y plane. Since the third dimension along z is available, we can
lift the path C of particle A above particle B, and then shrink it down to a
point, leaving particle B at all times outside the closed path C. .1 . We may
associate a “time” interval to this adiabatic process, such that C(t = 0) = C
(the initial path in the x − y plane), and C(t = 1) = 1 ( the position point
of A). The process through which A circles around B(rotation of 2π is thus
equivalent to a process which leaves particles unchanged, i.e. the identity, so
we can write
√
T =1
et donc
E = 1,
where the last equation is written symbollically, meaning that E has two
eigenvalues e1 = exp(2iπ) = 1 and e2 = exp(iπ) = −1 . This superselection
rule shows that in 3D all point particles may be separated in two classes, depending on the behaviour of their wave function upon interchange of identical
particles. e1 = 1 corresponds to bosons and e2 = −1 to fermions.
This piece of reasoning is not valid anymore when the particle dynamics
is restricted to 2D (2 space dimensions). In this case the path C cannot be
shrunk to a point without crossing the particle position B. One says that a
1
This would be impossible if B were an infinite line in direction z
80
Chern-Simons Theories and Anyon Physics
path which encloses another particle (particle B) is not in the same homotopy class as a path which encloses none (and can be shrunk to a point. In
2D, paths can be classified by the number of enclosed particles, or by the
number of times it winds around a given particle. From an algebraic point of
view, the physical requirement is that physical quantities must be invariant
by a 2π rotation. This requirement does not apply to the wave functions,
since only probabilities are eventually observable. Therefore the eigenvalues
λ of the 2π rotation operator R(2π) may be any number of modulus unity,
such as eiαπ with α real(whence the name anyon! for particles with generalised statistics). It is easy to see that eigenstates of R(2π) with different
eigenvalues λ 6= λ′ are orthogonal. In fact no local observable can connect
states corresponding to different α values. States corresponding to different
λ values are said to belong to different superselection sectors. Schematically,
(anti-) commutation rules for the relevant fields must be generalized
ψ(r1 )ψ(r2 ) = ±ψ(r2 )ψ(r1 ) ⇒ ψ(r1 )ψ(r2 ) = eiαπ ψ(r2 )ψ(r1 ),
(5.13)
where απ is called the statistical angle.
What about the Pauli principle? Contrary to bosons, fermions occupation
of a state cannot exceed 1. In terms of fields, this is expressed by
2ψ(r)ψ(r) = 0,
for fermionic fields at the same point r1 = r2 = r, where we use equation
(5.13) with the minus sign for fermions. In general, for arbitrary angle α ,
one finds
1 − eiαπ ψ(r)ψ(r) = 0.
(5.14)
Thus ψ(r)ψ(r) 6= 0 if and only if ψ(r)ψ(r) 6= 0, as is well known for bosons.
When α 6= 0 mod(2), we have necessarily ψ(r)ψ(r) = 0, and equation (5.14)
is interpreted as generalized Pauli principle.
5.2.1
Anyons and Chern-Simons theories
We now analyse the statistical properties of the fields ψCS (r) which result
from the Chern-Simons transformation, using the known properties of electronic fields ψ(r). Defining
τ (r) ≡
Z
d2 r′ θ(r − r′ )ρ(r′ ),
(5.15)
Statistical Transmutation – Anyons in 2D
81
to simplify notation in the following expressions one has
ψCS (r1 )ψCS (r2 ) = eiφ̃τ (r1 ) ψ(r1 )eiφ̃τ (r2 ) ψ(r2 )
= eiφ̃τ (r1 ) eiφ̃τ (r2 ) e−iφ̃τ (r2 ) ψ(r1 )eiφ̃τ (r2 ) ψ(r2 ). (5.16)
With the help of the Hausdorff formula,
eA Be−A = B + [A, B] +
∞
X
1
1
[A, [A, B]] + ... =
Cn (A; B),
2
n=0 n!
(5.17)
where Cn (A; B) = [A, Cn−1 (A; B)] is defined by a recurrence relation, Cn=0 (A; B) ≡
B, and
[τ (r2 ), ψ(r1 )] =
Z
d2 r′ θ(r2 − r′ )[ψ † (r′ )ψ(r′ ), ψ(r1 )] = −θ(r2 − r1 )ψ(r1 ),
one finds eventually:
e−iφ̃τ (r2 ) ψ(r1 )eiφ̃τ (r2 ) = eiφ̃θ(r2 −r1 ) ψ(r1 ).
This yields for expression (5.16)
ψCS (r1 )ψCS (r2 ) = eiφ̃θ(r2 −r1 ) eiφ̃τ (r1 ) eiφ̃τ (r2 ) ψ(r1 )ψ(r2 )
and similarly, interchanging r1 ↔ r2 ,
ψCS (r2 )ψCS (r1 ) = eiφ̃θ(r1 −r2 ) eiφ̃τ (r2 ) eiφ̃τ (r1 ) ψ(r2 )ψ(r1 ).
With ψ(r1 )ψ(r2 ) = −ψ(r2 )ψ(r1 ), θ(r2 −r1 ) = θ(r1 −r2 )+π et [τ (r1 ), τ (r2 )] =
0, one finds
ψCS (r1 )ψCS (r2 ) = −eiφ̃π ψCS (r2 )ψCS (r1 ),
(5.18)
and also
†
†
ψCS (r1 )ψCS
(r2 ) + eiφ̃π ψCS
(r2 )ψCS (r1 ) = δ(r1 − r2 ).
(5.19)
Comparing those expressions to equation (5.13), one sees that φ̃ plays the
rôle of the statistical angle α, and Chern-Simons transformations are found to
allow changing particle statistics. Notice moreover that the choice φ̃ = 2s + 1
transforms fermions into bosons, while φ̃ = 2s does not change the particle
statistics, as is the case for the CF theory discussed above.
82
Chern-Simons Theories and Anyon Physics
5.2.2
Fractional charge and fractional statistics
The topics discussed in the previous section are directly related to the Berry
phase, which is a “geometrical” phase the particle wave function acquires
when the particle is adiabatically displaced along a path in parameter space.
An example of Berry phase is that due to the Aharonov-Bohm phase, which
appears when a particle with charge e∗ follows a path ∂Σ = C enclosing a
surface Σ
e∗ I
e∗ Z 2
Γ=−
dr · ACS (r) = −
d rBCS (r),
(5.20)
h̄ ∂Σ
h̄ Σ
where the gauge field is that of the Chern-Simons transformation. In this
case the Berry phase is an operator, because the “magnetic field” BCS is
proportionnal to the density operator. Within the mean field approximation
equation (5.10), BCS = hφ̃nel /e, one finds
Γ = 2π
e∗
φ̃N (Σ),
e
(5.21)
where N (Σ) is the number of electrons enclosed within the surface Σ.
We now ditinguish three cases:
• In the first case (the most simple one), the particle moving on the path
C around a Laughlin liquid is an electron of charge e∗ = e, it acquires
a phase which is a multiple of Γ = 2π φ̃. When φ̃ is an integer, as is the
case for Laughlin’s theory, this pahse remains a multtiple of 2π.
• When it is a quasi-particle with a fractional charge, e.g. e∗ = e/φ̃, the
acquired phase acquise is again a multiple of Γ = 2π.
• The most interesting case is that when there is one (or a number of)
particle(s) added to the Laughlin liquid within the surface Σ, within
the path followed by the quasi-particle. Remember that the quasi particle at z0 has a wave function, in Laughlin’s theory, which is obtained
Q
by multiplying by the factor j (zj − z0 ). In terms of Chern-Simons
transformations de Chern-Simons, this can be modelled by the transformation
R 2 ′
′
′
UV (r) = eiq d r θ(r−r )ρV (r ) ,
where ρV (r) is the density of quasi-particles with vorticity q = ±1.
For a single quasi-particle at r0 , one would have ρV (r) = δ (2) (r − r0 ).
Just as in the case of the Chern-Simons transformation, in contrast
Statistical Transmutation – Anyons in 2D
83
with the Laughlin (or Jain) wave function, this transformation attaches
a singular phase to the quasi-particle, without attaching the proper
modulus of the zero.
In terms of gauge transformation, one has ACS (r) → ACS (r)−(h̄/e)∇f (r)
when the wave function transforms as ψ(r) → exp[if (r)]ψ(r). The total vector potential is thus
ACS (r) → ACS (r) −
h̄q Z 2 ′
∇ d r θ(r − r′ )ρV (r′ ),
e
and the relationship between the magnetic field and the densities (5.9)
now writes
hq
h
BCS (r) = [∇ × ACS (r)]z = − φ̃ρ(r) + ρV (r).
e
e
(5.22)
The first term gives rise to the same Berry phase Γ as in the case of a
quasi-particle circling around a Laughlin liquid enclosed within Σ, and
the second term adds a phase ∆Γ dueto the presence of possible quasiparticles within Σ. Suppose that we have exactly one quasi-particle at
position rV ∈ Σ, ce qui donne
e∗
q
e∗ Z 2 hq
,
d r δ(r − rV ) = 2π q = 2π
∆Γ =
h̄ Σ
e
e
2s + 1
(5.23)
for the case where e∗ = e/(2s + 1) ( Laughlin quasi-particle). One sees
thus that cahrge fractionalisation generates engendre également une
fractional statistic in 2D, since the statistical angle associated with the
Berry phase ∆Γ is
q
e∗
.
(5.24)
α=q =
e
2s + 1
Looking for experimental evidence for fractional statistics is an active field
of research to this day. The quasi-particle fractional charge has already been
observed in tunneling experiments [45] as weel as in shot noise experiments
[46]. In the latter, one brings the two Hall bar edges close to one another by
applying an adequate gate voltage on electrodes. When the two edges are
sufficiently close to one another, a quai-particle may be back-scatterd from
one edge to the other , and its charge gives rise to a characteristic shot noise.
The measured quasi-particle charge ν = 1/3 is e/3 [46], and at ν = 2/5 their
charge is e/5.
84
Chern-Simons Theories and Anyon Physics
Because of the close relationship between fractional charge and fractional
statistics, it is no easy task to find separate evidence for each. Various devices
have been recently proposed for ν = 5/2, which seems to correspond to a
state with non abelian statistics [48]. A discussion of these ideas is beyond
the scope of these lectures.
Chapter 6
Hamiltonian theory of the
Fractional Quantum Hall Effect
The theory of FQHE follows nowadays distinct and complementary paths.
Following Laughlin [12, 20], one appoach concentrates on writing down manybody wave functions, and studying their properties by numerical means, such
as exact diagonalisations, Monte-Carlo computation, Density Matrix Renormalization Group, and so forth. However, the most powerful computer to
date cannot easily handle the large Hilbert space involved with more than
12 particles. Tricks may allow to simulate up to 24 particles. Such methods
often allow to reach definite conclusions about the relative stability of states
with different symmetries. Sometines, however, the thermodynamic limit is
not available, and doubts linger about the final conclusions of computations
on small size particles clusters. It is useful, both for a more transparent
understanding of the physics at hand, and for possible comparison with numerics, to be able to conduct analytical approaches to the theory in the
thermodynamic limit. Even though the accuracy may be less satisfactory
than for exact results on small number of particles, it is interesting to have
a theory where finite size effects do not blur the conclusions. Such an analytical theory in the thermodynamic limit is the Hamiltonian approach, the
first challenge of which is to attack the degeneracy problem.
We have discussed some aspects of Chern-Simons theories in the previous
chapter. We notice that the transformation (5.2) deals with the kinetic part
of the Hamiltonian only, not with the interaction term, which is invariant
under Chern-Simons transformations. However, as we discussed in chapter
4, the FQHE is due to a lifting of the fractionally occupied Landau level
85
86
FQHE Hamiltonian theory
degeneracy by interactions among electrons. In fact the model of electrons
restricted to a single LL occupation [equation (4.17)] involves the interaction
term only. There seems to be a contradiction in the theory. This criticism to
Chern-Simons theory is however too severe. Indeed, the Chern-Simons theory
aims at replacing the true repulsive Couomb interaction by a statistical
interaction expressed by the concept of flux attachment to electrons. This
generates a singularity in the N particle wave function when two particles
attempt to sit at the same point in space [see equation (5.12)]. The physical
notion behind the Chern-Simons manipulation is that the non perturbative
many-body effect of Coulomb interactions is to generate collective effects in
the form of flux tubes attached to electrons, in such a way as to renormalize
the effective external magnetic field. The latter renormalisation results in
the degeneracy lifting for fractional filling of the LLL which is a key factor to
account for the FQHE. Chern-Simons theories have thus to be credited with
an interesting step forward.
A difficulty remains however in the mean field version (5.10) which renormaises the magnetic field, B → B ∗ = B(1 − φ̃ν): the energy separation
between the new levels (LL∗ ) is h̄ωC∗ = h̄eB ∗ /m. This energy scale which
involves the electronic mass m cannot be correct. The physics imposes (4.17)
the energy scale e2 /ǫlB , which does not depend on m. It is thus appropriate
to seek a more satisfatory theory beyond mean field, which should yield the
correct energy scale in a natural fashion. Various approaches have been proposed such as a random phase approximation [16, 49] which renormalises the
mass m → m∗ to get the right energy scale in the limit ν → 1/2 (p → ∞).
An alternative approach, which is discussed in this chapter, is the FQHE
Hamiltonian theory [50, 51, 105]. We shall concentrate on the formulation
due to Murthy and Shankar [51].
6.1
Miscroscopic theory
This section deals with the connection between Chern-Simons theories and
the effective model described by equation (4.17), in the framework of a microscopic formulation of the Hamiltonian theory. The main focus is on the
treatment of the fluctuations of the vector potential, (4.17), using the a new
theoretical tool, i.e. a new quantum gauge field a(r). The latter object
amounts to resorting to new unphysical degrees of freedom, which have to be
removed in a suitable way, i.e. by imposing constraints on the new system.
Microscopic theory
87
The various steps involved in this approach are fairly involved technically, so
we try and provide as much physical insight as possible. Readers with less
interest in the mathematical details may directly go to section 6.2 where the
end product, the effective theory is discussed in simpler terms. That section
does not rely on the microscopic theory developped in the present one.
6.1.1
Fluctuations of ACS (r)
Using first quantisation 1 the Chern-Simons hamiltonien (5.5) is written as
a sum over particles j = 1, ..., Nel
ĤCS =
1 X
[pj + eA∗ (rj ) + eδACS (rj ) + ea(rj )]2 .
2m j
(6.1)
The mean value of the Chern-Simons vector potential has been absorbed
in an effective “external” vector potential, A∗ (r) = A(r) + hACS (r)i, and
fluctuations δACS (r) have been singled out. They are connected to density
fluctuations through equation (5.9). Equivalently, in Fourier space, (h̄ ≡ 1)
δACS (q) =
2π φ̃
δρ(q)e⊥ ,
e|q|
(6.2)
where δρ(q) = j exp(−iq·rj )−nel and e⊥ = iez ×ek , with ek = q/|q|2 . Although the Hamiltonian 6.1looks like that of a non interacting particle Hamiltonian, the presence of δACS (q) introduces the full many body character of
the problem because δρ(q) depends on all electron co-ordinates. The last
term in equation (6.1) represents a new transverse gauge field, ∇ · a(r) = 0,
which has been introduced, for the time being, artificially. Its quantum character, which corresponds to the quantum character of the operator δρ(q) is
ensured by the introduction of its conjugate (longitudinal) field P(r), with
the canonical commutation rules
P
[a(q), P (−q′ )] = iδq,q′ ,
(6.3)
where, in Fourier space
a(q) = a(q)e⊥ ,
1
2
P(q) = iP (q)ek .
We use here for simplicity, a first quantisation approach.
δACS (q) is indeed transverse to q in order to recover the correct direction for its curl
88
Hamiltonian theory of the FQHE
We have artificially enlarged the Hilbert space with the degrees of freedom
of this new gauge field. The physical sub-space is that of states |ϕphys i which
are annihilated by a(q),
a(q)|ϕphys i = 0.
(6.4)
What is the advantage of this formal operation? Just as momentum px is
the translation generator in direction x (see section 2.2.2), the conjugate field
P(q) is a translation generator for vector potentials. Therefore one may use
the transformation
P
′
′
(6.5)
Ua = ei q′ P(−q )δACS (q )
to get rid of the Chern-Simons vector potential fluctuations by a vector potential translation
Ua† a(q)Ua = a(q) − δACS (q),
Ua† a(r)Ua = a(r) − δACS (r),
(6.6)
The latter equation is derived using (6.3) and Baker-Hausdorff formula(2.41).
Because of the constraint (6.2), the transformation is also under the action
of the gradient operator(the momentum in r representation), since

Ua = exp i
2π φ̃ X
e
P (−q′ )
q′
One gets therefore

1 
|q′ |
X
j

′
e−iq ·rj − nel  .
[−i∇j + eA∗ (rj ) + eδACS (rj ) + ea(rj )] Ua
= Ua
"
= Ua
"
2π φ̃ X
1
−i∇j + eA (rj ) + ea(rj ) +
∇j
P (−q) e−iq·rk
e
|q|
k
∗
#
#
2π φ̃
−i∇j + eA (rj ) + ea(rj ) +
P(rj ) ,
e
∗
and for the transformed Hamiltonian
ĤCP
"
#2
1 X
2π φ̃
P(rj )
=
pj + eA∗ (rj ) + ea(rj ) +
2m j
e
.
(6.7)
The index CP means that the Hamiltonian is transformed in a basis of Composite Particles. In second quantized notation, fields ψCS (r) = Ua ψCP (r).
Notice that the constraint (6.4) is also transformed to
Microscopic theory
89
"
#
2π φ̃
a(q) −
δρ(q) |ϕphys i = 0.
e|q|
(6.8)
The physical interpretation of this constraint is as follows: instead of treating density fluctuations exactly (or, equivalently, those of the Chern-Simons
vector potential), one describes them by the new quantum field a(q).3 Since
a(q) · P(q′ ) = 0 (remember that a(q) is a transverse field and P(q) is a
longitudinal one), the last two terms of the Hamiltonian (6.8) represent a
harmonic oscillator coupled to the sector of charged particles in a field B ∗ ,
which is the meaning of the first two terms . To be sure, the Hamiltonian
contains three terms, ĤCP = ĤB ∗ + Ĥosc + Ĥcoupl : that of charged particles
in an effective magnetic field B ∗ (this is the expression of the Chern-Simons
theory at the the mean field approximation level ,
ĤB ∗ =
1 X
[pj + eA∗ (rj )]2 ,
2m j
(6.9)
that of a harmonic oscillator, which represents the density fluctuations
Ĥosc =
and a coupling term
Ĥcoupl
1 X
2m
j

e2 a2 (rj ) +
2π φ̃
e
!2

P 2 (rj ) ,
(6.10)
#
"
1 X
2π φ̃
=
P(rj ) .
[pj + eA∗ (rj )] · ea(rj ) +
m j
e
(6.11)
Let us first neglect the coupling term, and discuss the model with the first
two terms only. The diagonalization of the Hamiltonian with the coupling
term will be treated in the next section. Consider the harmonic oscillator
P
term. It can be re-written, using ρ(r) = j δ(r − rj ) = nel + δρ(r),
Ĥosc
3

1 Z 2
2π φ̃
=
d rρ(r) e2 a2 (r) +
2m
e
!2

P 2 (r)
This procedure is not new in theoretical physics: in the context of path integrals,
there is some similarities with the Hubbard Stratonovich transformation. The theory of
one-dimensional electron sytems resorts to bosonisation, where bosons represent charge or
spin fluctuations. Indeed, Murthy and Shankar themselves were influenced, in their own
words, by the treatment of plasmons in an interacting Fermi liquid, as initially proposed
by Bohm and Pines [53].
90
Hamiltonian theory of the FQHE

nel X  2 2
2π φ̃
≃
e a (q) +
2m q
e
!2

P 2 (q) .
In the last step, we have neglected terms of order 3 in density fluctuations,4
which are taken into account at the level of the harmonic approximation
O(δρ(q)2 ). Now introduce the ladder operators for the harmonic operator
which obey
"
#
2π φ̃
ea(q) + i
A(q) ≡ q
P (q) ,
e
4π φ̃
1
h
i
A(q), A† (q′ ) = δq,q′ .
The Hamiltonian (6.10) now writes
Ĥosc = ω0
X
q
(6.12)
(6.13)
A† (q)A(q),
(6.14)
with the characteristic frequency
ω0 = 4π φ̃
nel
= ν φ̃ωC .
2m
(6.15)
The oscillator ground state is the usual gaussian
χosc
e2 X 2
a (q) ,
= exp −
4π φ̃ q
!
(6.16)
up to a normalisation factor.
In the absence of a coupling term, the N particles wave function is the
product of the oscillator wave function, and that of N charged particles in
a field B ∗ : ψCP ({rj }) = χosc ({rj })φp ({rj }), where φp ({rj }) is unique (non
degenerate) for ν ∗ = p. Due to the constraint (6.8), the oscillator wave
function can be written as:
χosc
"
#
φ̃ X
2π
= exp −
δρ(−q) 2 δρ(q) .
2 q
|q|
The exponent can be regarded as a Hamiltonian for charged particles interacting with a 2D Coulomb potential, ṽ(q) = 2π/q 2 ↔ v(r) = − ln |r|. We
4
Remember that a(q) ∼ δρ(q),because of the constraint (6.8).
Microscopic theory
91
recover here the 2D single component plasma , as discussed in section 4.2.
This leads in real space to the expression (with lB ≡ 1)
χosc ({rj }) = e
φ̃
2
= C
P
j,k
Y
k<j
R
d2 rd2 r ′ [δ(r−rj )−nel ] ln |r−r′ |[δ(r′ −rk )−nel ]
|rk − rj |φ̃ e−φ̃ν
P
k
|rk |2 /4
,
(6.17)
where C is a proportionnality constant , and we used nel = ν/2π. Notice [see
equation (5.12)], that the singular phase has already been attached through
the Chern-Simons transformation and the the N electrons wave function thus
writes, in complex coordinates notation ,
ψ({zj }) =
Y
k<j
(zk − zj )φ̃ e−φ̃ν
P
k
|zk |2 /4
φp ({zj }).
(6.18)
When φ̃ = 2s, i.e. for fermionic statistics we retrieve the Laughlin (4.30), as
well as Jain’s wave function (4.66), up to the LLL projection. The last step,
in order to complete the comparison to those wave functions, requires the
correct magnetic length to be inserted in the gaussian. To this end, notice
that φp ({zj }) represents the N partciles wave function for an integer filling
∗
∗
factor ν ∗ = p in a field B
q = [∇ × A (r)]z . The characteristic magnetic
∗
length inφp ({zj }) is lB
= h̄/eB ∗ , the gaussian factor in the wave function
is thus
!
"
#
X |zk |2
X |zk |2
= exp −(1 − ν φ̃)
,
exp −
∗2
k 4lB
k 4lB
2
∗2
where we used lB
/lB
= B ∗ /B = (1 − ν φ̃). In the end, the gaussian factor in
the electronic wave function (6.18) is then
exp −
X |zk |2
k
2
4lB
!
,
as expected for electrons in a magnetic field B.
6.1.2
Decoupling transformation at small wave vector
We are left now with the coupling term between particles (6.9)and oscillators
(6.10), the latter term representing collective density modes q. We only
sketch here the main steps of the decouplig derivation for small wave vectors,
92
FQHE Hamiltonian theory
|q|lB ≪ 1. The interested reader is refered for more details to the MurthyShankar review, especially to cond-mat/0205326v2.
The total Hamiltonian (without the interaction term), now writes
ĤCP =
X Πj,+ Πj,−
j
2m
+ ω0
X
q
A† (q)A(q) + θω0
q
Xh
i
c† (q)A(q) + c(q)A† (q) ,
q
(6.19)
where θ = π φ̃/4πnel , c(q) = j q̂− Πj,+ exp(−iq · rj ) et Πj,± = pj,± +
eA∗± (rj ). Vector operators are written here in complex notation, V± = Vx ±
iVy , et q̂± = ek · ex ± iek · ey is the unit vector in the direction of q, in complex
notation. To the usual hrmonic oscillator commutation rules (6.13), we have
to add
[Πi,− , Πj,+ ] = 2eB ∗ δi,j
(6.20)
P
and
[c(q), c† (q′ )] = −2eB ∗
X
j
′
e−i(q−q )·rj + O(q) ≃ −2eB ∗ δq,q′ ,
(6.21)
where the last equation is analogous to a Random Phase Approximation,
which is again equivalent to neglecting terms of order 0(q3 ): density fluctuations are taken care of at the gaussian approximation level. Corrections to
equation (6.21) would be of higher order than O(δρ(q)2 ).
In Hamiltonian (6.19), the coupling between particles described by operators Πj,± [or c(q) et c† (q)], and the oscillator fields A(q) or A† (q) is
linear. The canonical transformation which decouples the Hamiltonian has
the following form:
iλS0
U (λ) = e
(
= exp λθ
Xh
q
†
†
)
i
c (q)A(q) − c(q)A (q)
,
(6.22)
Where we want eventually to chose the “flow” parameter λ in a convenient
way, leaving it undetermined for the time being. An operator transformed
through (6.22), Ω(λ) = exp(−iλS0 )Ω(λ = 0) exp(iλS0 ), may be derived from
the derivative
dΩ
(6.23)
= −ie−iλS0 [S0 , Ω] eiλS0 .
dλ
For operators c(q) and A(q), which occur in Hamiltonian (6.19), this leads
to the flow equations
dA(λ, q)
= −θc(λ, q)
dλ
et
Microscopic theory
93
dc(λ, q)
µ2
= − A(λ, q),
dλ
θ
with µ2 ≡ 2eB ∗ nel θ2 = 1/2ν ∗ , integrated, with initial conditions c(q) =
c(λ = 0, q) et A(q) = A(λ = 0, q), as
A(λ, q) = cos(µλ)A(q) −
θ
sin(µλ)c(q),
µ
(6.24)
and
µ
sin(µλ)A(q) + cos(µλ)c(q).
(6.25)
θ
This transformation may be interpreted as a rotation in Hilbert space, which
mixes the c(q) degrees of freedom (particles) and A(q) degrees of freedom (oscillators). The transformed Hamiltonian is thus derived inserting
equations(6.24) and (6.25) in Hamiltonian, (6.19), and from the transformation of its first term with the same method (integration of the differential
equation(6.23)). The end result contains many terms, and, even though the
calculation is straightforward, we only show here its global form,
c(λ, q) =
ĤCP (λ) =
X Πj,+ Πj,−
2m
j
+
Xn
α(λ)c† (q)c(q)
(6.26)
q
h
io
+β(λ)A† (q)A(q) + γ(λ) c† (q)A(q) + c(q)A† (q)
.
Decoupling is achieved by chosing λ = λ0 such that the implicit equation
γ(λ = λ0 ) = 0
⇒
tan(µλ0 ) = µ.
(6.27)
is satisfied. The detailed derivation [51] yields moreover
β(λ0 ) = ωC
et
α(λ0 ) = −
1
,
2mnel
for the other parameters in Hamiltonian (6.26). Finally, the decoupled Hamiltonian , adding the interaction term, V [ρ(λ0 , q)](which depends on the transformed density), writes
ĤCP =
X Πj,+ Πj,−
j
+ωC
2m
X
q
−
X eB ∗
1 XX
Πj,+ e−iq·(rj −rk ) Πk,− +
2mnel j,k q
j 2m
A† (q)A(q) + V [ρ(λ0 , q)].
Some remarks are here in order:
(6.28)
94
FQHE Hamiltonian theory
• The oscillator frequency is given by the LL energy difference, ωC , in
agreement with Kohn’s theorem which states that collective density excitations in the limit q → 0 oscillate at this frequency, in the presence
of translation invariant interactions. In the strong field limit, the oscillators represent high energy excitations, which condense at T = 0 in
the lowest energy mode. The fourt term then becomes an unimportant
constant, which can be subsequently ignored.
• The third term indicates that particles posess a magnetic moment
e/2m.
• The sum on wave vectors is limited by the number of oscillators |q|≤Q =
nosc . This introduces a cut in Q at large wave vector. In principle, the
number of operators has not been specified. However, if one focuses on
diagonal terms j = k in the sum in the second term of equation (6.28),
this choice influences the effective mass m/m∗ = (1 − nosc /nel ) in
P
X Πj,+ Πj,−
j
2m∗
,
which lumps together the first and the second term. The natural choice
seems to be nosc = nel , which results in the vanishing of the kinetic
enrgy term (apart from the non diagonal term j 6= k). This is what
one expects for the dynamics of electrons projected on a single LL
[equation (4.17)]. The choice nosc > nel would lead to the unphysical
result of a negative effective mass, and nosc < nel would not make it
vanish. Another justification for the choice nosc = nel (ou Q = kF )will
be given later on, in the discussion of the effective theory (section 6.2).
• There remain non diagonal terms (j 6= k) in the kinetic part. With the
cut at Q = kF , those can be rewritten, using
X
|q|<Q
e−iq·(rj −rk ) = δ(rj − rk ) −
X
e−iq·(rj −rk ) ,
|q|>Q
and thus as a sum of a term which is zero for j 6= k and a term which is
only relevant at large wave vectors |q| > Q, and which may be neglected
at small wave vectors, |q|lB ≪ 1. Within this approximation, there is
no kinetic term any more.
6.2. EFFECTIVE THEORY AT ALL WAVE VECTORS
95
Eventually, the Hamiltonian (6.28), which describes the low energy dynamics, becomes
X eB ∗
ĤCP (λ) =
+ V [ρ(λ0 , q)].
(6.29)
j 2m
This result is fairly satisfactory, since it only contains the interaction term,
apart from a constant which plays no dynamical role. To complete the discussion, one must compute the transformed density, ρ(λ0 , q), as well as the
constraint, the form of which is also alterd by the transformation. The density is derived following the same procedure as for c(λ, q) and A(λ, q),, from
the integration of the differential equation (6.23), while the constraint is given
by
i
e|q| h
A(λ, q) + A† (λ, −q) .
ρ(λ, q) = q
4π φ̃
The final result [51] is, to lowest order in |q|lB
ρ(λ0 , q) =
X
j
2
e−iq·rj 1 − ilB
i
q ∧ Πj
c|q| h
A(q) + A† (−q) ,
+q
1+c
4π φ̃
(6.30)
where the parameter c is connected to the decoupling parameter, at ν ∗ = p,
c2 = cos2 (λ0 µ) =
pφ̃
.
pφ̃ + 1
(6.31)
As regards the constraint, one gets
"
#
2 q ∧ Πj
.
χ(λ0 , q)|ϕphys i = 0,
with χ(λ0 , q) =
e−iq·rj 1 + ilB
c(1 + c)
j
(6.32)
The constraint does not involve oscillators A(q), only particles. That is a
necessary condition for a complete decoupling of Hamiltonian (6.29). The
density operators (6.30) however still contain a contribution from oscillators,
but the latter vanish on the average when they condense in the ground state
hA(q)i = hA† (q)i = 0, as we assume is the case in the following.
X
6.2
Effective theory at all wave vectors
The connection with the model (4.17) becomes even clearer when one attempts to construct a theory at all wave vectors, which coincides with the
96
FQHE Hamiltonian theory
small wave vector theory in the limit |q|lB ≪ 1. This generalisation is based
on a rather daring piece of reasoning: suppose that expressions ( 6.30) and
(6.32) represent the first term in the expansion of an exponential. We would
then have 5
ρ̄(q) =
X
(e)
e−iq·Rj
et
χ̄(q) =
j
X
(v)
e−iq·Rj ,
(6.33)
j
with
Πy
Πx
= x+
,y −
1+c
1+c
!
Πy
Πx
R
et R = x −
,y +
.
c(1 + c)
c(1 + c)
(6.34)
The components of those new operators satisfy the commutation rules
(e)
h
i
2
X (e) , Y (e) = ilB
,
(v)
et
h
i
X (v) , Y (v) = −i
2
lB
.
c2
(6.35)
A comparison with equation (2.16) shows that we may interpret R(e) as the
guiding center of an electron while R(v) seems to be that of a second particle
with charge −c2 , in terms of the electronic charge. The associated densities
are automatically projected on the LLL, and the final model is 6
Ĥ =
1X
v0 (q)ρ̄(−q)ρ̄(q),
2 q
(6.36)
with the commutation rules for densities
!
q∧k
[ρ̄(q), ρ̄(k)] = 2i sin
ρ̄(q + k),
2
!
q∧k
χ̄(q + k),
[χ̄(q), χ̄(k)] = −2i sin
2c2
[χ̄(q), ρ̄(k)] = 0
et
χ̄(q)|ϕphys i = 0.
(6.37)
This is the same model as that discussed in section 4.1.2, where one has added
the constraint associated with the density χ̄(q) and its quantum algebra.
5
The bars over the symbols mean that we are dealing with generalised densities at all
wave vectors.
6
we discuss here the LLL, n = 0.To generalise the Hamiltonian theory to higher LL,
one needs only substituting the effective potential, v0 (q) → vn (q).
Effective theory
97
ez x Π lB2
y
(e)
R
(v)
R
x
Figure 6.1: Composite Fermions in teh Hamiltonian theory.
What is the physical content of the model, in the framework of CF picture, supposing that χ̄(q) describes the density of a second species of particle,
which we will call “ pseudo-vortex”? The “pseudo” in this expression indicates that this particle lives in a larger Hilbert space, and that projection
to the physical space is necessary, which is the case for states which are
annihilated by χ̄(q). Moreover, this particle, which does not appear in teh
Hamiltonian, has no dynamics.
• The electron and the pseudo-vortex guiding centers are at a distance
2
∼ |Π|lB
∼ lB away from one another, which gives rise to a dipolar
2
moment d = −eez × ΠlB
(figure 6.1).
• The CF may be pictured as a composite of an electron and a pseudovortex. Notice that the latter has been indirectly introduced by the
oscillator degrees of freedom , a(q) or A(q). The choice nosc = nel , as
we discussed earlier, may thus be interpreted as equating the number
of electrons and the number of pseudo-vortices. As a result there are as
many CF as electrons. The CF charge is the sum of the elctron charge
and of the pseudo-vortex one, , i.e. e∗ /e = −(1 − c2 ).
• The pseudo-vortex is an excitation of the electron gas, which is composed of the true elementary excitations. This can be checked on equation (6.34): the guiding centers of both particle species are expressed
98
FQHE and the Hamiltonian theory
in terms of electronic co-ordinates x, y and Πx , Πy . This is the physical
meaning of the constraint
χ̄(q)|ϕphys i = 0.
Notice that we could have started with the model (4.17) (electrons
restricted to a single LL). Instead of resorting to the oscillator gauge
field a(q) and its conjugate field as additional variable, we could have
introduced the field χ̄(q) in the effective model (4.17). The microscopic
theory discussed in the previous section is nevertheless useful, because
it has established the connection with Chern-Simons theories.
• The limit p → ∞ or ν → 1/2 yields a charge c2 = 1 for the pseudovortex. The CF at ν = 1/2 are electrically neutral, but they have a
2
dipolar moment d(ν = 1/2) = −ekF lB
[105, 54].
6.2.1
Approximate treatment of the constraint
In spite of the relative formal simplicity of the Hamiltonian theory (6.37), it
is difficult to treat the constraint explicitly in computations. For a simpler
treatment, Murthy et Shankar proposed a ”short-cut” which amounts to
replacing the projected density by a ”preferred combination”
ρ̄CF (q) = ρ̄(q) − c2 χ̄(q),
(6.38)
in the Hamiltonian. A priori, all combinations such as ρ̄γ (q) = ρ̄(q) − γ χ̄(q),
with arbitrary γ are equivalent because of the constraint. The advantage of
chosing γ = c2 is that the matrix elements of the projected density operator
obey hN |ρ̄CF (q)|0i ∝ q 2 in the limit q → 0. This is the condition for the
P
structure factor S(q, ω) = N |hN |ρ̄CF (q)|0i|2 δ(ω − EN ) to vary as q 4 at
small wave vector, which is demanded by the LLL projection [35, 91].
Notice that the preferred combination short-cut is a valid approximation
for gapped states, such as occur at ν = p/(2sp + 1). The constraint must
be explicitly dealt with, however, in the sudy of ν = 1/2, which has a compressible strange Fermi liquid ground state [16]. Another problem with the
preferred combination is that the Hamiltonian no longer commutes strictly
with the pseudo-vortex density. The model remains however weakly gauge
invariant because the commutator vanishes in the sub-space defined by the
constraint.
Microscopic theory
99
There is another algebraic argument in favor of the preferred combination,
i.e. γ = c2 : with this choice, the ρ̄(q) algebra (6.37) is correctly reproduced
to lowest order in q,
h
i
ρ̄CF (q), ρ̄CF (k)
≃ i(q ∧ k)ρ̄CF (q + k)
!
q ∧ k CF
≃ 2i sin
ρ̄ (q + k) + O(q 3 , k 3 ).
2
Higher order terms in q are suppressed in the Hamiltonian, because of the
gaussian in the effective potential (4.18). From now on, the ρ̄CF (q) operator
is interpreted as the CF density. Physically, modes associated with internal
CF structure are neglected in this approximation.
The CF basis is introduced by transforming variables R(e) et R(v) in CF
guiding center , R(CF ) , and cyclotron variable, η (CF ) . This transformation
must ensure that the new variables
satisfy commutation rules in terms of the
√
∗
CF magnetic length, lB
= 1/ 1 − c2 ,
h
i
∗2
ηx(CF ) , ηy(CF ) = −ilB
h
et
i
∗2
X (CF ) , Y (CF ) = ilB
,
(6.39)
in analogy with the electronic variables (2.16). The appropriate transformation is [51]
R
R(e) − c2 R(v)
et
=
1 − c2
R(e) = R(CF ) − η (CF ) c
⇔
c (e)
(v)
R
−
R
1 − c2
(v)
R = R(CF ) − η (CF ) /c. (6.40)
η (CF ) =
(CF )
et
As in the electronic basis (section 4.1), the CF density operator may be
written in second quantized form as
ρ̄CF (q) =
X
j,j ′ ;m,m′
hm|e−iq·R
(CF )
|m′ ihj|ρ̄p (q)|j ′ ic†j,m cj ′ ,m′ ,
(6.41)
where states |ji are associated to the operator η (CF ) and states |mi to R(CF ) .
The first matrix element in this expression is identical to that (4.9) deduced
in section 4.1.1, in terms of the CF magnetic length, for m ≥ m′ ,
−iq·RCF
hm|e
′
∗ |2 /4
−|qlB
|m i = e
s
m′ !
m!
∗
−iqlB
√
2
!m−m′
m−m′
Lm
′
∗ 2
|qlB
|
.
2
!
100
FQHE Hamiltonian theory
The second one writes , for j ≥ j ′ ,
hj|ρ̄p (q)|j ′ i ≡ hj|eiq·η
s
(CF ) c
− c2 f˜(q)eiq·η
(CF ) /c
!j−j ′
|j ′ i
∗
iq̄lB
c
∗ 2 2
√
e−|qlB | c /4
=
2
"
!
∗ 2
′
|qlB
c|
′
2
2
j−j ′
× Lj ′
− c2(1−j+j ) e−|q| /2c Ljj−j
′
2
j ′!
j!
(6.42)
∗ 2
|qlB
|
2
2c
!#
.
The gaussian f˜(q) = exp(−|q|/2c2 ) in the second term takes into account
the pseudo-vortex magnetic length, which is different from the electronic one.
The operators c†j,m and cj,m , with {cj,m , c†j ′ ,m′ } = δj,j ′ δm,m′ and {cj,m , cj ′ ,m′ } =
0, are respectively the CF creation and annihilation operators in state |j, mi =
|ji ⊗ |mi.
Because of the commutation rule (6.39) for the guiding center co-ordinates,
∗2
each state occupies a minimal surface 1/nB ∗ = 2πlB
, in analogy with the
electronic case. There exist thus AnB ∗ states per “ CF LL” j, and the filling
factor for CF LL, ν ∗ = nel /nB ∗ is connected to the electronic filling factor
through equation (4.68), ν = ν ∗ /(2sν ∗ + 1). When p CF levels are filled ,
ν ∗ = p, the ground state can be described by the average
hc†j,m cj ′ ,m′ i = δj,j ′ δm,m′ Θ(p − 1 − n),
(6.43)
with Heavyside function Θ(x) = 1 for x ≥ 0 and Θ(x) = 0 for x < 0. This is
no small progress. There was no way in the electron basis to define a starting
state from which a perturbation treatment might be conducted, except for
ν = n. In contrast, we only need ν ∗ = p, i.e. ν = p/(2sp + 1), in the CF
model.
6.2.2
Energy gaps computation
The ground state (6.43) can now be used to compute various physical quantities, such as quasi-particle gaps or activation gaps[51]. This is a simple task,
once the model is established. As the quasi-particle is a CF added in level p
when p CF LL are completely filled, its energy relative to the ground state
is
∆qp (s, p) = hcp,m Ĥc†p,m i − hĤi
=
(6.44)
p−1
X
X
1X
v0 (q)hp|ρ̄p (−q)ρ̄p (q)|pi −
v0 (q)
|hp|ρ̄p (q)|ji|2 ,
2 q
q
j=0
Effective theory
101
where averages, defined with respect to the ground state are computed as
Wick’s contractions(6.43). Similarly, the CF quasi-hole energy is in level
p−1
∆qh (s, p) = hc†p−1,m Ĥcp−1,m i − hĤi
1X
= −
v0 (q)hp − 1|ρ̄p (−q)ρ̄p (q)|p − 1i
2 q
+
X
v0 (q)
q
p−1
X
j=0
(6.45)
|hp − 1|ρ̄p (q)|ji|2 .
The activation gap , i.e. the energy needed to create a non interacting quasiparticle/hole pair,is
∆a (s, p) = ∆qp (s, p) + ∆qh (s, p).
(6.46)
The figure 6.2 shows the results for the activation gaps, compared to numerical computations, taking into account a finite sample width in direction
z. The finite width alters the effective potential, v0 (q) → v0 (q)f (q), where
various prescriptions have been put forward for the correcting factorf (q) :
Zhang and Das Sarma haved used Yukawa type potential, which leads to
f (q) = e−qλ ,
(6.47)
where λ is the width parameter [55]. In this picture two particles cannot get
closer
√ than this width: The distance r in the Coulomb potential is replaced
by r2 + λ2 . Alternatively , on may simulate the finite width by a parabolic confinement potential, which leads to a gaussian exp(−z 2 /4λ2 ) which
multiplies the wave function [56], and the correcting factor becomes
f (q) = eq
2 λ2
[1 − Erf(qλ)] ,
(6.48)
where Erf(x) is the error function. The general tendency of the factors which
take into account the sample finite width is to cause a lowering (in modulus)
of the characteristic Coulomb energy. This leads to a lowering of activation
gaps (se figure 6.2), and also of quasi-particle or quasi-hole gaps. Compared
to numerical results, the Hamiltonian theory overestimates activation gaps
by a factor 1, 4 to 2 for λ = 0, but the agreement becomes better for larger
λ values.
102
FQHE Hamiltonian theory
gaps d’activation
(a)
δ
0.25
'PMJ p=1'
'Hamiltonian Theory p=1'
'PMJ p=2'
'Hamiltonian Theory p=2'
0.20
0.15
0.10
0.05
0.00
0.0
0.5
1.0
1.5
2.0
2.5
3.0
gaps d’activation
λ
70x10
-3
'PMJ p=3'
Hamiltonian Theory p=3'
'PMJ p=4'
Hamiltonian Theory p=4'
60
50
40
δ
30
20
10
0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
λ
100x10
Activation Gaps
gaps d’activation
(b)
2/5
-3
Exact diag. 2/5
This theory 2/5
Exact diag. 3/7
This theory 3/7
80
60
40
20
0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
paramètre
de largeur
Width
parameter
b
Figure 6.2: After reference [51]. Activation gaps as functions of the sample width in
direction z, in units of e2 /ǫlB . (a) comparison between Hamiltonian theory and numerical
computation by Park et al. in the framework of Jain’s functions [57]. A Yukawa type
potential has been used to simulate the finite width [equation (6.47)]. (b) comparison
to exact diagonalisation results [56], in the approximation of a parabolic confinement
potential [equation (6.48)].
Effective theory
103
ν = 4/11
ν* = 1+1/3
Figure 6.3: Second generation Composite Fermions at ν ∗ = 1 + 1/3 (ν = 4/11). The
CF2 , formed in the partially filled CF level, is a bound state of a “first geberation” CF
and a CF vortex carrying two flux quanta. The CF in the lowest level are inert .
6.2.3
Self similarity in the effective model
Until now, we have focused the discussion on correlated electrons at filling
factors ν = p/(2sp + 1). We have seen that CF theory allows to understand
the FQHE at those filling factors in terms of quasi-particles, which are CF, in
the mean field approximation of the Hamiltonian theory. Indeed the latter
allows to discuss a non degenerate ground state at ν ∗ = p , which has p
completely filled CF levels. This was not possible within the electron model,
because of the huge degeneracy in the lowest LL. In that sense, the FQHE
of electrons may be interpreted as a CF IQHE at ν ∗ = p.
It is natural to ask what is the situation when the CF LL themselves have
fractional filling at ν ∗ 6= p – here again we face the huge degeneracy problem
in the CF model. How is this degeneracy lifted by the residual interactions
between CF? The motivation behind the question is the discovery of a FQHE
at a fraction ν = 4/11, which corresponds to a CF filling factor ν ∗ = 1 + 1/3.
Assume, as is evidenced by experiments that the state is fully spin polarised:7
In that case the first excited CF LL is 1/3 filled, and it is tempting to interpret
7
In the case of a partially spin polarised state a FQHE at ν = 4/11 may also be
understood in the framework of Halperin’s wave function [19].
104
FQHE Hamiltonian theory
this state in terms of a FQHE (a Laughlin state) of CF. The CF in the level
p = 1 would bind to a CF vortex carrying two additional flux quanta, giving
rise to a “second generation “ CF, see figure 6.3), in analogy with the CF
formation in the electron basis and the pseudo-vortex of the electronic liquid.
The Hamiltonian theory is a perfect framework to treat the case of a
partial filling of interacting CF level. Formally we may use the same approximations as in the deduction of the CF model in the electron basis restricted
to a single LL(section 4.1), i.e. only excitations in the same LL are taken
into account, and inter CF LL excitations belong to a higher energy sector.
This approximation is somewhat less justified in the last case because the
only relevant energy scale is e2 /ǫlB , both in the CF LL formation and in
the residual interactions. In fact the justification of this approximation will
appear later on, due to the appearance of a “small parameter” due to the
charge renormalisation. The restriction to the CF LL then yields for the
density operator
CF
¯
ρ̄CF
(6.49)
p (q) = Fp (q)ρ̄(q),
where
ρ̄¯(q) =
X
m,m′
hm|e−iq·R
CF
|m′ ic†p,m cp,m′
(6.50)
is the projected density operator of CF, and
FpCF (q) ≡ hp|ρ̄p (q)|pi,
(6.51)
in terms of matrix elements (6.42), is the “CF form factor” of level p. As
for the electronic form factor(4.12), it can be absorbed in the effective CF
interaction potential, which leads to
h
i2
CF
vs,p
(q) = v0 (q) FpCF (q)
=
∗2 2
2πe2 −q2 l∗2 /2
q 2 lB
c
Lp
e B
ǫq
2
"
(6.52)
!
− c2 e−q
2 /2c2
Lp
Then one finds for the interacting CF Hamiltonian
1 X CF
v (q)ρ̄¯(−q)ρ̄¯(q).
Ĥ CF =
2 q s,p
∗2
q 2 lB
2c2
!#2
.
(6.53)
As in the electronic case the commutator (6.39) between guiding center
components R(CF ) induces the operator algebra for ρ̄¯(q),
∗2
(q ∧ k)lB
¯
¯
[ρ̄(q), ρ̄(k)] = 2i sin
ρ̄¯(q + k).
2
!
(6.54)
Effective theory
105
This together with Hamiltonian (6.53), defines the interacting CF model.
The latter has the same structure as that for electrons restricted to a single
level– one has to replace the effective interaction potential by the potential
CF
∗
vs,p
(q) and to use the CF magnetic length, lB
. This self-similarity at the
level of the model structure can account, under certain conditions, for the
experimental Hall curve self-similarity, which was initially noticed by Mani
and v. Klitzing, using a scaling transformation [59]. The difference between
the spatial variation of the interaction potentials between CF and between
electrons in a LL n (4.18), indicates that this self-similarity of the Hall curve is
not an automatic by-product of the mathematical self similarity of the model.
Because of the latter, one expects the formation of incompressible quantum
liquids which would be the CF analogues of Laughlin liquids. Formally, such
liquids may be dubbed CF2 . The CF2 basis is deduced from the model
(6.53), in the same manner as CF are deduced from the electronic model
(6.36) and (6.37) : the Hilbert space is enlarged with the CF pseudo vortices
¯ (q), which carry 2s̃ flux quanta, and which have a charge
degrees of freedom χ̄
2
¯ (q)|ϕphys i = 0, for the
c̃ = 2p̃s̃/(2p̃s̃ + 1). This leads to a new constraint , χ̄
physical states |ϕphys i. The pseudo-vortex operator components satisfy the
algebra
!
q ∧ k ∗2
¯ (q), χ̄
¯ (k)] = −2i sin
¯ (q + k),
[χ̄
l
χ̄
(6.55)
2c̃2 B
induced by the commutation
rules
h
i for the pseudo-vortex guiding center comv−CF
v−CF
v−CF
∗2 2
ponents R
, X
,Y
= −ilB
/c̃ . In order to describe the CF2 ,
which is built of a first generation CF located at the guiding center R(CF )
and a pseudo-vortex at Rv−CF (se figure 6.3), one introduces a new pre2
¯ (q). The CF2 global charge is thus
ferred combination, ρ̄C F (q) = ρ̄¯(q) − c̃2 χ̄
2 ∗
2
2
ẽ = (1 − c̃ )e = (1 − c̃ )(1 − c ), in units of the electron charge −e. The
2
2
CF2 cyclotron variable and guiding center, respectively η C F et RC F , are
deduced from the CF guiding center and CF pseudo-vortex in the same manner as for CF [see equation (6.40)]. The new preferred combination, which
is interpreted as the CF2 density, is written in second quantized notation
2
ρ̄C F (q) =
X
j,j ′ ;m,m′
hm|e−iq·R
C2F
|m′ ihj|ρ̃p (q)|j ′ id†j,m dj ′ ,m′ ,
(6.56)
where operators d†j,m and dj,m , together with {dj,m , d†j ′ ,m′ } = δj,j ′ δm,m′ and
{dj,m , dj ′ ,m′ } = 0, are respectively the CF2 creation and annihilation operators in state |j, mi. The matrix elements in equation (6.56) are the same
106
FQHE Hamiltonian theory
Pseudo−potentiels m
V
0.4
0.40
0.30
0.3
0.03
électrons
0.02
0.2
0.004
0.01
0.1
0.002
FC
FC2
1
23
5
47
9
m
11
6
13 15
8
17
19
10
Figure 6.4: Pseudo-potentials for the electronic interaction (black curve), for CF (gray
curve) and for CF2 (clear gray curve ), in units of e2 /ǫlB . Notice the scale difference on
the z axis.
√
∗
∗
as q
for CF [Eqs. (6.41) and (6.42)] if we replace lB
→ ˜lB = lB
/ 1 − c̃2 =
1/ (1 − c̃2 )(1 − c2 ), the CF2 magnetic length, and c2 → c̃2 . A Quantum
Hall Effect would then be expected to appear for some CF filling factors
ν ∗ = p + p̃/(2s̃p̃ + 1), where the integer p̃ is the number of completely filled
CF2 LL. Such a QHE may be interpreted both as a CF FQHE [58] or as a CF2
IQHE [60]. The filling factors ν ∗ are related to the electronic filling factors
through relation (4.68), and ν ∗ = 1 + 1/3 corresponds thus to ν = 4/11.
The formalism thus described is generic and may be iterated for the next
CF generations. One finds in that manner a hierarchy of states which is
different from the Haldane and Halperin hierachies [37, 43], at filling factors
which are determined by the recurrence relation
νj = p j +
νj+1
,
2sj+1 νj+1 + 1
(6.57)
where sj+1 is the number of pairs of flux quanta carried by the pseudo-vortex
in the (j +1)-th generation of CF (CFj+1 ), and pj is the number of completely
filled FCj s levels. The FCj s IQHE is determined by νj = pj . Formally, the
electronic filling factor corresponds to j = 0 and thus ν0 = ν and ν1 = ν ∗ .
Equation (6.57) is a generalisation of the relation between electronic filling
factors and those of CF [Eq. (4.68)].
Although the recurrence formula 6.57 for CF hierarchical states suggests
a large number of FQHE, only a limited number are observable in practice.
Effective theory
107
Indeed, the FQHE due to first generation CF is retsricted to the two lowest
LL, n = 0 and 1, while in higher Landau levels, such liquid states compete
with electronic solids, as will be discussed in the next chapter. Apart from
this competition with other phases, one may state certain general stability
criteria for higher generation CF (CFj+1 ). The first one is clearly the stability
criterion for the ”parent” state, (CFj ), a necessary condition. For n = 2, for
example, the Laughlin liquid is not stable at ν̄ = 1/3, and CF2 are not formed
at ν̄ = 4/11. A more restrictive condition for the CFj+1 formation is given
by the form of the effective interaction potential of FCj ,
(j)
v{si ,pi } (q) =
j h
i2
2πe2 −q2 /2 Y
i
e
FsCi ,pFi (qli ) ,
ǫq
i=1
(6.58)
√
in terms of FCj magnetic length lj = 2sj pj + 1lj−1 . Expanded in Haldane
pseudo-potentials [see equation (4.54)],
Vmj =
X
q
(j)
v{si ,pi } (q)Lm (q 2 lj2 )e−q
2 l2 /2
j
,
> 1, 2, for a
the interaction must be sufficiently short ranged , i.e. V1 /V3 ∼
Laughlin state to be stabilized.
Pseudo-potentials with odd indices 8 are plotted in figure 6.4 for electrons
in LL n = 0, cf with s = p = 1 and CF2 with s = p = s̃ = p̃ = 1. Notice
the difference in scale on the energy axis: the interaction between CF is
roughly one order of magnitude smaller than that for electrons. This is easily understood when looking at the effective CF interaction potential (6.52),
which is globally reduced by the CF form factor , [FpCF (q)]2 ≃ (1 − c2 )2 , at
order O(q 0 ), compared to the potential between electrons. As two factors of
this type enter the expression for the CF2 effective interaction potential [see
equation (6.58)], the latter is again an order of magnitude smaller than that
between CF. In this sense, (1−c2 )2 ≤ 1/9 may be interpreted as the hierachical CF theory small parameter – as discussed earlier, this is a posteriori an
indication for the CF LL stability with respect to residual CF interactions.
A second remark is about the specific form of the CF (and CF2 ) interaction. Contrary to the electronic case, their pseudo-potentials do not vary
monotonically but exhibit a minimum at m = 3. A possible origin of this
8
Remember that only odd index pseudo-potentials matter in the case of fully spin
polarised electrons, because of their fermionic statistics (see section 4.2).
108
FQHE Hamiltonian theory
peculiarity is the CF dipolar character, due to their internal structure, as
already discussed at the beginning of section 6.2. Since the pair correlation
function of the Laughlin liquid (with s = 1) is maximum for that value of
the relative kinetic moment, one may expect this pseudo-potential form to
stabilise CF Laughlin liquids.
Chapter 7
Spin and Quantum Hall Effect–
Ferromagnetism at ν = 1
Until now we have by-passed all questions connected to the electron spin.
We have been satisfied with the notion that the Zeeman effect separates
LL in two spin branches, the energies of which are separated by a gap ∆z
(see figure 4.1). Within this picture, there would be no difference between
the IQHE at ν = 2n (both spin branches filled) and that at ν = 2n + 1
(only the lower spin branch filled ) – in both cases a plateau would be due
to localisation of additional electrons. The only difference would be the
magnitude of the excitation gap. Indeed, since ∆z ≃ h̄ωC /70, the two spin
branches are not resolved in weak magnetic fields, for which only the IQHE
at ν = 2n is observed. In that case, the system behaves as if each state was
doubly degenerate, with |n, m; σi (σ =↑, ↓). However this picture turns out
to be incorrect. Interactions between electrons have important effects, even
for ν = 2n + 1. A new form of quantum magnetism arises. That is the topic
of this chapter.
7.1
Interactions are relevant at ν = 1
Let us start with a discussion of the various energy scales. Notice first that ,
since Landau Level quantization only deals with orbital degrees of freedom,
the mass which enters the LL energy separation h̄eB/mb is the band mass,
mb = 0.068m for GaAs in terms of the bare mass m of the electron. The latter
determines the Zeeman gap ∆z = gh̄eB/m, since the Zeeman effect deals
109
110
Spin and Quantum Hall Effect – Ferromagnetism at ν = 1
with the electron spin, an internal degree of freedom. Moreover the effective g
factor for GaAs is g = −0, 4, which causes the Zeeman gap in this material to
be a factor roughly 70 smaller than the LL separation, as already mentionned
earlier. Expressed in Kelvins, the Zeeman gap is ∆z = 0, 33B[T]K, while the
LL separation is h̄ωC = 24B[T]K, where the magnetic field is measured in
Tesla.
On the other hand, the characteristic Coulomb energy is e2 /ǫlB =
q
50 B[T]K. For a field 6T, which corresponds roughly to filling ν = 1, one
finds thus ∆z ≃ 2K ≪ e2 /ǫlB ≃ 120K < h̄ωC ≃ 140K. Interactions
are therefore of the same order of magnitude as the LL separation, and are
more relevant than the Zeeman gap. They must be taken into account when
discussing effects connected to the electron spin, in particular at ν = 1, which
we will focus on in the remaining parts of this chapter.
The first problem is to understand why we observe a Quantum Hall Effect
at all at this filling factor. The Zeeman gap is so small that each state is
almost degenerate, so we might expect a macroscopic degeneracy at ν = 1 in
the kinetic part of the hamiltonian. Just as for the FQHE, interactions are
responsible for the lifting of this degenaracy. So we are led to this counter
intuitive idea that the IQHE at ν = 1 should rather be looked at as a special
case of FQHE.
7.1.1
Wave functions
Let us start with a two spin 1/2 particles wave function, at ν = 1. In
the symmetric gauge, the orbital part is built from the single particle wave
function, in the symmetric gauge φm (z) = z m , (neglecting normalisation
factors) with m = 0, 1 (2.28). As for the spin function, we have four possible
states for the coupling of two spin√1/2 particles: an antisymettric singlet ,
|S = 0, M = 0i = (| ↑↓i − | ↓↑)/ 2, and a symmetric triplet |S =√1, M i,
avec |S = 1, M = 1i = | ↑↑i, |S = 1, M = 0i = (| ↑↓i + | ↓↑i)/ 2 and
|S = 1, M = −1i = | ↓↓i. We are dealing with a problem without explicit
spin-dependent potentials. The interaction is SU(2)invariant, the total spin
is a good quantum number. Since the fermionic wave function must be
antisymmetric, we have
ψS=0 (z1 , z2 ) = φs (z1 , z2 ) ⊗ |S = 0, M = 0i
ψS=1 (z1 , z2 ) = φa (z1 , z2 ) ⊗ |S = 1, M i,
and
Interactions are relevant at ν = 1
111
with φs (z1 , z2 ) = z10 z21 + z20 z11 = z1 + z2 and φa (z1 , z2 ) = z10 z21 − z20 z11 =
z1 − z2 . The second choice with an antisymmetric orbital wave function
is energetically favourable if the interaction is sufficiently strongly repulsive
at short range, as is the case for the Coulomb interaction. Thus Coulomb
interactions, combined with the Pauli principle, create an exchange force
which aligns spins. This is the origin of ferromagnetism in transition metals.
It is important to realize that the ν = 1 Laughlin wave function, (4.30)
φν=1 ({zi }) =
Y
i<j
(zi − zj )e−
P
k
|zk |2 /4
(7.1)
is in fact the orbital part of a ferromagnetic N-particles wave function: (4.30),
φν=1 ({zi }) =
Y
i<j
(zi − zj )e−
P
k
|zk |2 /4
| ↑, ↑, .... ↑i.
(7.2)
This wave function is the lowest energy state if the Zeeman effect is strong.
At first sight, a spin excitation in this state would cost the Zeeman energy.
In fact, because of the exchange effect, a strong cost in spin flip-energy arises
even if the Zeeman effect vanishes (this can actually be done experimentally
by applying external pressure in Ga As samples.) Let us in fact imagine that
to be the case. Then there would be no reason for the total spin to be along
the z direction, because no external potential breaks the Hamiltonian SU(2)
symmetry (if the Zeeman effect vanishes). The most general spin function
describing the most general orientation for the total spin is
ψs ({θ, ϕ} ) =
Y
m
!
θ
θ
cos e−iϕ/2 e†m,↑ + sin eiϕ/2 e†m,↓ |0i,
2
2
(7.3)
where e†m,σ creates an electron in the state |n = 0, m; σi. We have chosen the
parametrisation in terms of two angles, θ et ϕ which respects the normalisation |σi = u| ↑i+v| ↓i avec |u|2 +|v|2 = 1, in order to establish the connection
between the SU(2) and O(3) description of the rotation group. This allows
to introduce immediately the magnetisation field at the m Landau site:


sin θm cos ϕm

n(m) =  sin θm sin ϕm 
,
cos θm
(7.4)
This will be useful to describe the low energy degrees of freedom within the
effective model (section 8.3).
112
Spin and Quantum Hall Effect – Ferromagnetism at ν = 1
The wave function in equation (7.2) corresponds to θ = 0. It has a
quantum number M = N/2 which is the z component of the total spin.
Therefore the total spin is S = N/2, since M ≤ S. Other states, characterized by θ 6= 0, have −N/2 ≤ M ≤ N/2 [ψs ({θ, ϕ}), with θ et ϕ arbitrary],
are obtained from the former by applying rotation operators in the SU(2)
representation (equation 7.3).
In the state described by equations 7:02b, each particle is surrounded by
an “exchange hole”, due to the Pauli principle when all spins are aligned.
This lowers the Coulomb energy per particle. For filling factor ν = 1
hVCoulomb i
π e2
=−
≈ 200K
N
8 ǫlB
r
(7.5)
This is an order of magnitude larger than the Zeeman splitting and is the
mechanism which strongly stabilizes the ferromagnetic state, and would do
so even if the Zeeman effect was zero.
Even though the same mechanism is at work in all simple and transition metals, the latter are not all ferromagnets, because the kinetic energy
dispersion relation opposes the ferromagnetic polarisation: the broader the
band, the larger the kinetic energy cost to produce a finite spin polarisation;
only transition metals with the most narrow d-band, Fe, Co and Ni exhibit a
ferromagnetic state. In the completely filled Landau level, the kinetic energy
is frozen; there is no kinetic energy cost in spin polarizing the interacting
electron gas. This is why the ν = 1 IQH state is the “best understood itinerant ferromagnetic state” in condensed matter physics. A last remark before
going over to the next section: the introduction of the spin degree of fredom
is a special aspect of a more general problem: that of the QHE in multicomponent sytems [74]. Multi-component systems include systems with spin
degrees of freedom, but also iso-spin such as layer index in a bilayer system,
or valley index in Si, AlAs or graphene.
7.2
Algebraic structure of the model with spin
In this section, we generalize the model of electrons restricted to a single
level , as introduced in section 4.1, to include an internal degree of freedom
with SU(2) symmetry. For the physical spin case, the interaction is SU(2)
Algebraic structure of the model with spin
113
′
invariant, but we may consider a more general case 1 v0σσ (q). For a global
rotation symmetry, v0↑↑ (q) = v0↓↓ (q) et v0↑↓ (q) = v0↓↑ (q). The former more
general interaction which may break the SU(2) symmetry will be useful in
the next chapter, where the layer index in a bi-layer will be viewed as an
iso-spin index s = 1/2, with | ↑i for a state in the upper layer and | ↓i for
the lower layer. The interaction Hamiltonian writes now
H=
1 X X σ,σ′
v (q)ρ̄σ (−q)ρ̄σ′ (q),
2 σ,σ′ q 0
(7.6)
where2
ρ̄σ (q) =
X
fm,m′ (q)e†m,σ em′ ,σ
(7.7)
m,m′
is the electron density with spin σ projected on the LLL , with fm,m′ (q) =
hm|f (q)|m′ i the matric element for the projected density operator matrix
element f (q) = exp(−q · R), in first quantised forme [see equation (4.7)].
The total electron density is thus written
ρ̄(q) = ρ̄↑ (q) + ρ̄↓ (q)
X X
fm,m′ (q)δσ,σ′ e†m,σ em′ ,σ′ ,
=
(7.8)
σ,σ ′ m,m′
where δσ,σ′ represents the identity 12×2 . In comparison with the case without
internal degree of freedom , the electron density is replaced by
f (q) → f (q) ⊗ 12×2 ∼ fm,m′ (q) ⊗ δσ,σ′ ,
where the right hand part is the matrix representation which is relevant
for the second quantised notation. Similarly, we find the spin densities by
replacing the identity by the SU(2) generators S µ = τ µ /2, where τ µ are the
Pauli matrices ,3 with [τ µ , τ ν ] = iǫµνσ τ σ /2 and (τ µ )2 = 1. We define
f µ (q) = f (q) ⊗ S µ ,
1
(7.9)
We have already seen that the interaction potential in a LL with arbitrary n is obtained by replacing the gaussian by the form factor, 4.12, exp(−q 2 /2) → [Fn (q)]2 , in the
expression for the effective potential
2
We omit for simplicity the index n = 0 in the electron operators for the LLL.
3
The greek indices refer to the 3D space directions x, y and z. Since we have a Euclidian
space, we do not specify co- or contra-variant vectors, and Einstein summation is the rule
for repeated indices. The symbol ǫµνσ is the unit antisymmetric tensor :1 for {µ, ν, σ} =
{x, y, z} and all cyclic permutations, −1 for all other permutations and 0 if any index is
repeated.
114
Spin and Quantum Hall Effect – Ferromagnetism at ν = 1
and the spin densities are written, in second quantized form, as
S̄ µ (q) =
X X h
σ,σ ′ m,m′
i
µ
†
fm,m′ (q) ⊗ Sσ,σ
′ em,σ em′ ,σ ′
(7.10)
In the case with no internal degree of freedom, the algebraic structure was
derived from equation (4.13) using the commutation rules in first quantisation
q ∧ q′
q ∧ q′
[f (q), f (q )] = 2i sin
f (q+q′ ) → [ρ̄(q), ρ̄(q′ )] = 2i sin
ρ̄(q+q′ ).
2
2
′
Now using the same procedure, we have to compute [S̄ µ , ρ̄(q′ )] and [S̄ µ (q), S̄ ν (q)],using
[f µ (q), f ν (q′ )] = f (q)f (q′ ) ⊗ S µ S ν − f (q′ )f (q) ⊗ S ν S µ
(7.11)
1
([f (q), f (q′ )] ⊗ {S µ , S ν } + {f (q), f (q′ )} ⊗ [S µ , S ν ])
=
2
and
[f µ (q), f (q′ )] = [f (q), f (q′ )] ⊗ S µ .
We have
(7.12)
q ∧ q′
[f (q), f (q )] = 2i sin
f (q + q′ ),
2
!
q ∧ q′
′
f (q + q′ )
{f (q), f (q )} = 2 cos
2
!
′
and
[S µ , S ν ] = iǫµνσ S σ ,
1 µν
{S µ , S ν } =
δ ,
2
which yields
q ∧ q′
ρ̄(q + q′ ),
(7.13)
[ρ̄(q), ρ̄(q )] = 2i sin
2
!
q ∧ q′
µ
′
[S̄ (q), ρ̄(q )] = 2i sin
S̄ µ (q + q′ )
and
(7.14)
2
!
!
′
′
q
∧
q
q
∧
q
i
δ µν sin
ρ̄(q + q′ ) + iǫµνσ cos
S̄ σ (q + q′ ).
[S̄ µ (q), S̄ ν (q′ )] =
2
2
2
(7.15)
′
!
7.3. EFFECTIVE MODEL
115
The equations (7.13)-(7.15) are the SU(2) extensions of the magnetic translation algebra (4.16).
The Hamiltonian (7.6) is written
H=
X
1X
vSU (2) (q)ρ̄(−q)ρ̄(q) + 2
vsb (q)S̄ z (−q)S̄ z (q),
2 q
q
(7.16)
with potentials
i
1 h ↑↑
v0 (q) + v0↑↓ (q)
2
i
1 h ↑↑
v0 (q) − v0↑↓ (q) .
2
(7.17)
The first term in the Hamiltonian (7.16) is SU(2) invariant, while the second
one, if non zero, breaks explicitly the SU(2) symmetry. In the remaining
parts of this chapter we discuss ferromagnetism in the physical spin case.
The interaction is then SU(2) invariant. Only the (small) Zeeman term
vSU (2) (q) =
HZ =
et
vsb (q) =
gh̄eB z
S̄ (q = 0)
m
breaks the Hamiltonian SU(2) symmetry.
Equation (7.14) exhibits a remarkable property: because of the non commutativity between spin and charge densities, the dynamics of both degrees
of freedom are coupled. One can handle spins by acting on charges, and vice
versa! This unusual property is connected to the quantum dynamics under
magnetic field, and has a simple expression because of the projection on a
single LL, which results in non commutativity of charge density operators
with non parallel wave vectors [ equation (7.13)]. Spin-charge entanglement
in the LL ferromagnetism is studied in more details in the followingsection.
7.3
Effective model
We discuss two types of spin excitations of the quantum ferromagnetic state:
(a) spin waves (magnons) [figure 7.1(a)] and (b) spin textures which have
a non zero topological charge (skyrmions) [figure 7.1(b)]. While spin waves
are the Goldstone modes, the energy of which goes to zero, in the limit of
no Zeeman effect, when the wavelength goes to infinity, skyrmions cannot be
continuously deformed to retrieve the ground state. They are topologically
stable objects, classified by an integer which is called their topological charge.
116
Spin and Quantum Hall Effect – Ferromagnetism at ν = 1
z
(a)
y
x
(b)
z
y
x
Figure 7.1: Excitations in the ferromagnetic state . (a) Spin waves (magnons). Such
an excitation can be continuously deformed to the ground state with the same topological
property of the magnetisation field, as can be seen on the Bloch sphere (right) which
represents a mapping of the spin configuration (bold face spins) in the plane on S2 – the
gray line can be continuously deformed to a point (b) Skyrmion with non zero topological
charge. This excitation has a flipped spin at the origin , r = 0, and the ferromagnetic
state is recovered at infinity |r| → ∞. Contrary to spin waves the mapping of this spin
configuration on the sphere (bold face spins)covers the whole S2 surface once.
Effective model
7.3.1
117
Spin waves
The procedure to derive the spin wave spectrum is analogous to well known
other cases in localized or itinerant electron ferromagnets: one studies the
time evolution of the spin lowering operator:
N
X
Sq− ≡
µ
e−iqµ rj Sj−
(7.18)
j=1
where rjµ are the components of the position operator for the jth particle.
For an excitation in the lowest LL, one has to project this operator, and we
obtain
i
X h
−
S̄q− =
fm,m′ (q) ⊗ S↓,↑
e†m,↓ em′ ,↑
(7.19)
m,m′
Then one computes the commutator of this operator with the Coulomb interaction Hamiltonian:
h
i
H, S̄q− = (1/2)
X
k6=0
h
v(k) ρ̄(−q)ρ̄(q), S̄q−
i
(7.20)
This is evaluated using the familiar commutator algebra. When applied to
the ground state, which is annihilated by ρ̄k one obtains
where
h
i
H, S̄q− |ψi = ǫq S̄q− |ψi
ǫq ≡ 2
X
−
e
|k|2
2
v(k) sin
2
k6=0
(7.21)
!
q∧k
.
2
This proves that S̄q− is an exact excited eigenstate of H with excitation
energy ǫq . In the presence of the Zeeman coupling, ǫq → ǫq + ∆. The only
assumption is that the ground state at filling factor ν = 1 is fully polarised.
The dispersion relation is quadratic in q at small q:
ǫ q ≈ ρs q 2
with
ρs ≡
1 X − |k|2
e 2 v(k)|k|2
2 k6=0
For very large q, sin2 can be replaced by its average value 1/2 so that
ǫq ≈
X
v(k)e−
|k|2
2
k6=0
The energy saturates at a constant value for q → ∞.
118
7.3.2
Spin and Quantum Hall Effect – Ferromagnetism at ν = 1
Skyrmions
For simplicity reasons, let us first discuss a topological excitation in a 2D
XY ferromagnet (spins are constrained to lie in a plane). In that case a
topological excitation is a vortex type defect which is labelled by the number
of times the spin direction rotates around the origin along a closed path
encircling the defect placed at the origin (figure 7.2). This number is the
topological charge, which does not depend on the path geometry, only on its
homotopy class. Consider the mapping of the circle S 1 , which parametrizes
the path in the physical plane around the defect we want to characterize, on
the circle S 1 which parametrizes the spin orientation in the 2D spin space.
The mappings which can be continuously deformed into one another form
homotopy classes. They form a group, the fundamental group, π1 (S 1 ) = Z.
The topological charge is Q ∈ Z.
For the XYZ ferromagnet, the parametrization of the most general spin
texture maps on the surface of the Bloch sphere, S 2 . The 2D plane is mapped
by stereographic projection on a S 2 sphere. Just as for the case of the
S 1 → S 1 mappings, the S 2 → S 2 mappings which can be continuously
deformed into one another can be classified in homotopy classes, which form
a homotopy group,
π2 (S 2 ) = Z.
Elements of the group, integers Q ∈ Z label different topological sectors.
Spin waves deform continuously to the ground state, they belong to the
topological sector with topological charge Q = 0, while skyrmions carry a
topological charge Q 6= 0 (figure 7.1).
A state with a certain spin texture |ψ[n(r)]i may be generated from the
ferromagnetic ground state by applying a spin rotation operator, using the
generators (7.10) of the magnetic translation algebra (with internal SU(2)
structure) [87]
"
|ψ[n(r)]i = exp −i
X
q
#
Ωµ−q S̄ µ (q)
|ψF M i,
(7.22)
where |ψF M i is the ferromagnetic state(7.2), with a uniform magnetization
along the z quantization axis. The functions Ωµq which enter expression (7.22)
are, up to a permutation of the x and y axis, the Fourier transforms of n(r),
Ω(r) =
X
q
Ωq e−ir·q = ez × n(r).
(7.23)
Effective model
119
Q = −1
Q=1
Figure 7.2: Topological excitation in the xy model. The topological charge Q is the
number of times the spin direction turns by 2π along a path (gray curve) which circles
around the defect center(black point. Left : topological excitation with charge Q = 1 –
spins turn in the same direction as the path indicated by the arrow. Right : topological
excitation with charge Q = −1.
This state looks like a coherent state [see section 2.2.2, especially equation
(2.34)]. Using the Hausdorff series expansion [equation (5.17)], we have for
the transform of operator S̄ µ (q)
eiŌ S̄ µ (k)e−iŌ = S̄ µ (k) + δSkµ ,
where we have defined Ō =
P
q
(7.24)
Ωµ−q S̄ µ (q) et
ii
1h h
(7.25)
Ō Ō, S̄ µ (k) + ...
2
If we limit the Hausdorff series expansion to second order, we find for the
average magnetization in the spin texture (7.22)
nel µ
n (r),
S µ [n(r)] = hS µ (r)i + hδS µ (r)i =
2
as expected for the O(3) magnetization field. This justifies the choice of the
function Ω(r) [equation (7.23)]. The technical details for the computation
leading to this result are not given here. They are closely analogous to those
we present in the following section for the computation of the charge density
induced by the spin texture.
h
i
δSkµ = i Ō, S̄ µ (k) −
7.3.3
Spin-charge entanglement
For a better understanding of the spin-charge entanglement, which we mentionned above in the discussion of the model algebraic structure (section 7.2),
120
Spin and Quantum Hall Effect – Ferromagnetism at ν = 1
we now compute the charge density change due to the spin texture (7.22),
iiE
1 Dhh
Ō, Ō, ρ̄(q) , (7.26)
2
where averages are defined with respect to the ferromagnetic ground state
with magnetization along z. Averages are computed using the commutation
rules (7.13)-(7.15) and
δρq ≡ heiŌ ρ̄(q)e−iŌ i − hρ̄(q)i ≃ i
iE
Dh
Ō, ρ̄(q)
−
hρ̄(q)i = nel δq,0
et
n
el
hS̄ µ (q)i = δ µz δq,0 .
2
(7.27)
(7.28)
One finds thus
h
i
Ō, ρ̄(q)
X
=
k
= 2i
h
X
k
iE
Dh
Ō, ρ̄(q)
⇒
i
Ωµ−k S̄ µ (k), ρ̄(q)
!
k∧q
sin
Ωµ−k S̄ µ (q + k)
2
= 0,
since the argument of the sine vanishes for q k k, and
Ō Ō, ρ̄(q)
h ′
i
′
k∧q
= 2i
sin
Ωµ−k′ Ωµ−k S̄ µ (k′ ), S̄ µ (q + k)
2
k,k′
X
= −
X
sin
k,k′
µ′ µν
+ 2ǫ
′
k′ ∧ (q + k)
k∧q
′
Ωµ−k′ Ωµ−k δ µµ sin
ρ̄(q + k + k′ )
2
2
k′ ∧ (q + k)
cos
S̄ ν (q + k + k′ ) .
2
So the average is
Dh
h
iiE
Ō Ō, ρ̄(q)
= −nel ǫ
µ′ µz
X
k
!
′
k∧q
Ωµk+q Ωµ−k .
sin
2
we get thus for the modified charge density (7.26), with ν = 2πnel ,
δρq
!
′
ν X µ′ µz
k∧q
=
ǫ
sin
Ωµk+q Ωµ−k
4π k
2
′
−ν X µ′ µz
ǫ (q ∧ k)Ωµk+q Ωµ−k
≃
8π k
′
−ν X µ′ µz
=
ǫ
[i(k + q)] Ωµk+q ∧ (−ik)Ωµ−k ,
8π k
(7.29)
Effective model
121
where we have expanded the sine in the second line, thus resorting to a long
wavelength limit, i.e. a slow space varying modulation of the spin density.
The Fourier transform back to real space yields the more compact result
−ν µ′ µz (2) µ′
ǫ ∇ Ω (r) ∧ ∇(2) Ωµ (r)
8π
−ν ij
=
ǫ n(r) · [∂i n(r) × ∂j n(r)] ,
8π
δρ(r) =
(7.30)
where roman indices, {i, j} = {x, y}, correspond to 2D space coordinates
– not to be confused with the three spin vector components n(r). Compare the result (7.30)to the so-called Pontryagin topological charge density (
Pontryagin index)
δρtop (r) =
1 ij
ǫ n(r) · [∂i n(r) × ∂j n(r)] .
8π
(7.31)
We see that the electric charge density is proportional to the Pontryagin
index, δρ(r) = −νδρtop (r). The topological charge is obtained by integration
over the physical plane
Z
d2 r δρtop (r) = q ∈ Z.
(7.32)
The electric charge of a topological excitation is thus
Q = eνµ.
(7.33)
As was the case for the Laughlin quasi-particle, a skyrmion excitation at
ν = 1/(2s + 1) carries a fractional electric charge, Q = ±e/(2s + 1), for
|µ| = 1. The connection to the Berry phase will be discussed in section 7.4.
7.3.4
Effective model for the energy
It is useful to set up a simple energy functional model for the energy of a
spin structure. The energy of the state (7.22) with a O(3) magnetization
field n(r), is computed in the same manner as the charge modulation, using
the long wavelength expansion,
δE = i
Dh
≃ −
Ō, H
iE
−
iiE
1 Dhh
Ō, H
+ ...
2
′
1 X
v0 (k)Ωµ−k′ Ωµ−k hCi,
4 q,q′ ,k
(7.34)
122
Spin and Quantum Hall Effect – Ferromagnetism at ν = 1
where we have defined
h
′
i
n
h
′
io
C = 2 S̄ µ (q′ ), ρ̄(−k) S̄ µ (q), ρ̄(k) + ρ̄(−k), S̄ µ (q′ ), S̄ µ (q), ρ̄(k)
q′ ∧ k
q∧k
′
sin
S̄ µ (q′ − k)S̄ µ (q + k)
2
2
′
q ∧ (q + k) q∧k
′
sin
ρ̄(−k), ρ̄(q′ + q + k)
−δ µµ sin
2
2
′
q ∧ (q + k) q∧k
µ′ µσ
cos
ρ̄(−k), S̄ σ (q′ + q + k) .
−2ǫ
sin
2
2
= 8 sin
The average of this expression is computed, with the help of the structure
factor
1
hρ̄(−q)ρ̄(q′ )i = δq,q′ s̄(q)
(7.35)
nel
and of
δ µz
1
(7.36)
δq,q′ s̄(q)
hρ̄(−q)S̄ µ (q′ )i =
nel
2
and
′
1 µ
δ µµ δ µz
µ′
′
δq,q′ s̄(q).
(7.37)
hS̄ (−q)S̄ (q )i =
nel
4
Finally we find
hCi = 2nel δµ,µ′ δq,−q′ sin
2
q′ ∧ k
[s̄(q) − δ µz s̄(q + k)] .
2
!
So we find for the energy (7.34)
q′ ∧ k
nel X
v0 (k)s̄(k) sin2
Ωµ−q Ωµq .
δE = −
2 q,k
2
!
(7.38)
In the long wavelength limit, the sine can be linearized, so we get the non
linear O(3) sigma model
δE =
ρS X
ρS Z 2
(−iq)Ωµ−q (iq)Ωµq =
d r[∇n(r)]2 ,
2 q
2
(7.39)
where the exchange stiffness
ρS = −
ν Z∞
dkk 3 v(k)s̄(k),
32π 2 0
(7.40)
7.4.
BERRY PHASE AND ADIABATIC TRANSPORT
is
ρS =
1
e2
√
16 2π ǫlB
123
(7.41)
at ν = 1, since s̄(k) = −1. Any magnetization variation incresases the energy
with respect to the ground state, which indeed corresponds to ∇n(r) = 0, i.e.
a ferromagnetic state with uniform magnetization. The energy dispersion at
small wave vector
ρS
(7.42)
ω(q) ≃ q 2
2
is precisely the Goldstone mode (spin wave) energy discussed earlier. The
complete analytic expression for the latter is [88] [89]
ω(q) =
r
q2
π e2
2
1 − e−q /4 I0
2 ǫlB
4
"
!#
,
(7.43)
which coincides with expression (7.42) in the limit q ≪ 1.
7.4
Berry phase and adiabatic transport
In the previous parts of this chapter, skyrmions, topological excitations of the
Quantum Hall ferromagnet, have been discussed on the basis of the Hamiltonian theory, using the commutator algebra of the projected density operators. A topological charge density (the Pontryagin density) is associated to
spin texture. The properties of the homotopy group Π2 (S2 ) ≡ Z indicate
that the electric charge carried by a topological defect is an integer in terms
of the electron charge, at ν = 1.
This result is intimately connected to the Berry phase notion [90, 91]
and to the notion of adiabatic transport in quantum mechanics. Consider
a quantum system, described by a Hamiltonian HR which depends on a set
of external controlable parameters. This set is represented by a vector in
parameter space R. We assume now that there exists a compact domain in
parameter space where the ground state is separated of all excited states by
a gap. What is the result of letting the system evolve slowly, with a slow
variation of R(t) along a closed loop in this part of parameter space, during
a time interval T ? We have
R(0) = R(T )
(7.44)
124
Spin and Quantum Hall effect – ferromagnetism at ν = 1
If the evolution along the closed path is sufficiently slow, such that h/T <<
∆min where ∆min is the minimum energy gap along the loop, the state evolves
adiabatically. This means that at all times, the system remains in the ground
(0)
state ΨR of the Hamiltonian HR(t) .
Each point R in parameter space is associated to a complete set of eigen
states:
(j)
(j) (j)
HR ΨR = ǫR ΨR .
(7.45)
The solution of the time dependent Schrodinger equation
ih̄
∂Φ(r, t)
= HR(t) Φ(r, t)
∂t
is then:
− h̄i
(0)
Φ(r, t) = ΨR(t) (r)eiγ(t) e
Rt
0
dt′ ǫ0R(t′ )
+
(7.46)
X
(j)
aj (t))ΨR(t)
(7.47)
j6=0
The adiabatic approximation amounts to neglecting the contribution of
excited states represented by the second term on the right hand side. This
becomes exact in the limit of of a very slow variation of R(t) as long as the
excitation gap remains finite. Everything is well known at this point, except
the ”Berry Phase” γ(t). γ(t) can be determined by requiring Φ(r, t) to obey
the time dependent Schrödinger equation. The LHS of equation 7.46, if we
neglect the aj (t) when j > 0 becomes:
"
#
i
i
∂Φ(r, t) h
∂
(0)
(0)
iγ(t) − h̄
ih̄
= −h̄γ̇(t) + ǫR(t) Φ(r, t) + ih̄Ṙµ
Ψ
(r)
e
e
R(t)
∂t
∂Rµ
Rt
0
(0)
dt′ ǫR(t′ )
(7.48)
The RHS of equation 7.46, within the same approximation is:
(0)
HR(t)
~ Φ(r, t) = ǫR(t) Φ(r, t)
(7.49)
Using the completeness relation
*
+
+
∞ ∂
E
X
∂
(j)
(0)
(0)
(j)
ΨR(t) | µ ΨR(t) .
ΨR (t)
=
Ψ
∂Rµ R
∂R
(7.50)
j=0
Here again the adiabatic approximation allows to neglect the contribution
of excited states, so that equation 7.48 becomes:
"
*
+
#
∂Φ
∂
(0)
(0)
(0)
ih̄
= −h̄γ̇(t) + ih̄Ṙµ ΨR (t)| µ ΨR(t) + ǫR(t) Φ
∂t
∂R
(7.51)
Berry phase and adiabatic transport
125
Equation 7.46 is satisfied if
γ̇(t) = iṘ
D
µ
*
(0)
ΨR(t) |
(0)
(0)
∂
(0)
ΨR(t)
µ
∂R
+
(7.52)
E
Thanks to the constraint ΨR |ΨR = 1 we know that γ̇ is real.
At first sight γ̇(t) could be chosen to vanish. In fact, for each R we have
a different set of eigen functions, and one may choose the ground state phase
arbitrarily. That is a kind of gauge choice in parameter space: γ̇(t) and γ in
that sense, are not gauge invariant. Chosing γ̇ = 0 is then a gauge choice.
What Berry [90] found is that this is not always possible. In certain cases
implying a closed path in parameter space, there is a finite gauge invariant
phase, the Berry Phase,
γBerry ≡
Z
T
0
γ̇dt = i
I
Γ
dR
µ
*
(0)
ΨR |
+
∂
(0)
ΨR .
µ
∂R
(7.53)
That quantity is ”gauge invariant” because the system returns to the
departure point in parameter space, so that the arbitrary choice of gauge at
the start has no consequence. This is analogous to electrodynamics when
the circulation of the vector potential is along a closed path, and equals the
enclosed magnetic flux, which is gauge invariant. In fact it is useful to define
the ”Berry connection” , A, in parameter space:
µ
A (R) = i
*
γBerry =
I
which leads to
(0)
ΨR |
Γ
∂
(0)
ΨR
µ
∂R
dRµ · Aµ (R)
+
(7.54)
(7.55)
The Berry phase is a purely geometric object, independent of the velocity
Ṙµ (t). It only depends on the path in parameter space. It is often easiest to
evaluate this expression using Stoke’s theorem, since the curl of A is gauge
invariant.
It is easy to check that in the case of electromagnetism and the AharonovBohm effect, the Berry connection, A is, up to a multiplicative factor, h̄q
(where q is the particle charge), the electromagnetic vector potential A:
q
Aµ (R) = + Aµ(R)
h̄
(7.56)
126
Spin and Quantum Hall effect – ferromagnetism at ν = 1
The Berry phase for a loop threaded by a magnetic flux φb is
γBerry =
qI
φb
dRµ Aµ = 2π
h̄
φ0
(7.57)
where φ0 is the flux quantum.
A second example is of interest for the quantum Hall ferromagnetism.
Consider a quantum spin coupled to a magnetic field, with a Hamiltonian:
~
~
H = −∆(t)
·S
(7.58)
~ The circuit in parameter space
The gap to the first excited state is h̄|∆|.
~ = 0 where the spectrum has a degeneracy. During
must avoid the origin ∆
the adiabatic evolution of the ground state, one has
D
(0)
(0)
~
Ψ∆
~ |S|Ψ∆
~
E
= h̄S
~
∆
~
|∆|
(7.59)
~ is defined by the polar angle θ and the
Thus if the orientation of ∆
~ >. For a spin S = 1/2, an
azimuthal angle φ, the same must be true for < S
appropriate set of states is:
cos 2θ
sin 2θ eiφ
|Φθ,φ >=
!
(7.60)
since these obey:
θ
θ
hΦθ,φ |S |Φθ,φ i = h̄S cos − sin2
2
2
z
2
and
D
!
= h̄S cos θ
E
hΦθ,φ |S x + iS y |Φθ,φ i = Φθ,φ |S + |Φθ,φ = h̄S sin θeiφ
(7.61)
(7.62)
~ around axis
What is the Berry phase in the case of a slow rotation of ∆
ẑ, at constant θ?
γBerry = i
=i
Z
0
2π
dφ
cos
θ
2
sin
θ −iφ
e
2
Z
0
2π
*
∂
dφ Φθ,φ | Φθ,φ
∂φ
0
i sin 2θ eiφ
!
= −S
+
Z
0
2π
(7.63)
dφ(1 − cos θ) (7.64)
7.5. APPLICATIONS TO QUANTUM HALL MAGNETISM
= −S
Z
0
2π
dφ
1
X
cos θ
d cos θ′ = −SΩ
127
(7.65)
where Ω is the solid angle subtended by the path as viewed from the origin
of the parameter space. This is precisely the Aharonov-Bohm phase one
expects for a charge −S particle traveling on the surface of a unit sphere
surrounding a magnetic monopole. The degeneracy of the spectrum at the
origin is precisely the cause for presence of the magnetic monopole [90]
The definition of the connection A implies the existence of a singularity
at the south pole, θ = π. A ”Dirac string” (i.e. an infinitely thin solenoid
carrying one flux quantum) is attached to the monopole. The singularity
would be attached to the north pole if we had chosen the basis
e−iφ |Φθ,φ >
(7.66)
In order to reproduce correctly the Berry phase in a path integral for the
spin the Hamiltonian of which is given by 7.58, the Lagrangian must be:
n
o
L = h̄S −ṁµ Aµ + ∆µ mµ + λ(mµ mµ − 1)
(7.67)
where m is the spin coordinate on the unit sphere, λ is a Lagrange multiplier
which enforces the length constraint, and the Berry connection A obeys:
~ ×A=m
∇
~
(7.68)
This Lagrangian reproduces correctly the equations for the spin dynamics
which describe its precession.
7.5
7.5.1
Applications to quantum Hall magnetism
Spin dynamics in a magnetic field
In the following, we show that the Lagrangian above allows to describe the
quantum spin dynamics in an effective field.
The equations of motion are:
Using 7.67 we have
d δL
δL
=
µ
dt δ ṁ
δmµ
(7.69)
δL
= −Aµ
δ ṁµ
(7.70)
128
and
Spin and Quantum Hall effect – ferromagnetism at ν = 1
δL
= −ṁν ∂µ Aν + ∆µ + 2λmµ ,
δmµ
(7.71)
∆µ + 2λmµ = Fµν ṁν ,
(7.72)
so that
where Fµν = ∂µ Aν − ∂ν Aµ
The previous section on the Berry phase shows we must chose:
Fµν = ǫαµν mα
(7.73)
~m
which is equivalent to ∇
~ ∧ A[m] = m. The equation of motion becomes:
δµ + 2λmµ = ǫαµν mα ṁν
(7.74)
Multiplying both members of equation 7.74 by ǫγβµ mβ , then applying on
both sides of this equation the identity: ǫναβ ǫνλη = δαλ δβη − δαη δβλ , we get:
~m
~
− ∆∧
γ
= ṁγ − mγ (ṁβ mβ )
(7.75)
The last term vanishes, because of the constraint on the length of m. Using Euler-Lagrange equations, we retrieve the spin precession equations in a
magnetic field.
Compare 7.67 with the Lagrangian of a particle of mass m, and charge
−e in a magnetic field with vector potential A:
1
L = mẋµ ẋµ − eẋµ Aµ
2
(7.76)
We see that the Lagrangian in 7.67 is equivalent to a Lagrangian of a zero
mass object, with charge −S, placed at m,
~ moving on a unit sphere containing
a magnetic monopole. The Zeeman term is analogous to a constant electric
~ which exerts a force S ∆
~ on the particle. The Lorentz force due to
field −∆,
the monopole field drives the particle on a constant latitude orbit on the unit
sphere. The absence of a kinetic term in ṁµ ṁµ in the Lagrangian indicates
that the particle has zero mass, and is in the lowest LL of the monopole field.
7.6
Application to spin textures
Consider a ferromagnet with a local static spin orientation m(r). When an
electron is displaced, one may assume that the strong exchange field forces
7.6. APPLICATION TO SPIN TEXTURES
129
the electron spin to follow the local orientation of m(r). If the electron has
a velocity ẋµ , the variation rate of the local spin orientation seen by the
electron is ṁν = ẋµ ∂x∂µ mν . This induces a non trivial Berry phase in the
presence of a spin texture. Indeed, the one particle Lagrangian contains an
additional term with a time derivative of first order, wich adds to the term
due to the field-matter minimal coupling term:
L′ = −eẋµ Aµ − h̄S ṁν Aν [m]
~
(7.77)
L′ = −eẋµ (Aµ + aµ )
(7.78)
The first term is the field-matter coupling, the second one gives rise to the
Berry phase. We have for the latter ∇m ∧ A = m.
~ That can be re-written,
∂
ν
ν
µ
using ṁ ≡ x˙ ∂xµ m . Then
with
!
∂
~
mν Aν [m]
2πa = φ0 S
∂xµ
µ
(7.79)
a is the Berry connection, a vector potential which adds to the magnetic field
vector potential. The curl of a thus contributes a ”Berry” flux which adds
to the magnetic field flux:
b = ǫαβ
∂aβ
∂xα
(7.80)
!
1
∂
∂
~
mν Aν [m]
α
β
∂x
∂x
2π
!
∂ ∂
~
mν Aν [m]
= φ0 Sǫαβ [
α
β
∂x ∂x
!
∂
∂mγ ∂Aν
ν
+
m
]
∂xβ
∂xα ∂mγ
= (φ0 S/2π)ǫαβ
The first term of the last equation vanishes by symmetry, which results in:
b = φ0 Sǫαβ
∂mν ∂mγ
(1/2)F νγ
∂xβ ∂xα
(7.81)
with F µν = ǫαµν mα . We used the symmetry ν ↔ γ in the last surviving term
.
With S = 1/2 one gets
b = φ0 ρ̃
(7.82)
130
Spin and Quantum Hall effect – ferromagnetism at ν = 1
with
1 αβ abc a
ǫ ǫ m ∂α m b ∂β m c
8π
1 αβ
=
ǫ m
~ · ∂α m
~ ∧ ∂β m
~
8π
ρ̃ ≡
(7.83)
(7.84)
We recognize in 7.83 the Pontryagin topological density.
If now, starting from a uniform magnetization, we deform the ground
state magnetization adiabatically into a spin texture, everything happens,
for orbital degrees of freedom, as if flux from b(r) was injected adiabatically.
In a quantum Hall state with ρii = 0 and ρxy = ν, the Faraday law then
causes this spin texture to attract (or repel) at the end of the process a
charge density
ν ρ̃. Since the skyrmion topological charge is an integer,
R
Qtop = d2 rρ̃(r) = integer,
the charge associated to a skyrmion in the IQHE is δρ = −νe × (integer).
We have thus recovered, as a result of the Berry phase, the result obtained
earlier by the Hamiltonian approach.
Chapter 8
Quantum Hall Effect in
bi-layers
8.1
Introduction
In the previous chapters, we have emphasized the importance of Coulomb interactions in the Quantum Hall Effect physics, including for the ν = 1 filling
factor of the LLL. Even if the Zeeman coupling vanishes, Coulomb interactions stabilize a ferromagnetic order, which has important consequences on
the excitations spectrum. Instead of a spin degenerate metal at νσ = 1/2,
we have a quantum Hall ferromagnetic state with a gap.
An analogous effect occurs for bilayers ( a system of two coupled layers),
where each layer has filling ν = 1/2. In that case, the rôle of spin is played by
the isospin index of each layer [87, 93, 94]. The analogy with the ferromagnetic monolayer system at ν = 1 will be extensively used in the following.
Quantum Hall bilayer physics is quite rich, and involves coupling between
layers at various equal, or different, filling factors. This chapter focuses on
the particular case of two layers at ν = 1/2, for which exciting results have
been obtained in the last few years.
Modern MBE techniques have allowed in the recent years to manufacture
2D electron gases with high mobility, in bi-layers or multi-layers structures
[95]. As shown on the figure 8.1, a bi layer is a system of two 2D electron
gases organized in parallel layers, at a distance d from each other which
is comparable with the magnetic length, and to the average distance between electrons in the layer (i.e. d ∼ 10nm ). We know that correlations
131
132
Quantum Hall Effect in bi-layers
W
W
2t
d
Figure 8.1: Sketch of the conduction band profile for a two dimensional electrons system
in a bi-layer. The order of magnitude for the width, as well as for the distance, of the two
layers is W ∼ d ∼ 10nm. In the presence of a tunneling term t, the band splits into a
bonding, and an anti-bonding band, with a separation ∆SAS = 2t.
are especially important at high fields, when electrons occupy the LLL only,
because the kinetic energy is then frozen out of the problem, and cannot oppose the ferromagnetic polarisation. The FQHE results from gap formation
between the ground state and excited states, resulting in an incompressible state. Theory predicts that gaps appear for certain fractional fillings in
the bilayer system when inter-layer interactions are strong enough [19, 96].
Such predictions have been backed by experiments [97]. Recently [98], theory has predicted that inter-layer correlations could induce unusual broken
symmetry states, with a new type of inter-layer coherence. This new interlayer coherence appears even in the absence of inter-layer tunneling, when
the coupling between layers is of purely Coulombic origin. What appears
here is excitonic superfluidity, which is the unexpected realization, in the
bi-layer physics, of the phenomenon of excitonic superfluidity predicted in
3D semiconductors since 1962 [99]. This phenomenon has been looked for
without unquestionable experimental success ever since [100] until it eventually appeared in a spectacular manner in the Quantum Hall bi-layer system
[87, 93, 94, 101, 102, 103, 104, 6].
8.2. PSEUDO-SPIN ANALOGY
8.2
133
Pseudo-spin analogy
We assume that the Zeeman effect saturates the “real” spin, so that it does
not play any rôle any more. Each layer is given a “pseudo-spin” label ↑ or +
for one layer, and ↓ or − for the other. In the situation we chose to discuss,
ν↑ + ν↓ = 1.
A state having inter-layer coherence is a state with ferromagnetic pseudospin order, in a direction defined by the polar angle θ, and azimuthal angle
φ. In the Landau gauge, such a state writes [see also equation (7.3)]
|ψi =
Y
k
!
θ
θ
cos c†k↑ + sin eiφ c†k↓ |0i .
2
2
(8.1)
In the state described by this wave function, each state k is occupied by one
electron, and has an amplitude cos(θ/2) to be in the ↑ layer, and amplitude
sin(θ/2) exp(iφ) to be in layer ↓. Physically the ratio between the squared
amplitude may be altered by applying a voltage between the layers, so as
to charge one layer at the expense of the other one, the total filling factor
remaining equal to 1. Remember that in the Landau gauge, a state k labels
2
a state localized on a line at guiding center position Xk ≡ klB
. We discuss
the following cases:
Spins along the z axis. For θ = 0 this wave function describes a spin alignment along the ẑ axis,
|ψz i = Πk c†k,↑ |0i = Πi<j (zi − zj ) |↑, ↑, ... ↑i .
The choice θ = 0 describes a situation where all particles are in the layer
labelled by ↑.
Spins along the x̂ axis. One has θ = π/2, φ = 0, which yields the state


 c† + c† 
|ψx i = Πk  k↑√ k↓  |0i .
2
Obviously, this wave function which describes a symmetric superposition of
electronic states in the two layers must have a low energy, compared to θ = 0,
as soon as the Coulomb energy plays a rôle, as we shall see later on, and when
both layers have the same potential.
Spins along a general direction in the xy plane. In that case, we have θ =
134
Quantum Hall bi-layers
π/2, φ 6= 0 and thus


 c† + eiφ c† 
|ψxy i = Πk  k↑ √ k↓  |0i .
2
That state, as the previous one describes a symmetric superposition of states
with equal amplitude in each layer, but the resulting pseudo-spin magnetization has been rotated by an angle φ with respect to the x axis
In any case, the total occupation of each k state is 1, but the layer index
for each electron is undetermined except when θ = 0. The amplitude for an
electron to be in layer ↑ is cos θ/2, the amplitude to be in layer ↓ is eiφ sin θ/2.
The most general choice is to have neither θ = π/2 nor 0. The total weight
remains equal to 1, since sin2 (θ/2) + cos2 (θ/2) = 1.
Even in the absence of quantum tunneling between layers (i.e. physical
transfer of electron between layers), quantum mechanics, with the superposition principle, allows to describe the possibility of the simultaneous presence
of an electron in both layers.
In the ferromagnetic monolayer situation (ν = 1), we have seen in the
previous chapter that in the absence of Zeeman coupling, the ferromagnetic
exchange coupling, due to its Coulomb origin, does not break the Hamiltonian
SU(2) symmetry. A different situation arises in the bi-layer case. In the next
section, we list the various physical parameters which give its originality to
the bi-layer pseudo-spin ferromagnetism
8.3
Differences with the ferromagnetic monolayer case
What are the main differences between the bi-layer physics (with ν↓ = ν↑ =
1/2) and the monolayer at ν = 1?
• When the two layers are far away from each other (d/lB ≫ 1), interactions between electrons in one layer and electrons in the other
are negligible. The layers are single layers with filling ν = 1/2, there
are two flux quanta per electron. A Composite Fermion construction,
whereby two flux quanta are attached to each electron (see chapter 4
and 6 and reference [16]) results in a problem where CF evolve in zero
effective magnetic field, B ⋆ = 0. This is a metallic state, CF organize
Differences with the ferromagnetic monolayer
135
in a strange Fermi Liquid, with circular Fermi surface, where each CF
carries a dipole [105]. Although this an interesting object in its own
right, we are not going to discuss this limit in these lectures. In any
case, there is a continum of particle-hole excitations above the Fermi
liquid ground state, and there is no QHE
• At a distance comparable to the magnetic length, i.e. d ∼ lB , intralayer and inter-layer Coulomb interactions, although different, become
comparable, especially if d → 0, in which case they become equal. Using the pseudo-spin analogy, we can write the interaction Hamiltonian
[see equation (7.6) in the previous chapter]
Hcoul =
1 X X σ,σ′
v (q)ρ̄σ (q)ρ̄σ′ (−q).
2 σ,σ′ q 0
(8.2)
we have therefore
v0↑↑ (q) = v0↓↓ (q) ≡ v A (q)
v0↑↓ (q) = v0↓↑ (q) ≡ v E (q)
where v A (q) [v E (q)] is the Fourier transform with respect to the planar coordinates of the intra-layer [inter-layer ]interaction between electrons. Neglecting the physical width of the sample , we have v A (q) =
(2πe2 /q) exp(−q 2 /2) et v E (q) = e−qd v A (q).
Letting vSU (2) (q) = [v A (q) + v E (q)]/2 and vsb (q) = [v A (q) − v E (q)]/2,
one may separate, in the interaction Hamiltonian, a part which is independent of the pseudo spin, v0 (q), and one which is not , as we have
in equations (7.16) and (7.17) of the previous chapter. One finds then
H=
X
1X
vSU (2) (q)ρ̄(−q)ρ̄(q) + 2
vsb (q)S̄ z (−q)S̄ z (q),
2 q
q
h
(8.3)
i
with S̄ z (q) = [ρ̄↑ (q) − ρ̄↓ (q)]/2. Note that H, S̄ µ (q = 0) 6= 0 for
µ = x, y, because of the second term, which breaks the SU(2) symmetry.
Since, for a finite separation between layers, we always have v A (q) >
v E (q), for all wave vectors, the second term in equation (8.3) creates
an easy magnetization plane, perpendicular to ẑ, in the bi-layer plane.
The pseudo-spin Hamiltonian symmetry is reduced from SU(2) to U(1).
136
Quantum Hall bi-layers
• A third difference with the mono-layer at ν = 1 arises from the interlayer tunnelling term. This term has the following expression:
T = −
t X †
ck,↑ ck,↓ + c†k,↓ ck,↑
2 k
= −t
X
S̄kx .
(8.4)
(8.5)
k
It acts as a pseudo-magnetic field applied along x̂. It stabilises the
pseudo-spin orientation in direction x̂. Indeed the symmetric combination of states in each potential
√ well (bonding combination) corresponds
−x would
to the spinor ψb = (1, 1)/ 2 (bonding state). The direction
√
correspond to an anti-bonding state, ψa−b = (1, −1)/ 2
It is feasible experimentally to have widely different values of t between
10−3 et 10−1 × e2 /ǫlB . Note that the tunnel term, since it creates a
preferred direction in the xy plane, breaks the U (1) symmetry in the
Hamiltonian.
• Apply a voltage bias between the layers. This generates a term −e(N↑ −
N↓ )V = −2eS z V which is analogous to a magnetic field applied along
the z quantization axis of the pseudo-spin. Minimizing the anisotropy
term added to the electric term, we see that V creates a charge unbalance between the two layers, which, in pseudo-spin language is a finite
value for S z = (N↑ − N↓ )/2
• One may apply a magnetic field Bk parallel to the plane of the bi-layer.
Such a parallel field has no effect (except in terms of Zeeman gap) in
the monolayer case. One may expect new effects in the bi-layer case,
which should be orbital effects (the real spin is expected to be entirely
polarised). Chosing a gauge for the associated vector potential, the
presence of Bk can be taken into account as follows:
Axk = 0
y

A|| =  Ak = 0 
,
Azk = Bx


(8.6)
where ẑ is the direction perpendicular to the layers. In that gauge, the
gauge invariant tunnel term becomes
2π
iφ
t → te
0
R d/2
−d/2
dzAzk
i2π Bxd
φ
= te
0
≡ teiQx ,
8.4. EXPERIMENTAL FACTS
137
with Q = 2πBd/φ0 = 2π/Lk , which implies
Lk =
φ0
.
Bd
in the presence of Bk , the total tunneling term becomes thus
T = t
X
eiQx c†k,↑ ck,↓ + e−iQx c†k,↓ ck,↑
k
h
i
= −t eiQx (S x + iS y ) + e−iQx (S x − iS y )
= 2tS cos(φ + Qx)
(8.7)
This term is thus equivalent to a field rotating around the x axis, uniform along y, which is causing the pseudo-spin magnetization to oscillate along x. In fact this term competes with the exchange term, as we
shall see later on.
• When a Bk field is applied, the tunnelling current between the layers
becomes
J↑↓ = it
X
k
eiQx c†k,↑ ck,↓ − e−iQx c†k,↓ ck,↑
h
(8.8)
i
= it eiQx (S x + iS y ) − e−iQx (S x − iS y ) = 2tS sin(φ + Qx).
8.4
8.4.1
Experimental facts
Phase Diagram
As discussed above, the energy difference ∆SAS = 2t between symmetric
and antisymmetric superpositions of layer states of the two wells may vary,
depending on the sample, from a few millidegrees to a few hundreds of degrees Thus ∆SAS may be much smaller, or much larger than the inter-layer
Coulomb interactions. Experiments are able to scan a large array of the
characteristic ratio ∆SAS /(e2 /ǫd), from a weak electronic correlation regime
to a strong one.
When the layers are far from each other (d ≫ lB ), there are no inter-layer
correlations, each layer is in the ν = 1/2 metallic ground state, there is no
138
Quantum Hall bi-layers
Figure 8.2: Phase diagram for the bi-layer QHE (after Murphy et al. ??). The samples
with parameters below the dotted line exhibit the IQHE and an excitation gap.
QHE. When the inter-layer separation decreases, an excitation gap is found
to appear, together with a quantized Hall plateau with σxy = e2 /h [94, 101].
If ∆SAS /Ec ≫ 1 (with Ec = e2 /(ǫd), this is fairly easy to understand, since
things look as if the two layers were like a single one, with total filling factor
ν = 1. All symmetric states are occupied, we have the usual QHE. A far
more interesting situation arises when the ν = 1 QHE is found in the limit
∆SAS → 0. In this limit, the excitation gap is clearly a collective effect,
since it may be as large as 20 K while ∆SAS < 1K. The excitation gap
survives in this limit because of a spontaneous breaking of the U (1) gauge
symmetry associated with the phase degree of freedom –the azimuthal angle φ
in expression (8.1) [87, 98]. Figure 8.2 shows the experimental determination
of the QHE part of phase diagram, below the dotted line. This change
from single particle to collective behaviour is analogous to the ferromagnetic
behaviour of a monolayer at ν = 1. In the latter case, the excitation gap
remains finite even when the Zeeman effect vanishes, because of the exchange
forces connected to the Coulomb interaction.
The remarkable fact is that the IQHE at ν = 1 survives when ∆SAS → 0
provided the inter-layer distance between layers is smaller than a critical
value d/lB ∼ 2. In that case, the gap is a purely collective effect due to
interactions. As we shall see, it is due to a pseudo-ferromagnetic quantum
Halll state, which posesses a spontaneous inter-layer coherence.
Experiments
139
Figure 8.3: Experiment by Murphy et al. [94]. The thermal excitation gap ∆ is plotted as a function of the magnetic field tilt angle, in a bilayer with small tunnel term
(∆SAS = 0.8K). The black dots correspond to filling factor ν = 1, the triangles to ν = 2/3.
The arrow shows the critical angle θc . The continuous line is a guide to the eye. The
dotted line is a rough estimate of the tunnel amplitude renormalized by the parallel magnetic field. This single particle effect exhibits a slow negative variation, compared to the
observed effect. The inset is an Arrhenious plot of the dissipation, measured by the longitudinal resistance. The low temperature activation energy is ∆ = 8.66K. The gap however
decreases sharply at a much lower temperature, roughly 0.4K.
8.4.2
Excitation gap
An additional indication of the collective nature of excitations is provided
by the excitation gap variation with temperature, as shown on figure 8.3.
The activation energy ∆ at low temperature is clearly larger than ∆SAS .
If ∆ was a single particle gap, one would expect an Arrhenius law up to
temperatures of the order of ∆/kB . Instead, the gap decreases sharply as
soon as A T ∼ 0, 4K. This suggests that the order responsible for the
collective excitation gap is vanishing .
8.4.3
Effect of a parallel magnetic field
Another experimental finding suggests strongly a collective order phenomenon: the strong sensitivity of the system to a relatively weak Bk magnetic
field, applied in a direction parallel to the layers plane. The figure 8.3 shows
that the activation gap decreases rapidly when B|| , the parallel component
140
Quantum Hall bi-layers
Figure 8.4: Example of electronic process in a 2D bi-layer, such that the flux of Bk
produces decoherence effects. In this process, an electron tunnels at point A from the upper
layer to the lower one. The electron pair thus created moves coherently, then annihilates
at point b where the particle tunnels in the other direction. The amplitude for such a
process depends on the flux of Bk through the path
.
of the field, increases. Assume that the electronic gas in each layer is stricly
2D (in other words neglect the physical width of the potential wells). Then
the orbital effect of Bk can only be due to electronic processes between the
layers with closed loops containing some flux from Bk . Such loops will cause
B|| to be felt if there is phase coherence over the whole loop. Such a loop is
shown on figure 8.4.
An electron tunnels from one layer to the other at point A, travels a
distance L|| , tunnels back to the departure layer, then back to point A.
The magnetic field parallel component, Bk , contributes to the amplitude of
this process a (gauge invariant) Aharonov-Bohm phase factor, exp(2πiφ/φ0 ),
where φ is the flux of Bk threading this circuit.
Such loops contribute significantly to correlations, since one observes a
rapid decrease of the activation gap as a function of Bk : the decrease is by
a factor 2 up to a critical field Bk∗ ∼ 0.8T , beyond which the gap remains
roughly constant. This value is remarkably small. Let Lk be the length such
that the flux through the loop is one flux quantum: Lk Bk∗ d = φ0 ⇔ Lk [Å] =
4, 14×105 /d[Å]Bk [T]). With Bk∗ = 0, 8T and d = 150Å, one has Lk = 2700Å,
i.e. roughly twenty times the average distance between electrons in a layer,
and thirty times the magnetic length corresponding to B⊥ . A significant
decrease of the excitation gap is already observed in a parallel field of 0.1T,
Experiments
141
Figure 8.5: (a) In a standard experiment, the Hall current is transported simultaneously
in both layers, without tunnel between layers. (b) It is possible to inject current in one
layer and to extract it in another. The tunneling current then behaves as a superfluid
current .
which implies enormous coherence lengths. This is again a hint of the strongly
collective nature of the observed order in quantum Hall bi-layers.
8.4.4
The quasi-Josephson effect
A spectacular experiment by Spielman et al. [102], confirmed the theoretical
ideas about excitonic superfluidity in bi-layers. In the standard transport
experiments on bi-layers, a current JHQ is injected in both layers simultaneously, and is also extracted from both layers simultaneously. In the experiment by Spielman et al., a current JHQ is injected in one layer, and extracted
from the other one (Fig. 8.5). Qualitative differences arise in the tunnel conductance when the ratio d/lB is varied, for example varying the electronic
density at constant filling of the LLL. Below a critical value of the ratio
(the critical value which corresponds to the transition line between the QHE
regime at ν = 1 and the metallic regime) a giant anomaly appears at zero
bias, as shown on fig. 8.6. The qualitative understanding of this experiment
is as follows: for d/lB ≫ 1, electronic liquids in different layers are uncorrelated. At zero inter-layer bias, the Coulomb repulsion between electrons in
one layer, and an electron in the other will inhibit the inter-layer tunneling
process of the latter: the zero bias conductance is strongly suppressed. Only
a finite bias, of the order of the Coulomb repulsion e2 /(ǫd), can supersede
the latter. When d/lB ≃ 1 a coherent state is established, such that an
142
Quantum Hall bi-layers
electron in one layer is bound to a hole in the other one at the same Landau
site. Coulomb repulsion is strongly suppressed by this collective structure
for inter-layer tunneling events, and the tunneling conductance increases by
two orders of magnitude.
At the time of this writing, as far as the author knows, there is yet no
general consensus on the intrinsic, or extrinsic character of the zero bias
conductance finiteness. Is it impossible, for fundamental reasons, to ever
observe a divergent conductance at zero bias, which would be the signature
of a complete analogy with the superconducting Josephson junction? Is the
conductance finiteness due to experimental limitations, (impurities, etc.),
or to the order parameter topological defects at finite temperature? Those
questions are still being discussed among specialists.
8.4.5
Antiparallel currents experiment
In order to check the ideas about bosonic superfluid exciton liquid in the bilayer system at ν = 1, one needs an experimental proof of electron-hole pair
transport. How can one couple to and detect electrically neutral objects by
electric transport? The solution is to notice that electron-hole pair transport
in a bi-layer implies an anti-parallel circulation of currents in different layers.
Experiments have allowed, in the last few years to get independent electric
connections to each layer [108]. It has thus been possible to inject equal
intensity currents with opposite flow direction in both layers, to test the
contribution of excitons to particle transport [6].
The figure 8.7 is a schematic representation of what one expects from
such an anti-aparallel current experiment. The two traces represent the expected Hall voltage in each layer, neglecting all quantum phenomena except
the excitonic condensation. Because of the Lorentz force, the Hall voltage is
proportional to the magnetic field. In a bi-layer system driven by two oppositely directed parallel currents, the Hall voltage will have opposite signs in
the two layers. If the two layers are sufficiently coupled, and the magnetic
field has the relevant intensity (so that ν↑ = ν↓ ≃ 1/2 the inter-layer electronhole pairs which form will carry anti-parallel currents. The Hall voltage in
both layers must then vanish, as suggested by the figure 8.7.
Two experimental groups have confirmed those predictions [104].
143
(a)
A)
NT=10.9 D)
(d)
B)
(b)
NT=6.9
NT=5.4
z
-7
-1
Tunneling
=1 /dV
(10 W )
Conductance
at nTdJ
Conductance
tunnel
Experiments
0.5
NT=6.4
(c)
C)
-5
0
5
-5
0
5
Tension
intercouches V (mV)
Interlayer Voltage (mV)
Figure 8.6:
Quasi-Josephson effect [102]. Plot of the tunneling conductance dJz /dV
as a function of voltage bias between the two layers, for various electronic densities, NT
in units of 1010 cm−2 . In the samples [from (a) to (c)] with larger electronic density (i.e.,
smaller lB ) the system does not exhibit any QHE, tunnelling processes are suppressed at
zero bias. In the low density sample (d) there is a finite tunneling conductance peak at
zero bias. The current at zero bias vanishes, contrary to the superconducting Josephson
junction current. Whence the expression ”quasi-Josephson effect”.
144
Quantum Hall bi-layers
tension de Hall
10
+ − + − + − +
− + − + − + −
+ − − + − + −
0
+ + − − + + −
ν=1
−10
5
10
champ magnétique
Figure 8.7: Antiparallel currents experiment (After ref. [6]). A Hall voltage measurement detects the exciton condensation. The two traces are schematic representations of
the Hall voltage in each layer when electric currents flow in opposite directions. Quantum
effects other than the excitonic condensation are ignored in this figure. When currents
flow in an uncorrelated manner between both layers, one must observe finite Hall voltages,
which balance the Lorentz force in each layer. As the currents flow in opposite directions,
Hall voltages must have opposite signs in each layer compared to the other. If exitonic
condensation occurs, in a certain span of magnetic field values, the opposite currents in
the layers will be carried by a uniform exciton current density in one direction. Since
excitons are electrically neutral, they are not submitted to Lorentz forces, and the Hall
voltage must vanish in both layers, as observed experimentally by Kellogg et coll. [104].
8.5. EXCITONIC SUPERFLUIDITY
8.5
145
Excitonic superfluidity
Within the pseudo-spin analogy(section 9.2), the Coulomb interaction between layers favours a ferromagnetic state with an easy magnetization plane
when the inter-layer distance is small enough to stabilize a correlated state.
The ground state wave function is then of the form [see equation (8.1)]


Y  c†k↑ + eiφ c†k↓ 
√
|ψφ i =
|0i .


2
(8.9)
k
In other words, θ/2 = π/4, the magnetization is in the bi-layer plane, and
one has hSz i = 0. The amplitude is equal for opposite pseudo-spin states,
which means that, for the time being, we consider a situation with zero bias
between the two layers. The total occupation of each k state is 1. When the
tunnel term t vanishes, φ has any value, provide it is the same all over the
bi-layer plane. When the system chooses a particular φ value, among the
continuous infinity of choices, the original U (1) symmetry of the Hamiltonian
(in the presence of the pseudo-spin anisotropy) is broken by the ground state.
In the limit of zero tunneling term, we have thus a one parameter family
of equivalent ground states, with the phase φ as parameter. This phase is
conjugate to the difference in particle number between the two layers. In
equation (8.9), the phase is well defined, but the number of particles of
each pseudo-spin (i.e. the number of particles is each layer) is completely
undetermined. Similarly, one may construct a state such that the phase is
undetermined, while the number of particles in each layer is specified exactly.
To do this, integrate 8.9 over the phase. This yields
|ψS z i =
Z
dφ −i(N↑ −N↓ )φ
e
|ψφ i .
2π
(8.10)
We obtain thus a wave function with exactly N↑ particles in the ↑ layer, and
N↓ = N − N↑ in the ↓ layer, N being the total number of guiding centers.
The angle φ and S z are canonical conjugate variables,
[φ, Sz = N↑ − N↓ ] = 1,
(8.11)
whence the uncertainty relation δ(N↑ − N↓ ) × δφ > 1.
Since we are dealing with a continuous broken symmetry, there must exist
a Goldstone mode the energy of which goes to zero in the limit of infinite
146
QHE bi-layers
wavelength. A state such that the phase varies in time and space may be
written as
i
Yh †
|ψφ i =
ck↑ + eiφ(Xk ,t) c†k↓ |0i ,
(8.12)
k
where φ is the superfluid phase of the system. The long wavelength superfluid
mode corresponds to equal intensity currents of opposite signs propagating
in the two layers.
To understand better why the state described by (8.9) breaks the gauge
symmetry associated to the charge difference between layers, consider the
θ
gauge transformation induced by the unitary operator U− (θ) = ei 2 (N↑ −N↓ ) .
This transformation acts on electron creation operators as
θ
U−† (θ)c†k↑ U− (θ) = e−i 2 c†k↑
(8.13)
.
(8.14)
U−† (θ)c†k↓ U− (θ)
i θ2
=e
c†k↓
The Hamiltonian is invariant under this transformation,
U−† (θ)HU− (θ) = H,
(8.15)
since [H, (N↑ − N↓ )] = 0, in the absence of inter-layer tunneling terms.
In contrast, expression (8.9) shows that the coherent phase exhibits a non
trivial order parameter.
D
E
D
E
nel iφ
S x (Xk ) ≡ c†k↑ ck↓ = S̄ x (Xk ) =
e ,
2
2
with the total density nel = 1/2πlB
. (Here I have defined the x̂ direction
as the arbitrary direction of the sponaneous pseudo-spin orientation in the
plane x, y).
This order parameter is not gauge invariant,
D
E
S x (Xk ) → U−† (θ)c†k↑ ck↓ U− (θ) = eiθ S x (Xk ) .
(8.16)
That is a more formal way to show that the state has less symmetry
than the Hamiltonian, and breaks the U (1) symmetry associated with the
conservation of the charge difference between layers N↑ − N↓ . 1
In a superconductor, the order parameter χ(r) = hc†↑ (r)c†↓ (r)i transforms in a non
trivial way under the gauge transformation associated with the total charge conservation,
Ũ+ (θ) = exp[iθ(N↑ + N↓ )/2]. The pseudo-spin bi-layer order parameter is invariant under
this transformation: this expresses simply the fact that the total particle number N↑ + N↓
is conserved in the excitonic superfluid.
1
Excitonic superfluidity
147
We can write an expression for the inter-layer tunneling current operator
as a function of position in space,
J↑↓,Xk = −it c†k↑ ck↓ − c†k↓ ck↑ ,
the average of which
(8.17)
hJ↑↓,Xk i = −it [S x (Xk )∗ − S x (Xk )] = t sin(φ).
This expression is similar to the Josephson current expression: it depends
only on the order parameter phase, not on the inter-layer voltage bias.
The pseudo-spin language expresses the conjugate character of phase and
charge difference between layers through the commutation relations of the
spin density operators. With the order parameter along x,
[S y , S z ] = iS x ≃ i.
As S y ∝ sin φ ≈ φ, this leads to [φ, S z ] = i. As a consequence, the current
associated to the phase gradient
Jzz =
2ρE
∇φ
h̄
is indeed the difference of the electric currents in the two layers.
An apparent conceptual difficulty is that the wave function (8.9) describes
a state where the difference between the layer charges fluctuates, while this
difference should be conserved in the limit t = 0 . This is analogous to the
superconducting BCS wave function, which has a fluctuating total number
of particles, while it is in fact strictly conserved for an isolated sample.The
solution of this apparent paradox is that each macroscopic piece of the sample
may be subdivided in smaller macroscopic parts, between which particle
exchanges are numerous and rapid, so that phase coherence is established
in each macroscopic part of the sample, the total particle number remaining
constant. Furthermore, in the thermodynamic limit, the ratio δN/N is of
order N −1/2 → 0.
A similar reasoning holds in the bi-layer case. Examine a slighlty more
complicated object than the order parameter,
D
GXk ,Xk′ = c†k↑ ck↓ c†k′ ↓ ck′ ↑
E
.
(8.18)
This object conserves the total particle number in each layer. It is equal to
hS̄ x (Xk )S̄ x (Xk′ )i, and it is non zero in the wave function (8.9).
148
QHE bi-layers
Notice that the wave function (8.9) is indeed an exciton condensate. To
see that, define the state |f erro ↑i as the state where all electrons are in the
Q
↑ layer , |f erro ↑i = k c†k↑ |0i. Then the state (8.9) may be re-written as
|ψφ i ≡
Y
k


1 + eiφ c†k↓ ck↑

 |f erro ↑i .
√
2
(8.19)
This can be again re-written in a form reminiscent of a bosonic coherent
state,
Y
exp eiφ b†k |f erro ↑i ,
(8.20)
|ψφ i ≡
k
where b†k = c†k↑ ck↓ is the excitonic boson. This is the reason why one may
speak of a ”coherent” state (see section 2.2.2). One also speaks of ”spontaneous phase coherence”, when the tunneling term is absent. Indeed in
that case the coherent state is entirely due to Coulomb interactions. On the
contrary, when the tunneling term is finite, the symmetric combination of
layer states is the most stable, even in the absence of interactions. This is
analogous to the magnetization induced by an external magnetic field in the
case of ”real” spins.
8.5.1
Collective modes – Excitonic condensate dynamics
As mentionned above, a consequence of the breaking of a continuous symmetry by the phase coherence is the existence of the collective excitation
mode (Goldstone mode) the energy of which goes continuously to zero as the
wavelength goes to infinity. The Hamiltonian formalism was used above to
derive collective mode energies in the ferromagnetic monolayer case. Here we
use the Lagrangian formulation, with the inclusion of the Berry connexion
term discussed in the previous chapter. The Lagrangian which describes the
long wavelength physics, in the absence of applied inter-layer voltage bias,
and with zero tunneling term, is
ν Z 2
L =
d rṁ · A[m]
(8.21)
2
4πlB
Z
i
ρA
ρE h
− d2 r β(mz )2 +
.
|∇mz |2 +
|∇mx |2 + |∇my |2
2
2
Excitonic superfluidity
149
Coefficients β, ρA and ρE may be evaluated with a microscopic approach, as
we have seen in section 8.3.2 Let us write the Euler-Lagrange equations of
motion,
d δL
δL
=
.
(8.22)
µ
dt δ ṁ
δmµ
Here the ground state is taken with the (vector) order parameter of length
1 aligned along the x̂ axis. For small variations of the order parameter away
from x̂, one may linearize, considering only first order deviations in my and
mz . m = [1 − O(m2y + m2z ), my , mz ], and one chooses the Berry connexion
A = (0, −mz /2, my /2), which yields
ν
δL
ν mz
,
=
Ay [m] = − 2
2
δ ṁy
4πlB
4πlb 2
ν ṁz
δL
=
+ ρE ∆my ,
2
δmy
4πlB
2
(8.23)
and
ν
ν my
,
Az [m] = − 2
2
4πlB
4πlb 2
ν ṁy
+ ρA ∆my − 2βmz ,
=
2
4πlB
2
δL
δ ṁz
δL
δmz
=
(8.24)
where ∆ = ∇ · ∇ is the Laplacian. In Fourier space, applying 8.22, one finds
the system of linear equations
iω
4π 2
q
ρE
ν
4π
(2β
ν
+ q 2 ρA )
−iω
!
my
mz
!
= 0.
(8.25)
So finally the collective mode dispersion relation is given by
4π
ω (q) =
ν
2
2
(2β + q 2 ρ2A )q 2 ρE .
(8.26)
When d = β = 0, and ρA = ρE = ρ0 one retrieves the collective mode
(pseudo-spin wave ) of the ferromagnetic SU (2) phase,
ω(q)|B=0 =
2
4π 0 2
ρq .
ν
The expansion in gradients of mz is not stricly correct, because the long range nature
of he Coulomb interaction induces a non local term which we do not take into account
here. The latter term is smaller than the terms considered here.
150
QHE bi-layers
Q=+1/2
v=+1
Q=+1/2
v=−1
Q=−1/2
v=+1
Q=−1/2
v=−1
Figure 8.8: Four meron ”flavours”. With two possibilities for the choice of the vorticity,
and two additional ones for the choice of the pseudo-spin orientation at the vortex core,
merons have a topological charge Q = ±1/2, and exist in four possible ”flavours”.
The mass term β 6= 0 changes qualitatively the collective mode dispersion,
which becomes linear in q at small q,
lim ω(q)|β6=0
q→0
4π q
=
2βρE q .
ν
That is analogous to the bosonic superfluid collective mode (with weak repulsive interactions). But here the order parameter represents the condensation
of neutral bosons, which carry no charge.
8.5.2
Charged topological excitations
For a system in the same universality class as that of the 2D XY model, there
must exist a Kosterlitz-Thouless (KT) transition at TKT = (π/2)ρS /kB . The
essence of this transition is the ionisation (dissociation) of vortex-antivortex
pairs. In our case, the order parameter symmetry group is U (1), but the
pseudo-spin direction is not confined to the xy plane, so that the pseudo-spin
vortex is in fact a ”meron” , which may be considered as a half skyrmion.
The system order parameter in the presence of a vortex at the origin has
the approximate following form
q
m = ± 1 − m2z cos θ,
q
1 − m2z sin θ, mz (r)
,
(8.27)
Excitonic superfluidity
151
where the ± sign refers to the vorticity (left or right) and θ is the azimuthal
angle of the position vector r. At large distance from the meron center, mz (r)
tends to zero to minimise the capacitive energy. At the vortex core, however,
we have mz = ±1, mx = my = 0, to avoid the large energy cost of a core
singularity.
The local topological charge is computed using the Pontryagin density
expression [see equation (7.31)]
δρ = −
1 ij
ǫ (∂i m × ∂j m) · m.
8π
With expression (8.27), this density writes
δρ(r) =
1 dmz
.
4πr dr
The total charge is Q = d2 rδρ(r) = 21 [mz (∞) − mz (0)]. For a meron, the
spin at the core is either ↑ or ↓, and gradually gets oriented in the xy plane
as the distance from the core increases. It lies in the xy plane far from the
meron core. The topological charge is thus ±1/2 depending on the core spin
polarity.
The general result for the topological charge is
R
Q=
1
[mz (∞) − mz (0)] nv
2
(8.28)
where nv is the vortex winding number. The electric charge is ±νe/2, half
that of a skyrmion, which comes as a support of the meron as a half skyrmion,
as mentionned above.
One may write a meron variational wave function. The simplest one is


†
†
M
E
Y
c
+
c
 m,↑ √ m+1,↓  |0i .
ψnv =+1,−1/2 =
2
m=0
(8.29)
In this expression, c†m,↑(↓) creates an electron in layer↑ (↓), in the state of
angular momentum m in the LLL, and M is the corresponding moment on
the sample edge. The vorticity is +1, since far from the core, the spinor is
√
χ(θ) = (1/ 2)
eiθ
1
!
,
152
QHE bi-layers
where θ is the polar angle of the vector r. The charge is +1/2 because an
electron has been suppressed at the center, in the ↓ layer: all states have 1/2
occupation, except m = 0 which is empty. The meron charge can be changed
without changing the vorticity, as we see with the wave function


†
†
M
E
Y
c
+
c
†
 m,↑ √ m+1,↓  |0i .
ψnv =+1,+1/2 = c0,↓
2
m=0
This state has charge −1/2, because an electron has been created in the state
m = 0 in the ↓ layer.
It is useful to examine a meron pair wave function, to check wether the
meron is a half skyrmion. Examine the case of a pair of merons with opposite
vorticities, but equal charges, placed at points z̄1 and z̄2 . The following wave
function seems to obey our requirements,
ψλ =
eiφ (zj − z̄1 )
(zj − z̄2 )
Y 1
j
√
2
!
Φf erro ,
(8.30)
j
where φ is an arbitrary angle and ()j is a spinor for the j-th particle.
At large distance from z̄1 and z̄2 , the spinor for each particle becomes
eiφ
1
zj
!
.
(8.31)
This corresponds to a fixed spin orientation in the xy plane, with an angle φ
with the x axis. Vorticity is thus zero. By construction, the spin orientation
is purely ↑ for an electron at z̄2 , and purely ↓ for an electron at z̄1 . Moreover,
the net charge must be νe since, asymptotically, the factor zj is the same
as for the Laughlin quasi-particle in the spin polarised state. For symmetry
reasons, one might think that a charge νe/2 is asociated to each localised
state near z̄1 or z̄2 .
The fact is that this wave function (8.30) is nothing but a different representation for the skyrmion! Choose z̄1 = λ and z̄2 = −λ, and suppose for
simplicity that the asymptotic orientation of spins is in the x direction, so
that φ = 0. Now rotate all spins by a global rotation around the ŷ axis, with
an angle −π/2. Using
π
1
exp i σ y √
4
2
zj − λ
zj + λ
!
,
Excitonic superfluidity
153
one finds the variational skyrmion wave function. The previous wave function
is well adapted to the U (1) symmetry, because t describes spins oriented
mainly in the xy plane.
8.5.3
Kosterlitz-Thouless transition
The presence of topological defects of the vortex type may spoil the phase
coherence of the XY ground state. This may happen at zero temperature,
because of quantum fluctuations, if the distance between layers exceeds a
critical distance d∗ . Here we are discussing thermal effects.
The effective model at finite temperature is given by
ρS Z 2
E=
d r |∇φ|2 .
2
For typical experimental parameter values in the AsGa bi-layers, the HartreeFock estimate of the exchange stiffness ρS goes from 0, 1K to 0, 5K.
The Kosterlitz-Thouless transition is due to ionisation of vortices in the
XY model, at a temperature TKT approximately given by the exchange stiffness ρS . Free vortices induce a discontinuous renormalisation of the exchange
stiffness, which vanishes at TKT . The classical action generates a logarithmic
interaction between vortices.
A meron gas has an energy of the form
E = M Ecore − 2πρS
M
X
i<j
ni nj ln
M
X
e2
Rij
qi qj
+
,
Rcore
4ǫRij
i<j
(8.32)
where Ecore is the meron core energy, Rcore its size, and Rij is the separation
between the i-th and the j-th meron. The last term is new. It is specific of the
QHE bi-layer physics: it is due to Coulomb interactions between the merons
fractional charges. qi = ±1 is the electric charge (±e/2) sign, of the i-th
meron. The origin of the logarithmic term is not, as in the superconducting
case, the kinetic energy stored in the supercurrents. It comes from the loss
of exchange energy due to the phase gradients associated to vortices. The
Coulomb interaction is irrelevant at TKT because it decreases faster with
distance than the logarithmic interactions. It may cause a shift of TKT ,
but the transition is not qualitatively altered. The phase diagram however
becomes richer, with chiral phases, with an order parameter hni qi i where
vorticity and electric charge are no longer independent[109].
154
QHE bi-layers
v=−1
ξ
Λ
v=+1
Figure 8.9: In the presence of an inter-layer tunneling term, meron pairs of opposite
vorticities are bound by a string, or domain wall, of length Λ and characteristic width ξ.
The meron confinement energy varies linearly with Λ.
8.5.4
Effect of the inter layer tunneling term
As already discussed above, an inter-layer tunneling term breaks the U (1)
Hamiltonian symmetry
Hef f =
Z
t
ρs
|∇φ|2 −
cos φ .
dr
2
2πρ2
2
(8.33)
Here ρs is the iso-spin exchange stiffness, which may be computed microscopically with the same techniques used to compute the equivalent parameter
in the ”true” ferromagnetic case. For a finite t value, the collective mode
acquires a mass (just as spin wave in a SU (2) ferromagnet acquire a mass
in an external field, because of the Zeeman effect). Quantum fluctuations
are thereby decreased, which explains the upwards curvature of the phase
transition line in the phase diagram.
The tunneling term, because it breaks the U(1) symmetry, and gives a
larger energy cost to vortex pairs configurations, destroys the KT transition.
Excitonic superfluidity
155
To lower the energy, the system deforms the spin deviations in domain walls,
or strings, which connect vortex cores, as shown on figure 8.9. Spins are
oriented in direction x, which is imposed by the tuneling term everywhere,
except in the wall region, where they rotate quickly of 2π. The wall energy is
proportional to its length Λ, so that we have a vortex confinement mechanism
analogou to quark confinement in elementary particles such as hadrons or
mesons. The line tension (the energy per unit length) may be estimated by
examining the infinitely long domain wall parallel to the y axis. The optimal
form is given in that case by
φ(r) = 2 sin
−1
x
tanh
ξ
!
,
(8.34)
1/2
2
where the characteristic wall width is ξ = (2πlB
ρS /t)
thus (see [110])
!1/2
8ρs
tρS
=
.
T0 = 8
2
2πlB
ξ
. The line tension is
(8.35)
If the wall is long enough (Λ ≫ ξ), the total energy of a segment of length
Λ will be approximately
Epair = 2Ecore +
e2
+ T0 Λ.
4ǫR
(8.36)
Minimising, we conclude that Epair is optimal for Λ = Λ′0 = (e2 /4ǫT0 )1/2 ,
whence
!1/2
e2 T0
′
Epair = 2Ecore +
.
ǫ
Thus, except for meron core energies, the charge gap at fixed d (i.e. at
1/4
1/2
fixed ρs ), is proportional to T0 ∝ t1/4 ∼ ∆SAS . This is in contrast with
the free electron case. The exponent 1/4 is small. The charge gap increases
quickly as soon as the tunneling term is non negligible. The cross-over regime
between the pseudo-spin meron pair textures and the domain wall texture is
established at a finite t value.
8.5.5
Combined effects of a tunnel term and a parallel
field Bk
We have seen that the vector potential corresponding to a parallel field Bk
may be chosen as A = (0, 0, Az|| = Bx), where z is the direction perpendicular
156
QHE bi-layers
to the layers. Equation (8.7) shows that the expression of the tunneling
matrix element changes with the field, so as to respect gauge invariance:
instead of a constant phase φ = 0 (spin alignment along the x axis, as we saw
in section 9.3), one finds a spatial variation of the phase, φ → φ − Qx. The
tunneling term competes with the spin stiffness one. The latter is minimized
for a uniform magnetization, while the former favours a rotating one, when
Bk 6= 0,
)
Z (
St
2
ρS |∇φ| −
H=
cos(φ − Qx) d2 r .
(8.37)
2
2πlB
This Hamiltonian, known in the 2D physics of commensurate-incommensurate
transitions as the Pokrovski-Talapov model, has a phase diagram structure
which depends on the relative values of ρs and Q.
If Q → 0, the energy is minimised by a phase φ = Qx, the exchange
energy loss being ρs Q2 . One has then a commensurate state (the term phase
here is used for the order parameter phase φ, not to be confused with a
”thermodynamic phase”) : for any x, the order parameter phase is locked at
the value dictated by the periodic potential minima.
When Q increases, minimising the periodic term with a linear variation of
the phase becomes too costly in exchange energy. The conflict between those
two terms results in the appearance of solitons which are phase defects. The
latter which are solutions of a Sine Gordon equation, express the compromise
between the ”elastic energy” (the exchange term, quadratic in Q, and the
periodic ”potential energy”, (the sinusoidal tunneling term). The limiting
~ ∼ 0 and the average value of cos(φ − Qx) vanishes.
behaviour is when ∇φ
For Bk larger than a critical value Bkc periodically ordered topological defects
start being formed, in a uniaxial 2D anisotropic environment.
At zero temperature, the critical value is
Bkc
2lB 2t 1/2
= B⊥
.
πd πρs
With ∆SAS = 0.45K, one finds Bkc ≈ 1, 3T, slightly larger than the observed
value 0.8T. The corresponding value of Lk is large.
In the commensurate phase, the order parameter tumbles more and more
rapily as Bk incrases, since φ = Qx. In the incommensurate phase, the system state becomes roughly independent of Bk , so that the excitation gap
saturates at a fixed value. In the presence of the tunneling term, the lowest energy charged excitations are meron pairs with opposite vorticities and
Excitonic superfluidity
157
equal charges (i.e. ±1/2, connected to one another by a domain wall with
a constant line energy. For Bk = 0, the energy is independent of the wall
orientation. The effect of Bk is more clearly seen with a variable change.
Let ϕ(r) = φ(r) − Qx. This variable is constant in the commensurate phase,
and varies in the incommensurate one. In terms of this new variable, the
Hamiltonian becomes
H=
Z
(
)
i
ρS h
t
2
2
dr
(∂x ϕ + Q) + (∂y ϕ) −
cos ϕ .
2
2
2πlB
2
(8.38)
Thus Bk defines a preferred direction of this problem. Domain walls align
in this direction and involve a phase change, in terms of ϕ, with a preferred
sign (negative for Q > O).
One can show that the energy per unit soliton length of the wall, i.e. the
line tension, decreases linearly with Q, and thus with Bk , i.e.


Bk
T = T0 1 − ∗  ,
Bk
where T0 is given by equation (8.35). There is a transition when T becomes
negative. We have seen in section 7.5.5 that the charge excitation
q gap is
given by the vortex pair energy with an optimal separation Λ = e2 /4ǫT .
The equation (8.36) for the meron pair energy is equally affected by the T0
renormalisation, which yields
Epair ≃ 2Ecore +
s

1/2
Bk
e2 T0 
1 − ∗
ǫ
Bk
.
Thus, as Bk increases, the line tension decreases and the line gets longer.
On the whole, the energy is lowered. Far into the incommensurate phase,
the inter layer tunneling term becomes negligible. Therefore, the ratio of the
charge gap at Bk = 0 and that when Bk → ∞ should be roughly
∆0
t 1/4 (e2 /ǫlB )1/2 t1/4
.
=
≃
3/4
∆∞
tcr
8ρs
Using typical values for t and ρs , the former expression yields values between
1.5 and 7, in qualitative agreement with experiment.
158
8.5.6
QHE bi-layers
Effect of an inter-layer voltage bias
What is the tunneling curent in the presence of an inter-layer voltage bias?
The total tunneling current is
It ∝ et
Z
n
d2 r ei[φ(r)+Qx] − e−i[φ(r)+Qx]
n
o
= F eiφ(r) qy =0,qx =Q
o
n
o
− F e−iφ(r) (8.39)
qy =0,qx =−Q
(8.40)
where F{f (r)}|q is the 2D Fourier transform of f (r) at wave vector q.
Experimental results show that the tunneling current vanishes at zero
inter-layer voltage bias, so that the current can be computed perturbatively.
To second order in t, one has
It (V ) =
2πet2 L2
[S(Q, eV ) − S(−Q, −eV )] .
h̄
(8.41)
where S(q, h̄ω) is the fluctuations spectral density of the opeator eiφ at wave
vector q et and frequency ω, i.e. the transform of heiφ(r,t) e−iφ(0,0) i. A striking
prediction follows: when disorder is weak, the spectral density, and thus
It (V ) exhibit a peak centered at
eV = h̄ωQ
where ωQ is the collective frequency at wave vector Q. Thus as Bk varies,
the conductance peak position varies according to the low energy collective
mode dispersion. The parallel field only allows tunneling events between
states which differ by their momentum Q. Energy conservation ensures that
the state energies differ by eV . This has been fully confirmed by experiment
[111]. See figures 8.10, and 8.11. A transport experiment allows thus a direct
measurement of the collective mode dispersion relation, which is found to be
linear in Q, as predicted by theory.
As shown in figure 8.10, the tunneling current in the presence of a parallel field Bk exhibits a peak which corresponds directly to the collective mode
dispersion of the superfluid phase. The figure shows the tunneling conductance at T = 25mK for an electronic density of 5.2 × 1010 cm−2 , for a series
of parallel magnetic field values between Bk = 0 and 0.6T. The insert is a
blow-up of curves for Bk values between 0, 07 and 0, 35T. The dots indicate
the position of satellite resonances for dI/dV . (After [111]
Excitonic superfluidity
159
2.5
-7
1.5
10 Ω
-1
-6
-1
dI/dV (10 Ω )
2.0
-200
0
200
B|| = 0
V (µV)
1.0
0.5
B|| = 0.6T
0.0
-200
0
200
V (µV)
Figure 8.10: Experimental determination of the Goldstone mode dispersion in a transport experiment [111]. The tunneling current in the presence of a parallel field Bk exhibits
a peak which corresponds directly to the collective mode dispersion of the superfluid
phase. The figure shows the tunneling conductance at T = 25mK for an electronic density
of 5, 2 × 1010 cm−2 , for a series of parallel magnetic field values between Bk = 0 and 0.6T.
The insert is a blow-up of curves for Bk values between 0, 07 and 0, 35T. The dots indicate
the position of satellite resonances for dI/dV . (After reference [111]
160
QHE bi-layers
eV* (meV)
0.2
0.1
0
0
10
20
6
30
-1
q (10 m )
Figure 8.11:
Goldstone mode dispersion determined by Spielman et al. [111] [111].
Energy eV ∗ of the resonance peaks, as a function of the wave vector Q = eBk d/h̄ in
the presence of a parallel magnetic field, for different electronic densities (cross : nel =
6.4 × 1010 cm−2 ; squares : nel = 6.0 × 1010 cm−2 ; black dots : nel = 5, 2 × 1010 cm−2 . The
dotted line is a theoretical estimate by Girvin for the Goldstone mode dispersion at small
q [87]. The continuous line is a guide for the eye, and corresponds to a collective mode
velocity of 1, 4 × 104 m/s.
In figure 8.11, the energy eV ∗ of the resonance peaks is shown as a function of the wave vector Q = eBk d/h̄ in the presence of a parallel magnetic
field, for different electronic densities (cross : nel = 6.4 × 1010 cm−2 ; squares
: nel = 6.0 × 1010 cm−2 ; black dots : nel = 5.2 × 1010 cm−2 . The dotted
line is a theoretical estimate by Girvin for the Goldstone mode dispersion at
small q [87]. The continuous line is a guide for the eye, and corresponds to
a collective mode velocity of 1.4 × 104 m/s.
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