Download A conformal field theory approach to the fractional quantum Hall

University of Amsterdam
MSc Physics
Track Theoretical Physics
Master Thesis
A Conformal Field Theory Approach to
the Fractional Quantum Hall Effect in Graphene
by
B.A. van Voorden
10193685
July 2016
60 ECTS
September 2015 - July 2016
Supervisor:
Prof.dr. C.J.M. Schoutens
Examiner:
Dr. P.R. Corboz
Institute of Physics
Institute for Theoretical Physics Amsterdam
Table of Contents
Page
Scientific Abstract
1
Introduction
The fractional quantum Hall effect in graphene
Structure of the thesis
Acknowledgements
2
2
3
4
Chapter 1. Anyons and Non-Abelian Statistics
1.1. Bosons, fermions and anyons
1.2. Non-abelian anyons
5
5
6
Chapter 2. The Quantum Hall Effect
2.1. Experiments
2.2. Theoretical description of the integer quantum Hall effect
2.3. Topology
2.4. Theoretical descriptions of the fractional quantum Hall effect
2.4.1. Laughlin wave functions
2.4.2. Composite particles
2.4.3. Moore-Read states
2.4.4. Read-Rezayi states
2.4.5. Halperin states for multiple components
8
8
10
11
11
11
12
13
13
13
Chapter 3. Graphene
3.1. Electronic properties
3.2. The integer quantum Hall effect
3.3. The fractional quantum Hall effect
15
15
16
17
Chapter 4. Mathematical Preliminaries
4.1. Conformal Field Theory
4.2. Group Theory
4.3. Affine Lie algebras
19
19
21
24
Chapter 5. The Conformal Field Theory Approach
5.1. Laughlin
5.2. Moore-Read
5.3. Read-Rezayi
5.4. Multiple component wave functions
25
25
26
27
29
Table of Contents
Chapter 6. SU(4) Wave Functions
6.1. Generalized Halperin wave function
6.2. Construction of the SU(2)k,M NASS states
6.2.1. Quasiholes
6.3. Construction of the SU(4)k,M states
6.3.1. SU(5) root and weight space
6.3.2. (k,M)-clustering
6.3.3. Quasiholes
31
31
32
36
38
38
41
44
Chapter 7.
45
Four Component Polarized Wave Functions
Chapter 8. Numerical Analysis
8.1. FQHE wave functions on the sphere
8.2. Exact diagonalization
8.3. Numerically obtaining the theoretical wave functions
8.4. Analysis
48
48
49
50
52
Chapter 9. Quasihole State Counting
9.1. Fields and fusion rules
9.2. Degeneracy factors
9.3. Counting results
9.3.1. Example 1
9.3.2. Example 2
9.3.3. Example 3
54
54
56
57
58
59
59
Summary and Discussion
61
Appendix A.
62
Wave Functions for Charged Particles in a Magnetic Field
Bibliography
64
Populair-wetenschappelijke samenvatting
67
1
Scientific Abstract
In recent years, measurements of the quantum Hall effect in graphene
have shown a unique QHE landscape. At low energies, the Landau
levels in graphene are approximately SU(4) symmetric, which raises
the possibility for new sets of FQH states to occur. In this thesis,
the SU(2)-symmetric non-abelian spin singlet wave functions are
generalized to SU(4)-symmetric non-abelian singlet states using a
conformal field theory description of FQH states. This new set of
wave functions describes the FQHE of k-clustered SU(4)-symmetric
particles at filling fractions ν = (4k)/(4kM + 5), where M determines
if the particles are fermionic or bosonic. For k ≥ 2, the quasiparticle
excitations above the ground state behave as non-abelian anyons,
which are of great interest due to their possible use in a fault-tolerant
topological quantum computer. For the k = 2 wave function on the
sphere, the zero energy states are calculated after the insertion of
extra flux quanta, for a select number of cases.
2
Introduction
The fractional quantum Hall effect in graphene
One of the main subjects that condensed matter theory is concerned with is the classification
of different states (or phases) of matter when many particles are brought together. Until
a few decades ago, it was thought that all quantum mechanical phases can be described
using Landau’s symmetry-breaking theory. This theory states that different phases have
different symmetries and that a phase transition occurs when one of those symmetries is
broken. Moreover, each phase can be characterized by a local order parameter that has a
finite expectation value. A clear example is the phase transition between a ferromagnetic and
an antiferromagnetic phase, where the symmetry is the rotational symmetry and the order
parameter is the magnetization. When the antiferromagnetic phase transitions to the ferromagnetic phase, the rotational symmetry is broken and the magnetization becomes nonzero.
However, it was discovered in the 1980’s that there exist states of matter that can not be
described using this theory. These new phases can have the same symmetry properties, but
are of a different topological order. There are no local order parameters that describe the
different topological phases, but sometimes a global order parameter can be defined. The
fractional quantum Hall effect, first measured in 1982, was the first realization of such a
topological phase of matter. In the quantum Hall effect, the conductance of the system
is quantized at special values of the applied magnetic field perpendicular to the system.
The different states are characterized by the filling fraction ν. For integer values of ν, the
quantum Hall effect can be explained by considering single particle physics, but this is
impossible when ν is a (rational) fraction. This fractional quantum Hall effect is inherently
a macroscopic many body phenomenon and is ‘the’ system showing a strongly correlated
topological effect.
One of the interesting properties of fractional quantum Hall states is that their excitations
can have a fractional charge. These fractionally charged excitations obey fractional statistics,
meaning that they are neither bosons nor fermions, but so called anyons. In itself, this
occurrence of quasiparticles with completely different statistics than all other known particles
is already interesting, but for certain states there is also the possibility that the anyons
are non-abelian anyons. Non-abelian anyons are of great interest for the field of quantum
computing, because they can, in theory, be used to make a fault-tolerant quantum computer. In the last decade there has been a renewed interest in the fractional quantum Hall
effect due to its measurement in graphene. Graphene is an almost perfect two dimensional
hexagonal grid of carbon atoms, with many remarkable properties. The quantum Hall effect
in graphene is also special, in that the structure of the energy bands is different than in
any other measured material, due to the occurrence of “relativistic electrons” that have an
approximate SU(4) symmetry. Furthermore, the integer quantum Hall effect in graphene
has reliably been measured at room temperature, while all other systems before needed to
be cooled to almost zero Kelvin.
Introduction
To theoretically explain the fractional quantum Hall states, the Coulomb force between
the many particles needs to be taken into account, which makes the construction of the
wave functions analytically impossible. By making educated guesses, many wave functions
describing the FQHE at different fractions have been proposed. The properties of these
theoretical states can be compared to the states measured in experiments and they can
also be analyzed using numerical methods. In many cases, these idealized wave functions
approximate the real wave functions to a high degree. Multiple states have been proposed to
describe the different FQHE states in graphene, often generalizations of previously created
states.
The goal of this thesis is to construct a set of SU(4)-symmetric fractional quantum Hall
states with non-abelian excitations, that can possibly occur in graphene. The derivation of
these wave functions relies on an analysis using conformal field theory and group theory in
analogy with an analysis done before for non-abelian SU(2) spin singlet states [1].
Structure of the thesis
I wrote this thesis with the intention that any master student theoretical physics can follow
the subjects discussed. However, the field of fractional quantum Hall effects is diverse
and comprehensive and I can not include many interesting aspects. Therefore, I chose to
only highlight the most useful and applicable aspects of quantum Hall effects, graphene,
conformal field theory and group theory. I did include references to many review articles
and books about specific subjects for readers interested in a more thorough understanding
of the subjects discussed in this thesis.
The structure of this thesis is as follows. The first few chapters are used to give an introduction
to the different aspects that are important to understand the methods and the results. In
chapter 1, an overview of anyons will be given. It will be explained what they are and how
they can occur in nature, their possible use in quantum computation will also be mentioned.
An introduction to the quantum Hall effect, both experimentally and theoretically will
then be given in chapter 2. Chapter 3 will give an introduction to graphene and it will
specifically focus on the quantum Hall effect in graphene. Finally, the introductory chapters
will be terminated by chapter 4 on the mathematical preliminaries, namely conformal field
theory and group theory. The use of conformal field theory in describing quantum Hall
wave functions is the subject of chapter 5, which is followed by the actual construction
of the new SU(4) symmetric wave functions in chapter 6. In chapter 7, a set of polarized
four component states is created by combining previous wave functions. The numerical
analysis of these theoretical quantum Hall wave functions will be described in chapter 8.
The SU(4) wave functions are further analyzed in chapter 9 by counting the degeneracy of
the states with an excess number of flux quanta. At the end of the thesis, appendix A can be
found containing more in depth calculations of the quantum Hall wave functions, which were
deemed unnecessary to place in the body of the thesis. The bibliography with all sources
cited in this thesis can be found after that. This thesis is concluded by a (Dutch) popular
science abstract, meant to make the content of this thesis accessible for the general public.
3
4
Introduction
Acknowledgements
First and foremost, I’d like to thank Kareljan Schoutens for his role as supervisor during the
master project. I’d also like to thank Phillipe Corboz, for acting as the second examiner of
this thesis. Also a big thanks to Nicolas Regnault, who guided me with the usage of the
DiagHam package and even used his time to write some new functions specifically for this
thesis. And of course I’d like to thank my friends and family for their support.
5
CHAPTER 1
Anyons and Non-Abelian Statistics
In this first chapter, a short introduction will be given in the theory behind anyons and
non-abelian statistics. They will play a central role in the results of this thesis. For a
thorough review of anyons, non-abelian statistics, their relation to quantum Hall states and
their application to quantum computation I refer to two review articles by Stern [2, 3], a
review article by Nayak et al. [4] and a book by Pachos [5], which all serve as the sources
for this chapter.
1.1. Bosons, fermions and anyons
All particles encountered in nature are either fermions or bosons, meaning that their
symmetry-properties under the exchange of two of these particles are different. Fermions
are antisymmetric and therefore the multi particle wave function acquires a minus sign
when exchanging two particles. Bosons are symmetric, so exchanging two particles does not
change the wave function. This difference between the two types has huge implications for
the properties of fermionic and bosonic systems, most notably the Pauli exclusion principle:
multiple bosons can occupy the same quantum state, but fermions can not.
A two particle system is described by some wave function ψ(r1 , r2 ) with r1 and r2 the
positions of the two particles. When these two particles are identical, exchanging the two
particles should result in the same wave function, because physically the system is the same.
However, the wave function could have obtained a phase θ,
ψ(r1 , r2 ) → eiθ ψ(r2 , r1 ) .
(1)
When exchanging the particles back to the original configuration in the same (counter)clockwise
motion, the wave function once again acquires the same phase θ. However, the original wave
function has to be retrieved in this case, so there is the constraint 2θ = 2nπ, where n is an
integer. The two possible solutions modulo 2π correspond to the fermionic and bosonic cases,
fermions have θ = π and bosons θ = 0. This phase θ is called the statistical angle, because
it describes the statistical behaviour of the particles. However, this argument is not valid
in two dimensional systems, which follows from the topology (actually, the first homotopy
groups) of these spaces. In three dimensions, the winding of one particle around the other
is topologically trivial, because the loop can always be contracted to a point. As a result,
the wave function has to return to the original wave function when braiding one particle
around the other. This is not the case in two dimensions, the path of one particle around
the other can never be contracted to a point, because the loop will ‘get stuck’ behind the
second particle. Therefore, the wave function does not have to be the same when exchanging
two particles twice. The statistical angle θ is thus not restricted to any value and can be
anything. Particles with a statistical angle θ 6= {0, π} are therefore called anyons.
6
Chapter 1. Anyons and Non-Abelian Statistics
1.2. Non-abelian anyons
An important consequence of this arbitrary value of θ is that the exchange of two particles is
no longer necessarily an abelian operation. In a system with N particles there is a difference
between first exchanging particles (1,2) and then (2,3) or the other way around, see figure 1.
The exchange of particles is therefore no longer described by the permutation group SN , but
by the braid group BN . When the system is described by a degenerate set of g states, the
elements of the braiding group act as g × g unitary matrices on the space of the degenerate
states. If these matrices do not commute, the particles have non-abelian braiding statistics.
Braiding two particles around each other thus results in a nontrivial rotation in the space of
degenerate states.
Figure 1. The braiding operators that exchange particles (1,2) and (2,3) are non-abelian.
The position of the particles is on the horizontal axis and time on the vertical axis. The
braided world lines of the particles are topologically inequivalent in the two figures. Figure
obtained from Nayak et al. [4]
When two or more particles are close together, they can be mathematically described as one
particle with the combined statistics of the two original particles. For instance, two electrons
(with θf = π) can act as one composite particle with bosonic statistics (θb = 2θf = 2π). An
example of this phenomenon is the pairing of electrons in Cooper pairs in a superconductor.
In the case of anyons, this behaviour is even more interesting, because the combined statistical
angle of two anyons is in general once again a new anyonic statistical angle. For example,
two anyons with angle θ = π/3 form another type of anyon with their combined statistical
angle equal to 2π/3. The effect is that if there is one anyon in a system, there will also be
other types of anyons present. The process of combining two anyons φa and φb together is
called the fusion of the particles and it obeys certain fusion rules,
φa × φb =
X
Nabc φc ,
(2)
c
where the Nabc are just integers. The particles are abelian if Nabc 6= 0 for only one c, but if
there is more than one nonzero Nabc for at least one combination of an a and a b, then the
particles are non-abelian.
As said before, all particles usually encountered in nature are either fermions or bosons,
which is unsurprising since we live in a three dimensional space. However, in certain materials
the physics are effectively described by a two dimensional space. The real particles (electrons,
protons, neutrons, photons) that are present still live a three dimensional space and are
thus fermionic or bosonic, but the excitations of the system in the form of quasiparticles
or quasiholes (hereafter, these two terms are used interchangeably) are confined to the
two dimensional systems and can indeed, in theory, behave as anyons. Anyonic particles
are thus an emergent property of certain condensed matter systems. The most prominent
Chapter 1. Anyons and Non-Abelian Statistics
systems exhibiting these anyonic quasiparticles are fractional quantum Hall liquids. There
are theories describing the wave function of the ground state of certain fractional quantum
Hall liquids with non-abelian anyons as their excitations. Interferometer experiments have
been carried out concluding that the anyons indeed exist in these states [6] and there is
evidence that some of these states are indeed non-abelian [7]. It should however be noted
that any experimental data surrounding these excitations are controversial. It is widely
agreed upon that the excitations of these quantum Hall states can have fractional charges,
but the non-abelian nature of quasiparticles is still debatable.
One of the reasons that there is an interest in anyons is their possible application in a
topological quantum computer. Currently, one of the big problems in the construction of
a conventional quantum computer is that the computations are prone to errors due to the
hardware. It is crucial in a quantum computer to keep track of the states of the qubits. However, small local perturbations in the system can influence the states of the qubits and render
the computations invalid. There exist quantum error correction protocols that decrease the
number of faulty calculations, but they are not perfect. One of the possible solutions to this
problem is to make a fault tolerant (topological) quantum computer by using non-abelian
anyons as the qubits. The space in which the calculations take place is then the degenerate
ground state and the quantum computer operators are the braiding operators. This system
is immune to decoherence, because local perturbations necessarily have no nontrivial matrix
elements within this Hilbert space. Moreover, these systems are also fault tolerant towards
errors in the operators and measurements themselves, due to their topological nature. In
conventional quantum computers there can exist errors in the operators, a rotation of π/2 + δ
instead of π/2 for example. The braiding of particles is necessarily a discrete operations:
either two particles have been braided, or they have not. The measurements in topological
quantum computers involve bringing multiple anyons together, but the exact location of the
measurement does not matter, because the calculations are only based on the braiding of
the particles and not on their coordinates. In conclusion, anyons are an ideal candidate to
build a fault tolerant quantum computer, although a realization of one is far from achieved
at this time.
7
8
CHAPTER 2
The Quantum Hall Effect
In this chapter, a short overview of the quantum Hall effect will be given. The integer and
fractional quantum Hall effects will be described and an overview of multiple theoretical
wave functions will be given. Before going into the more in depth theoretical descriptions of
the integer and fractional quantum Hall effect, a short historical overview of the important
experimental discoveries will be given in the first section. The information in this chapter
was obtained from varying sources, including review articles by Jain [8], Goerbig [9] and
Stormer et al. [10] and books by Ezawa [11] and Fradkin [12].
2.1. Experiments
In the late nineteenth century, the classical Hall effect was discovered by Edwin Hall. The
classical Hall effect describes the phenomenon that a transverse voltage is measured when a
magnetic field is applied perpendicular to a current flowing through a conducting material,
see figure 2. The occurrence of the Hall voltage can be explained by the deflection of the
electron paths due to the Lorentz force. This results in a net charge difference between
the two edges of the material. This results in an electric field between the two edges and
a measurable potential difference, the Hall voltage. There is no net current flowing in the
perpendicular direction in a steady state, meaning that the forces due to the transverse
electric field and the perpendicular magnetic field precisely cancel each other. The Hall
voltage can then be calculated to be VH = (IB)/(ned) and the corresponding Hall conductance σ = en/B, where e is the electron charge, n the electron density, B the magnetic field
strength, I the electric current and d the thickness of the material. Measurements of the Hall
conductance are used to accurately calculate the electron density in materials. Interestingly,
the Hall voltage explicitly depends on the sign of the charge of the charge carriers, which
is useful to determine if the charge carriers in a material are electrons or holes. Another
Figure 2. Setup of a measurement of the Hall effect in a 2D material. The white arrow
represents the magnetic field. Figure taken from the lecture notes by Goerbig [9].
Chapter 2. The Quantum Hall Effect
application of the Hall effect is in devices measuring the strength of magnetic fields.
Almost exactly a hundred years later, in 1980, the quantization of the Hall effect in 2D
electron gases was discovered by von Klitzing, Dorda and Pepper [13] in a metal-oxidesemiconductor field effect transistor (MOSFET) at ultracold temperatures (Helium cooled)
and with strong magnetic fields (∼ 15 − 20 T). One of the most prominent features of this
quantization is the occurrence of plateaus in the transversal Hall conductivity with the
simultaneous disappearance of the longitudinal conductivity when varying the strength of the
magnetic field B, see figure 3. The values of the magnetic field for which these phenomena
occur are given by B = (ne h)/(eν), where ν is called the filling fraction of the material, the
reason of which will be explained later. The Hall conductance is then given by σ = νe2 /h.
Originally, these plateaus were only discovered at integer values of ν, thereby leading to
an almost exact quantization of the Hall conductance in multiples of e2 /h. This quantum
mechanical version of the Hall effect is therefore called the integer (or integral) quantum
Hall effect (IQHE) owing to the Hall conductance being quantized at integer quanta of e2 /h.
2
As a sidenote, the fine structure constant α is equal to µ20 c · eh . The magnetic constant µ0
and the speed of light c are exactly defined in SI-units, which means that the fine structure
constant just depends on e2 /h. The quantum Hall effect can therefore be used to accurately
calculate the value of α, which was actually the goal of the original paper of von Klitzing et
al [13].
Figure 3. Measurements of the Hall resistance and transversal resistance as a function
of the magnetic field. The line is the classically expected value of the Hall resistance. The
many visible integer and fractional states are indicated. Figure from Eisenstein and Stormer
[14].
Two years later, in 1982, Tsui, Stormer and Gossard [15] discovered that the plateaus in
the Hall voltage not only exist at integer values of ν, but that a plateau is also present
at ν = 1/3. They had discovered the first fractional quantum Hall effect (FQHE). Since
9
10
Chapter 2. The Quantum Hall Effect
then, the FQHE has been observed at many other rational fractions (of the form p/q
where p and q are integer) of the filling factor, see figure 3. The integer and fractional
QHE has not only been measured in MOSFETs, but also in other systems including GaAsAlGaAs hetero junctions, bilayers set-ups and, recently, graphene [16, 17, 18, 19]. The
fractional quantum Hall effect has been measured most prominently in the lowest Landau
level (see next section), but has also been measured for a small set of fractions in higher levels.
Interestingly, the quantum Hall effect does not only occur for electrons trapped in 2Dinterfaces between materials, but also for bosons in rapidly rotating Bose-Einstein condensates.
For a comprehensive review article on this specific subject, I refer to the review articles of
Cooper [20] and Viefers [21].
2.2. Theoretical description of the integer quantum Hall effect
The theoretical understanding of the physics behind the integer and fractional QHE has lead
to many papers in the years since its discovery. The IQHE can be understood in a single
particle picture for the electrons moving in a two dimensional plane with a perpendicular
magnetic field [22]. A short overview of these single particle wave functions will be given
in this section, for a comprehensive derivation of this theory, see appendix A. Crucial in
the understanding of the IQHE is the formation of highly degenerate Landau levels in the
energy spectrum of an electron in a magnetic field at energies En = (n + 1/2)~ωc , where n is
the Landau level index and ωc = eB/m is the cyclotron frequency. The single particle wave
functions can be calculated for all Landau levels, but they have a particularly nice form for
the lowest Landau level (LLL) in the symmetric gauge,
1
2
2
LLL
q
ψm
= √
z m e−|z |/(4lB ) .
(3)
2
( 2)m 2πlB
In this equation the zi are the complex coordinates
p x + iy of the electrons, m is the angular momentum index of the state and lB = h/(2πeB) is the magnetic length (also
written as l0 , both notations are used in this thesis). The probability distribution forms
an annulus around z = 0, whose radius depends on the angular momentum of the state.
The energy of all these states is however the same, since they are within the same Landau level.
The degeneracy of each Landau level due to the possible angular momenta states depends on
the strength of the magnetic field. More precisely, the density of states is the inverse of the
2 . The density of states is thus B/Φ , where
area that one ψm occupies, which is ∆S = 2πlB
D
ΦD is the (Dirac) magnetic flux quantum, ΦD = h/e. The number of states in a Landau
level is then equal to the number of total flux quanta Nφ = B/ΦD in the system. The filling
factor, the ratio of the number of electrons and the number of states in one Landau level, is
equal to the ν from before. The integer QHE can then be understood in this picture: if ν is
an integer, an even number of Landau levels are completely filled. Therefore, an energy gap
exists between the highest filled Landau level and the lowest empty level. If the energy scale
of this gap is larger than the typical energy scales in the material, all charged excitations will
be gapful and the result is an incompressible superfluid: the quantum Hall fluid. Since the
degeneracy of the Landau levels is equal to the number of flux quanta in a system, tuning
the magnetic field strength to the special values B = (ne h)/(eν) creates a quantum Hall
fluid. The reason that the quantum Hall effect is experimentally measured for a small range
of values around these theoretical values of the magnetic field is due to weak disorder effects
in the material, which breaks the Lorentz symmetry. In the energy level diagram, these
impurities create extra states in between the Landau levels. Instead of instantly going to the
Chapter 2. The Quantum Hall Effect
next Landau level, an extra electron will therefore occupy these localized impurity states
and hence the QHE is extended to a small range of B-values.
The many particle wave function of the completely occupied lowest Landau level is given by
the Slater determinant of all single particle LLL states ψm ,
P
Y
2
2
ψ(z1 , . . . zN ) = N
(zi − zj ) ei j |zj | /(4lB ) .
(4)
i<j
In this equation, N is a normalization constant. For every electron there is one magnetic
flux quantum present in the system. By removing a particle, or adding a magnetic flux, a
quasihole is created carrying electrical charge e.
2.3. Topology
An important theoretical discovery was made by Thouless, Kohmoto, Nightingale and de
Nijs [23], namely that the filling fraction ν is a topological invariant, equal to the Chern
number. The measurement of the quantum Hall effect was proof that topological phases do
exist in real physical systems. This discovery would lead to the research field of topological
materials, such as topological insulators and topological superconductors. As a result of
the bulk-boundary correspondence, the topological nature of the quantum Hall effect leads
to the appearance of chiral band gap crossing edge states. As a consequence, an electric
current flows along the edge in only one specific direction. For a review article on topological
insulators I refer to Hasan and Kane [24].
2.4. Theoretical descriptions of the fractional quantum Hall effect
The one body description of the IQHE is not sufficient to explain the FQHE. In the one
body picture, a fractional filling factor corresponds to a partially filled energy level. An
additional electron can thus be placed in an unoccupied state in the same Landau level
as the other electrons. Furthermore, low energy excitations are now possible within the
same Landau level. There is thus no apparent band gap present in the system, which is
needed to explain the observed incompressible FQH state. The single particle picture can
therefore not describe the FQHE and many body effects between the electrons, mainly the
Coulomb interaction, should be taken into account. This is however no simple task and in
the years since the experimental discovery of the FQHE there have been many attempts to
construct variational wave functions that approximate the observed states. Different theories
lead to sets of possible wave functions for certain filling fractions. There are often multiple
promising wave functions for one filling fraction that possibly describe the state, often with
differing characteristics such as their excitation spectrum, spin polarization, etc. It is often
theoretically hard to distinguish which of these wave functions is the one realized in systems
and only experiments can verify the viability of a wave function. Furthermore, it is also
possible that other phases of matter are energetically favoured over a FQH state so that the
FQHE does not occur at all at the theoretically expected filling fraction. In the rest of this
section the most prominent theories of the fractional quantum Hall effect wave functions
will be discussed.
2.4.1. Laughlin wave functions. The first theoretical explanation of a FQHE was
Laughlins guess of the wave function [25] to explain the occurrence of the ν = 1/3 state in
the experiments of Tsui [15]. The Laughlin wave function acts as a building block in many
other theories and shows many of the characteristics that also underlie these other theories.
11
12
Chapter 2. The Quantum Hall Effect
Laughlins trial wave function can be easily generalized to filling factor ν = 1/m, in which
case it is written as
P
Y
2
2
L
ψm
=
(zi − zj )m e− i |zi | /(4lB ) .
(5)
i<j
This wave function should be multiplied with an appropriate normalization constant, but it is
often omitted for clarity. The Gaussian factor at the end is usually also omitted, and just the
factor (zi − zj )m is called the Laughlin function. This trial wave function is only sensible for
odd m, because it is a wave function for the fermionic electrons, so it should be antisymmetric
under the exchange of any two particles. The particles are strongly repelled from one another,
due to the m-th power of the relative coordinates between all pairs of particles. This behaviour also ensures that the state is incompressible, because it does not break any continuous
spatial symmetries. When expanded, equation (5) is a summation of many terms of the form
lN
z0l0 z1l1 . . . zN
, hence it is a polynomial in the coordinates of all the particles and so the wave
function is an angular momentum eigenstate of the system, see also appendix A. The angular
momentum of one particle is li and the total sum of all the exponents li in one term is the
total angular momentum, equal for all terms in the expansion. The maximum exponent that
one coordinate can have is (N − 1)m, because there are (N − 1) pairings with the other
coordinates in which one specific coordinate occurs and each pairing has exponent m. The
maximum exponent of a coordinate is equal to the number of flux quanta, because it is the
highest state in one Landau level. Hence, the Laughlin wave function of order m has filling
N
1
fraction ν = (N +1)m
≈ m
for N 1. The Laughlin wave function has a certain elegance
in that it is almost universal, the only system dependent parameter is the magnetic length lB .
The quasihole/quasiparticle excitations of this wave functions are made by slightly shifting
the number of fluxes. Since this number is related to the exponents of the zi , this means
that all particles are able to occupy one higher angular momentum state when one extra flux
quantum is added. The wave function (5) should then be multiplied with a factor (zi − z0 )
for all particles i in order to increase the maximum exponent of the particles with one. The
quasihole wave function is thus
P
Y
Y
2
2
ψ L,qh =
(z − z )
(z − z )m e− i |zi | /(4lB ) .
(6)
i
m
i
0
j
k
j<k
The charge of this quasihole is 1/m, which can be deduced from the extra charge that is
necessary to keep the filling fraction the same as before (ν = Ne /Nφ → ∆Ne = ∆Nφ /m).
The fractional charge of the excitations has been confirmed in experiments [26] and the
fractional statistics are also claimed to be observed [6]. In conclusion, the Laughlin wave
function describes the ν = 1/m fractional quantum Hall effect and has quasihole excitations
that are fractionally charged.
2.4.2. Composite particles. In an attempt to unify the integer and fractional quantum
Hall effect, Jain [27] showed that the occurrence of the FQHE of strongly interacting electrons
at fractions ν = 1/(2p ± 1) (where p is an integer) can mathematically be described as
the integer quantum Hall effect for weakly interacting composite particles. By attaching p
flux quanta to each electron, a composite boson (if p is odd) or fermion (p even) is created.
In this picture, the quantum Hall liquid can be viewed as the Bose condensation of the
composite bosons or the IQHE of the composite fermions. For example, in the ν = 1/3
state, a composite boson is created by attaching 3 fluxes to each electron. There are no
flux quanta left over, so the result is a gas of N bosons. Alternatively, the attachment of 2
flux quanta to each electron results in N composite fermions and N unattached flux quanta.
13
Chapter 2. The Quantum Hall Effect
These composite fermions then experience a reduced magnetic field corresponding to the
filling fraction ν ∗ = 1. Consequently, the FQHE for electrons corresponds to the IQHE for
the composite particles. The excitations of these fractional QH states can be shown to also
be quasiholes that are fractionally charged and obey fractional statistics. Hoewever, their
statistics are still abelian.
2.4.3. Moore-Read states. The first wave function with non-abelian excitations was
formulated by Moore and Read [28]. Their wave function, derived from a conformal field
theory description of the QHE, describes the FQHE at filling fractions ν = 1/m, but now for
m even. The wave function is equal to the Laughlin wave function multiplied by a Pfaffian,
Y
PN
1
2
2
MR
ψm (z1 , . . . , zN ) = Pf
(zi − zj )m e− i=1 |zi | /4 l0 .
(7)
zi − zj
i<j
The Pfaffian Pf(Mij ) is the antisymmetrization of the matrix elements Mij ,
Pf(Mij ) = A(M1,2 , . . . , MN −1,N )
N/2
Y
X
1
MP (2i−1)P (2i) ,
sgn(P )
= N/2
2 (N/2)! P ∈S
i=1
(8)
N
where SN is the symmetric group of N elements and sgn(P ) is the sign of the permutation
P . The Pfaffian is completely antisymmetric, so the Laughlin factor has to be symmetric
to obtain an overall antisymmetric
wave function. Furthermore, the Pfaffian can also be
p
calculated by Pf(M ) = ± det(M ). The effect of the Pfaffian is that there is a pairing
inequality between different sets of particles, because the Pfaffian divides out certain factors
of (zi − zj ). As a result, certain pairs of particles repel each other less than others, because
the exponent of their pairing factor (zi − zj ) has been lowered. Consequentially, the particles
will be paired together, similar to Cooper pairs in a superconductor. The Pfaffian factor
does not influence the filling fraction, because it does not change the maximum exponent of
a coordinate. So ν is equal to the ν calculated for the Laughlin wave function: ν = 1/m.
Moore and Read showed that the excitations of the Moore-Read state are non-abelian anyons,
these results will be discussed in the conformal field theory description in section 5.2. The
Moore-Read wave function is the leading candidate to describe the FQHE at ν = 5/2 [29],
which can be seen as two completely filled, inert Landau levels and one half filled Landau
level described by the Moore-Read state. The ν = 5/2 FQHE can therefore be the first
system in which non-abelian particles are actually observed, but it is in competition with
multiple other possible wave functions, some of which have abelian excitations. Experimental
evidence has not yet been conclusive on the actual observed wave function, but there is
evidence that the non-abelian quasiparticles are actually present in the system[7].
2.4.4. Read-Rezayi states. The MR-wave function can be generalized to allow the
electrons to be clustered together in groups of k electrons, as was shown by Read and Rezayi
[30]. Their wave functions are the exact zero energy ground state of certain k + 1 body
Hamiltonians using projection operators on certain states. The Moore-Read wave function
(7) is retrieved for k = 2. Just as for the MR-state, these wave functions will be discussed in
the chapter on the CFT description of quantum Hall effects, specifically in section 5.3.
2.4.5. Halperin states for multiple components. The wave functions considered
so far have all been for quantum liquids with only one constituent. Even though electrons
have a spin, the one component picture is justified in many situations due to the polarization
of the spin degree of freedom as a result of the Zeeman-effect. When incorporating two
14
Chapter 2. The Quantum Hall Effect
components, such as the two electron spin states or two layers of the material, the Laughlin
wave function (5) can be generalized into Halperin wave functions [31],
PN/2
Y
Y
Y
0
2
2
2
H
ψm,m
(zi − zj )m (wi − wj )m
(zi − wj )n e− i=1 (|zi | +|wi )/4l0 . (9)
0 ,n ({zi , wj }) =
i<j
i<j
i<j
Here, the zi are used as the coordinates of the first component and the wi as the coordinates
of the second component. The Gaussian part of the wave function is written here once
more explicitly to show how this part changes when there are multiple components, but
will usually still be omitted. The parameters m, m0 and n give the (possibly different)
strengths for the intralayer and interlayer interactions. The behaviour of the wave function
is strongly dependent on these three values, as will be seen later. For example, it is an
SU(2) symmetric spin singlet when the parameters obey the relation m = m0 = n + 1. The
Halperin wave function allows the values of the exponents of the intercomponent and the
two intracomponent interactions to be different, leading to a filling fraction that depends on
these parameters. A generalization of this wave function to more than two components can
also be made [32] and is applicable in situations with more than one quantum number or
with both a spin and a layer degree of freedom:
Gen.Halp.
−
ψc,{m
= φL{mi } φint
{nij } e
i ,nij }
Pc
p=1
P Nc
j=1
2
|zjp | ,
(10)
where c is the number of components. The term φL{mi } describes the interactions of the
particles within the same component and is given by a product of Laughlin wave functions,
L
φ{mi }
Np
c Y
Y
=
(zip − zjp )mp .
(11)
p=1 i<j
The term φint
{nij } gives the interactions between the different components,
φint
{nij }
=
Np Nq
c Y
Y
Y
(zip − zjq )npq .
(12)
p<q i=1 j=1
The strengths of all the interactions are given by the integers mi and nij and they can be
conveniently written as a matrix M with Mij = nij and Mii = mi . This matrix can then
be used to calculate certain properties of the state. The number of fluxes seen by each
component should be the same. For a certain component
P i, the maximum exponent (and
thus the number of fluxes) is given by Nφ,i = mi Ni + j nij Nj , or equivalently in matrix
~φ = M ·N
~ . Inverting this relation and dividing by Nφ , the result is the filling fraction
form: N
for each component: (ν1 , . . . , νk )T = M −1 · (1, . . . , 1). The total filling fraction is simply the
sum of these component filling fractions. This is only possible if M is an invertible matrix,
which poses the condition that M , a (k × k)-matrix, has to be of rank k. Or equivalently: all
filling fractions should be independent of each other. If this is not the case, the dependent
filling fractions can be combined to make a smaller k 0 × k 0 matrix that is invertible and the
filling fraction can then be calculated with this reduced matrix.
15
CHAPTER 3
Graphene
Most measurements related to the quantum Hall effect were traditionally done in MOSFETs
or GaAs/AlGaAs heterostructures, where the interface of the two materials realized an
almost perfect two dimensional electron system exhibiting the QHE at low temperatures.
In the last couple of years, there has been a huge interest in the QHE in graphene, a
naturally two-dimensional system consisting of carbon atoms arranged in a hexagonal lattice.
This two dimensional system would be an ideal place to study the quantum Hall effect for
multiple reasons. Firstly, graphene allows for easier access to measurements, because it is
not necessary to measure at the interface of two three dimensional materials and the IQHE
occurs at room temperature. Secondly, the Landau level structure in graphene is different
than all conventional quantum Hall systems, leading to new phases. The QHE in graphene
was indeed discovered experimentally in 2009, almost simultaneously by two different groups:
Du et al. [18] and Bolotin et al. [19] In this chapter the electronic properties of graphene,
with a strong focus on the quantum Hall effect, will be explained. Furthermore, an overview
of recent experiments will be given. The sources for this chapter include a book by Bernevig
[33] and review articles of Castro Neto et al. [34], Das Sarma et al. [35], Goerbig [36] and
Goerbig and Regnault [37].
3.1. Electronic properties
Graphene is a two dimensional honeycomb structure consisting of carbon atoms, see figure
4. Inside one unit cell are two inequivalent sites, called A and B, that form two triangular
sublattices of only A- and B-sites. Each site is connected to three nearest neighbours with
the vectors δi and six next nearest neighbours with the vectors ai , whose absolute value is
the lattice spacing a. The Brillouin zone in reciprocal space can be taken to be either a
parallelogram of the two reciprocal lattice vectors bi or the hexagon with corners at K and
K 0 , which are the so-called Dirac points of graphene.
Figure 4. The graphene lattice in real and reciprocal space with the elementary direction
vectors indicated, figure from Bernevig [33].
16
Chapter 3. Graphene
These two Dirac points are the reason of some of the extraordinary properties of graphene. A
simple model for the electronic interactions on the graphene lattice is a tight binding model
P
with only nearest neighbour (NN) hopping: H = t hi,ji c†i cj , where the sum is taken over
all nearest neighbours. Writing it explicitly as creation/annihilation operators on the two
different sites and Fourier transforming these operators, the eigenvalues of this Hamiltonian
give the following band structure:
Figure 5. The band structure of the tight binding nearest neighbour hopping Hamiltonian as a three dimensional surface plot and a two dimensional contour plot..
In the figure above, it can be seen that this model leads to two energy bands that are
symmetric around the zero energy level and most notably touch each other at the two corners
of the Brillouin zone, the Dirac points. The effective Hamiltonian around these two points
gives the low energy behaviour, because the Fermi level lies in between the two bands. The
~
Hamiltonian expanded around these points is H = ξ~vF (qx σx + qy σy ), where ~q = ~k − K
~ 0 ) is the momentum expanded around the Dirac points, vF = (3ta)/(2~) is the Fermi
(or K
velocity and ξ = ±1 denotes the valley pseudospin (or isospin), which is 1 for the point K
and -1 for K 0 . The energy eigenvalues expanded around the Dirac points give the dispersion
relation E = λ~vF |q|, where λ = ±1 is the upper/lower energy band. This linear dispersion
is equal to that of relativistic particles obeying the Dirac equation. The electrons at these
points are therefore said to be relativistic and the name “Dirac points” is justified.
3.2. The integer quantum Hall effect
When graphene is placed in a sufficiently strong magnetic field, the quantum Hall effects
will occur. Just as is done in appendix A for the QHE of the 2DEG, the Hamiltonian of a
single particle in graphene in a strong magnetic field can be obtained by the substitution
p → p + eA = P (the Peierls substitution). The Hamiltonian expanded around the
Dirac points then becomes H = ξ~vF (Px σx + Py σy ). Similar to equation (144) in the
†
appendix, the P√
’s can be written in terms
√ of creation/annihilation operators a and a as
†
†
Px = ~(a + a)/( 2lB ), Py = ~(a − a)/( 2lB ). The Hamiltonian in terms of these operators
then becomes
√
2~vF
0 a
H=ξ
.
(13)
a† 0
lB
Solving the eigenvalue equation Hψn = En ψn with the spinors
√ ψn = (un , dn ) leads to the
†
equations ξωadn = En un and ξωa un = En dn , where ω = ( 2~vF /lB ). Inserting the un
Chapter 3. Graphene
of the first equation into the second equation results in ω 2 a† a dn = En2 dn . So dn is an
eigenstate of the number operator a† a with (integer) eigenvalue n and corresponding energies En2 = nωn2 . The un -part of the spinor also depends on n according to the first equation.
√
The energies are thus given by En = λω n, where λ = ±1 denotes the positive and negative
√
−1
energy solution corresponding
to
the
conduction
and
valence
bands.
Since
ω
∼
l
∼
B,
B
√
the energies depend on B instead of B as in the previously encountered 2DEG. More
importantly, the dependence on the quantum number n is also different.
Not only does the
√
level spacing become relatively smaller for higher n due to En ∼ n, but there is also a
B-independent zero energy state at n = 0. Moreover, the energy levels do not only have
the usual Landau level degeneracy, but each level is also fourfold more degenerate due to
the spin and isospin degrees of freedom, that both commute with the Hamiltonian. This
makes graphene an approximate SU(4)-symmetric system, where the particles can be in
the four components (spin, isospin) = (↑, +), (↑, −), (↓, +) and (↓, −). This also means
that the filling fraction dependence on n in graphene is different than in the 2DEG, namely
ν = ±2(2n + 1). The ν-difference between two levels is thus a multiple of 4 and the IQHE
(sometimes also called the relativistic quantum Hall effect due to the relativistic electrons)
occurs when ν = 2, 6, 10, etc. The shift for the n = 0 Landau level can be explained because
this n = 0 LL is half filled. Two of the levels will lie slightly below the zero energy level
because of small symmetry breaking terms, such as a small Zeeman effect.
The IQHE in graphene was first measured in 2005 for the theoretical expected filling fractions
ν = 2, 6, 10, . . . [16, 17]. Not long thereafter, in 2006, the IQHE was also measured at filling
fractions outside the expected values, namely at ν = 0, ±1, ±4 [38] and in later experiments
also at other integer ν. The occurrence of these states can be explained by small symmetry
breaking terms that split the fourfold degeneracies. This breaking of the symmetry can
be caused by outside forces, most notably the Zeeman effect, or the intrinsic interactions
between the electrons themselves. In the experimentally obtainable systems, the energy
scales of the intrinsic effects are a factor of ten bigger than the extrinsic effects. The intrinsic
effects are therefore more likely to be the explanation of these states. One of the proposed
causes of this effect is the phenomenon of fractional quantum Hall ferromagnetism [39], an
exchange effect where the Coulomb interaction energies are minimized by a polarization
of the electrons, leading to the completely filled subband of one or more components, and
hence an IQHE, at any integer filling fraction. The excitations of these kind of states are
topological quasiparticles called skyrmions.
3.3. The fractional quantum Hall effect
The first measurements of the FQHE in graphene have been reported in 2009 [18, 19] at
ν = 1/3. In recent years, the FQHE has also been measured at many other fractions
[40, 41, 42], see figure 6 for the results of such a study. The experimentally realized FQH
states are often different than those observed in the usual semiconductor based materials.
Certain fractions measured before in other materials turn out to be absent and other fractions
are much stronger or weaker than expected. This shows the need for new theories to explain
the FQHE in graphene and also gives rise to the possibility of the occurrence of new states
that were not observed before. The Coulomb interaction commutes with the spin of the
particles and is only weakly broken by the valley (iso)spin, so the FQHE in graphene can
also be approximated to be SU(4) symmetric at low energies. Some theories have been put
forward to suggest that some of these unexpected results are a consequence of different
17
18
Chapter 3. Graphene
symmetry breaking terms [43, 44, 45, 46, 47]. These studies result in phase diagrams as
functions of the strengths of the different symmetry breaking terms. Depending on the
values of these terms, the FQHE phases in graphene can be described by charge-density
waves, Kekulé distortions or (anti)ferromagnets. Furthermore, some studies report on the
ability to control the different FQHE phases by tuning the effective electron interactions, see
for example [48] for a theoretical background. If there are multiple possible states close to a
certain energy, then it is possible with these methods to break the competition between these
states and stabilize the system into a desired FQH phase. These quantum phase transitions
between different (un)polarized states have indeed been reported [42].
Figure 6. A measurement of the inverse compressibility dµ/dn in graphene as a function
of the filling fraction and the magnetic field. Red areas are incompressible, blue areas are
compressible. The filling fractions of the many FQHE states are indicated. Figure obtained
from Feldman et al.[42]
This short overview of the measurements of the FQHE in graphene shows that it has a rich
structure. The FQHE at many filling fractions has been observed and it is clear that there
is a strong competition between different quantum phases at certain fractions. By tuning
the parameters of the system, the desired phases can be stabilized. In this thesis, a certain
series of SU(4) wave functions with non-abelian excitations will be constructed and due to
the rich structure of the FQHE phases in graphene, it is possible that there will be some
regime in which these created wave functions are stable. The theoretical wave functions for
the SU(4)-symmetric FQHE states in graphene will be discussed in chapter 6.
19
CHAPTER 4
Mathematical Preliminaries
4.1. Conformal Field Theory
In this section, a short overview of conformal field theory will be given, with a strong focus
on the elements applicable for this thesis. There are many resources that give a thorough
overview of conformal field theory. Ginsparg [49] and Fradkin [12] are the main sources of
this chapter.
A conformal field theory (CFT) is a field theory that is invariant under conformal transformations. This means that the metric transforms with a scale factor, thereby preserving the
angle between two vectors,
0
gµν (x) → gµν
(x0 ) = Ω(x)gµν (x) .
(14)
The four main operations that have this property are translations, rotations, dilations and the
so-called special conformal transformations. In 2 + 1 dimensions these transformations have
far reaching consequences for field theories. It is first of all natural to write the coordinates
in a complex notation: z = x + iy and z̄ = x − iy, because the conformal mappings in two
dimensions are then (anti-)holomorphic functions. The most important fields in a CFT are
the primary fields, transforming under conformal transformations as
∂w h ∂ w̄ h̄
φ(z, z̄) →
φ(w, w̄) ,
(15)
∂z
∂ z̄
where h, h̄ are the conformal weights of the primary field. Their sum, h + h̄ = ∆, is called
the scaling weight or dimension of the field. The transformation property above heavily
constraints the expectation value of two of these primary fields, it has to have the form
D
E
c12
φ1 (z1 )φ2 (z2 ) =
,
(16)
(z1 − z2 )2h
where c12 is a constant, h = h1 = h2 and only the chiral (z̄-independent) part is considered.
If the two fields in the correlator do not have the same conformal weight, the result will be
zero. The three-point correlator is also heavily restricted,
D
E
c123
φ1 (z1 )φ2 (z2 )φ3 (z3 ) = h123 h231 h132 .
(17)
z12 z23 z13
Here, the short hand notations z12 = z1 − z2 and h123 = h1 + h2 − h3 have been used. The
form for higher order correlators is not completely fixed by the transformation rules.
An important concept in CFT is the operator product expansion (OPE). When two fields in
a correlator are near each other, they will behave singular and can act as if the fields have
fused together in one or more other fields. This information is encoded in the OPE of these
two fields,
X
lim φi (z)φj (w) =
Cijk φk (z) (z − w)−∆i −∆j +∆k ,
(18)
z→w
k
20
Chapter 4. Mathematical Preliminaries
where the sum is taken over all the primary fields in the theory. The constants Cijk are
called the structure constants. The two fields will thus fuse together to form one or more
other fields times a singular coefficient that depends on the conformal dimensions of all three
fields present in the fusion process. The fusion rules themselves can be written down more
simply in the form
X
φi × φj =
Nijk φk .
(19)
k
The fusion process is both associative and commutative, and results in an algebra of the
constants Nijk , called the Verlinde algebra [50]. A field φi is called a simple current if for all
other fields φj , the fusion φi × φj results in only one field. An example of a simple current is
the identity (or vacuum) field I. If the OPE of two fields contains multiple nonzero structure
constants, there will be multiple fusion channels and the fusion is called nontrivial. These
nontrivial fusion rules are important for this thesis, because fields with nontrivial fusion
rules correspond to non-abelian particles. This property can make calculating the correlator
of multiple fields hard, because there can be many different ways in which all the fields fuse
together. A necessary condition for a correlator to be nonzero is that there has to be at
least one fusion path resulting in the identity field, because only the expectation value of
the identity field is nonzero. If, for example, the fusion rule for some fields is ζ × ζ = ξ, then
hζζi = 0, because these two fields do not fuse to the identity. In general, the correlator is a
sum of all the different ways in which the fields can be fused together,
D
E X
φ1 (z1 ) . . . φN (zN ) =
Fi (z1 . . . zN ) .
(20)
i
Here, the Fi are all the possible resulting (holomorphic) functions and they are called
conformal blocks. A consequence is that a correlator is not always just equal to one holomorphic function, but can be seen as a vector in the space of the many possible resulting
functions. It can be shown that displacing the zi in the fields on the left hand side results in
the conformal blocks transforming into each other, this is the braiding of the conformal blocks.
A special OPE is that of a primary field φ with the energy-momentum tensor T , given by
T (z)φ(w) =
h
1
φ(w) +
∂φ(w) + . . .
2
(z − w)
(z − w)
.
(21)
The dots at the right hand side of this formula represent all nonsingular terms, they are
usually omitted in the OPE, because they play no role in the calculation of correlators. The
OPE (21) is often used to determine the conformal dimension h of a field.
In this thesis, the most useful primary fields are the vertex operators V (z) = eiαφ(z) . Here,
the φ(z) are chiral bosonic fields, meaning that they have no z̄ dependence. The correlator
of two of those φ’s is hφ(z)φ(w)i = − ln(z − w). Using formula (21) with these fields, the
conformal dimension of the vertex operator can be calculated to be α2 /2. The correlator of
two of these vertex operators with different conformal dimensions is then
D
E
eiαφ(z) eiβφ(w) = (z − w)αβ δα,−β .
(22)
The correlator of vertex operators that do not obey α = −β can also be nonzero with the
use a little mathematical trick,
D
E Y
V (z1 )V (z2 ) . . . V (z∞ ) =
(zi − zj )αi αj .
(23)
i<j
21
Chapter 4. Mathematical Preliminaries
This relation is only true if the V (z∞ ) has an α equal to minus the sum of all the other α’s,
the wave function is multiplied by an appropriate power of this z∞ and the value of z∞ is
sent to infinity. This field at infinity then acts as a background charge. This trick directly
follows from the three-point correlator,
D
E
2
lim z (α+β) eiαφ(x) eiβφ(y) e−i(α+β)φ(z)
z→∞
2
z (α+β)
= lim
z→∞ (x − y)−αβ (x − z)α(α+β) (y − z)β(α+β)
(24)
= (x − y)αβ ,
where the second line follows from the first one by using the general form of three point
correlators for primary fields (17). In the remainder of this thesis, this extra background
field will in general not be written down and the correlator
D
E
eiαφ(z) eiβφ(w) = (z − w)αβ
(25)
will just be used. It should however be remembered that this is not actually the complete
correlator and that there exists some background field that keeps the correlator nonzero.
4.2. Group Theory
The construction of the SU(4)-symmetric wave functions depends on group theoretical arguments. Just as the CFT section above, the most important concepts of group theory that are
applicable to this thesis will be summarized in this section. It will naturally have a strong
focus on the SU(N )-groups, since this thesis focuses on these groups. The content from this
section is mainly derived from the textbook “Lie Algebras in Particle Physics” by Georgi [51].
A Lie algebra consists of a set of generators Xi , the commutators of which are linear
combinations of all generators,
X
[Xi , Xj ] = i
fijk Xk ,
(26)
k
where the sum over k is taken over all elements Xk in the group. The fijk are called the
structure constants of the group. These algebra rules and names are very reminiscent of the
fusion rules of the CFT mentioned in the previous section.
The groups SU(N ) are the Lie groups of N × N unitary matrices with determinant 1. As a
result of these constraints, they consist of N 2 − 1 elements and their rank is N − 1, which
means that a maximum of N − 1 elements are mutually commutative. Since SU(N ) is
semisimple and compact, the number of invariant operators (the Casimir operators) is equal
to the rank.
The two most useful representations of SU(N ) are the first fundamental representation
and the adjoint representation. The first fundamental representation consists of traceless,
Hermitian, complex N × N matrices. Examples of first fundamental representations are the
Pauli matrices for SU(2) and the Gell-Mann matrices for SU(3). The adjoint representation
consists of (N 2 − 1) × (N 2 − 1) dimensional matrices Ta , whose elements are defined to be
the structure constants themselves: (Ta )bc = −ifabc . These matrices act on the space where
the group elements are the basis vectors by Ti Xj = [Xi , X
Pj ] = ifijk Xk . It can be shown that
they also obey the same commutation rules: [Ta , Tb ] = c ifabc Tc .
22
Chapter 4. Mathematical Preliminaries
When taking the first fundamental representation, the subset of mutually commutating generators Hi (or equivalent: the subset of generators that can be diagonalized simultaneously)
is called the Cartan subalgebra. The number of generators in the Cartan subalgebra is equal
to the rank of SU(N ), N − 1. They are physically relevant, because they lead to the labelling
of states in multiplets of the quantum numbers. After diagonalizing the Hi , the eigenvectors
of these matrices are just simply the vectors e1 = (1, 0, 0, . . .), e2 = (0, 1, 0, . . .), etc. They
are conveniently ordered from highest to lowest eigenvector, based on the index with a 1. It
will later be shown that the N 2 − N matrices that are not in the Cartan subalgebra act as
raising and lowering operators between all of these eigenvectors. Acting with element Hi on
one eigenvector returns the eigenvalue of that eigenvector, which is called the weight µi . The
vector consisting of all µi belonging to one eigenvector is called the weight vector of that
eigenvector. There are N − 1 matrices Hi , so the dimension of the weight vectors is also
N − 1, which is one dimension lower than the dimension of the eigenvectors themselves. A
weight vector is called positive (negative) if the first nonzero component is positive (negative).
The roots are the weights of the adjoint representation. The dimension of the root vectors
is still N − 1 (has to be equal to the rank), but the number of roots is now equal to the
number of elements in SU(N ). The root vectors of the basis vectors that correspond to
matrices within the Cartan subalgebra are zero vectors, because Hi |Hj i = [Hi , Hj ] = 0.
The zero root vectors thus have a degeneracy of (N − 1). The other N 2 − N matrices have
nonzero root vectors: Hi |Eα i = αi |Eα i. Taking the complex conjugate of this expression
and using that Hi is Hermitian shows that Eα† = E−α . Calculating Hi E±α |µi (where |µi is
the state with weights µ) results in (µ ± α)Eα |µi, so the elements that are not in the Cartan
subalgebra behave as raising and lowering operators on the weight states. This also shows
that the root vectors α can be conveniently calculated by the differences of the weights µ.
Identical to the weight vectors, a root vector is called positive if the first nonzero entry is
positive. The positive root vectors are then defined to be the raising operators and the
negative roots the lowering operators.
A simple root is a positive root that can not be written as the sum of other positive roots.
The simple roots are linearly independent and they are complete, which implies that there
are (N − 1) (the dimension of the root vectors) simple roots. If a weight state is annihilated
by all simple roots, it is a highest weight state of the representation. The simple roots can
be used as a basis for the root space. In general, the simple roots are not orthonormal, this
information is encoded in the (N − 1) × (N − 1) Cartan matrix A with matrix elements
Aij = 2
αi · αj
.
αj · αj
(27)
α
j
The weights for which µi · αj ·α
= δij are called the fundamental weights and they form a
j
basis for the weight space. From the definition it is clear that the fundamental weights and
the simple roots are dual to each other. The basis with the fundamental weights as basis
vectors is also called the Dynkin basis and is often the basis used to describe the roots and
vectors of an algebra. The Dynkin labels of the simple roots are just the rows of the Cartan
matrix. Since all the simple roots are by definition independent, the rank of the Cartan
matrix is N − 1 and is therefore invertible. The inverse of the Cartan matrix, G, can then
be used as a metric in the space where the weights and roots are given in the Dynkin basis.
The raising and lowering operators can be used to construct the allowed combinations
of multiple combined SU(N ) representations, or in a physical viewpoint: to construct
23
Chapter 4. Mathematical Preliminaries
the allowed wave functions of a many particle state. The raising or lowering operator on
a multiple particle state is equivalent to the sum of the one particle raising/lowering operators.
A simple example of an SU(N ) group is SU(2), which is used in physics to describe spin 1/2
particles. It consists of three elements, the spin directions Sx , Sy and Sz , and there is only
one Casimir operator, the total spin S 2 . The fundamental representation is given by the
three Pauli-matrices, only one of which can be diagonalized simultaneously. The Cartan
subalgebra then only consists of one Pauli matrix, for example σz , with eigenvectors (1,0)
and (0,1). The corresponding weights are then just the values ±1. The roots of SU(2) are
then ±2 (the differences between the weights, corresponding to σ± ) and 0 (corresponding
to the one matrix in the Cartan subalgebra). The more complex cases of SU(3) and SU(5)
will be derived in a later chapter when constructing the SU(2) and SU(4)-symmetric wave
functions.
Combinations of the other two Pauli matrices can indeed act as the raising and lowering
operators on the two eigenvectors: E21 = σ+ and E12 = σ− . The two body states can then
be constructed by starting from a highest weight state ψ1 = e1 e1 , where the first vector
corresponds to the first particle and the second to the second particle. Acting with the
lowering operators on these states results in
E21 (e1 e1 ) = (E21 e1 ) e2 + e1 (E21 e2 )
√
= e2 e1 + e1 e2 ≡ 2 ψ2
√
1
E21 ψ2 = √ 2 e2 e2 ≡ 2 ψ3
2
E21 ψ3 = 0 .
(28)
√
Here, the factors of 2 are introduced to normalize the states. The two particles are thus
combined in a triplet state. When acting with the total spin operator on these three states,
the result will always be 1, which shows that the total spin is an invariant Casimir operator.
Acting with σz results in 1, 0 and -1. The two spin 1/2 particles combined can thus be
described by a spin 1 particle. There is however a fourth state that is also allowed, it can be
made by taking a state orthogonal to ψ2 and is given by e1 e2 − e2 e1 . The operators E12 and
E21 destroy this state, so it is a spin 0 singlet.
The possible resulting states when combining multiple SU(N) particles can be easily deduced
pictorially with the help of Young tableaux. In these pictures every particle is represented
by one box
with dimension N . A proper Young tableau consists of rows and columns
of these boxes, where the number of boxes in every row is at most the number of boxes in
the row above it and the number of boxes in a column is at most the number of boxes in
the column to the left of it. Boxes in the same row represent a symmetric combination of
these particles and boxes in a column the antisymmetric combination. When adding a box
to a tableau, the new tableaux consist of all possible combinations that result in a proper
tableau. If a column has N boxes, it can be removed from the diagram, For example, the
combination of two boxes is
⊗
=
⊕
.
(29)
So the combination of two particles results in a symmetric and an antisymmetric state, just
as expected.
24
Chapter 4. Mathematical Preliminaries
The dimension of the representation corresponding to a tableau can be calculated by finding
all permutations of the boxes resulting in a unique state, but can be calculated more easily
by the following rules. The dimension D = F/h, where F and h are determined as follows.
For F the box in the top left corner gets assigned a factor N , then the factors assigned to
the other boxes can be found by adding 1 for every box you go to the right and subtracting
1 for every box you go lower. Then F is the product of all of these factors. In order to
calculate h, every box gets a factor assigned that is equal to the number of boxes to the
right of it plus the number of boxes below it plus 1 for the box itself. Then h is again the
product of these two. For example, a proper diagram for 5 particles in SU(4) is
4 5 6
3 4
4 3 1
2 1
.
(30)
The second diagram has the F -factors filled in and the third diagram the h-factors. This
diagram thus has dimension (4 · 5 · 6 · 3 · 4)/(4 · 3 · 1 · 2 · 1) = 60. Combining many boxes of
the N -dimensional irrep thus results in a sum over many irreps within SU(N ). For example,
the combination of four SU(4) particles gives
15 ⊕ 2·20
200 ⊕ 35 ⊕ 3·45
45 .
4 ⊗ 4 ⊗ 4 ⊗ 4 = 1 ⊕ 3 ·15
(31)
The different representations can also be denoted as a vector of their values of the Casimir
operators Λi which are the same as the highest weight of the representation. For SU(N ),
there are N − 1 Casimir operators, so these vectors have N − 1 indices. For example, in
SU(2) the representations can be denoted by just one number. In that case, (0) is identified
with the identity, (1) with the first fundamental representation corresponding to a spin 1/2
particle, (2) with the adjoint representation etc. For SU(3), two indices are necessary, for
example 1 ≡ (0, 0), 3 ≡ (1, 0), 3̄ ≡ (0, 1) and 6 ≡ (2, 0).
4.3. Affine Lie algebras
All SU(N )-groups have a finite number of generators, but the combination of the fundamental
representations give an infinite number of higher representations. The affine Lie algebras
transform this infinite number of representations to a finite one by effectively introducing
a cutoff of the allowed representations. The notation for these groups is SU(N )k , where k
indicates the maximum value of the sum of the Dynkin labels that is still allowed in the
algebra. For example, in SU(3)2 the only allowed representations are (0,0), (1,0), (0,1),
(1,1), (2,0) and (0,2). This also means that the root lattice is defined up to k times the
simple roots. The combination of two representations within this affine Lie algebra is then
calculated as for the usual Lie algebra, but any resulting representations that are not within
the group are just discarded. The result of this procedure is that this algebra is now closed
under the combination of the representations in the group.
25
CHAPTER 5
The Conformal Field Theory Approach
In this chapter, the fractional quantum Hall wave functions, discussed in chapter 2, will
be derived from the conformal field theory approach. In this CFT language, the particles
will be identified with the fields of the CFT and the wave functions will then be derived by
calculating the correlators of all fields. The methods described in this chapter will then be
used in the following chapter to construct new wave functions. The main sources of this
chapter are the book Field Theories of Condensed Matter Physics by Fradkin [12], and the
PhD-thesis of Ardonne [52].
5.1. Laughlin
The Laughlin wave function was already discussed in section 5.1, where it was shown that
for N particles the FQHE at filling fraction ν = 1/m (with m odd to get an antisymmetric
wave function) can be written as (5)
PN
Y
2
2
L
ψm
(z1 , . . . , zN ) =
(zi − zj )m e− i=1 |zi | /4l0 .
(32)
i<j
The Laughlin wave function is often written without the Gaussian factor and is in that case
implied. This wave function is equivalent
to the expectation value of N conformal field
√
theory vertex operators Ve (zi ) = ei mφ(zi ) , as was first shown by Moore and Read [28], using
equation (23)
*N
+
Y
Y √
√
2
mN
lim z∞
ei mφ(zi ) e−i mN φ(z∞ ) =
(zi − zj )m .
(33)
z∞ →∞
i=1
i<j
The Laughlin wave function can thus be expressed as the CFT√correlator of many vertex
operators with conformal dimension h = m/2 and with charge m. The vertex operators
in the CFT correspond to the locations of the electrons in the Laughlin wave function and
these operators will therefore be called electron operators.
This CFT language also allows the insertion of other vertex operators, but with some
restrictions. Inserting an extra vertex √
operator exp(iαφ(w)) in the correlator, the final result
√
α
gets extra terms of the form (z − w) m . To keep this wave function physical, α m has to
be an integer, because interchanging two particles should only give an
√ overall factor of ±1.
The physically allowed vertex operators are thus restricted to α = n/ m, with n an integer.
√
The simplest situation is when n = 1, the corresponding vertex operator Vqh (w) = eiφ(w)/ m
is called the quasihole operator for reasons explained below. Inserting one quasihole vertex
operator in the correlator of equation (33) results in
D
E
L
ψm
(w, z1 , . . . , zN ) = Vqh (w)Ve (z1 ) . . . Ve (zN )
Y
Y
(34)
=
(zi − w) (zi − zj )m .
i
i<j
26
Chapter 5. The Conformal Field Theory Approach
When compared to equation (6), it is clear that inserting the quasihole operator in the
correlator indeed resulted in the Laughlin wave function with an added quasihole. Adding
more than one quasihole operator to the correlator not only adds interactions between the
quasiholes and the particles, but also between the quasiholes themselves,
D
E
Vqh (w1 ) . . . Vqh (wM )Ve (z1 ) . . . Ve (zN )
Y
Y
1 Y
=
(wa − wb ) m
(zi − wa ) (zi − zj )m .
i,a
a<b
(35)
i<j
These interactions between the quasiholes have an exponent of 1/m, meaning that the
exchange of two of these quasiholes does not simply return the wave function times ±1,
because there is a branch cut. The quasiholes have fractional statistics π/m and a fractional
charge of e/m. The scaling dimension ∆ = h + h̄ leads to a phase factor exp(2πi∆) upon
rotation over 2π, the conformal dimension ∆ √
can thus be identified with the spin. Since
h = α2 /2 for exp(iαφ), the operator with α = m represents a bosonic particle for m even
and a fermion for m odd. All other values of α give fractional statistics and thus describe
an anyon. In conclusion, the quasiholes of the Laughlin wave function are anyons, but they
are abelian, since their fusion rules are trivial.
5.2. Moore-Read
The Moore-Read wave function (7), discussed in chapter 2.4.3, is a model wave function for
1
a quantum Hall liquid at filling fraction ν = m
at even m and it is equal to the Laughlin
wave function multiplied by a Pfaffian factor,
1
L
ψm
zi − zj
Y
PN
1
2
2
= Pf
(zi − zj )m e− i=1 |zi | /4l0 .
zi − zj
MR
ψm
(z1 , . . . , zN ) = Pf
(36)
i<j
In the previous section it was shown that the Laughlin wave function can be calculated
with the help of vertex operators corresponding to the particles. The Pfaffian factor can,
however, not be generated by these Laughlin vertex operators, but it can be expressed as a
correlation function of fermionic Majorana fields χ. The two point correlation function of
these Majorana fields is hχ(z)χ(w)i = 1/(z − w), they have conformal weight h = 1/2. The
correlation function of an even number of χ’s can then be calculated using Wick’s theorem,
which gives the totally antisymmetrized (because the fields are fermionic) combinations of
all the fields, indeed resulting in the Pfaffian factor,
Pf (zi , . . . , zN ) =
*N
Y
+
χ(zi )
.
i=1
For example, the correlator of four of these Majorana fields is
(37)
27
Chapter 5. The Conformal Field Theory Approach
D
E D
ED
E D
ED
E
χ(w)χ(x)χ(y)χ(z) = χ(w)χ(x) χ(y)χ(z) − χ(w)χ(y) χ(x)χ(z)
D
ED
E
+ χ(w)χ(z) χ(x)χ(y)
1
1
1
−
+
(w − x)(y − z) (w − y)(x − z) (w − z)(x − y)
1
= Pf
for(a, b) ∈ (w, x, y, z) .
a−b
=
(38)
Since the Moore Read wave function is the product of this Pfaffian and the Laughlin wave
function, it can be written as the product of the two corresponding CFT correlators, that
can then be combined,
L
ψ MR = Pf (zi , . . . , zN ) ψm
(zi , . . . , zN )
+
*N
+* N
Y
Y √
=
χ(zi )
ei mφ(zi )
=
i=1
*N
Y
(39)
i=1
χ(zi ) e
√
i mφ(zi )
+
.
i=1
The MR
wave function is thus equal to a correlator of the electron operators Ve (z) =
√
i
mφ(z)
χ(z) e
and this correlator can be split in a neutral sector of Majorana fermions and
a charge sector of chiral bosons. These Majorana fermions occur in the CFT of the two
dimensional classical Ising model or, equivalently, the one dimensional quantum Ising model.
Another type of fields present in this CFT are the spin fields σ. In√the MR wave function,
they correspond to the quasihole operators: Vqh (w) = σ(w) ei/(2 m) φ(w) . The σ’s have
conformal dimension ∆σ = 1/16, which can be deduced from the fact that the wave functions
should be analytic in the coordinates of the electrons. As seen above, the fusion of the
parafermions χ is trivial, but the fusion of the spin fields is not: σ × σ = 1 + χ and ψ × σ = σ.
These nontrivial fusion rules result in multiple different fusion paths when fusing many
σ-fields and thus to a degenerate state. This has the consequence that the quasiholes over
the Moore-Read state are non-abelian anyons, as was claimed earlier in this thesis.
5.3. Read-Rezayi
The Read-Rezayi wave functions are generalizations of the Moore-Read wave function (36).
In the MR-case, the electron operators included a parafermion ψ (change in notation: χ → ψ),
that fused together with one other ψ to form the identity. For the RR wave functions this
is generalized to let k ψi ’s fuse to the identity, the MR wave function is then retrieved by
setting k = 2. This generalization results in the electrons being clustered in groups of k
electrons. The describing CFT is called a Zk parafermion CFT and it contains the primary
fields ψ0 = I, ψ1 , ..., ψk−1 and the spin fields σ1 , ..., σk−1 . The fusion rule between two of
these parafermion fields is ψa × ψb = ψa+b where a + b has to be taken modulo k. With the
introduction of these
fields, the electron operator for the RR wave functions is written as
√
i M +2/k φc (z)
Vel (z) = ψ1 (z) e
with M indicating the exponent of the Laughlin wave function.
This wave function has filling fraction ν = k/(kM + 2). For all parafermions to be able to
fuse to the identity, the number of electrons N has to be a multiple of k. The correlator of
28
Chapter 5. The Conformal Field Theory Approach
many electron vertices is then
D
ΨRR
k,M = V1 V2 . . . VN
E
=
*N
Y
+
ψ1 (zi )
i=1
Y
(zi − zj )M +2/k .
(40)
i<j
The first correlator on the right hand side is, in general, problematic to calculate, because it
does not consist of primary fields. There do exist two different schemes that are proven to
construct this exact wave function. They both make explicit use of the (k, M )-clustering
properties. The first of these two methods, obtained by Read and Rezayi themselves [30], will
be called the χ-method, the second method, derived by Cappelli, Georgiev and Todorov [53],
will be named the η- or colour-method. Both methods will be used later in the construction
of the SU (4)-symmetric wave functions, so an explanation of these methods will now be given.
For M = 0, the N coordinates zi are divided in p groups of k coordinates each, such that
N = pk. The coordinates get an extra index denoting the number of the group to which
they belong. Then, to each pair of groups a factor χa,b is assigned in the following manner:
χa,b = (z(a−1)k+1 − z(b−1)k+1 ) (z(a−1)k+1 − z(b−1)k+2 )
(z(a−1)k+2 − z(b−1)k+2 ) (z(a−)k+2 − z(b−1)k+3 )
(41)
. . . (zak − zbk ) (zak − z(b−1)k+1 ) .
So every coordinate in group number a is paired with two other coordinates in group number
b. The Read-Rezayi wave function can then be constructed by multiplying all possible
χ-factors and symmetrizing them,


N/k
Y

ΨRR
χab  ,
(42)
k =S
a<b
where S means the symmetrization over all electron coordinates. The equivalence of (40) and
(42) was shown by Gurarie and Rezayi [54]. The wave function in equation (42) explicitly
shows the clustering of the electrons at order k, as can be seen in a quick example. For 6
electrons and k = 2 there are three groups of two electrons each: (z1 , z2 ), (z3 , z4 ) and (z5 , z6 ).
The product then becomes
N/k
Y
χab = χ12 χ13 χ23
a<b
= (z1 − z3 )(z1 − z4 )(z2 − z4 )(z2 − z5 )(z1 − z5 )(z1 − z6 )
(43)
(z2 − z6 )(z2 − z5 )(z3 − z5 )(z3 − z6 )(z4 − z6 )(z4 − z5 ) .
All permutations of the six coordinates are then added to this term and the total is averaged
to get the final wave function. In formula (43) every coordinate is paired up with four other
coordinates, meaning that this term becomes 0 when these four specific coordinates coincide.
But for each zi there is one other zj with which it is not paired, so these two electrons are
allowed to have the same position. Due to the symmetrization in the wave function, there is
always one nonzero term when two electrons are at the same coordinates. However, when
three coordinates have the same value, all terms in the total wave function become zero,
so this method does clearly induce the clustering of electrons at order k = 2. For M > 0
the same procedure can be used as described
Q above,Mbut the final wave function has to be
multiplied by the Laughlin wave function (zi − zj ) .
29
Chapter 5. The Conformal Field Theory Approach
The second method of constructing the RR wave function, which will be referred to as the
η-method or the colour-method, is to divide all N electrons into k groups of N/k electrons
each, where N/k has to be an integer. Each group is then seen as a different ‘colour’ electron
and for each group a factor ηa can be assigned, where
Y
ηa =
(zia − zja )2 .
(44)
i<j
This η is equal to the Laughlin wave function at order 2 for the zi inside group a. The whole
Read-Rezayi wave function is then retrieved by multiplying the η-factor of each colour and
once again symmetrizing this result,
" k
#
Y
ΨRR
ηa .
(45)
k =S
a=1
Because interchanging two electrons within the same group does not change the η-factor,
the symmetrization is only taken over the electrons in different clusters . The same example
as before, with N = 6, k = 2, now results in the two groups (z1 , z2 , z3 ) and (z4 , z5 , z6 ). Then
2
Y
ηa = (z1 − z2 )2 (z1 − z3 )2 (z2 − z3 )2 (z4 − z5 )2 (z4 − z6 )2 (z5 − z6 )2 .
(46)
a=1
The same reasoning as above in the χ-method leads to the conclusion that this wave function
also vanishes if more than three particles have the same coordinate, in other words it is also
clustered at order 2. Using numerical methods, it was checked that equations (42) and (45)
result in the same wave function.
quasiholes in this theory are created by the quasihole
√ The i/ k
M +2/k φ (w)
c
operator Vqh (w) = σ1 (w) e
, where the σ1 is the smallest charge spin field
in the CFT. Just as in the MR wave function, these quasiholes are non-abelian because of
their nontrivial fusion rules.
5.4. Multiple component wave functions
The set of Halperin wave functions for FQH states with two components was discussed in
chapter 5.4, they are given by (9)
PN/2
Y
Y
Y
2
2
2
0
H
ψm,m
(zi − zj )m (wi − wj )m
(zi − wj )n e− i=1 (|zi | +|wi | )/4l0 . (47)
0 ,n ({zi , wj }) =
i<j
i<j
i<j
A previously encountered example of a Halperin wave function is the Laughlin wave function
with multiple quasiholes (35), it is the Halperin wave functions with parameters (m, 1/m, 1)
Y
Y
Y
L
ψm
(z1 , . . . , zn , w1 , . . . wm ) =
(zi − zj )m (wi − wj )1/m (zi − wj ) .
(48)
i<j
i<j
i,j
It was already shown in the section on Laughlin wave functions
that this wave function
√
√
could be obtained from the correlator of CFT operators ei mφ(z) for electrons and eiφ(w)/ m
for quasiholes. This construction, with two different operators of the form eiαφ , is only valid
for Halperin states of the form (m1 , m2 , n = m1 m2 ).
Another important series of Halperin states are the SU(2)-symmetric spin singlet states
(n + 1, n + 1, n), with abelian excitations. These wave functions can be retrieved from a
CFT correlator by splitting the wave function in a “charge” and a “spin” sector and then
finding operators for these two sections. This can be done in the following way:
30
Chapter 5. The Conformal Field Theory Approach
H
ψn+1,n+1,n
=
Y
Y
Y
(zi − zj )n+1 (wi − wj )n+1 (zi − wj )n
i<j
=
Y
i<j
n+1/2
(zi − zj )
i<j
Y
i<j
(zi − wj )
n+1/2
i<j
Y
(wi − wj )n+1/2
i<j
Y
Y
Y
×
(zi − zj )1/2 (zi − wj )1/2 (wi − wj )−1/2
i<j
i<j
*N
Y √
=
ei n+1/2
=
i=1
i
e
√
n+1/2 φc (wi )
i=1
i=1
*N
Y
φc (zi )
i<j
N0
Y
e
i
√
+* N
Y √
ei 1/2
0
φs (zi )
N
Y
e
√
+
1/2 φs (wi )
i=1
N0
√ Y
n+1/2 φc (z)+φs (z)/ 2
i=1
+
√
√ i
n+1/2 φc (w)−φs (w)/ 2
−i
e
(49)
i=1
D
E
= V↑ (z1 ) . . . V↑ (zN )V↓ (w1 ) . . . V↓ (wN ) .
√
√ i
n+1/2 φc (z)±φs (z)/ 2
The vertex operators V↑↓ (z) = e
thus consist of a field representing
the charge and a field representing the spin of the particles, where the ± in front of the
spin field determines whether the particle has spin up or down. This method of writing the
wave function as a correlator of vertex operators with multiple different fields can be used to
write any Halperin wave function of order (m, m0 , n) as a CFT correlator. Moreover, this
method can easily be applied to generalizations of Halperin wave functions for more than
two components, as will be discussed in the next chapter.
31
CHAPTER 6
SU(4) Wave Functions
In this chapter, the focus will shift to wave functions for the completely SU(4)-symmetric
states, possibly occurring in graphene. The goal will be to construct a new set of wave
functions that has non-abelian quasiholes. In the first section, the generalized Halperin wave
functions for four components will be quickly discussed. These states are natural candidates
for the four component wave functions, but their excitations are abelian. In section 6.3 a
new set of SU(4) wave functions will be created that does have non-abelian excitations. The
states are a higher symmetry generalization of the SU(2) NASS states, so as an illustrative
example, the SU(2) NASS states will first be derived in section 6.2.
6.1. Generalized Halperin wave function
The Halperin wave functions (47) are easily generalized for systems with more than two
components [32], which was already discussed in subsection 2.4.5. This generalized Halperin
wave function is written as
ΨSU(N ) =
−
φL{mi } φint
{nij } e
PN
j=1
j 2
kj =1 zkj P Nj
,
(50)
where the φL{mij } describes the interactions of the particles within the same component and
is given by a product of Laughlin wave functions,
φL{mi } =
Nl
N Y
Y
(zi − zj )ml
(51)
l=1 i<j
and the φint
{nij } defines the interactions between particles in different components and is given
by
φint
{nij } =
Np Nq
N Y
Y
Y
(zip − zjq )npq .
(52)
p<q i=1 j=1
Here, N is the N in SU(N ) and Ni is the number of particles in component i. The strengths
of the interactions are given by the integers mi and nij , which are often given in the form
of a matrix M with Mij = nij and Mii = mi . This matrix can then be used to calculate
certain properties of the state, such as the filling fractions. By changing the number of
particles in each component or changing the values of the interaction strengths mi and nij ,
many different polarized and unpolarized states can be created. This procedure was done
for the four component wave functions in reference [32] for a select number of cases.
The total number of particles in each component determines the total Sz , Iz and Pz values.
If the first component describes the particles with (spin, isospin, polarization) = (↑, +, +),
the second (↑, −, −), the third (↓, +, −) and the fourth (↓, −, +), then the quantum numbers
and occupation numbers in terms of each other are
32
Chapter 6. SU(4) Wave Functions
1
(NT + Sz + Iz + Pz )
4
1
Sz = N1 + N2 − N3 − N4
N2 = (NT + Sz − Iz − Pz )
4
(53)
1
Iz = N1 − N2 + N3 − N4
N3 = (NT − Sz + Iz − Pz )
4
1
Pz = N1 − N2 − N3 + N4
N4 = (NT − Sz − Iz + Pz ) .
4
This convention for the names of the different components will be used in the remainder of
the thesis, although the total number of particle will mostly be called N , but it was written
here as NT to avoid confusion with the N from SU(N ).
NT = N1 + N2 + N3 + N4
N1 =
This generalized Halperin wave function can be obtained form a CFT by using multiple
primary fields φi inside the vertex operators Va = exp(ikµa φµ ). These vertex operators have
conformal dimension 1/2 · k · k = kµ Gµν kν /2, where G is the metric tensor of the space. The
wave function (50) can then be obtained in the familiar way by calculating the correlator
of many of these vertex operators. The correlator of two vertex operators Va (z) and Vb (w)
results in the exponent Mab = ka · kb = kaµ Gµν kbν for the coordinate pairing (z − w)Mab .
6.2. Construction of the SU(2)k,M NASS states
The theory behind the CFT approach to SU(4)-symmetric NASS wave functions is a generalization of the theory of the SU(2) symmetric non-abelian spin singlet (NASS) wave
functions, created by Ardonne and Schoutens [1, 52]. As an illustrating example of the
method, the theory behind the SU(2) symmetric NASS-state will be explained in this section.
The SU(2) wave functions clustered at order k, will be created by starting from the affine
SU(3)k algebra. Each affine algebra has a Lie subalgebra whose rank is 1 lower, so SU(2)k
is a subalgebra of SU(3)k . This discussion starts by considering the SU(3) group. The
information from section (4.2) on general SU(N ) groups immediately provides that the
SU(3)-space is eight dimensional and thus has eight root vectors. These eight root vectors
will correspond to the eight simple currents in the CFT. Two of those root vectors have to
be zero vectors, the other six are the raising and lowering operators between three weights.
The dimension of the roots and weights is two, so there will be two simple roots and thus
two invariant directions. The two directions will correspond in the CFT to the two fields
necessary to make the wave function and they will be identified as the charge field φc and
the spin field φs . The roots and weights thus get a spin and a charge direction and the
~ The
weights and nonzero roots are identified with vertex operators of the form exp(i~kn · φ).
two zero roots correspond to the simple currents ∂φc and ∂φs . The task is now to explicitly
find the values for the weights, roots and thus the vertex operators.
A simple check to see that the roots correspond to the simple currents is their dimension.
Simple currents need to have conformal dimension 1, so the inner products of the simple
P
currents have the constraint that ~ki · ~ki = µν kiµ Gµν kiν = 2. Using the Dynkin basis to
denote the vectors in the root-weight space, the Cartan matrix and the metric (inverse
Cartan-matrix) of SU(3) are
1 2 1
2 −1
ASU(3) =
GSU(3) =
.
(54)
−1 2
3 1 2
33
Chapter 6. SU(4) Wave Functions
In this case, the solutions of the constraint ~k · ~k = 2 are the six vectors k1,2 = ±(2, −1),
k3,4 = ±(1, 1) and k5,6 = ±(−1, 2), which exactly correspond to the roots v1...6 of SU(3) in
the Dynkin basis. The weights themselves,
in the basis of the√eigenvalues of the Cartan
√
√
matrices, are given by w1 = (1/2, 1/(2 3)), w2 = (−1/2, 1/(2 3)) and w3 = (0, −1/ 3).
All weights and roots are depicted in figure 7 below.
v5
w2
v2
−1/2
3
v7 ,v8
−1
1/2
−1
3
2
w1
1
p
2 3
p
v3
p
−1/2
v1
1/2
1
w3
v4
(a) All weights of SU(3)
p
−
2
3
v6
(b) All roots of SU(3)
Figure 7. The weights and roots of SU(3) as indicated in the text, the axes are the
eigenvalues of the Cartan matrices.
From figure 7b, it is clear that there are three positive roots (v1 , v3 and v6 ), and two simple
roots (v3 and v6 ; v1 is equal to the sum of those two simple roots). In order to identify the
electrons with these roots, a charge and spin axis are defined such that some roots form an
SU(2) spin subspace. In this thesis, v1 = (2, −1) and v3 = (1, 1) were chosen to be the spin
singlet of the electrons. These two vectors can be written as kc ± ks , showing that they depict
particles with the same charge but with opposite spin. The solution is kc = (3/2, 0) and
ks = (−1/2, 1). The inner products of kc and ks with themselves are 3/2 and 1/2 respectively
and ks ·kc = 0. See figure 8 below for the root diagram with the spin and charge axes inserted.
The corresponding vertex operators for the up and the down electron in the CFT can thus
be written as
q
q
~
~
Ve,↑↓ (z) = eik↑↓ ·φ(z) = e
i
3
φ (z)±
2 c
1
φ (z)
2 s
,
(55)
where the plus (minus) sign is used in the vertex operator for the spin up (down) electrons.
The correlator of many of these electron operators is then
*
+
Y
Y
Y
Y ↑
Ve↑ (zi↑ )
Ve↓ (zj↓ ) =
(zia − zja )2
(zi − zj↓ ) .
(56)
i
j
a=↑↓,i<j
i,j
This wave function is equal to the Halperin wave function (47) with n12 = 1 and both mi = 2.
This result is only correct if the number of up and down electrons is the same, otherwise
there would be some “leftover” spin fields in the correlator, resulting in an amplitude of
zero. Therefore, the same number of up and down vertices is necessary in order to cancel
out the spin fields. It is possible to get a nonzero value for an unequal number of fields, but
34
Chapter 6. SU(4) Wave Functions
&KDUJHD[LV
6SLQD[LV
v5
v2
v3
v7 ,v8
v4
v1
v6
Figure 8. The roots of SU(3) with the spin and charge axes. The roots v3 and v1 are
identified as the spin up and spin down electron operators.
this would require the addition of an extra background field for the spin, which should be
avoided. The filling fraction ν for this wave function is given by
1
2
ν SU(2) = 2 = .
(57)
kc
3
Twice the total conformal dimension of these operators is 1/2 + 3/2 (= spin part + charge
part) = 2. So the case handled above was for bosonic particles, because the conformal
dimension is an integer. In order to describe fermions, the conformal dimension should
become half integer. To achieve a half integer conformal dimension, the values of the inner
products of the k-vectors should be changed. This can be done by stretching the whole
diagram in the charge direction. The spin direction stays untouched in order to maintain
the SU(2)-spin symmetry.When writing the total conformal dimension as h = (2 + M )/2,
the situation is bosonic for even M (because the conformal dimension is an integer) and
fermionic for odd M (half integer). So without changing anything in the spin-part, the
equation for the conformal dimension becomes
kc2 + ks2 = 2h
1
kc2 + = 2 + M
2
3
kc2 = + M .
2
The vertex operator of the electrons then has to become
i
Ve,± (z) = e
q
3+2M
2
φc (z)±
q
(58)
1
φ (z)
2 s
.
(59)
The filling factor is also altered to
SU(2)
νM
=
1
2 2 2
2
=
= , , ,... .
kc2
2M + 3
3 5 7
(60)
These operators are still unclustered, so the excitations will still act abelian. If the particles
in the liquid are clustered together at order k, the filling fraction for M = 0 has to become
35
Chapter 6. SU(4) Wave Functions
νk = k νk=1 . To get a correlator of vertices that results in this filling fraction, an extra
parafermion field should be introduced. This parafermion ψ is of order k, meaning that
ψ × . . . × ψ = I for k ψ’s. The vertex operator of the electrons is then
Ve,± (z) = ψ(z) ei(kc φc (z)+k±s φs (z)) .
(61)
As long as M = 0, the filling fraction should now be k times bigger, so ν = 2k/3. Since the
filling fraction was equal to the inverse of the inner product of kc , kc2 has to be 3/(2k). The
whole diagram is thus effectively stretched by a factor k, which also leads to ks2 = 1/(2k).
The total conformal dimension (hψ + hc + hs ) still has to be unity, so the conformal dimension
of the ψ has to be
hψ + hc + hs = 1
1
3
+
=2
(62)
2hψ +
2k 2k
1
→ hψ = 1 − .
k
Keeping this k and reintroducing M , the same steps as above can be followed to get the
fermionic version of the vertex operators:
kc2 + ks2 + 2h2ψ = 2 + M
1
1
+ 2(1 − ) = 2 + M
kc2 +
2k
k
2M k + 3
2
.
kc =
2k
Accordingly, the vertex operator for general (k, M ) is
Ve,± (z) = ψk (z) e
i
q
q
2kM +3
φc (z)±
2k
1
φ (z)
2k s
(63)
.
(64)
Hence, the very general filling fraction of the SU (2)-symmetric wave functions, clustered at
order (k, M ) is
(
2 4 6
2k
SU (2)
3 , 3 , 3 , . . . for M =0 (bosons)
νM,k =
=
.
(65)
2 4 2
2M k + 3
, , , . . . for M =1 (fermions)
5 7 3
The general correlation function with the same number of spin up and spin down electrons
can then be calculated to be
*
+ *
+
Y
Y
Y
Y
H
1 L
Ve↑ (zi )
Ve↓ (wj ) =
ψ(zi )
ψ(wj )
ψ2,2,1 (zi , wj ) k ψM
(zi , wj ) , (66)
i
j
i
j
H
ψ2,2,1
L the Laughlin wave funcwhere
is the (2, 2, 1)-Halperin wave function (47) and ψM
tion (32). The coordinates zi are used for the spin up particles and the wj for the spin
down particles. For the (k = 1, M = 0) case this NASS wave function simplifies to the
Halperin(2,2,1) wave function, which, as said earlier, has no non-abelian excitations. The
clustering of the electrons at order (k ≥ 2) is thus a crucial step in the construction of the
non-abelian states. This is also evident from the fact that there are no parafermions in the
vertex operators for k = 1.
The wave function (66) has the same general structure as the Read-Rezayi wave function
(40) with a correlator of parafermions times an explicit wave function. The parafermion
correlator results in many conformal blocks, which makes the calculation hard. Luckily, it
also has the same kind of solution to circumvent this problem. Ardonne [55] showed that
36
Chapter 6. SU(4) Wave Functions
the wave function can also be generated by a modified version of the χ-method explained in
section 5.3. This modified χ-method works as follows.
The two spin components are both divided in clusters of k particles. The same χab -factor as
in equation (41) is assigned to each combination of clusters within the same component:
χp,p
a,b = (z(a−1)k+1 − z(b−1)k+1 ) (z(a−1)k+1 − z(b−1)k+2 )
(z(a−1)k+2 − z(b−1)k+2 ) (z(a−)k+2 − z(b−1)k+3 )
(67)
. . . (zak − zbk ) (zak − z(b−1)k+1 ) ,
where the exponent (p, p) means that both the first and the second coordinate in each term
belong to component p ∈ {↑, ↓}. Each combination of particles in different components gets
the factor
p
q
p
q
p
q
χp,q
=
z
−
z
z
−
z
.
.
.
z
−
z
(68)
a,b
ak
bk
(a−1)k+1
(b−1)k+1
(a−1)k+2
(b−1)k+2
assigned. In the case of SU(2), there are only two components, so the only relevant
intercomponent χ-factor is χ↑↓
ab . Equation (68) is written here with the more general
exponents (p, q), because the same expressions will be used later in the four component
SU(4) situation. The complete wave function can then be obtained from these χ-factors as
follows,


Y
Y
SU(2)
 L ↑ ↓
ψk,M (zi↑ , zj↓ ) = S 
χpp
χ↑↓
(69)
ab
ab ψM (zi , zj ) .
p,a<b
a,b
The same reasoning as in section 5.3 of the Read-Rezayi wave functions leads to the conclusion
that these wave functions are inherently clustered at order k: when more than k electrons
of the same component come together, every χ-factor in the completely symmetrized wave
function becomes zero.
6.2.1. Quasiholes. In the section above, the vertex operators for the electrons were
assigned to the roots of the diagram. The quasihole operators can be assigned to the
weights of the diagram using the same construction of spin and charge directions. For the
k = 1, M = 0 case, the two quasiholes with spin up and down are to be w2 = (−1, 1) and
w3 = (0, −1) (in the Dynkin basis, see also figure 9). The third weight only has a charge
part and not a spin part. In terms of (kc , ks ) they become w2 = − 31 kc + ks and − 13 kc − ks ,
with k i · k i = 1/6 for the charge vector and 1/2 for the spin vector. The factor for the spin
direction is the same as in the electron operator, showing that the electrons and holes have
the same spin, as was required. The charge is negative and only 1/3 of the electrons, this
shows that these operators are indeed quasihole operators. The vertex operators of these
two quasiholes in terms of the spin and charge fields are then
Vqh,↑↓ (w) = e
i
√1 φc (w)± √1 φs (w)
6
2
.
(70)
The correlator of an electron and a quasihole operator with the same spin is hVe,↑↓ (z)Vqh,↑↓ (w)i =
(z − w), and with different spin just the identity. This is as expected, because the quasihole
operators let the ↑↓-electrons occupy a higher angular momentum state. This also means
that the spin up quasiholes actually add another empty spin up electron state (and the same
with the spin down quasiholes). Although it should be noted that the first correlator should
vanish, because there is no background spin to cancel the spin fields of the vertex operators.
The total spin of the vertex operators thus should be zero, which constraints the number of
electrons and quasiholes to Ne,↑ + Nqh,↑ = Ne,↓ + Nqh,↓ .
37
Chapter 6. SU(4) Wave Functions
&KDUJHD[LV
6SLQD[LV
w2
w1
w3
Figure 9. The weights of SU(3) with the spin and charge axes. The weights w2 and w3
are identified as the spin up and the spin down quasihole operators.
When reintroducing M , the correlators between the electrons and quasiholes above should
remain unchanged in order not to violate the physical condition that the electrons have to
behave non-singular. The spin direction remains the same when transforming the root-weight
diagram, but the charge part is stretched out. The electron vertex operator becomes (59),
which means that the (k = 1, M ) quasihole vertex operator has to become
k=1,M
Vqh,±
(w) = e
i
√ √1
2 3+2M
φc (w)± √1 φs (w)
.
2
(71)
When clustering the electrons at order k, the complete diagram is stretched and the electron
vertex operator has become (64). The same procedure for the quasiholes gives the quasihole
vertex operator for (k, M ),
k,M
Vqh
(w)
= σk (w) e
√i
2k
√ 1
φ (w)±φs (w)
2kM +3 c
,
(72)
where the parafermion σk has been introduced to get the correct conformal dimension back
(∆σ = (k − 1)/(k(k + 3))). The charge of these quasiholes is 1/(2kM + 3), which can either
be seen from the factor in front of φc or by calculating the correlator of two quasihole vertex
operators. The general correlator resulting in the SU(2)-symmetric case with clustering
(k, M ) is then
*
+
Y
Y
Ve↑ (zi↑ ) Ve↓ (zi↓ )
Vqh↑ (wj↑ ) Vqh↓ (wj↓ ) =
i
j
*
Y
i
+
ψi
Y
σj
Gen.Halp.
ψMij
(73)
zi↑ , zj↓ , wk↑ , wl↓
.
j
With the generalized Halperin wave function (see section 6.1) with M -matrix


 ↑ 
z
2/k + M 1/k + M 1/k 0
1/k + M 2/k + M

 z↓ 
0
1/k
 in basis 

M =
 1/k
 w↑  ,
0
a
b 
0
1/k
b
a
w↓
(74)
38
Chapter 6. SU(4) Wave Functions
where a = (2/k + M )/(2kM + 3) and b = −(1/k + M )/(2kM + 3).
6.3. Construction of the SU(4)k,M states
The method described in section 6.2 above of obtaining SU(2)-symmetric spin singlet wave
functions by identifying the roots and weights of SU(3) with vertex operators in a CFT will
be applied in this section to construct SU(4)-symmetric singlet wave functions. The vertex
operators will now be identified with some of the roots and weights of SU(5) and therefore
this section will start by a description of this root-weight space.
6.3.1. SU(5) root and weight space. The group SU(5) is generated by 24 (= 52 − 1)
elements, four of which are in the Cartan subalgebra. From the theory in section 4.2 it follows
that there are five weights, four zero-roots and twenty non-zero roots, all four dimensional.
The weights in the basis of the eigenvalues of the chosen elements in the Cartan subalgebra
are


−1/2
0√
√
 1/ 12 
 −2/ 12
√ 
√
w2 = 
w3 = 
 1/ 24 
 1/ 24
√
√
1/ 40
1/ 40



0
0



0√ 
0
 .

w5 = 
w4 = 


 −3/ 24 
0
√
√
−4/ 40
1/ 40


1/2
√
 1/ 12 

w1 =  √ 
1/√24 
1/ 40






(75)
Unfortunately, these weights can not be depicted easily, because they are four dimensional.
Their structure is however comparable to the weight structure of SU(3), just in two extra
dimensions. The weight w5 is also equal to minus the sum of the four other weights. The
Cartan matrix and the metric matrix (the inverse Cartan matrix) of SU(5) are given by

ASU(5)

2 −1 0
0
 −1 2 −1 0 

=
 0 −1 2 −1 
0
0 −1 2

GSU(5)

1
2 
 .
3 
4
(76)

0
 0 

k4 = 
 −1  .
2
(77)
4
1
3
= 
5 2
1
3
6
4
2
2
4
6
3
The four simple roots in the Dynkin basis are


2
 −1 

k1 = 
 0 
0


−1
 2 

k2 = 
 −1 
0

0
 −1 

k3 = 
 2 
−1


All six other positive roots can be made by certain summations of these four simple roots.
The complete root system of the positive roots of SU(5) is as follows,
39
Chapter 6. SU(4) Wave Functions
(1, 0, 0, 1)
(1, 0, 1, −1) (−1, 1, 0, 1)
(1, 1, −1, 0) (−1, 1, 1, −1) (0, −1, 1, 1)
(2, −1, 0, 0) (−1, 2, −1, 0) (0, −1, 2, −1) (0, 0, −1, 2)
Figure 10. The root diagram of SU(5).
In this diagram, the vectors in the lowest row are the four simple roots and the vectors in
the rows above consist of the sums of the simple roots beneath them.
The roots need to be identified with the simple currents of the CFT. The root-weight space
has dimension four, so there are necessarily four fields φi in order to span the complete space.
Just as before, the simple currents ∂φ are identified with the zero roots of the diagram and
~ From a physics
the other 20 roots are identified with the vertex operators Vj = exp(i~kj · φ).
standpoint, the SU(4)-symmetric electrons are described by their charge C and the three
quantum numbers spin S, isospin I and the polarization spin P . The first step in identifying
the different particles with the roots is to define a charge axis. The four left most vectors of
the root system above were chosen to be the charge sector, explicitly


 




1
1
1
2
 0 
 0 
 1 
 −1 







α
=
(78)
α1 = 
α
=
α
=
2
3
4
 0  .
 0 
 1 
 −1 
−1
0
0
1
The inner products of these four vectors are αi · αj = 1 + δi,j . These four roots then have
to be written as kc + kδ,i , where kc is the charge vector and kδ,i the difference between the
charge vector and the root αi . The charge vector is found by averaging the four roots αi
and is


5
4
 0 

kc = 
 0  .
0
The four vectors kδ,i are then
 1 
 1
−4
−4
 0 
 0


kδ,1 = 
 0  kδ,2 =  1
1
−1





− 14
 1 

=
 −1 
0

kδ,3
(79)

3
4

 −1 

kδ,4 = 
 0  .
0
(80)
The important quantities are all the inner products between these vectors,
kc · kc = 5/4
k c · k δi = 0
kδ,i · kδ,j = δij − 1/4 .
(81)
The second inner product shows that the charge vector is perpendicular to all the difference
vectors, which is indeed required. The third inner product shows that the four kδ -vectors
are not completely independent. Furthermore, they have all the same length (the inner
product with themselves are all the same) and the angles between the vectors are all the
same (all inner products kδ,i · kδ,j are the same). From these inner product statements it
follows that the kδ,i are the corner stones of a tetrahedron in a three dimensional subspace.
40
Chapter 6. SU(4) Wave Functions
Within this subspace, three orthogonal directions have to be defined to represent the spin,
isospin and polarization fields. From the physical view point, the four vertices should have
spin ↑ or ↓, isospin + or − and polarization + or −. For the particles in the fundamental
4 representation, as they were defined at the beginning of this chapter, the value of the
polarization depends on the spin and isospin values: P = + if (S, I) = (↑, +) or (↓, −) and
P = − in the other two cases. The four vectors representing the four different particles
should therefore be written as
k1
k2
k3
k4
= kc + kδ,1
= kc + kδ,2
= kc + kδ,3
= kc + kδ,4
= kc + ks + ki + kp
= kc + ks − ki − kp
= kc − ks + ki − kp
= kc − ks − ki + kp .
(82)
The tetrahedron structure of the spin/isospin/polarization subspace has been embedded in
a cube, whose sides align with the directions of the spin, isospin and polarization vectors,
see figure 11. Inserting the values for the kδ,i from equation (80) in the equations above, the
solutions for the vectors ks , ki , kp are






−1/4
1/4
−1/4
 0 
 1/2 
 −1/2 





ks = 
k
=
k
=
(83)
i
p
 1/2 
 −1/2 
 0  .
0
1/2
1/2
k1
k3
3RODUL]DWLRQ
k4
k2
6SLQ
SLQ
,VRV
Figure 11. The four vectors assigned to the four different particle types in the rootweight space.
Their inner products are given by ka · kb = δa,b /4, showing that these three vectors are indeed
orthogonal and all have the same length. Now that the ~kj of the four different particles
41
Chapter 6. SU(4) Wave Functions
j ∈ {1, 2, 3, 4} are known, the vertex operators can be written down as
Vj (z) = e
i
q
5
φ (z)+
4 c
q
1~ ~
k ·φ(z)
4 j
,
(84)
~ = (φs , φi , φp ). By calculating
with ~kj = {(1, 1, 1), (1, −1, −1), (−1, 1, −1), (−1, −1, 1)} and φ
the correlator of multiple vertex operators, it is shown that the filling fraction is again equal
to the inverse of the kc inner product with itself. So the SU(4)-symmetric spin singlet wave
function has ν = 4/5. Before giving a general expression for the correlators with these
vertices, the theory will be extended by reintroducing the (k, M )-clustering from the NASS
states.
6.3.2. (k,M)-clustering. Twice the conformal dimension of the operators (84) is
2 = 2, which is bosonic since the conformal dimension is an integer. Once again, a
+ kδ,j
factor M is added to this conformal dimension, so that it becomes a fermionic theory for
odd M and a bosonic theory for even M . The charge direction once again (see equation
(58)) has to be transformed to get this nonzero M result:
kc2
kc2 + kδ2 = kc2 +
−→
kc2
3
=2+M
4
(85)
5
= +M .
4
SU(4)
Consequentially, the filling fraction has changed to νM
= 4M4+5 . In the same way as in
equation (62), the k-clustering factor is introduced through a parafermion field of order k,
with conformal dimension
2hψ +
3
1
5
+
= 2 −→ hψ = 1 − .
4k 4k
k
(86)
Combining the k and M as in equation (63), kc becomes
1
3
2(1 − ) + kc2 +
=2+M
k
4k
5 + 4kM
−→ kc2 =
.
4k
(87)
The filling fraction for general (k, M ) is therefore given by
(
4 8 12
4k
SU(4)
5 , 5 , 5 , . . . for M=0 (bosons)
νM,k =
=
4 8 12
4kM + 5
, , , . . . for M=1 (fermions)
.
(88)
9 13 17
On a small tangent, a comparison of this SU(4) filing fraction with that of the SU(2) case
2k
(ν = 2k+3
) suggests that for general N the filling fraction of (k, M ) clustered SU(N ) states
is given by
(N − 1)k
SU(N )
νk,M =
.
(89)
(N − 1)kM + N
Back to the SU(4) states, the vertex operators of the four different electrons with general
(k, M ) can be written as
k,M
Ve,j
(z)
i
= ψk (z) e
q
5+4kM
4k
~
φc (z)+ √1 ~kj ·φ(z)
4k
.
(90)
42
Chapter 6. SU(4) Wave Functions
The spin parts of the correlator of two particles of the same component is
D
E
±i
~
~
√i ~
√i ~
√
kj ·φ(z)
kj ·φ(w)
φs (z) √±i φs (w)
4k
4k
4k
4k
e
= e
e
e
× (φs → φi ) × (φs → φp )
h
= (z − w)
1
4k
i3
(91)
= (z − w)
3
4k
.
The correlator of each spin-direction results in an exponent of 1/(4k), because all different
’spin’-directions have the same sign for the coordinates z and w. When calculating the
correlator of two particles in different components, only one of the three directions will have
the same sign (see equation (82)), so the correlator is then
h
i
~
~
−1 2
−1
1
√i ~
√i ~
k ·φ(z)
k ·φ(w)
e k j
= (z − w) 4k (z − w) 4k = (z − w) 4k .
e k i
(92)
Actually, those correlators would be zero, because not all fields cancel each other out. A
correlator of these vertex operators is only nonzero when an equal number of particles from
each component is present, otherwise there will always be some left over spin, isospin and/or
polarization fields. A fully written out correlator of N particles (where N is of course
divisible by 4) is then
+
* N/4
+ *
+*
+*
Y √i ~kq ·φ~s (z q )
YY
Y
Y iq 5+4kM φ (z q )
q
q
c
i
i
4k
V (z ) =
ψ (z )
e
e k
q
q
i
q i=1
i
q,i
=
*
Y
q,i
+
ψq (ziq )
YY
ziq
−
zjq
3
4k
=
+
ψq (ziq )
=
+
ψq (ziq )
ziq − zjq
0
5+4kM
4k
q<q 0 i,j
Y Y
YY
q i<j
q,i
*
Y
ziq − zj
ziq
−
0
zjq
−1
4k
(93)
q<q 0 i,j
q i<j
*
Y
5+4kM Y Y 4k
q
q i<j
q,i
×
q,i
YY
ziq − zjq
2 +M Y Y k
ziq − zjq
0
1 +M
k
q<q 0 i,j
1
k
q
Gen.Halp.
L
ψm
(z
)
ψM
(ziq ) ,
i
i =2,nij =1
q,i
Gen.Halp.
where ψ
is the multicomponent generalized Halperin wave function (50) and ψ L the
Laughlin wave function (32). To get from the first to the second row for the correlator of
the spin/isospin/polarization fields, equations (91) and (92) were used. This result is simple
for the case k = 1, M = 0: The ψ-part and the Laughlin part then disappear, so the wave
function is given by the familiar multicomponent wave function with indices mi = 2, nij = 1.
With the help of the matrix Mij the filling fraction can be calculated to be


1/5
 1/5 

~ν = M (nij )−1 · ~1 = 
(94)
 1/5  .
1/5
The filling fraction of each component is 1/5 and the total filling fraction is νtot = 4/5, which
is indeed equal to the filling fraction (88).
For the cases (k = 1, M 6= 0) the wave functions only get multiplied by an extra Laughlin
wave function of order M . For k 6= 1 the calculation of the wave functions from equation
Chapter 6. SU(4) Wave Functions
(93) gets a lot harder, because the ψ-correlator is in general difficult to calculate exactly.
Instead of calculating this correlator directly, a solution is also given by generalizations of
the χ and η-methods discussed before. For the χ factor, this generalization has already been
given in the section on NASS-states 6.2, the only difference is an increase in the number
of components. As a quick reminder, the method is as follows. To get the (k, M ) wave
function (93), the coordinates z p belonging to each component p are divided into groups of
k coordinates. Thereupon, the following factors are assigned to each combination of groups
within the same component and between different components:
p
p
p
p
p
p
χpp
ab = (z(a−1)k+1 − z(b−1)k+1 ) (z(a−1)k+1 − z(b−1)k+2 ) (z(a−1)k+2 − z(b−1)k+2 ) . . .
p
q
p
q
χpq
ab = (z(a−1)k+1 − z(b−1)k+1 ) (z(a−1)k+2 − z(b−1)k+2 ) . . . .
The wave function (93) is retrieved by multiplying all possible χ- factors and symmetrizing
this product over all coordinates within the same component (so symmetrize all ’up’-electrons
first, then the ’down’ etc.). As a last step, this term is multiplied with the Laughlin wave
function at order M to obtain


Y pp Y pq
L
ψ=S
χab
χab  ψM
(zip ) .
(95)
p,a<b
p<q,a,b
This equation is the same as (69) but with more than two components. Although this
method works, the general form is not directly insightful. It would be useful to have a
generalization of the η-method described in section 5.3.
The “colour” or η-method works for a general number of components as follows. For each
component, the corresponding coordinates are divided into k groups with N/(4k) coordinates
each (or equivalent: all coordinates are divided into k groups with the same number of
coordinates for each component per group). Each of these groups can be seen as a different
“colour” variant of the particle. Then a factor (zi − zj )2 is assigned to all combinations of
particles within the same component and with the same colour (the same as in equation
(45)) and a factor (zi − zj ) for all particles with the same colour but in different components.
No factors are assigned to combinations of different coloured particles. All these factors
all multiplied with each other and then this term is completely symmetrized. Finally, it is
L to obtain the complete wave function
multiplied with the Laughlin wave function ψM


2 Y Y p
p
p
q
L
 ψM
ψ=S
za,i − za,j
za,i
− za,j
(zip ) .
(96)
p,a,i<j
p<q,a,i,j
Here, the a denotes the colour of the particle and p the component. This wave function does
indeed have the correct clustering properties.
For k = 1 it can be seen that both methods give the same generalized Halperin (mi = 2,
nij = 1) wave function, but for k > 1 (the cases with non-abelian excitations) it is harder to
see that both methods result in the same wave function. It was explicitly checked numerically
that for k = 2 the two different methods result in the same wave function using two layers (p
and q) and 4 particles in each layer. According to these tests, the χ and η methods indeed
result in the same wave function. For cases where the number of terms become much higher,
such as for example N = 6, k = 2, the required computation time grows quickly, which is a
result of the symmetrization procedure.
43
44
Chapter 6. SU(4) Wave Functions
6.3.3. Quasiholes. Similar to the method in the NASS section 6.2, the quasiholes of the
theory will be identified with the roots of the SU(5) space. The weights of SU(5) (75) therefore
first have to be rewritten from the Dynkin basis to the basis of the charge, spin, isospin and
polarization vectors (83). The Dynkin-basis is written as (w1 , w1 + w2 , w1 + w2 + w3 , ....), so
each weight is in this basis (0, ..., −1, 1, 0, ..., 0). Starting with the (k = 1, M = 0)-situation,
the five weights in terms of the charge and spin fields are
4
kc
5
1
1
w2 = − kc + ks + ki − kp
w3 = − kc + ks − ki + kp
5
5
(97)
1
1
w5 = − kc − ks − ki − kp .
w4 = − kc − ks + ki + kp
5
5
This result is luckily as was expected: compared to the particles, the quasiholes carry a
smaller (negative) charge, but the same spin quantum numbers. Notice that the quantum
numbers of w5 are the opposite of the quantum numbers of k1 (see (82)), and similarly
w4 , w3 and w2 are the opposites of k2 , k3 and k4 . This is also reflected in their inner products:
k1 · w5 = −1 and the inner product with all other wi is zero. It is therefore logical to rename
the weights to reflect this behaviour:
w1 =
w5 ↔ w1
w4 ↔ w2
With this renaming, the quasihole vertex operator of component j is
i
Vqh,j (w) = e 2
~
√1 φc (w)+k~j ·φ(w)
5
,
(98)
~ are the same as in (84). Notice that in this vertex operator all
where the vectors ~kj and φ
fields have gained a minus sign, just as in the NASS states before. This was done to assure
that the correlator of a particle and a quasihole of the same component results in a factor
(z − w). The sign changes can be justified, because the creation of a hole is equivalent to the
removal of a particle with the opposite quantum numbers.
The same procedure as before results in the quasihole operator for general (k, M )
k,M
Vqh,j
(w)
= σk (w) e
√i
4k
√
1
~
φ (w)+k~j ·φ(w)
4kM +5 c
.
(99)
As can be seen, the charge of the quasiholes is 1/(4kM + 5). Once again, the parafermions
σk have nontrivial fusion rules, so the quasiholes are non-abelian. These fusion rules and the
resulting degenerate excited states of the fusion of multiple quasiholes, will be discussed in
chapter 9.
45
CHAPTER 7
Four Component Polarized Wave Functions
All wave functions considered in the previous chapter are unpolarized, their total Sz , Iz
and Pz are all zero. In the calculations of the correlators, the number of particles in each
component had to be the same in order to assure that the correlator would become nonzero.
Otherwise, the three spin-like fields would not sum to zero, leading to the necessity of an
unwanted ‘spin background’. In this section, wave functions will be considered that have
polarized states but do not require a spin background. By combining two SU(2)-NASS wave
functions, an SU(2)×SU(2)-symmetric state will be created that can be polarized in the
broken spin-direction of the original SU(4) state.
The ground state wave functions considered in this chapter can be written in the most
general case as
SU(2)
SU(2)
ψ(k1 k2 M1 M2 ) = ψk1 ,M1 ψk2 ,M2 ,
(100)
where the values of ki and Mi can be freely chosen in both NASS wave functions. If the first
ψ has vertex operators with a spin field and the second vertex operators with an isospin field,
then the third spin direction (already called the polarization spin) can become polarized. The
number of up and down electrons and also the number of isopin + and isospin − electrons
should still be equal. Evaluating this wave function with the results of formula (66) gives
D
E
D
E
ψ(k1 k2 M1 M2 ) = Ve1 z1↑ Ve2 z2↑ . . .
Ve1 z1+ Ve2 z2+ . . .
k1 ,M1
k2 ,M2
(101)
D
ED
E
1 1
k1
k2
H
↑↓
H
±
L
↑↓
L
±
= ψ... ψ...
ψ221 (z )
ψ221 z
ψM1 (z ) ψM2 (z ) .
In this expression are four different types of electrons (z ↑ , z ↓ , z + and z − ), that have either a
spin or an isospin. To get back to the SU(4) picture of electrons with both spin and isospin,
each coordinate in the first correlator should be matched with a coordinate in the second
one. For example, for a coordinate with spin ↑ and isospin +
rename
zi↑ = zj+ −−−−→ zk↑+ .
(102)
There are N coordinates in total, that can now be written as the four different components
zi↑↓,± . An important point is that there is a freedom to choose how many spin ↑ coordinates
are matched with isospin + or isospin −. Let N↑+ denote the number of z ↑+ coordinates
and similar for the other combinations. These numbers have the following restrictions:
N↑+ + N↑− + N↓+ + N↓− = N
N↑+ + N↑− = N/2
N↑+ + N↓+ = N/2
N↓+ + N↓− = N/2
(103)
N↑− + N↓− = N/2 .
The first equation follows from the fact that the total number of particles should remain
equal to N , the other four equations follow from the restriction that within each correlator
exactly half the particles have (iso)spin up and other half (iso)spin down. Note that the
first equation can easily be obtained from the other four. This system, with four variables
46
Chapter 7. Four Component Polarized Wave Functions
and four equations, results in one free choice of a parameter. Let N↑+ ≡ R be the chosen
variable, then N↓− = R and N↑− = N↓+ = N/2 − R.
The filling fraction ν of wave function (101) does not depend on the parafermion correlators,
so they can be neglected in the calculation of ν. After choosing the value of R and writing
out the Halperin and Laughlin wave functions, the total wave function (upto a minus sign)
becomes
Y pp 2 + 2 +M Y pq 2 + 1 +M Y pq 1 + 2 +M Y pq 1 + 1 +M
(zij ) k1 k2
,
(104)
(zij ) k1 k2
(zij ) k1 k2
(zij ) k1 k2
pq
where zij
= (zip − zjq ), p and q ∈ {↑ +, ↑ −, ↓ +, ↓ −} and M = M1 + M2 . The first product
is over all coordinate pairs with the same spin and isospin, the second over all pairs with
equal spin but different isospin, the third over equal isospin and different spin and the fourth
when both quantum numbers are different. This formula is once again in the form of a
generalized Halperin state with corresponding M -matrix (see section 6.1)




f22 f21 f12 f11
↑+
 f21 f22 f11 f12 


 with basis  ↑ −  ,
M =
(105)
 f12 f11 f22 f21 
 ↓+ 
f11 f12 f21 f22
↓−
with fab = ka1 + kb2 + M . By inverting this matrix, the filling fractions of the different
components can be found. They are, in terms of N↑+ = R,
ρ↑+
1
2R
ν↓− = ν↑+ =
= 3
3
nB
N
k1 + k2 + 2M
(106)
ρ↑−
1
2R
= 3
1−
.
ν↓+ = ν↑− =
3
nB
N
k1 + k2 + 2M
The total filling fraction is then equal to
X
2
2k1 k2
νtot =
νi = 3
=
.
3
3k1 + 3k2 + 2k1 k2 M
k1 + k2 + 2M
i
(107)
This νtot is R-independent. The spin and isospin polarizations are of course still 0, because
the correlators have an equal number of (iso)spin up and down electrons. The P -polarization
has become
8N↑+
2
P = ν↓− + ν↑+ − ν↑− − ν↓+ =
−
,
(108)
CN
C
2
with C = (k1 k2 )/(3k1 + 3k2 + 2k1 k2 M ). The result is that −2
C < P < C , note that these
limits are equal to ± νtot .
47
Chapter 7. Four Component Polarized Wave Functions
The νtot for different values of k1 , k2 and M are given in the tables below.
M =0
2 3
(k1 , k2 ) 1
1
1/3
4/9 2/3
2
3
1/2 4/5 1
M =1
(k1 , k2 )
1
2
3
1
1/4
2
4/13 2/5
1/3 4/9 1/2
3
M =2
(k1 , k2 )
1
2
3
1
1/5
2
4/17 2/7
1/4 4/13 1/3
3
(109)
48
CHAPTER 8
Numerical Analysis
The subject of this chapter is the numerical analysis of the FQHE wave functions. Theoretically, many different wave functions resulting in the FQHE at a given filling fraction can be
created, but it is hard to theoretically predict which of these states occurs in reality, if the
FQHE even occurs at all. Experimental measurements can provide proof of the occurrence of
certain wave functions, but as a first approach, numerical analysis can also be used to test the
viability of the states. An important goal of this research project was to not only create the
new non-abelian SU(4) wave functions, but to also numerically analyze them. Unfortunately,
no results could be obtained in the end due to the high computational time needed. Despite
this complication, the basis of the numerical analysis will be explained in this chapter with
special attention for the created SU(4) wave functions. The programs/languages used during
this project were Python, Mathematica with the LieART-plugin and the DiagHam package
created by Nicolas Regnault [56].
8.1. FQHE wave functions on the sphere
The fractional quantum Hall wave functions live in a two dimensional plane, but this is not
the best geometry for numerical purposes, mainly because the plane has no boundaries. For
numerical analyses, bounded geometries are used, such as the disc, sphere or torus, each
having its own strengths and weaknesses. For the remainder of this thesis, the spherical
geometry will be used. The wave functions on the plane should be projected onto this
sphere. The ring shaped orbits in the plane (at a certain radius) become orbits on the
sphere at constant latitude. In the plane, the rings with a certain radius correspond to
states with a certain angular momentum (see also appendix A), so the angular momentum
of the states on the sphere increases with increasing latitude. In the lowest Landau level, the
angular momentum of a wave function on the plane equals the exponent of z. The highest
exponent of a coordinate in the expansion of a many particle state thus corresponds to the
highest angular momentum for a particle and can therefore be identified with the orbit at
the highest latitude. To create these orbits on a sphere, a monopole is needed with strength
(flux number) equal to the number of orbits minus one [57]. The number of fluxes of the
monopole does not equal the number of fluxes in the plane, the difference is the shift S. In
the thermodynamic limit, this S can be neglected, but it is significant for the numerical
analysis of a few particles. Taken into account that the filling fraction is ν = N/Nφ , the
number of fluxes on the sphere is Nφ,sphere = Nφ,plane /ν − S. The exact value of S is wave
function dependent.
The exponent of the zi -factor thus corresponds to the orbit on the sphere. A second
quantization occupation basis can then be used to describe the wave functions as the
superposition of (un)occupied orbits. To illustrate this method, the Laughlin wave function
49
Chapter 8. Numerical Analysis
for three particles with ν = 1/3 is
ψ3L (z1 , z2 , z3 ) = (z1 − z2 )3 (z1 − z3 )3 (z2 − z3 )3
= z30 z23 z16 − z20 z33 z16 + 3 z2 z32 z16 + . . .
.
(110)
In the expansion of this wave function, the highest power of z1 will be 6, followed in the
same term by z2 with exponent 3 and z3 with exponent 0 (or the other way around). In
this term, one particle occupies the sixth orbital, another particle the third orbital and
the last one the zeroth. This can be written in the occupation basis as (1001001), where
each number gives the occupation number of that state (so 1 particle in the zeroth orbital,
0 in the first, etc.), this is also called the root of the wave function. These roots can be
used to generate all other terms in the expansion [58, 59]. The number of fluxes necessary
to create these orbits is 6, because there are 7 orbits. This is not equal to the number of
fluxes in the plane, Nφ,plane = 3(1/3)−1 = 9, so the shift S for the Laughlin wave function is 3.
In general, the necessary number of fluxes on the sphere can be deduced by counting the
maximal power of a certain coordinate in the expanded wave function. For the SU(4)k,M
wave functions, there are N particles, divided into n = N/4 particles per component, where
each component is once more divided into k “colour” groups of n/k electrons per group.
The total (k, M ) SU(4) wave function is given by (96),


2 Y Y
SU(4)
p
p
p
q
L
 ψM
ψk,M = S 
za,i
− za,j
za,i
− za,j
.
(111)
p,a,i<j
p<q,a,i,j
n
k
In the first product, the z is paired with the − 1 other coordinates with the same colour
and component and gets a power of 2, so in total an exponent of 2(n/k − 1). In the second
product, it is paired with the n/k coordinates with the same colour, but different component,
each with a power of 1. Since there are three other components, z gets a total exponent of
3 n/k from this term. Finally, in the Laughlin wave function it is paired with all N − 1 other
coordinates with a power of M . The maximum power of z, which is equal to the number
of fluxes on the sphere, is thus 2(n/k − 1) + 3 n/k + (N − 1)M = N ν −1 − (M + 2), with
ν = 4k/(4kM + 5). From this equation the shift S = 2 + M is obtained, which is used to
determine the necessary number of fluxes at a certain N .
8.2. Exact diagonalization
In the previous section it was shown how to translate the theoretical constructed wave
functions to the sphere geometry. The goal of this section is to elucidate how these wave
functions can be compared to realistic states. The two most important techniques are
calculating the overlap of the wave functions and the comparison of the entanglement spectra
of the two states.
The general idea is to use an exact diagonalization of the Hamiltonian with the Coulomb
force to obtain all eigenstates and energies of the system. The basis used to construct the
Hamiltonian matrix (or Hilbert space) is exactly the occupation basis of the previous section,
which are angular momentum eigenstates. The angular momentum li of the individual
particles lies in the range {−L, L}. The zeroth orbit corresponds to the −L state, the first
orbit to the −L + 1 state, etc. This means that L = Nφ /2. The angular momentum is a good
quantum number for the Coulomb force, because this interaction is rotationally invariant
and thus preserves the total angular momentum. Therefore, the interaction energy can be
50
Chapter 8. Numerical Analysis
calculated with these angular momentum eigenstates and the values only depend on relative
angular momentum between two states. The Hamiltonian can be written as
X
H=
hm1 , m2 |V |m3 , m4 i a†m1 a†m2 am3 am4 ,
(112)
m1 +m2
=
m3 +m4
where
hm1 , m2 |V |m3 , m4 i =
2L X
l
X
hL, m1 ; L, m2 |l, mi hl, m|L, m3 ; L, m4 i V n (l) .
(113)
l=0 m=−l
Here, the first two terms are Clebsch-Gordan coefficients and V n (l) is the Haldane pseudopotential, which represents the interaction energy of two electrons in Landau level n with
relative angular momentum 2L−l. The formula for the Haldane pseudopotentials in graphene
is given by [60]
Z ∞
dq
V n (m) =
q V (q) [Fn (q)]2 Lm (q 2 ) e−q ,
(114)
2π
0
where V (q) is the momentum space Coulomb potential V (q) = 2πe2 /( lB q) with the
dielectric constant. The Lm (x) arehthe Laguerre polynomials.
The form function Fn (x)
i
x2
x2
x2
1
is given by L0 ( 2 ) for n = 0 and 2 Ln ( 2 ) + Ln+1 ( 2 ) for n > 0. In the nonrelativistic
2
situation, the form functions are given for all n by Ln ( x2 ), so in the lowest Landau level the
pseudopotentials of graphene are the same as in the usual 2DEG, but they are modified in
the higher Landau levels. In the case of multiple components, the pseudopotentials between
the different components can be different due to extra interaction terms. Details of these
multiple component pseudopotentials, including the SU(4) case in graphene, can be found in
an article by Davenport and Simon [61].
The interaction Hamiltonian for the chosen number of particles and fluxes can thus be
constructed using these pseudopotentials. The exact diagonalization of this Hamiltonian
then gives all energies and eigenstates of the system.
8.3. Numerically obtaining the theoretical wave functions
The theoretical wave functions can in some cases be generated by choosing suitable pseudopotentials. For example, the Laughlin wave function is the exact ground state of the situation
where V 0 (1) = 1 and all other V ’s are zero. Similarly, the Halperin wave functions ({mi , nij })
are the exact ground states of a model interaction with Vijl = 1 for l < nij and otherwise
zero. For other states, it is sometimes convenient to write the interaction Hamiltonian in
the form of projection operators to certain states [57], of which the wave function under
consideration is once again the unique zero energy eigenstate. The SU(2) NASS state with
k = 2 and M = 1 is the zero energy ground state of such a Hamiltonian [55],
X
X
3Nφ
3
H=U
Pijk
− 3,
+V0
Pij (Nφ , 0) ,
(115)
2
2
i<j<k
i<j
where Pij(k) (L, S) means the projection to the state with angular momentum L and spin S
for a two or three particle interaction. It will now be explained in detail how this Hamiltonian uniquely generates the SU(2) NASS states as the zero energy eigenstates. Then, this
Hamiltonian will be generalized to the case for SU(4), which can be used to numerically
create the SU(4) NASS wave functions.
51
Chapter 8. Numerical Analysis
The second projection operator demands that the zero energy eigenstate has no component
with L = Nφ and S = 0. As mentioned earlier, the maximum angular momentum for one
particle is Nφ /2, so a total angular momentum of Nφ for two particles means that both
particles should be in the highest orbit. In the actual wave function, this means that they
both have exponent 0. Therefore, in order to have zero energy, the wave function should
have no term with more than one coordinate with exponent zero. This implies that for
every combination of i and j there has to be a factor (zi − zj ), because then every term in
the expansion has at least exponent 1 for one of the two coordinates. Every zero energy
eigenstate of this Hamiltonian thus has a Laughlin (M = 1) factor, which indeed appears in
the NASS wave function. The spin value of this projected state should be 0, since the total
wave function should be antisymmetric and the spatial part is already symmetric.
The total wave function for a zero energy state can thus be written as ψ = ψ̃ ψ1L , where ψ̃ is
the bosonic M = 0 part of the total wave function. The NASS state under consideration
should be clustered at order k = 2, so three particles at the same position should be forbidden.
The three body projection operator forces any zero energy state to have this property. In
the expansion of the Laughlin factor ψ1L (z1 . . . zN ) are terms of the form
N −1
z10 z21 . . . zN
=
N
X
zii−1 .
(116)
i=1
If three coordinates are at the maximum angular momentum Nφ0 /2 in the M = 0 situation (so
they all have a zero exponent), they will be placed at the angular momenta Nφ /2, Nφ /2 − 1
and Nφ /2 − 2 in the M = 1 case after multiplication by the terms of the form (116). For
example: (z10 z20 z30 . . .)M =0 · (z10 z21 z32 . . .)M =1 = (z10 z21 z32 . . .). Therefore, using the projection
operator on the M = 1 case of total angular momentum 3Nφ /2 − 3 is the same as projecting
it to the situation for M = 0 with three particles at maximal angular momentum. This part
of the wave function should be symmetric, since the Laughlin factor is already antisymmetric
and the total wave function should be antisymmetric. The spin-part of the wave function
should also be symmetric, because the spatial part is already symmetric (all three particles
are in the same orbit). For three SU(2)-particles, this means that they should be projected
on the total spin 3/2 state. In conclusion, the three body projection operator in equation
(115) forces any zero energy eigenstate to have no clustering at order 3 or higher.
It can be shown that the terms in the expanded SU(2) NASS (k = 2, M = 1) wave functions
have exactly this structure. The ψM =0 part of the zero energy states is forced to have no
terms zi0 zj0 zk0 for all values of i, j and k, but all terms where one of the three particles has
exponent
These terms are exactly the expansion of the SU(2) wave function
Q 1 are allowed.
p
p 2
factor p,a,i<j (zia
− zja
) . By dividing all coordinates within the same component in two
groups (colours) and then writing all terms (zi − zj ) for all pairs with the same colour, a
maximum of only one coordinate in each colour can have exponent 0. Since there exist two
different colours (at k = 2), the maximum number of zero-exponent coordinates in every term
is indeed two. The term should be squared to obtain a symmetric wave function. However,
it is still possible to get two maximum angular momentum coordinates for
Qeach component,
p
q
so there have to be intercomponent terms that prohibit this. The terms p,q,a,i,j (zai
− zaj
)
exactly give this behaviour, this part should be antisymmetric, hence is unnecessary for
this term to be squared. These two factors are both only symmetric for coordinates within
the same colour, so the complete expression should be symmetrized over all coordinates in
52
Chapter 8. Numerical Analysis
different colours to get the full symmetric wave function.
The final result is indeed equal to the SU(2) NASS state for (k = 2, M = 1), so it is the
unique zero energy eigenstate of the Hamiltonian (115). For the generalization to the SU(4)
case, not much changes. The SU(4) wave function still obeys (k, M )-clustering, so the same
projection operators as before can be used as the appropriate Hamiltonian for the SU(4)
wave function at (k = 2, M = 1). The only difference is in the spin part of the operators,
because there are now three spinlike quantum numbers. For the same reasons as in the
SU(2) case, the two body operator should project to an antisymmetric spin state and the
three body operator to a symmetric spin state.
The dimensions of the symmetric and antisymmetric representations of multiple SU(4)particles are calculated with Young tableaux,
⊗
⊗
⊗
=
⊕
=
⊕2
⊕
(117)
The completely horizontal tableaux are the symmetric representations and the completely
vertical tableaux are the antisymmetric representations. These Young tableaux give the
dimension of the symmetric representation of three particles as 20 (with the actual representation named 20
20”) and the antisymmetric representation for two particles as 6. Therefore,
the projection Hamiltonian for the SU(4) symmetric (k = 2, M = 1) wave functions has to
be
X
X
3Nφ
H=U
Pijk
− 3, 20
20” + V 0
Pij (Nφ , 6 ) .
(118)
2
ij
ijk
To actually project it to a specific state, it is sufficient to project them to the highest weight
state of each representation, which are (Sz , Iz , Pz ) = (1,0,0) for 6 and (3/2, 3/2, 3/2) for
20
20”). In conclusion, the Hamiltonian that has the (k = 2, M = 1) SU(4) wave functions as
the unique zero energy states is
X
X
3Nφ
3 3 3
Hk=2,M =1 = U
Pijk
− 3, , ,
+V0
Pij (Nφ , 1, 0, 0)
(119)
2
2 2 2
ij
ijk
8.4. Analysis
The exact diagonalization of the Hamiltonian (119) results in its energies and eigenstates.
The eigenstate corresponding to the zero energy eigenvalue is the SU(4) NASS wave function. This eigenstate can then be compared to the eigenstates of the Coulomb interaction
Hamiltonian. First of all, the overlap of these two states can be calculated. This gives an
indication of how well the two states describe the same wave function. It should however
be noted that in the many particle limit, the overlap always tends to go to zero. Moreover,
even two states with different characteristics can, by chance, have a significant overlap.
Another method of comparing the states is the entanglement spectrum [62, 63, 64, 65]. The
general idea of this method is to divide the sphere in two parts, A and B, and use a Schmidt
decomposition of the wave functions in terms of basis states of these two subspaces,
X
|ψi =
e−ξi /2 ψiA ⊗ ψiB .
(120)
i
These ξi give the entanglement spectrum of the states, which contains information about
the topological properties of the system. Specifically for the model wave functions created
53
Chapter 8. Numerical Analysis
using the CFT approach, the lowest ξi -values correspond to the structure of the underlying
CFT. Therefore, a comparison of the entanglement spectra of the theoretical states with
the entanglement spectrum of the exact diagonalization of the Coulomb interaction gives an
indication which theoretical states best describe the real wave functions.
For both the fermion and boson situation, the following minimum particle and flux numbers
would be necessary to obtain the states with non-abelian excitations:
(k, M )
ν
2, 1
2, 0
8/13
8/5
Ne Nφ statistics
16
16
23
8
fermions
bosons
(121)
In order to create the complete nontrivial ground state wave function, a minimum of 16
particles is necessary. First of all, the number of particles of each component has to be
equal to get total a singlet state. Then secondly, for each component at least two particles
are necessary to be able to cluster them together in pairs. Finally, to get the part of the
wave function with the interactions between particles with the same colour and the same
component, at least two of these particles are necessary. So in total, the minimum number of
particle necessary is 4 × 2 × 2 = 16. Unfortunately, the Hilbert space of these combinations
of N and Nφ for the bosonic case has a dimension of 2.011.136.227, the analysis of this
space would take weeks of calculations1 and the fermionic Hilbert space is even bigger.
Consequentially, no actual numerical analysis results could be obtained.
1Personal communications with N. Regnault
54
CHAPTER 9
Quasihole State Counting
The insertion of an extra flux quantum results in the creation of a quasihole. If these
quasiholes are non-abelian, there will be multiple fusion paths when fusing them together,
leading to a ground state degeneracy. In this chapter, the possible ground states for N
particles with an excess flux of ∆Nφ will be calculated for the SU(4) NASS states at k = 2.
This will be done for the states on the sphere geometry, because these results can in principle
be compared with results from numerical analysis. First, the fusion rules of the CFT
operators will be given in the section below. The theoretical information in this chapter is
based on [55], in which the quasihole counting was done for the SU(2) NASS states.
9.1. Fields and fusion rules
The fields in the SU(5)2 CFT used for the construction of the SU(4)-symmetric spin singlet
states were already given in chapter 6. These fields correspond to the four different particles
and the four different quasiholes. In this section, these fields will be examined once more.
Their fusion rules and the resulting combined fields will be given explicitly. Once again, the
root system of SU(5) (figure 10) is as follows.
(1, 0, 0, 1)
(1, 0, 1, −1) (−1, 1, 0, 1)
(1, 1, −1, 0) (−1, 1, 1, −1) (0, −1, 1, 1)
(2, −1, 0, 0) (−1, 2, −1, 0) (0, −1, 2, −1) (0, 0, −1, 2)
Figure 12. The root system of SU(5).
The four particle fields were identified with the four leftmost vectors in this diagram and
the quasiholes with the weights. All fields in this CFT can be written as φΛ
λ , where
Λ = (Λ1 , Λ2 , Λ3 , Λ4 ) gives the SU(5)-representation in the Dynkin basis (equivalent to
the highest weight state) and λ = (λ1 , λ2 , λ3 , λ4 ) gives the quantum numbers within the
representation, as introduced by Gepner [66], see also [67]. Both vectors have four indices in
the case of SU(5)2 . There are some identifications of different fields, due to the structure
of the affine Lie algebras. First of all, all points in the root diagram are defined up to two
times a simple root vector (for general clustering at order k, the root lattice would have
been defined up to k times the simple root vectors). This leads to the identifications
Λ
Λ
ΦΛ
(λ1 , λ2 , λ3 , λ4 ) = Φ(λ1 +4, λ2 −2, λ3 , λ4 ) = Φ(λ1 −2, λ2 +4 ,λ3 −2 ,λ4 )
Λ
= ΦΛ
(λ1 , λ2 −2, λ3 +4, λ4 −2) = Φ(λ1 , λ2 , λ3 −2, λ4 +4) .
(122)
Furthermore, fields in one representation are linked to fields in other representations according
to the rule [68]
(Λ , Λ , Λ , Λ )
(2−Λ −Λ −Λ −Λ4 , Λ1 , Λ2 , Λ3 )
1
2
3
Φ(λ11, λ22, λ33, λ44) = Φ(λ1 +2,
λ 2 , λ 3 , λ4 )
= ...
.
(123)
55
Chapter 9. Quasihole State Counting
This rule continues for the other Λi in a cyclic manner. Two fields are fused together using
the fusion rules
X
00
Λ0
ΦΛ
ΦΛ
(124)
λ × Φλ0 =
λ+λ0 ,
Λ00 =Λ0 ×Λ
Λ00
Λ0
where the
= × Λ means the usual product of the representations and the λ + λ0 should
be taken modulo two times the simple root vectors, according to equation (122). For the four
fields ψi , their λ’s equal the left most roots of the root diagram and their Λ = (0, 0, 0, 0) = 1 .
Since 1 × 1 = 1 , the Λ of the fusion product of any combination of ψ-fields is still (0, 0, 0, 0).
Using these fusion rules, all unique combinations of the four ψi -fields can be found, they are
(0,0,0,0)
I = Φ(0,0,0,0)
(0,0,0,0)
ψ1 = Φ(2,−1,0,0)
(0,0,0,0)
(0,0,0,0)
ψ13 = Φ(−1,1,1,−1)
(0,0,0,0)
ψ123 = Φ(2,−2,2,−1)
(0,0,0,0)
ψ2 = Φ(1,1,−1,0)
ψ12 = Φ(−1,2,−1,0)
(0,0,0,0)
ψ1234 = Φ(−1,2,−2,2)
(0,0,0,0)
ψ3 = Φ(1,0,1,−1)
(0,0,0,0)
ψ23 = Φ(0,−1,2,−1)
(0,0,0,0)
ψ24 = Φ(0,−1,1,1)
(0,0,0,0)
ψ234 = Φ(1,1,−2,2)
(0,0,0,0)
ψ4 = Φ(1,0,0,1)
(0,0,0,0)
ψ34 = Φ(0,0,−1,2)
(125)
(0,0,0,0)
ψ14 = Φ(−1,1,0,1)
(0,0,0,0)
ψ124 = Φ(2,−2,1,1)
(0,0,0,0)
ψ134 = Φ(2,−1,−1,2)
The fusion rules of these fields are implemented in their names. Fusing two fields of the
same type results in the identity (ψi × ψi = I), and the fusion of two fields with different
indices results in the field with the combined indices of both original fields (ψi × ψj = ψij ),
where an index is removed if it appears twice (ψi × ψijk = ψjk ). Notice that the fields ψijk
and ψ1234 are not within the root system of SU(5), figure 12. However, they do correspond
to the roots in a higher representation (for example, the representation (2,0,0,0)), found by
using the identifications of formula (124).
The quasiholes correspond to exactly the same λ, but have Λ = (1, 0, 0, 0) = 5 . The fusion
of a σ and a ψ field is trivial and results in just another σ-field. However, the fusion of two
σ-fields is nontrivial and also results in fields with other Λ than the original fields. The
easiest example is the fusion of two quasiholes: 5×5 = 10+15. When fusing many of these
fundamental representations together in SU(5)2 , a total of fifteen different representations
can be reached. However, due to the identification rules of equation (123), many of these
representations can be identified with each other. Consequentially, there are three different
“families” of fields, namely
Φ1 with representations 1 , 15
15, 15
15, 50
50, 50
50,
Φ2 with representations 5, 5, 40
40, 40
40, 75
75,
(126)
Φ3 with representations 10
10, 10
10, 24
24, 45
45, 45
45.
The first family contains the ψ-fields and the identity field, the second family the σ-fields
and an identity-like field ρ and the third family contains a novel type of fields, which will be
called τ -fields. The fusion rules for the quasiholes and particles can then be put in terms of
these families,
Φ1 × Φ1 = Φ1
Φ1 × Φ2 = Φ2
Φ1 × Φ3 = Φ3
Φ2 × Φ2 = Φ1 + Φ3
Φ2 × Φ3 = Φ2 + Φ3
Φ3 × Φ3 = Φ1 + Φ2 + Φ3 .
(127)
56
Chapter 9. Quasihole State Counting
9.2. Degeneracy factors
When there are multiple quasiholes present in the system, there will be a degeneracy due to
the nontrivial fusion rules of these quasiholes. In this section, the degeneracy as a function of
the number of particles and quasiholes will be discussed. There are two different degeneracy
factors that result in the overall degeneracy. Firstly, there is a degeneracy due to the multiple
fusion channels of the non-abelian quasiholes. Secondly, there is an orbital degeneracy for
the particles on a sphere. One can not simply multiply these two factors, because the orbital
degeneracy depends on the fusion path taken.
The fusion path degeneracy can be depicted in a Bratteli diagram, see below in figure 13. In
this diagram, the bottom left point represents an identity field. Then in each step to the
right, the previous state is fused with one extra quasihole operator (in family Φ2 ) according
to the rules of equations (124) and (127). Due to the nontrivial fusion rules, there will be
multiple fusion paths that lead to most points in the diagram. The upper and lower rows
represent the identity and ψ-fields of family Φ1 and they can result in a nonzero correlator.
If the final fusion product of the quasiholes is a ψ-field, the correlator can only be nonzero
when there is a leftover ψ-field from the electron operators. The number of different fusion
paths to the points in family Φ1 is also indicated in figure 13. As a consequence of the
rank-level duality of the groups SU(5)2 and SU(2)5 , their Bratelli diagrams will be identical.
For SU(2)k , k quasiholes have to be fused together to end up at a whole particle ψ. Therefore,
the number of rows in the Bratteli diagram will be k + 1 = 6 for the case of SU(2)5 . This is
indeed the same diagram that can be obtained by the SU(5)2 fusion rules of the previous
section.
1
5
>•
>•
>•
>•
>•
>•
>•
>•
>•
>•
19
= •
!
= •
>•
>•
>•
•
•
•
•
1
1
2
5
= •
!
!
!
= •
•
14
221
= •
!
= •
!
= •
>•
66
: •
#
; •
!
= •
!
= •
!
!
= •
•
42
Φ2
$
: •
#
; •
#
$
: •
•
Φ1
Φ3
Φ3
Φ2
Φ1
131
Figure 13. The Bratteli diagram of SU(5)2 , or equivalently SU(2)5 . The numbers on
the upper and lower rows denote the number of fusion paths leading to that point. The Φi
on the right show which SU(5)-representations exist in that row. In each step, the previous
states are fused with one quasihole.
Similar to the SU(2) NASS states, whose counting formulas are given in [55], the total
number of states as a function of the total particle number N and the number of added
fluxes ∆Nφ can be calculated by splitting the intrinsic (fusion path) degeneracy numbers
and multiplying these by the relevant orbital degeneracy factor. The total degeneracy can
then be calculated with
4 N −F
X n1 n2 n3 n4 Y
m
m
+ nm
k
#SU(4) (N, ∆Nφ , k) =
. (128)
F1 F2 F3 F4 k
nm
Ni ,nj ,Fk
m=1
57
Chapter 9. Quasihole State Counting
The first term is the split degeneracy factor and the second term is the orbital degeneracy of
the states. This orbital degeneracy is given by a product of binomial coefficients, where the
numbers Fm can be interpreted as the number of unclustered particles of that component
[55, 69]. There are multiple restrictions on the values of the
P Ni , ni and Fi within formula
(128). First of all,Pthe total number of particles is fixed by i Ni = N and the total number
of quasiholes by i ni = n = 4k ∆Nφ . Secondly, the state must be a singlet, which means
that the total spin, isospin and polarization spin should be zero. This places the following
conditions on the particles:
Spin:
N1 + n1 + N2 + n2 = N3 + n3 + N4 + n4
Isospin:
Polarization:
N1 + n1 + N3 + n3 = N2 + n2 + N4 + n4
N1 + n1 + N4 + n4 = N2 + n2 + N3 + n3 .
(129)
And finally, each upper index in a binomial coefficient should be an integer, so this restricts
the Fi to have specific values that depend on the Ni and ni .
The actual formula for the split degeneracy factors from equation (128) has yet to be
determined. It is to be expected that it has a similar structure as the SU(2) NASS-states, in
which these factors for the case k = 2 are given by
n +F n +F 1
2
2
1
n1 n2
2
k
=
.
(130)
F1 F2 2
F1
F2
Furthermore, the factor should be consistent with the Bratteli diagram. Summing over all Fi
for a given number of quasiholes should return the same number of fusion paths as obtained
from the diagram,
X
n1 n2 n3 n4
= f (n1 + n2 + n3 + n4 ) ,
(131)
F1 F2 F3 F4 2
F1 ,F2 ,F3 ,F4
where f (n) gives the number of fusion paths from the Bratelli diagram 13. This number
can be found as a recursive formula or by matrix multiplication. Noting that the number of
paths to a point is equal to the sum of the points one row higher and lower in the previous
step, a matrix with only offdiagonal elements can be used to give the number of fusion paths
to every point in the next step. By starting from the very first step, with a 1 in the lowest
row and 0 everywhere else, and then applying the matrix n times, the number of fusion
paths to every point in the diagram is
 0
 1
 0
f (n) =  0

0
0
1
0
1
0
0
0
0
1
0
1
0
0
0
0
1
0
1
0
0
0
0
1
0
0
0
0
0
0
1
1
n  0
  0
  0
  0
 
0
1



 .

(132)
9.3. Counting results
Despite the absence of the correct counting formulas, all states arising from the ground state
at a certain N and ∆Nφ can be calculated by hand. This procedure was done for a select
number of cases by Kareljan Schoutens. In the section below, the general procedure and
some of the results will be given.
58
Chapter 9. Quasihole State Counting
9.3.1. Example 1. The simplest example is the situation with N = 4, k = 2, M = 0 at
ν = 8/5. For the ground state, the number of fluxes should be Nφ = N/ν − S = 8N/5 − 2 =
1/2, where S is the shift on the sphere. To get to an integer number of fluxes, 1/2 flux
quantum should be added, creating n = 4k ∆Nφ = 4 quasiholes. There are many different
combinations of (N1 , N2 , N3 , N4 ) and (n1 , n2 , n3 , n4 ) that satisfy the restrictions given in
the previous section. These possibilities are
(N1 , N2 , N3 , N4 ) = (2, 2, 0, 0)6 , (2, 1, 1, 0)12 , (1, 1, 1, 1)1
(133)
and the corresponding quasihole numbers are (2 − N1 , 2 − N2 , 2 − N3 , 2 − N4 ). The subscript
denotes the number of permutations that give an equivalent state. For example, the first
possibility is valid for N1 = N2 = 2, but also for N1 = N3 = 2, etc. Then the colour method,
described in section 6.3.2, can be used to create the wave functions of these states by first
dividing the coordinates into two colour groups, which can often be done in multiple different
ways, see equation (134) for how this is done in this example. Then the wave functions
can be generated from these groups by using the construction method mentioned before. A
specific choice of the groups should be disregarded if it leads to a wave function in which
one of the coordinates has an exponent higher than Nφ (= 1 in this example). In equation
(134), this happens for the first choice of the groups of (2,2,0,0) and also the first choice
for (2,1,1,0), because some terms get an exponent of 2. If it possible to multiply the wave
function with an extra zij without violating the condition that each exponent is at most Nφ ,
then these states correspond to higher angular momentum states. For the current example,
this is impossible for all terms, so every combination is an L = 0 singlet.
groups
(z1 , z2 )(x1 , x2 ) −→ (z1 − z2 )2 (x1 − x2 )2
(2, 2, 0, 0)6 −−−−→
(z1 , x1 )(z2 , x2 ) −→ (z1 − x1 )(z2 − x2 )
groups
(2, 1, 1, 0)12 −−−−→
(1, 1, 1, 1)1
groups
−−−−→
(z1 , z2 )(x1 , y1 ) −→ (z1 − z2 )2 (x1 − y1 )
(z1 , x1 )(z2 , y1 ) −→ (z1 − x1 )(z2 − y1 )
(134)

 (z1 , x1 )(y1 , w1 ) −→ (z1 − x1 )(y1 − w1 )
(z1 , y1 )(x1 , w1 ) −→ (z1 − y1 )(x1 − w1 )

(z1 , w1 )(x1 , y1 ) −→ (z1 − w1 )(x1 − y1 )
The z-component of the spin, isospin and polarization spin of these excitations can be
calculated by using the quantum numbers of the individual particles (see equation (53)), for
example Sz = n1 + n2 − n3 − n4 . The goal is then to find how these states fall in multiplets
of both the angular momentum and the SU(4)-spins. This procedure starts by noting that
the (2,2,0,0) state is the highest weight state of the representation 200 = (0, 2, 0) of SU(4).
The 200 -representation includes precisely 6 terms of the form (2,2,0,0), 12 terms of the form
(2,1,1,0) and 2 terms (1,1,1,1), which can be found by starting from the fact that the four
states (1, 0, 0, 0) are the 4-representation and then combining these states to obtain the
higher representations. There is an extra (1,1,1,1) left in formula (134), so the representation
1 = (0, 0, 0) also occurs once. There are no other states left and in conclusion, the counting
results for the situation with N = 4, ∆Nφ = 1/2, k = 2 are
1 200
L=0 1
1
(135)
59
Chapter 9. Quasihole State Counting
9.3.2. Example 2. The next example is once again for N = 4, but now with the
insertion of ∆Nφ = 3/2. Then n = 12 and the allowed combinations of the Ni become:
(4, 0, 0, 0)4
(3, 1, 0, 0)12
(2, 2, 0, 0)6
(2, 1, 1, 0)12
(1, 1, 1, 1)1 .
(136)
Some of these combinations allow higher angular momentum states. The full process of
obtaining the results will not be given here, but as an example, the (2,2,0,0) case is
groups
(z1 , z2 )(x1 , x2 ) −→ (z1 − z2 )2 (x1 − x2 )2
(2, 2, 0, 0)6 −−−−→
(137)
(z1 , x1 )(z2 , x2 ) −→ (z1 − x1 )(z2 − x2 ) .
The maximum exponent of a coordinate is now allowed to be 2 (= Nφ ). No extra coordinate
can be added in the first line, because all coordinates already have a maximum exponent of
2. In the second line, the first term is (z1 − x1 ), this term could be multiplied by either 1,
z1 + x1 or z1 x1 and is therefore an L = 1 state. The same is true for the second term and
the complete line thus has angular momentum multiplets of 1 ⊗ 1 = 0 ⊕ 1 ⊕ 2. When adding
coordinates to the wave functions, it should be taken into account that the symmetrization
over the coordinates returns a unique new state. It is, for example, sometimes also possible
to multiply the states with (z1 − x1 ) instead of (z1 + x1 ) to obtain a new L = 0 state, but
this is impossible in the (2, 2, 0, 0)-example above due to this symmetrization. The allowed
angular momentum states for all combinations of Ni are found to be
(4, 0, 0, 0)4 → 0
(3, 1, 0, 0)12 → 0 ⊕ 1
(2, 1, 1, 0)12 → 2×0 ⊕ 3×1 ⊕ 2
(2, 2, 0, 0)6 → 2×0 ⊕ 1 ⊕ 2
(1, 1, 1, 1)1 → 3×0 ⊕ 6×1 ⊕ 2×2
.
(138)
As can be seen, certain angular momentum multiplets can occur multiple times. Matching
all of these states with the SU(4) representations results in the following table,
15 200 45 35
L=0
L=1
L=2
0
1
0
1
0
1
0
1
0
1
0
0
(139)
9.3.3. Example 3. The final example given in this thesis is the situation with N = 8,
∆Nφ = 1 and k = 2, leading to n = 8. The derivation will not be given here, but the final
counting results for this situation are
1 15 200 45 45 84 175 105
L=0
L=1
L=2
L=3
L=4
1
0
1
0
1
0
1
1
1
0
1
0
1
0
0
0
1
0
0
0
0
1
0
0
0
1
0
1
0
0
0
1
0
0
0
1
0
0
0
0
(140)
61
Summary and Discussion
In this thesis, the conformal field theory approach to fractional quantum Hall effect wave
functions was applied to generate a new set of SU(4) singlet wave functions for particles
in four different components. These wave functions are generalizations of the SU(2) NASS
wave functions. In these states, the particles are either fermions or bosons depending on
the parameter M and are clustered at order k. For k ≥ 2, the quasiholes of these states are
non-abelian anyons. The wave functions can be expressed by assigning a different “colour”
to each particle in a cluster, then assigning terms to each combination of particles with
the same colour and finally symmetrizing the complete expression. The ground state wave
function is



k
Y
2 Y 
Y
SU(4)
p
p
p
q
L

 ψM
ψk,M = S
za,i
− za,j
za,i
− za,j
.
(141)


a=1
p,i<j
p<q,i,j
Here, S is the symmetrization operator, a denotes the colour ({1, . . . , k}), p and q the
L
component and ψM
is the Laughlin wave function of all coordinates. The filling fraction
of these states is given by ν = (4k)/(4kM + 5). The insertion of extra flux quanta creates
quasiholes in the system. The non-abelian statistics of the quasiholes result in degenerate
states after the fusion of multiple quasiholes. No explicit formula for this degeneracy has
been obtained in this thesis, but for a few examples the SU(4) representations and angular
momenta of the degenerate states on the sphere were calculated.
These SU(4)-symmetric wave functions could be interesting for the FQHE in systems that are
approximately SU(4)-symmetric, such as graphene. The Landau levels in graphene are, as a
first approximation, fourfold more degenerate due to the approximately SU(4)-symmetric
electrons. The energy scales of symmetry breaking terms are in experimentally realized
situations smaller than the Landau level energy gaps. Therefore, the FQHE in graphene
can, at low energies (without LL-mixing), be regarded as approximately SU(4)-symmetric.
Moreover, measurements of the FQHE in graphene have shown that many observed fractions
exhibit phase transitions between (iso)spin (un)polarized states. By tuning the parameters
of the system, the competition between the different states can be broken and a desired
phase can be stabilized. All these elements combined raises the possibility that the created
SU(4)-symmetric wave functions could actually exist in some region of the phase space of
graphene. The FQHE has been measured at ν = 4/9, which could be described by the
(k = 1, M = 1) SU(4) NASS wave function.
In principle, the created wave functions could be analyzed using numerical methods. The
overlap of these theoretical wave functions with the exact diagonalization of the Coulomb
interaction can be calculated and the entanglement spectra of the exact and theoretical
states can be compared. Unfortunately, the computational time necessary for even the
smallest nontrivial situation was too big to obtain any results. More sophisticated analytical
techniques, such as matrix-product-states [70], might result in smaller computation times
and thus an obtainable numerical test for these wave functions.
62
APPENDIX A
Wave Functions for Charged Particles in a Magnetic Field
In this appendix, the wave function of a charged particle in a magnetic field will be derived.
This derivation is based on multiple sources, including Ezawa’s book [11] and lecture notes
of Goerbig [9] and Jain [71].
The Hamiltonian of a single particle in a two dimensional space with coordinates (x, y) in a
perpendicular magnetic field ∇ × A = B ẑ is
1
1
H=
(p + eA)2 ≡
(P 2 + Py2 ) ,
(142)
2mb
mb x
where the Pi are the covariant momenta and mb the (band) mass. The coordinates of the
particle can be written as the sum of motionless guiding centre coordinates plus moving
relative coordinates. The guiding centre coordinates, that have to commute with the
Hamiltonian, are then defined by:
Py
Px
X ≡x+
Y ≡y−
.
(143)
eB
eB
With these two operators a new set of operators can be created,
lB
a = √ (Px + iPy )
2~
1
b= √
(X − i Y )
2lB
lB
a† = √ (Px − iPy )
2~
1
b† = √
(X + i Y ) ,
2lB
(144)
q
~
where lB = eB
is the magnetic length, which gives the radius of the motion of the electron
in the ground state around it’s guiding centre. These operators obey the usual creation and
annihilation operator anticommutation relations:
[a, a† ] = [b, b† ] = 1
[a, b] = [a† , b† ] = 0 .
(145)
These operators thus act as raising and lowering operators on the states and both have a
quantum number assigned to them: a† a |ψn,m i = n |ψn,m i and b† b |ψn,m i = m |ψn,m i. In
terms of these operators, the Hamiltonian can be simply written as
H = (a† a + 1/2) ~ωc ,
(146)
eB
where ωc = m ~l2 = m
is called the cyclotron frequency. This is the Hamiltonian of
b
b B
a harmonic oscillator with corresponding quantized energy levels, called Landau levels,
En = (n + 1/2)~ωc , where n ∈ N. The a and a† operators act as raising and lowering
operators of the Landau levels. Each level is highly degenerate, since the Hamiltonian (146)
does not depend on the b and b† operators.
Before continuing, the coordinates of the particle are rewritten in a complex notation as
z = x + iy = r exp(iθ) and z̄ = x − iy = r exp(−iθ). A gauge is also chosen to be Ax = By/2
and Ay = −Bx/2, which is called the symmetric gauge. All previously introduced operators
63
Appendix A. Wave Functions for Charged Particles in a Magnetic Field
are then also be written in these complex coordinates, specifically the a-operators and
b-operators have become
i
1
1
i
a† = √
z + 2 lB ∂¯z
z̄ − 2 lB ∂z
a= √
2 2 lB
2 2 lB
(147)
1
1
1
1
†
¯
b =√
b= √
z̄ + 2 lB ∂z
z − 2 lB ∂z .
2 2 lB
2 2 lB
∂
The z component of the angular momentum is given by −i~ ∂θ
= −~(b† b − a† a) ≡ ~(m − n) ≡
~l, where l = −n, −n + 1, . . ., which is equal to m in the lowest Landau level. The value of
l within a Landau level is raised (lowered) by applying b† (b). Thus the a-operators act as
ladder operators for the Landau level and the b-operators as ladder operators of the angular
momentum.
The wave function of a generic state is given by
s
E
1
|n, mi =
(a† )n (b† )m 0, 0 ,
n!(m + n)!
(148)
where the state |0, 0i is the vacuum: a |0, 0i = 0 and b |0, 0i = 0. These two equations are
solved by
1
2
2
|0, 0i = q
e−|z |/4lB .
(149)
2
2πlB
In the lowest Landau level (LLL), the single particle wave functions are thus simply given by
2
2
ψm = (b† )m |0, 0i = Cm z m e−|z |/4lB ,
(150)
q
−1
3m
m·
2
with the normalization constant Cm = m! · 2 2 · lB
2πlB
. An important observation
is that this wave function is a Gaussian times a power of z, where the exponent of z gives the
angular momentum of the state. For the two cases m = 2 and m = 6, this wave function is
plotted in figure 14. As can be seen in this figure, the probability function of wave function
150 is a ring which becomes wider with increasing m, this was expected, since the angular
momentum also becomes higher while the cyclotron frequency remains constant.
(a) ψn=0,m=2
(b) ψn=0,m=6
Figure 14. The single particle wave functions (150), the scale of both subfigures is the same.
64
Bibliography
[1] E. Ardonne and K. Schoutens, “New Class of Non-Abelian Spin-Singlet Quantum Hall States,”
Phys. Rev. Lett., vol. 82, no. 25, pp. 5096–5099, 1999.
[2] A. Stern, “Anyons and the quantum Hall effect-A pedagogical review,” Annals of Physics,
vol. 323, no. 1, pp. 204–249, 2008.
[3] A. Stern, “Non-Abelian states of matter.,” Nature, vol. 464, no. 7286, pp. 187–193, 2010.
[4] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, “Non-Abelian anyons and
topological quantum computation,” Reviews of Modern Physics, vol. 80, no. 3, pp. 1083–1159,
2008.
[5] J. K. Pachos, Introduction to Topological Quantum Computation. Cambridge University Press,
2013.
[6] V. J. Goldman, F. E. Camino, and W. Zhou, “Realization of a laughlin quasiparticle interferometer: Observation of anyonic statistics,” AIP Conference Proceedings, vol. 850, no. August,
pp. 1341–1342, 2006.
[7] R. L. Willett, C. Nayak, K. Shtengel, L. N. Pfeiffer, and K. W. West, “Magnetic-field-tuned
aharonov-bohm oscillations and evidence for non-abelian anyons at \nu=5/2,” Physical Review
Letters, vol. 111, no. 18, pp. 1–10, 2013.
[8] J. K. Jain, “Composite Fermion Theory of Exotic Fractional Quantum Hall Effect,” Annual
Review of Condensed Matter Physics, vol. 6, no. 1, pp. 39–62, 2015.
[9] M. O. Goerbig, “Quantum Hall Effects,” arXiv preprint, p. 102, 2009.
[10] H. L. Stormer, D. C. Tsui, and A. C. Gossard, “The fractional quantum Hall effect,” Reviews of
Modern Physics, vol. 71, no. 2, pp. S298–S305, 1999.
[11] Z. F. Ezawa, Quantum Hall Effects: Field Theoretical Approach and Related Topics. World
Scientific Publishing, 2000.
[12] E. Fradkin, Field Theories of Condensed Matter Physics. Cambridge University Press, 2013.
[13] K. V. Klitzing, G. Dorda, and M. Pepper, “New method for high-accuracy determination of
the fine-structure constant based on quantized hall resistance,” Physical Review Letters, vol. 45,
no. 6, pp. 494–497, 1980.
[14] J. Eisenstein and H. L. Störmer, “The fractional quantum Hall effect,” Science, vol. 248, no. 4962,
pp. 1510–1516, 1990.
[15] D. C. Tsui, H. L. Stormer, and A. C. Gossard, “Two-Dimensional Magnetotransport in the
Extreme Quantum Limit,” Physical Review Letters, vol. 48, no. 22, pp. 1559–1562, 1982.
[16] Y. B. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum
Hall effect and Berry’s phase in graphene,” Nature, vol. 438, no. 7065, pp. 201–204, 2005.
[17] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V.
Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,”
Nature, vol. 438, no. 7065, pp. 197–200, 2005.
[18] X. Du, I. Skachko, F. Duerr, A. Luican, and E. Y. Andrei, “Fractional quantum Hall effect and
insulating phase of Dirac electrons in graphene,” Nature, vol. 462, no. 7270, pp. 192–195, 2009.
[19] K. I. Bolotin, F. Ghahari, M. D. Shulman, H. L. Stormer, and P. Kim, “Observation of the
fractional quantum Hall effect in graphene,” Nature, vol. 462, no. 7270, pp. 196–199, 2009.
[20] N. Cooper, “Rapidly rotating atomic gases,” Advances in Physics, vol. 57, pp. 539–616, nov
2008.
[21] S. Viefers, “Quantum Hall physics in rotating BoseEinstein condensates,” Journal of Physics:
Condensed Matter, vol. 20, no. 12, p. 123202, 2008.
Bibliography
[22] R. B. Laughlin, “Quantized Hall conductivity in two dimensions,” Physical Review B, vol. 23,
no. 10, pp. 5632–5633, 1981.
[23] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. Den Nijs, “Quantized hall conductance
in a two-Dimensional periodic potential,” Physical Review Letters, vol. 49, no. 6, pp. 405–408,
1982.
[24] M. Z. Hasan and C. L. Kane, “Topological Insulators,” Physical review letters, vol. 98, no. 10,
p. 23, 2010.
[25] R. B. Laughlin, “Anomalous quantum Hall effect: An incompressible quantum fluid with
fractionally charged excitations,” Physical Review Letters, vol. 50, no. 18, pp. 1395–1398, 1983.
[26] R. De-Picciotto, M. Reznikov, M. Heiblum, V. Umansky, G. Bunin, and D. Mahalu, “Direct
Observation of a Fractional Charge,” Nature, vol. 389, no. September, p. 3, 1997.
[27] J. K. Jain, “Composite-Fermion Approach for the Fractional Quantum Hall Effec,” Physical
Review Letters, vol. 63, no. 2, pp. 199–202, 1989.
[28] G. Moore and N. Read, “Nonabelions in the fractional quantum hall effect,” Nuclear Physics B,
vol. 360, no. 2-3, pp. 362–396, 1991.
[29] R. L. Willett, “The quantum Hall effect at 5/2 filling factor,” Reports on Progress in Physics,
vol. 76, p. 076501, jul 2013.
[30] N. Read and E. H. Rezayi, “Beyond paired quantum Hall states: parafermions and incompressible
states in the first excited Landau level,” Physical Review B, vol. 59, p. 11, sep 1998.
[31] B. I. Halperin, “Multicomponent wave functions,” Helv. Phys. Acta, vol. 56, p. 75, 1983.
[32] M. O. Goerbig and N. Regnault, “Analysis of a SU(4) generalization of Halperin’s wave function
as an approach towards a SU(4) fractional quantum Hall effect in graphene sheets,” Physical
Review B, vol. 75, p. 241405, jan 2007.
[33] B. A. Bernevig, Topological Insulators and Topological Superconductors. Princeton University
Press, 2013.
[34] A. H. Castro Neto, N. M. R. Peres, K. S. Novoselov, A. K. Geim, F. Guinea, and A. Neto, “The
electronic properties of graphene,” Reviews of Modern Physics, vol. 81, no. 1, pp. 109–162, 2009.
[35] S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, “Electronic transport in two-dimensional
graphene,” Reviews of Modern Physics, vol. 83, no. 2, pp. 407–470, 2011.
[36] M. O. Goerbig, “The quantum Hall effect in graphene a theoretical perspective,” Comptes
Rendus Physique, vol. 12, no. 4, pp. 369–378, 2011.
[37] M. O. Goerbig and N. Regnault, “Theoretical aspects of the fractional quantum Hall effect in
graphene,” Physica Scripta, vol. T146, no. 4, p. 014017, 2012.
[38] Y. Zhang, Z. Jiang, J. P. Small, M. S. Purewal, Y. W. Tan, M. Fazlollahi, J. D. Chudow, J. A.
Jaszczak, H. L. Stormer, and P. Kim, “Landau-level splitting in graphene in high magnetic
fields,” Physical Review Letters, vol. 96, no. 13, pp. 1–4, 2006.
[39] K. Nomura and A. H. MacDonald, “Quantum Hall Ferromagnetism in Graphene,” Physical
Review Letters, vol. 96, no. 25, p. 256602, 2006.
[40] C. R. Dean, a. F. Young, P. Cadden-Zimansky, L. Wang, H. Ren, K. Watanabe, T. Taniguchi,
P. Kim, J. Hone, and K. L. Shepard, “Multicomponent fractional quantum Hall effect in
graphene,” Nature Physics, vol. 7, no. 9, p. 5, 2010.
[41] B. E. Feldman, B. Krauss, J. Smet, and a. Yacoby, “Unconventional Sequence of Fractional
Quantum Hall States in Suspended Graphene,” Science, vol. 337, no. 6099, pp. 1196–1199, 2012.
[42] B. E. Feldman, A. J. Levin, B. Krauss, D. Abanin, B. I. Halperin, J. Smet, and a. Yacoby,
“Fractional Quantum Hall Phase Transitions and Four-Flux States in Graphene,” Physical Review
Letters, vol. 111, no. 7, p. 076802, 2013.
[43] M. Kharitonov, “Phase diagram for the quantum Hall state in monolayer graphene,” Physical
Review B, vol. 85, no. 15, p. 155439, 2012.
[44] D. Abanin, B. E. Feldman, A. Yacoby, and B. I. Halperin, “Fractional and integer quantum Hall
effects in the zeroth Landau level in graphene,” Physical Review B - Condensed Matter and
Materials Physics, vol. 88, no. 11, pp. 1–17, 2013.
[45] I. Sodemann and A. H. MacDonald, “Broken SU(4) Symmetry and the Fractional Quantum
Hall Effect in Graphene,” Physical Review Letters, vol. 112, no. 12, p. 126804, 2014.
65
66
Bibliography
[46] F. Wu, I. Sodemann, Y. Araki, A. H. MacDonald, and T. Jolicoeur, “SO(5) symmetry in the
quantum Hall effect in graphene,” Physical Review B, vol. 90, no. 23, p. 235432, 2014.
[47] A. C. Balram, C. Tke, A. Wójs, and J. K. Jain, “Fractional Quantum Hall Effect in Graphene:
Quantitative Comparison between Theory and Experiment,” pp. 1–15, 2015.
[48] Z. Papić, R. Thomale, and D. A. Abanin, “Tunable electron interactions and fractional quantum
Hall states in graphene,” Physical Review Letters, vol. 107, no. 17, pp. 2–6, 2011.
[49] P. Ginsparg, “Applied Conformal Field Theory,” arXiv:hep-th/9108028, vol. 88, nov 1988.
[50] E. Verlinde, “Fusion rules and modular transformations in 2D conformal field theory,” Nuclear
Physics B, vol. 300, pp. 360–376, jan 1988.
[51] H. Georgi, “Lie Algebras In Particle Physics: from Isospin To Unified Theories,” 1999.
[52] E. Ardonne, A conformal field theory description of fractional quantum Hall states. PhD thesis,
University of Amsterdam, 2002.
[53] A. Cappelli, L. S. Georgiev, and I. T. Todorov, “Parafermion Hall states from coset projections
of abelian conformal theories,” Nucl. Phys. B, vol. 599, no. 3, pp. 499–530, 2001.
[54] V. Gurarie and E. H. Rezayi, “Parafermion Statistics and Quasihole Excitations for the Generalizations of the Paired Quantum Hall States,” vol. 61, no. 8, p. 13, 1998.
[55] E. Ardonne, N. Read, E. H. Rezayi, and K. Schoutens, “Non-abelian spin-singlet quantum Hall
states: wave functions and quasihole state counting,” Nucl. Phys. B, vol. 607, no. 3, pp. 549–576,
2001.
[56] N. Regnault, “DiagHam.” http://nick-ux.org/diagham/wiki/.
[57] F. D. M. Haldane, “Fractional quantization of the hall effect: A hierarchy of incompressible
quantum fluid states,” Physical Review Letters, vol. 51, no. 7, pp. 605–608, 1983.
[58] B. A. Bernevig and F. D. M. Haldane, “Model fractional quantum hall states and jack polynomials,” Physical Review Letters, vol. 100, no. 24, pp. 1–4, 2008.
[59] B. A. Bernevig and N. Regnault, “Anatomy of abelian and non-abelian fractional quantum hall
states,” Physical Review Letters, vol. 103, no. 20, pp. 1–5, 2009.
[60] V. M. Apalkov and T. Chakraborty, “Fractional quantum hall states of dirac electrons in
graphene,” Physical Review Letters, vol. 97, no. 12, pp. 1–4, 2006.
[61] S. C. Davenport and S. H. Simon, “Multiparticle pseudopotentials for multicomponent quantum
Hall systems,” Physical Review B, vol. 85, no. 7, p. 075430, 2012.
[62] G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, “Entanglement in quantum critical phenomena,”
Phys. Rev. Lett., no. 1, p. 5, 2002.
[63] A. Kitaev and J. Preskill, “Topological entanglement entropy,” Physical Review Letters, vol. 96,
no. 11, pp. 2–5, 2006.
[64] H. Li and F. D. M. Haldane, “Entanglement Spectrum as a Generalization of Entanglement
Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect
States,” Physical Review Letters, vol. 101, no. 1, p. 010504, 2008.
[65] O. S. Zozulya, M. Haque, and N. Regnault, “Entanglement signatures of quantum Hall phase
transitions,” Physical Review B - Condensed Matter and Materials Physics, vol. 79, no. 4, pp. 1–8,
2009.
[66] D. Gepner, “New conformal field theories associated with lie algebras and their partition
functions,” Nuclear Physics, Section B, vol. 290, no. C, pp. 10–24, 1987.
[67] E. Ardonne and K. Schoutens, “Wavefunctions for topological quantum registers,” Annals of
Physics, vol. 322, no. 1, pp. 201–235, 2007.
[68] D. Gepner, “Field identification in coset conformal field theories,” Physics Letters B, vol. 222,
no. 2, pp. 207–212, 1989.
[69] V. Gurarie and E. Rezayi, “Parafermion statistics and quasihole excitations for the generalizations
of the paired quantum Hall states,” Physical Review B, vol. 61, pp. 5473–5482, feb 2000.
[70] M. P. Zaletel and R. S. K. Mong, “Exact matrix product states for quantum Hall wave functions,”
Physical Review B, vol. 86, no. 24, p. 245305, 2012.
[71] J. K. Jain, “Composite Fermions And The Fractional Quantum Hall Effect: A Tutorial,” 2011.
67
Populair-wetenschappelijke samenvatting
Conforme veldentheorie aanpak voor
het fractionele kwantum Halleffect in grafeen
Wanneer een elektrisch stroom geleidend materiaal in een magneetveld wordt geplaatst,
ontstaat er een spanning loodrecht op de stroomrichting. Deze spanning wordt de Hallspanning genoemd. Onder normale omstandigheden neemt de Hallspanning lineair toe met
de sterkte van het magneetveld: wordt het magneetveld twee keer zo sterk, dan wordt
de Hallspanning twee keer zo groot. Dit klopt echter niet meer als het materiaal bijna
tot het absolute nulpunt wordt gekoeld en er zeer sterke magneetvelden worden gebruikt.
In experimenten is gemeten dat de Hallspanning VH als functie van de sterkte van het
magneetveld B niet meer continu is, maar gekwantiseerd wordt voor speciale waarden van B.
Dit komt doordat kwantummechanische (de natuurkunde van het allerkleinste) effecten een
belangrijke rol gaan spelen. Dit nieuwe effect wordt daarom het kwantum Halleffect genoemd.
De verschillende toestanden die kunnen ontstaan worden aangegeven door een getal ν (de
vulfractie), dat een geheel getal of een breuk kan zijn. Als ν heeltallig is, kan het kwantum
Halleffect goed verklaard worden door alleen te kijken hoe één deeltje in een magneetveld
zich gedraagt, maar dat is niet mogelijk als ν een breuk is. Voor dit fractionele kwantum
Halleffect moet gekeken worden naar de interacties die de vele elektronen in het materiaal met
elkaar hebben, maar dit is theoretisch gezien een moeilijke opgave. In de kwantummechanica
worden golffuncties gebruikt om te beschrijven welke eigenschappen deeltjes hebben en hoe
ze zich gedragen. Voor het fractionele kwantum Halleffect zijn meerdere golffuncties bedacht
voor verschillende waarden van ν. Deze golffuncties zijn niet exact, maar ze benaderen
de echte toestand. In dit verslag wordt een nieuwe golffunctie geconstrueerd voor een
bepaalde serie van fractionele kwantum Halltoestanden. Dit wordt gedaan door de deeltjes
te identificeren met velden in een bepaalde wiskundige theorie, de zogenaamde conforme
veldentheorie. In deze theorie kunnen golffuncties gemaakt worden door te berekenen hoe de
velden zich gedragen als ze samengebracht worden.
De golffuncties van de grondtoestanden (de laagste energietoestanden van het systeem)
die in dit verslag worden beschreven, hebben twee belangrijke eigenschappen waarmee ze
zich onderscheiden van andere fractionele kwantum Halleffecten. Ten eerste hebben ze
meer symmetrie-eigenschappen (wiskundig gezegd: ze hebben SU(4)-symmetrie) dan de
eerder genoemde golffuncties. Hierdoor kunnen de elektronen effectief gezien worden als
vier verschillende soorten deeltjes die allerlei verschillende interacties met elkaar hebben.
Dit resulteert in toestanden die er anders uitzien dan de situatie met slechts één of twee
verschillende deeltjes. Ten tweede hebben deze golffuncties de speciale eigenschap dat er
niet-abelse quasideeltjes ontstaan als de sterkte van het magneetveld wordt opgevoerd. Nietabelse quasideeltjes gedragen zich anders dan normale deeltjes wanneer deze deeltjes om
elkaar heen worden gedraaid. Deze eigenschap is interessant, want in theorie zouden deze
quasideeltjes gebruikt kunnen worden in een kwantumcomputer om berekeningen te maken
68
Populair-wetenschappelijke samenvatting
die niet verstoord kunnen worden door verstoringen in het systeem.
Grafeen is een voorbeeld van een materiaal waarin deze golffuncties de werkelijkheid zouden
kunnen benaderen. Grafeen is een materiaal bestaande uit een “kippengaas”-rooster van
koolstofatomen van slechts één laag atomen dik. Er is grote belangstelling voor het kwantum
Halleffect in grafeen, dat sinds enkele jaren geleden gemeten kan worden in experimenten.
Grafeen is interessant om experimentele redenen, want het integere kwantum Halleffect komt
in grafeen voor bij kamertemperatuur in plaats van bij temperaturen net boven het absolute
nulpunt. Daarnaast zijn de elektronen in het kwantum Halleffect in grafeen bij benadering
SU(4)-symmetrisch, wat ervoor zorgt dat het kwantum Halleffect andere eigenschappen heeft
dan in de meeste materialen. De golffuncties die in dit verslag gemaakt zijn, zouden dus een
benadering kunnen zijn van toestanden die in grafeen voorkomen.