Half-integer Hall response in
topological insulators
S.J. Bosman
supervisor: Prof. dr. K. Schoutens
Instituut voor de Theoretische Fysica Amsterdam
Universiteit van Amsterdam
A thesis submitted for the degree of
Master of Science Physics, theoretical track
August 25, 2011
2
Abstract
This thesis studies the Hall response of a topological insulator. First
we study the theoretical framework of topological materials, by study-
ing the adiabatic geometry of quantum systems. Secondly we derive
the relativistic Dirac theory of the surface of the three-dimensional
topological insulator Bi2Se3. Subsequently the Lie algebra of this the-
ory is studied and we calculate the Hall response. This response con-
sists of two contributions: a parity-normal- and a parity-anomalous
contribution. By considering a magnetic domain wall on a plane,
cylinder and sphere we construct a topological Thouless pump that
induces the parity-normal contribution to the Hall response. We also
construct a topological pump for the parity-anomalous contribution
of the Hall response by considering a mass domain wall on a plane.
Sal Jua Bosman, 22nd of August 2011
Contents
Contents ii
List of Figures v
Nomenclature viii
1 Introduction 1
1.1 Motivation and context . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Research questions . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Topological Phases of Matter 6
2.1 Bulk-Boundary correspondence . . . . . . . . . . . . . . . . . . . 7
2.2 Adiabatic Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 Hilbert Space geometry . . . . . . . . . . . . . . . . . . . . 13
2.2.2 Spectral flow . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.3 Z-Thouless pump of the iqHe . . . . . . . . . . . . . . . . 20
2.2.4 Z2-Thouless pump of the Shindou model . . . . . . . . . . 23
2.2.4.1 Topological materials with ν0 ∈ Z2/Z . . . . . . . 24
2.2.4.2 Topological materials with ν0 ∈ Z2 . . . . . . . . 25
2.2.4.3 An example of a Z2-system: The Shindou model 26
2.3 Bundle Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.1 Z: The TKNN-integer as topological invariant . . . . . . . 28
2.3.2 Chern classes and numbers . . . . . . . . . . . . . . . . . . 35
2.4 Periodic system of Topological matter . . . . . . . . . . . . . . . . 38
ii
CONTENTS
2.4.1 Hamiltonian classes . . . . . . . . . . . . . . . . . . . . . . 38
2.4.2 Homotopic classification . . . . . . . . . . . . . . . . . . . 41
2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3 The strong topological insulator 42
3.1 Derivation of the surface theory . . . . . . . . . . . . . . . . . . . 42
3.1.1 (3 + 1)d Topological insulator prototype: Bi2Se3 . . . . . 43
3.1.2 Band inversion of opposite parity . . . . . . . . . . . . . . 43
3.1.3 Topological non-trivial band inversion . . . . . . . . . . . . 46
3.1.4 Model Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 48
3.1.5 Effective surface Hamiltonian . . . . . . . . . . . . . . . . 50
3.2 Dirac Operator in (2+1)D . . . . . . . . . . . . . . . . . . . . . . 53
3.2.1 Dirac equation on flat space . . . . . . . . . . . . . . . . . 53
3.2.1.1 Perturbations . . . . . . . . . . . . . . . . . . . . 54
3.2.2 Lorentz- and Poincare group in 2+1D . . . . . . . . . . . . 56
3.2.2.1 Lorentz transformations . . . . . . . . . . . . . . 56
3.2.2.2 The Lie algebra: so(1, 2) . . . . . . . . . . . . . . 57
3.2.2.3 The Casimir operator & representations . . . . . 58
3.2.2.4 The Poincare algebra and its representations . . . 60
3.2.3 Dirac equation on curved space . . . . . . . . . . . . . . . 62
3.2.3.1 Spin connection formalism . . . . . . . . . . . . . 63
3.2.3.2 Dirac equation in isothermal coordinates . . . . . 66
3.3 Hall Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.3.1 Hall response from the Berry Curvature . . . . . . . . . . 68
3.3.2 Hall response from the Schwinger proper-time representation 69
3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4 The toy model collection of Landau problems 77
4.1 Overview of the models and field configurations . . . . . . . . . . 77
4.1.1 Relation between the Dirac and Schrodinger operator . . . 79
4.2 Landau problem on the plane . . . . . . . . . . . . . . . . . . . . 82
4.2.1 Non-relativistic . . . . . . . . . . . . . . . . . . . . . . . . 82
4.2.1.1 Zero field . . . . . . . . . . . . . . . . . . . . . . 83
iii
CONTENTS
4.2.1.2 Constant field . . . . . . . . . . . . . . . . . . . . 84
4.2.1.3 Constant field of compact support . . . . . . . . 87
4.2.1.4 Field with a single magnetic domain wall . . . . . 89
4.2.2 Relativistic Landau problem . . . . . . . . . . . . . . . . . 100
4.2.2.1 Zero field . . . . . . . . . . . . . . . . . . . . . . 100
4.2.2.2 Constant field . . . . . . . . . . . . . . . . . . . . 101
4.2.2.3 Constant field of compact support . . . . . . . . 104
4.2.2.4 Field with magnetic domain wall . . . . . . . . . 105
4.3 Landau problem on the sphere . . . . . . . . . . . . . . . . . . . . 108
4.3.1 Non-relativistic . . . . . . . . . . . . . . . . . . . . . . . . 111
4.3.1.1 Zero field . . . . . . . . . . . . . . . . . . . . . . 111
4.3.1.2 Constant field . . . . . . . . . . . . . . . . . . . . 112
4.3.1.3 Field with magnetic domain wall . . . . . . . . . 116
4.3.2 Relativistic . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.3.2.1 Zero field . . . . . . . . . . . . . . . . . . . . . . 121
4.3.2.2 Constant field . . . . . . . . . . . . . . . . . . . . 123
4.3.2.3 Field with magnetic domain wall . . . . . . . . . 125
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5 Mass domain walls in topological insulators 131
5.1 Mass domain wall on the plane . . . . . . . . . . . . . . . . . . . 131
5.1.1 Klein-Gordon solution . . . . . . . . . . . . . . . . . . . . 132
5.1.2 Dirac solution . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.1.3 Adiabatic Cycles . . . . . . . . . . . . . . . . . . . . . . . 140
5.1.4 Further ideas . . . . . . . . . . . . . . . . . . . . . . . . . 141
6 Conclusions & Outlook 143
6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
Bibliography 148
iv
List of Figures
2.1 Examples of the boundary operator on three manifolds with differ-
ent dimension. The third example, M3, is a slap of material with
a cavity inside. Therefore the boundary consists of two closed
surfaces that are disjoined and nested. . . . . . . . . . . . . . . . 8
2.2 Schematic diagram of a Berry connection on the U(1)-line bundle
over the parameter space X. The connection defines a horizontal
lift of the base manifold into the fiber bundle. By tracing a closed
path in X one obtains a Berry phase, eiϕ, the holonomy of the
connection on manifold X. . . . . . . . . . . . . . . . . . . . . . . 14
2.3 These diagrams illustrate the spectral flow of two systems as a
function of τ . The left picture is a trivial insulator, the right
picture is the spectral flow of a fictional non-trivial insulator with
a Z-valued topological order parameter. . . . . . . . . . . . . . . . 17
2.4 With use of a gauge transformation a system can be cut, such that
its Hamiltonian becomes independent of the threading flux, and
the flux is incorporated in the boundary conditions on the cut. . . 19
2.5 Three equivalent diagrams of measuring the Hall effect. . . . . . 20
2.6 Schematic diagram of the iqHe system during the removal of one
flux quantum. The electrons are adiabatically lowered one orbit,
and effectively the edges get polarized. . . . . . . . . . . . . . . . 21
2.7 Schematic diagram of the IQHE system during the removal of one
flux quantum. The electrons are adiabatically lowered one orbit,
and effectively the edges get polarized. In a corbino disc the orbits
with label m are of fixed radius r. . . . . . . . . . . . . . . . . . . 23
v
LIST OF FIGURES
2.8 Diagram of the states of the Shindou model during a single adia-
batic cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.9 Schematic diagram of the Shindhou model Shindou [2005] Fu and
Kane [2006], we see that the Kramer’s doublets switch partner
during the adiabatic cycle, thereby causing spin accumulation at
the end of the sample. . . . . . . . . . . . . . . . . . . . . . . . . 28
2.10 The original domain M is cutted along c1 and c2, which results in
the new domain M . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.11 The ten symmetry classes according their basic symmetries. . . . 40
3.1 (a) Crystal structure of Bi2Se3. The quintuple layer is indicated
with the red box. (b) Top view along the z-direction. (c) Side-view
of the quintuple layer. The figure is from Liu et al. [2010]. . . . . 44
3.2 Schematic figure of the origin of the band structure inversion.
There are three steps taken into consideration: I) hybridization
of Bi and Se orbitals, II) formation of bonding/anti-bonding, III)
crystal field splitting and IV) Spin-orbit coupling. The figure is
from Liu et al. [2010]. . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3 The STM tunneling spectra for the surface of Bi2Se3 in a magnetic
field up to B = 11T . The resonance peaks are peaks in the density
of states and therefore the signal of a Landau level. The figure is
from Liu et al. [2010]. . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4 Possible contours in the complex plane of k0 for the eigenvalues
±E0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.5 Disorder-averaged Hall conductivity σxy as a function of the filling
fraction ν for the dimensionless disorder strength: a) 0.4, b) 0.7 c)
1.1 The figure is from Nomura et al. [2008]. . . . . . . . . . . . . 75
vi
LIST OF FIGURES
4.1 An overview of the two field configurations of the magnetic field.
The left column is the field corresponding to a sphere immersed in a
constant magnetic field, leading to a kinked field configuration with
respect to the area two-form of the immersed surface. The right
column corresponds to a constant field configuration with respect
to the area two-form itself, leading to a monopole configuration on
the sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.2 An overview of all the toy models with constant magnetic fields,
including spectra, wavefunctions of the lowest Landau level and
ground state degeneracy. . . . . . . . . . . . . . . . . . . . . . . . 79
4.3 Diagram of the domain wall field configuration and relevant func-
tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.4 Illustration of the three types of classical orbits. . . . . . . . . . . 91
4.5 The effective potential for a particle in Landau gauge, V (x, k) =(k +B0w ln(cosh(x/w))
)2
At k = 0 there is a minimum, that for
k ≤ 0 branches in two parts. . . . . . . . . . . . . . . . . . . . . . 93
4.6 Wavefunctions and spectrum of the infinite well with a delta func-
tion gδ(x) in the middle. . . . . . . . . . . . . . . . . . . . . . . . 94
4.7 Diagram of the bandsplitting δε as a function of momentum k in
three orders of magnitude of α = B0/w, α ∈ 0.1, 1.0, 10. . . . . 96
4.8 Diagram of the spectrum of the two lowest Landau levels for B = 1
and w = 1 and as a function of k. . . . . . . . . . . . . . . . . . . 98
4.9 a) For every Landau level the system has two edge modes. b) Under
threading a flux through the cylinder two electrons are transported
to the domain wall. The ends of the cylinder have a charge of +Q,
which are a superpositions of the odd and even states. c) Under
the mapping w = ln(z) the cylinder becomes a plane, if one shrinks
the domain wall to z = 0 we recover the standard picture of the
iqHe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.10 Both diagrams of the potential of the relativistic case (right) and
its non-relativistic counter-part (left). The parameters are: k = 6.
and α = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
vii
LIST OF FIGURES
4.11 Spectrum of the n ∈ −1, 0, 1 Landau level of the massless Dirac
system on a plane. . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.12 a) The system has an odd number of edge modes. b) Threading a
flux attracts a single electron to the domain wall. c) For the plane
this implies the half-integer qHe. . . . . . . . . . . . . . . . . . . 109
4.13 Spectrum of the Schrodinger Hamiltonian on the sphere with a
magnetic field with a domain wall. The parameters are R = 1,
w = 1 and nφ = 40. . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.14 The spectrum including the unphysical domain where m < 20. . . 119
4.15 Illustration of the process of an adiabatic flux insertion. . . . . . . 120
4.16 Spectrum of the n ∈ −1, 0, 1 Landau level of the massless Dirac
system on a sphere. We used nφ = 40, R = 1 and w = 1. . . . . . 127
4.17 Dirac spectrum on the sphere including the unphysical domain. . 128
4.18 Eigenspectrum of the same system, with unmentioned parameters.
This figure is from Lee [2009] . . . . . . . . . . . . . . . . . . . . 128
4.19 Illustration of a flux insertion of the relativistic system on the sphere.129
4.20 Illustration comparing the relativistic Hall effect versus its non-
relativistic counterpart. . . . . . . . . . . . . . . . . . . . . . . . . 129
5.1 Density plots and spectra for the solutions of the Klein-Gordon
equation for mass amplitudes a ∈ ±√
2,±√
6,±2√
3. . . . . . . 134
5.2 The spectra of the Dirac equation for mass amplitudes a ∈ ±1,±2,±3.136
5.3 The spectra of the Jackiw-Rebbi states for a < 0 and a > 0. . . . 138
5.4 Diagram of the mass-domain wall and the single-way propagating
current. It turns out that the sign of the mass determines the
chirality of the orbits of the particles. . . . . . . . . . . . . . . . . 139
5.5 Spectral flow of a cylinder with a mass domain wall on it. Depend-
ing on the sign of a we get an opposite Hall response. . . . . . . . 140
5.6 Spectral flow of the a relativistic system on the plane derived from
the domain wall. . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
viii
Nomenclature
Units & conventions
c = 1 speed of light
~ = h2π
= 1 reduced Planck’s constant .
e = 1 electroncharge
Φ0 = hce
= 2π~ce
= 2π magnetic flux quantum
m = 2 (electron) mass, unless otherwise stated
Spaces and groups
G group G
H subgroup H
T target space of a non-linear sigma model (NLσM)
M manifold parametrizing a physical system
ds spatial dimension of a physical system, thus dim(M) = ds + 1
∂M boundary of a physical system parametrized with manifold M
L2(M) Hilbert space, a complex inner product space, with inner product:
〈φ|ψ〉 =∫Mφ†ψ
X parameter space of a Hamiltonian
L2(M)×X expanded Hilbert space, the Hilbert space combined with the parameter space
of the system, despite notation this does not have to be a simple product space.
TpM tangent space of Mat point p
g Lie algebra of Lie group G : g ' TeG
P (X, Y ) fibre bundle with X as base space and Y as fibre and with projection:
π : P 7→ X
πn(G) n− th homotopy group of group G
ix
LIST OF FIGURES
A abelian (Berry) connection one form
Aab non-abelian (Berry) connection one form
F (Berry) curvature two-form
chn(F ) n− th Chern character of (Berry) curvature F
Chn(F ) n− th Chern number of (Berry) curvature F
Sets & matrices
Z set of all integers: . . . ,−2,−1, 0, 1, 2, . . . N set of natural numbers: 1, 2, 3, . . . R set of real numbers
Z/2Z cyclic group of two elements: Z/2Z = −1, 1Z2 the set of integers modulo 2 : Z2 = 0, 1
In identity matrix of dimension n
σi & τi Pauli matrices: σx, σy, σz
γi gamma-vector matrices:
γ1 = σx ⊗ τx, γ2 = σy ⊗ τx, γ3 = σz ⊗ τx, γ4 = I2 ⊗ τy, γ5 = I2 ⊗ τzγij gamma-bi-vector matrices :
i, j ∈ x, y, z : γij = εijkσk ⊗ I2, γi4 = σi ⊗ τz, γi5 = −σi ⊗ τy, γ45 = I2 ⊗ τx
Physical objects & operators
πi dynamical momentum operator: πi = −i∂i − AiH(g) Hamiltonian with parameters g
H N ×Nmatrix of the first quantized Hamiltonian
Z partition function
S[φ] action of the field φ
|φ〉 bosonic physical state, or physical state described by a Schrodinger Hamiltonian
φ element of the Hilbert space belonging to the state |φ〉|ψ〉 fermionic physical state, or physical state described by a Dirac Hamiltonian
ψ element of the Hilbert space belonging to the state |ψ〉χ upper component of a spinor
x
LIST OF FIGURES
ξ lower component of a spinor
|Ω〉 ground state of a many-body system
|ua(k)〉 Bloch wavefunction of band a and momentum k
|u•a (k)〉 Bloch wavefunction of a filled band a
|vi (k)〉 Bloch wavefunction of a empty band i
σH Hall conductance
〈σH〉 flux-averaged Hall conductance
C particle-hole-conjugation operator
T & Θ time-reversal operator
S sub-lattice-conjugation operator or chiral operator
Λi time-reversal invariant momentum i
PΘ(Λi) time-reversal polarization of Λi
Special functions
θ(x) Heavyside step function
bxc floor function
sign(x) sign function
Γ(x) gamma function
Hn(x) Hermite polynomial of order n
Y ml (x) spherical harmonics
Pml (x) associated Legendre polynomial
P(α,β)n (x) Jacobi polynomial of order n
YmnJ (x) monopole harmonics of monopole charge n
Complex coordinates
z z = x+ iy
z z = z − iy∂ ∂ = 1
2(∂x − ∂y)
∂ ∂ = 12(∂x + ∂y)
A A = 12(Ax + iAy)
A A = 12(Ax − iAy)
xi
Chapter 1
Introduction
1.1 Motivation and context
The question of quantization, whether and why phenomena are quantized, has
been one of the main themes in physics. One origin of this debate between con-
tinuum theories and quantized discrete theories is from Greece where the atomists
challenged the worldview of Heraclitos. In the 17th century Newton and Huygens
debated whether the nature of light was either particle- or wave-like. In the first
half of the 20th century this debate was mathematically settled in the quantum
mechanical wave-particle duality, however its conceptual foundation and physical
origin are still puzzling.
In the 1930s Dirac introduced topological considerations in the physics dis-
course Dirac [1931]. The mathematical language of topological invariants is very
well suited to study questions of quantization, since it relates continuous quanti-
ties with discrete ones. The first physical realizations of topology in physics were
the Aharanov-Bohm effect and the integer quantum Hall effect (iqHe).
Untill the discovery of the iqHe, phases of matter were classified according
to their broken symmetries in the Landau-Ginzburg paradigm. Wherein for ex-
ample a solid is distinguished from a gas by broken translational invariance, or
superconductivity by the broken U(1)-gauge symmetry. The iqHe system was
1
the first example that could not be classified according to this paradigm. It took
more then twentyfive years to find, and most notably recognize, other topological
phases of matter, but in the very recent years a complete topological periodic
table has emerged. Kitaev [2009]; Ryu et al. [2010].
This thesis studies one of these recently discovered materials, dubbed the
strong topological insulator (TI). This three-dimensional material is in the bulk
insulating, whereas it has a metallic surface. In 2007 this exotic phase was pre-
dicted to exist in the semiconducting alloy Bi1−xSbx Fu and Kane [2007]. Soon
this system was realized and the prediction was confirmed Hsieh et al. [2008b].
One of the most striking properties of this material is the half-integer qHe.
The Hall conductance is defined as:
σH = νe
Φ0
with: Φ0 =h
e. (1.1)
This formula says that the system adiabatically transports ν electrons per
inserted flux quantum Φ0, where ν is the filling fraction and often denotes the
number of Landau levels underneath the Fermi level. For the topological insulator
it was found that:
σH = (n+ 1/2)e
Φ0
with: Φ0 =h
e. (1.2)
This implies that the filling fraction is ν = n + 1/2, with n the number of
Landau levels under the Fermi energy. In this thesis we study this Hall response.
This form of the Hall response does not tell the whole story, and the subtleties
in this response form the foundation of this thesis.
At first sight this response reflects the nature of the Landau level structure of
a relativistic electron gas, as discussed in chapter 4. However in chapter 3 we shall
see that there are two possible contributions to this Hall response, which has some
interesting aspects. Secondly this Hall response raises questions in consideration
to the process of adiabatically pumping electrons. Naively one would expect that
this Hall response implies that the electrons somehow become fractionalized in
2
order to accommodate for the half-integral Hall response. A widespread believe
in the field is that it is impossible for a non-interacting system to have excita-
tions with a fractional charge1. This poses a paradox, because the half-integer
Hall response generates fractional excitations.
The solution which was conjectured by Qi et al. [2008], relied on the fact
that the surface of a TI is necessarily compact. Therefore a flux quantum has
to penetrate the surface twice, because ∇ · B = 0. This implies that any extra
flux quantum that is inserted penetrates the surface twice, thereby doubling the
Hall response back to integer values. This conjecture was numerically studied
and confirmed on the sphere by Lee [2009], whose interesting article stimulated
the main theme of this thesis.
The main results of this study are the following:
• The fact that the Hall response consists out of two possible contributions
was known in the high-energy physics literature Schakel [1991], but is not
mentioned in the topological insulator literature. On the one hand the
discussion of the Hall response is usually based on graphene, which is sim-
ply divided by four to mod-out the spin-valley degeneracy. Whereas on
the other hand the literature derives the Hall response from the parity
anomaly. In this study we re-examine the results from the high-energy lit-
erature, which combines both approaches, and their consequences for the
Hall response of the topological insulator.
• The spectrum for a Schrodinger and Dirac particle in a magnetic field with
a domain wall is analytically analyzed and numerically solved on the plane,
the cylinder and sphere. We show that the states are odd and even super-
positions of 50% probability of being on one side of the domain wall. This
suggests that the resulting Hall response is not robust against disorder.
• This superposition is for particles far away from the domain wall, mere an
academic fact than a physical fact. At a certain momentum along the do-
1To our knowledge this has never been formalized in a theorem.
3
main wall this superposition starts to manifest itself as a band splitting,
due to lowering of the energy by hybridization into odd and even superpo-
sitions in both Landau domains. This suggests that the band splitting is a
measure for the entanglement of states of two adjacent Landau-domains.
• In the spectra for small number of flux quanta (nφ = 40), we clearly see
the degeneracy of the Landau level, and an increased band-splitting (e.g.
entanglement) for higher Landau levels.
• We explicitly solve a planar Dirac particle with a mass containing a recti-
linear domain-wall: m(x) = a tanh(x). The resulting spinors are expressed
in terms of the (generalized) Legendre polynomials. We show that besides
the massless Jackiw-Rebbi state, Jackiw and Rebbi [1976], there are bac1number of bound states with quantized mass, localized at the domain wall.
• We construct a topological Thouless pump that explicitly shows the spectral
flow of the parity-anomalous Hall response.
• Finally we speculate that this mass-domain wall model could be interesting
for explaining the fermionic mass generations in the Standard model in 3 +
1 + 1 dimensions. One dimension is ’compactified’, since the states become
localized at the domain wall and the degree of freedom in the direction
perpendicular to the domain wall is transformed into an effective mass term.
This could also explain the fact that the parity of the universe is broken.
1.2 Research questions
The research questions whereupon this thesis is based are the following:
• What precise form does the Hall response of the strong topological insulator
has?
• In what kind of set up could one construct a topological Thouless pump
that registers the half-integer Hall response?
1Here bxc = floor(x) denotes the largest integer not greater than x
4
• What is the nature of the half-integral excitation caused by the Hall re-
sponse? Is it a part of an entangled pair or is it a truly half-integral excita-
tion like in the Su Shrieffer Heeger model with open boundary conditions?
1.3 Structure
Chapter 2 is a general introduction into topological phases of matter. It com-
mences with the relation between the bulk properties and the topological-protected
edge states. Subsequently the main focus is the adiabatic geometry of the Hilbert
space. By constructing adiabatic cycles we prove that the Hall response corre-
sponds to a topological invariant for the iqHe. We also study an example of the
Z2-topological invariant characterizing the TI, but in a more intuitive setting.
The chapter is concluded with a concise discussion of the novel classification of
topological materials.
Chapter 3 focusses on the strong topological insulator. From the bulk Hamil-
tonian we derive the (2 + 1)d Dirac theory describing the surface. Subsequently
we study the symmetries of this theory in some detail. We follow up by setting
the stage for studying the surface theory on curved surfaces, by essentially cou-
pling the theory to a static gravitational background. The chapter is concluded
by the derivation of the half-integer Hall response of the surface theory.
Chapter 4 is the core of this thesis. Here we explicitly study a collection
of toy models of Landau problems. These are (2 + 1)d systems with different
types of perpendicular magnetic fields, including domain walls. We compare the
relativistic versus the non-relativistic Hall response for the planar, spherical and
cylindrical geometries.
Chapter 5 is a short chapter that studies the effect of a mass-domain wall on
the surface of a topological insulator. We suggest a way of pumping adiabatically
particles from and to the domain wall.
This thesis is concluded with some conclusions and a discussion.
5
Chapter 2
Topological Phases of Matter
Gapped many-body systems can either be sensitive to boundary conditions or
not. In cases where the gapped system is insensitive to the boundary conditions
all electronic phenomena will be local, which is one way of defining a (trivial)
insulator Kohn [1964]. One enters the realm of topological materials when the
many-body physics is sensitive to the boundary conditions. These materials are
endowed with gapless, extended states at the boundary, which are protected
against disorder as long as the bulk remains gapped and the generic symmetries
of the Hamiltonian are preserved. In the simplest case the boundary conditions
define the interface of the system with the vacuum, which is a trivial insulator.
Such interfaces act as a phase transition, of topological order, and thereby display
critical phenomena, such as gapless modes.
Phase transitions of topological materials cannot be characterized by the sym-
metry breaking mechanism of an order parameter in the Landau-Ginzburg pic-
ture. Topological phases are characterized by topological invariants, defining a
topological order parameter, being an element of either Z or Z2. Interfaces be-
tween materials of different order, ν ∈ Z or ν0 ∈ Z2, define quantum phase
transitions.1 They are the topological equivalents of the phase transitions at spa-
tial interfaces such as the interfacing surface of a liquid-gas transition. Examples
are the edge states of the integer quantum Hall effect (iqHe) and the spin quan-
1These phenomena remain at T = 0 and are therefore quantum phase transitions.
6
2. Topological phases of matter
tum Hall effect (sqHe). The topological equivalents of phase transitions in the
time domain, driven by turning a knob, such as the magnetization as a function
of the background field, are for example the tunneling process in a single electron
transistor. Another important example is when one considers a quantum Hall
system with a certain number of filled Landau levels. If one tunes the magnetic
field, such that the number of filled levels change, the topological order parameter
changes, and thereby the Hall response.1
This chapter introduces the basic elements of topological phases of matter
with emphasis on the Z- and Z2-order parameters. We commence with introduc-
ing the special relationship between the bulk and the boundary of topological
materials, known as the bulk-boundary correspondence. Subsequently we discuss
the adiabatic picture, where the topological invariants are the outcome of con-
sidering the spectral flow of states under cycles in the parameter or phase space.
The chapter is concluded with the topological equivalent of Mendelev’s periodic
table of elements.
2.1 Bulk-Boundary correspondence
As Shou Chen Zhang puts it, with topological materials some form of a total
derivative is always present somewhere. In this picture the bulk-boundary cor-
respondence is nothing less then a fancy way of the fundamental theorem of
calculus: ∫ b
a
∇(f(x))dx = f(b)− f(a). (2.1)
The result of the integral depends solely on the value of f(x) on the boundary
of the domain, if f(x) remains bounded within the domain. Something similar is
present in topological materials, the degrees of freedom at the boundary of a ma-
terial are invariant under certain changes (e.g. perturbations) of the bulk, they
are topologically protected. The condition that f(x) remains bounded within the
domain, is translated to the condition that the bulk-gap remains finite.
1This shall become clear later for readers unfamiliar with the qHe.
7
2. Topological phases of matter
Suppose we want to study a material that can be parametrized with a mani-
fold1 M of spatial dimension ds, where s denotes spatial. We define the boundary
operator ∂ on M as follows:
∂ : M 7−→ ∂M. (2.2)
The boundary satisfies dim(∂M) = ds − 1 and is often not simply connected as
illustrated in figure (2.1). From this simple illustration we can already see some
Figure 2.1: Examples of the boundary operator on three manifolds with differentdimension. The third example, M3, is a slap of material with a cavity inside.Therefore the boundary consists of two closed surfaces that are disjoined andnested.
important properties of general boundaries of M . In ds = 2 the boundary is a
collection of disjoint 1-dimensional lines isomorphic with S1, where the number of
disjoint elements is 1 + g, where g enumerates the connectedness of M . In ds = 3
the boundary is a closed, oriented surface of genus g, denoted Σg. Possibly this
surface is accompanied with a collection of surfaces nested in Σg, resulting from
cavities in the bulk. Because we chose M to be connected the nesting is maxi-
1smooth, connected, orientable, compact.
8
2. Topological phases of matter
mally one level deep, regardless how complicated the material is we chose to study.
The requirement that the physics of topological materials is sensitive to the
boundary conditions, causes that the states on ∂M of a material M are, by
definition, extended and long-ranged. This implies that the modes on ∂M are
unaffected by the phenomenon of Anderson localization Ryu et al. [2010]. This
phenomenon is an unavoidable effect that drives the physical response local,
thus insensitive to the boundary conditions, in the presence of disorder. Dis-
order can be implemented as terms in the Hamiltonian that break (translational)
symmetries, which model defects and impurities in real world materials. Since
we consider non-interacting systems of electrons we can describe ∂M by second
quantized Hamiltonians:
H =∑a,b
ψ†a Hab ψb, ψa, ψb = δab, (2.3)
where we label the single-particle states with a, b and ψa (ψ†a) denotes the
fermionic annihilation (creation) operator. For such a regularized system, where
we have no continuous state labels, Hab is an N ×N matrix describing the first
quantized Hamiltonian of the system. As we shall see later these random first
quantized Hamiltonians can be classified to their adherence to certain reality
conditions, because they are in bijection with Cartan’s symmetric spaces, which
was found by Altland and Zirnbauer [1997]. But let us first concentrate on the
topological origin of the degrees of freedom at the boundary.
Say we study a disordered material that has a boundary ∂M of dimension
ds − 1. Since we are only interested in the long-range physics, at scales much
larger than the mean-free path, we can describe the random Hamiltonian Hab
with a non-linear sigma model (NLσM). The method to find the correct model
for the disordered system is the fermionic replica method and can be found in
Altland and Simons [2006]. The non-linear sigma model is defined by D real
scalar fields φa(xµ), where a ∈ 1, . . . , D and µ ∈ 0, . . . , ds − 2. The fields
9
2. Topological phases of matter
define a differentiable mapping from the base space ∂M to the target space T :
φ : ∂M 7→ T, (2.4)
The D-dimensional target space characterizes the class of the random Hamil-
tonian and is the compact quotient space of classical Lie groups G/H, with H
the maximal subgroup of G. The partition function of the model is defined as:
Z =
∫G/H
Dφ e−S[φ], (2.5)
where the path integral is integrated with the Haar measure over the complete
compact Lie group. Since the target space is a Lie group we have a notion of
distance in the group, parametrized by the metric gab(φ). We denote the metric
of the base space ηµν(x), despite notation this is not necessarily a flat Minkowski
space. For this partition function we have the action:
S[φ] =1
λ
∫∂M
gab(φ)ηµν(x)∂µφa ∂νφb, (2.6)
where λ is the coupling constant. Note that ds = 1, for theories including
time, and ds = 2 for time-independent systems one has a special case because λ
is dimensionless.
Two homotopy groups1 of the target space determine whether an extra term
can be added to the action, which is independent of the metric, hence topolog-
ical, and has no adjustable coupling parameter Heinzner et al. [2005]; Schnyder
et al. [2008]. In disordered systems the topological term can govern the long-
range physics because other terms suffer from Anderson localization, whereas
the topological term evades it. If we return to our system with base space M ,
which has spatial dimensions ds, and the effective action of the boundary ∂M is a
non-linear σ-model with target space G/H, then a Wess-Zumino-Witten (WZW)
1For readers unfamiliar with the mathematics of homotopy groups we refer to Hatcher[2002].
10
2. Topological phases of matter
term is allowed if:
πds(G/H) = Z. (2.7)
In this case the ds-th homotopy group of the target space generates the Z-
valued topological order parameter of the system. The classical example of such
a system is the iqHe, where the WZW-term leads to a lower dimensional descen-
dant topological θ-term.
The other possible term generates the Z2-order parameter and is allowed if:
πds−1(G/H) = Z2.1 (2.8)
To appreciate the physical meaning of this abstract scheme let us recall what
a homotopy group does. In topology one studies space without the notion of dis-
tances and angles (e.g. a metric). The first homotopy group, π1(G/H), effectively
partitions all possible paths in the space G/H into equivalence classes of loops
(e.g. S1) that are continuously deformable into each other2. For example on a
torus one has two possible ways to wind around, therefore the π1(Torus) = Z×Z.
The higher homotopy groups are generalizations of π1: for πn(G/H) one studies
whether mappings of Sn 7→ G/H, are continuously deformable into one another.
So if one adds a term of topological origin, the path integral becomes partitioned
into disjunct sectors. This partitioning is protected by topology. One can en-
vision that impurities are perturbations of the manifold, such that for example
a sphere becomes a very pimpled potato, but unless it becomes a very pimpled
torus the system remains in the same topological sector (e.g. order). Thus if
there are degrees of freedom related to this topological term, it is robust against
disorder.
In topological materials the physics in the bulk is gapped, but the boundary is
endowed with gapless modes, which are protected by topology. The nature of the
1A third term, known as the Pruisken term, named after our inspiring proffesor, is possibleto add, but the evasion of Anderson localization for a Pruisken term is only accomplished forcertain values of the θ parameter, and so is not a valid term in this generalized scheme.
2The group property is easily found by considering the concatenation of the loops.
11
2. Topological phases of matter
gap of the bulk can have various origins: a standard band insulator, by Landau
quantization, a superconducting gap, etc. Despite the bulk gap, the system is still
sensitive to the boundary conditions, and therefore evading Anderson localization
at the boundary. This means that it is impossible to remove the degrees of
freedom living on the boundary by small perturbations or disorder. This gives
non-trivial long-range physics on the surface of the system. These gapless modes
are critical, which imply scale invariance and are thereby described by a conformal
field theory. All these highly non-trivial properties at the boundary can only
arise in relation to the bulk physics, this relation is known as the bulk-boundary
correspondence.
2.2 Adiabatic Cycles
In the last section we saw that homotopies in physical systems are important.
The homotopical perspective on a quantum mechanical system naturally arises
in situations where one can consider the Hamiltonian as an operator dependent
on a parameter τ ∈ X, where X denotes the parameter space of the Hamiltonian.
This parameter space denotes the space of knobs available on our Hamiltonian,
which we can dial infinitely slowly. In what follows we use the adiabatic assump-
tion, which implies that as we skim through the parameter space, the system
remains in the same, although changing, eigenvalue. For this assumption it is of
course necessary that eigenvalues do not cross. This problem, which is a branch
of mathematics on its own, is for our purposes captured with the no-crossing
theorem von Neumann and Wigner [1929].
Say we have a system living on a space M , described by the state |ψ〉 ∈ L2(M).
The total space wherein the state lives is the standard Hilbert space times its
parameters space: L2(M) × X. We shall call this space the extended Hilbert
space, and see that it is not necessarily a simple product space1. Now one can
consider to drive adiabatically a physical state at τ in the parameter space, |ψτ 〉,along a closed path τ := γ(λ) ∈ X. We assume that the eigenvalues En(τ)
are isolated and non-degenerate: the adiabatic approximation. The adiabatic
1In the sense that for example S2 6= S1 × S1.
12
2. Topological phases of matter
transportation of states through the parameter space is the quantum version of
parallel transport and its holonomy for a closed path is the famous Berry phase:
iϕ(t) = i
∫ t
0
〈ψγ(λ)|d
dλ|ψγ(λ)〉 dλ = i
∮γ
〈ψτ |∇τ |ψτ 〉 · dτ, (2.9)
where∇τ is the gradient in the parameter space X. Sometimes one encounters
that the space parametrizing M is used as a parameter space as well, for example
in the Aharonov-Bohm effect, but for now we leave it aside.
2.2.1 Hilbert Space geometry
To set up our Hilbert space geometry we recall that a physical state, |ψτ 〉 is an
equivalence class of vectors, ψτ , in the Hilbert space:
|ψτ 〉 ≡ [ψτ ] =
eiϕψτ | eiϕ ∈ U(1) and ψτ ∈ L2[M ]
. (2.10)
Thus at every point τ ∈ X we have a U(1)-degree of freedom and it defines a
U(1)-bundle P (X,U(1)) over the parameter space. The projection π : P 7→ X is
given as:
π
(eiϕ|ψτ 〉
)= τ. (2.11)
The bundle projection π induces a mapping dπp : TpP 7→ TpX between tangent
spaces. A connection is a mapping A : TP 7→ ker(dπ). Thus a vector v ∈ TpPis projected by the connection into the tangent space of the fiber, which is in our
case the Lie algebra of U(1). As shown in figure (2.2) a connection decomposes
the tangent space of the fiber bundle into a horizontal and a fiber part:
TpP = ker(dπp)⊕ Tπ(p)X = ker(dπp)⊕ ker(Ap), (2.12)
where we remark that if the fiber is a Lie group that ker(dπp) ' g, its algebra.
In this sense the connection A lifts a vector field on the parameter space, X, to a
vector field on P , known as a horizontal lift. To conclude our remarks note that a
connection on P gives an identification of fibers π−1(τ1) and π−1(τ2) by solving the
horizontal lift of a path connecting τ1, τ2 ∈ X, which defines parallel transport.
13
2. Topological phases of matter
TpP
p
π(p) = τ
U(1)
0 2π
2π
0
ψ0
ψ2πeiϕ
π−1(τ) ≃ U(1)
ker(dπp) ≃ Im(Ap)
P (X,U(1)
X ≃ S1
Tπ(p)X ≃ ker(Ap)
Figure 2.2: Schematic diagram of a Berry connection on the U(1)-line bundleover the parameter space X. The connection defines a horizontal lift of the basemanifold into the fiber bundle. By tracing a closed path in X one obtains a Berryphase, eiϕ, the holonomy of the connection on manifold X.
For more details see Reshetikhin, Stone and Goldbart [2009] and Nakahara [2003].
To conclude we see that the Berry phase defines the geometry of the Hilbert
space in the sense that it defines how a physical state is parallel transported
through the parameter space. Therefore we can define on the parameter space
a covariant derivative, with a connection one-form for every state of the Hilbert
space. Stricly speaking each state has its own one-form, but since we assume
there are no eigenvalue crossings, all states are transported by the same connec-
tion, and we drop the usual state label ψτ and replace it with the ground state:
|Ω〉.
This means that the integrant in the formula for the Berry phase is naturally
interpreted as a connection one-form:
iϕ = i
∮γ
〈Ω|∇τ |Ω〉 · dτ = i
∮γ
Aadτa, (2.13)
where we defined the Berry connection on the bundle of states over the space of
14
2. Topological phases of matter
parameters X, with basis τa as:
A = Aadτa = 〈Ω|∇τ |Ω〉dτ = 〈Ω| (d|Ω〉). (2.14)
Finally we give its coordinate representation in the parameter space as:
Aa(τ) = 〈Ω| ∂∂τa|Ω〉. (2.15)
As with normal geometry we can take the exterior derivative of this one-form
to find the curvature two-form:
F = dA = (d〈Ω|) ∧ (d|Ω〉) =
(∂〈Ω|∂τa
)(∂|Ω〉∂τ b
)dτa ∧ dτ b. (2.16)
This curvature is often referred to as adiabatic curvature, and is expressed for
the U(1) bundle in component form:
Fab = ∂aAb − ∂bAa. (2.17)
By using Stokes’ theorem we can write the Berry phase, as an integral over the
curvature of the area enclosed by γ on the parameter space X.
iϕ = i
∮γ
Aadτa = i
∫S
F (τ) · dS (2.18)
The Berry phase for a closed loop is a gauge invariant quantity, since it is mea-
surable quantity. From figure (2.2) we see that a global gauge transformation on
X just vertically shifts the section trough the bundle. Suppose we make a local
gauge transformation on the parameter space of the form:
|Ω〉 = e−iΛ(τ)|Ω〉, (2.19)
then the Berry phase of an open-path on X transforms into iϕ(t) = i(ϕ(t)+Λ(t)−Λ(0)). If we consider a closed-path we see that Λ(T )−Λ(0) = 2πm,m ∈ Z for the
Berry phase to remain gauge invariant. This reflects the fact that π1(U(1)) = Z.
Gauge transformations, the change of the connection A, that change the winding
number are known as non-trivial gauge transformations.
15
2. Topological phases of matter
2.2.2 Spectral flow
Now we shall connect the previous two paragraphs. Here we shall see that the
holonomy of paths in the parameter space of the Hamiltonian generate the topo-
logical invariants Z and Z2. Intuitively the picture is quite clear. Suppose we
study a many-body non-interacting system of fermions that is governed by the
eigenvalue problem:
H(τ)|ψτ 〉 = E(τ)|ψτ 〉. (2.20)
If we assume that τ ∈ S1, the spectrum is subject to the constraint Spec(H, τ) =
Spec(H, τ mod 2π). Subsequently one can consider the system undergoing an
adiabatic process where the parameter is tuned from τ : 0 7→ 2π. Such a cycle
induces a mapping of the set of eigenstates, |ψτ 〉, of the form:
Γ : |ψτ 〉 7→ |ψτ+2π〉. (2.21)
After an adiabatic cycle the eigenvalues of H return to their original value, but
the so called spectral flow of the eigenstates necessarily does not. The holonomy
of an eigenstate can be such that the state is coupled to another eigenvalue in
the spectrum of the Hamiltonian. Such a process is called a topological Thouless
pump with τ as pumping parameter. The holonomy mappings Γ define homotopi-
cal equivalences between the in- and out-states of a cycle and define topological
quantum numbers. Systems in different equivalence classes cannot be deformed
adiabatically into each other and are considered as different topological phases
of matter. This is the physics realization of the inequivalent paths under the
consideration of deformation retracts in homotopy theory.
After all these, perhaps somewhat too formal statements, it is time for an
very explicit example:
16
2. Topological phases of matter
Figure 2.3: These diagrams illustrate the spectral flow of two systems as a func-tion of τ . The left picture is a trivial insulator, the right picture is the spectralflow of a fictional non-trivial insulator with a Z-valued topological order param-eter.
Example Particle on a ring
Consider a particle on a ring of unit radius threaded by flux Φ ∈ R. The
magnetic field in cylindrical coordinates is given as:
Bz = zdA = z1
r
(∂(rAφ)
∂r− ∂Ar
∂φ
). (2.22)
If we set Aφ = Φ/Φ0 the magnetic field, (i.e. flux density), of the form:
B =1
r
Φ
Φ0
z. (2.23)
The total flux through the ring is found as:
Φ =
∫B · dA (2.24)
=
∫ 2π
0
rdθ
∫ 1
0
dr1
r
Φ
Φ0
(2.25)
= Φ, (2.26)
because in our units the magnetic flux quantum is Φ0 = 2π. The Hamiltonian
17
2. Topological phases of matter
becomes:
H = (−i∂φ +Φ
Φ0
)2. (2.27)
The solutions are subject to the condition ψ(0) = ψ(2π), leading to the solu-
tions:
ψn(φ) =1√2πeinφ, En = (n+
Φ
Φ0
)2, with n ∈ Z,Φ ∈ R. (2.28)
First we note that if we add adiabatically a flux quantum (i.e. Φ 7→ Φ + 2π)
the states all move one eigenstate. Now let us try to adiabatically change the
magnetic field and solve its spectral flow explicitly by using a Berry connection.
However in this picture the wave functions are independent of the magnetic field,
whereas the spectrum is. The deceptive property of wave functions is that they
are not gauge invariant. To make the wave function explicitly dependent on B
consider the following (local) gauge transformation:
ψ 7→ eiΛ(φ)ψ = e−i(Φ/Φ0)φψ (2.29)
A 7→ A+ ∂φΛ =Φ
Φ0
− Φ
Φ0
= 0. (2.30)
In this gauge transformed picture we have the following Hamiltonian and
boundary conditions:
− ∂2φψ = Eψ (2.31)
e−i(Φ/Φ0)·0ψ(0) = e−iΦψ(2π), (2.32)
with corresponding solutions:
ψn(φ) =1√2πeiφ(n+(Φ/Φ0)), E = (n+
Φ
Φ0
)2, with n ∈ Z,Φ ∈ R. (2.33)
This magic gauge transformation has a nice interpretation, which we shall
use more often. Effectively we cut open our ring, so that the system has no flux
threading through its centre. This is reflected by the fact that the Hamiltonian
becomes independent of the magnetic field. At the cut we have twisted boundary
18
2. Topological phases of matter
conditions, reflecting the phase picked up by encircling flux, as depicted in figure
(2.4).
φ = 0φ = 2π
ψ(0)
ψ(0) = ψ(2π)
Φ
ψ(0) = e−iΦψ(2π)
e−iΦψ(2π)
Figure 2.4: With use of a gauge transformation a system can be cut, such thatits Hamiltonian becomes independent of the threading flux, and the flux is incor-porated in the boundary conditions on the cut.
With our new wave functions it is possible to find the Berry connection of
adiabatically varying the magnetic field:
A = 〈ψ|∂Φ|ψ〉 = iφ
2π. (2.34)
Now consider adiabatically driving a magnetic flux quantum through the ring:
Φ 7→ Φ + 2π:
iϕ = i
∮S1
AdB = −i∫ 2π
0
iφ
2π= φ. (2.35)
Under this process the states are subject to the map:
Γ : ψn(φ) 7−→ e+iφψn(φ) =1√2πeiφ(n+Φ/Φ0+1) = ψn+1(φ). (2.36)
To conclude we have explicitly seen that by threading a flux quantum through
the ring we induce an adiabatic involution of all the states by one as depicted in
figure (2.3), depending on the direction of the flux.
Now that we have a basic idea of topological phases it is time to study quan-
titatively its features according to two examples, the first is the integer quan-
tum Hall effect (iqHe) and the second is the time-reversal invariant insulator in
d = 1 + 1.
19
2. Topological phases of matter
2.2.3 Z-Thouless pump of the iqHe
The iqHe system was the first topological phase of matter discovered in 1980,
which was accompanied by other phases as late as 2005. As we shall see later
that these phases can be characterized by their discrete symmetries. The iqHe
is a d = 2 + 1 system subject to a perpendicular magnetic field. Therefore time-
reversal symmetry, particle-hole symmetry and chiral symmetry are absent. The
iqHe system is discussed extensively in Ezawa [2000], Girvin [1999], Prange and
Girvin [1987], and here we shall recall some results to understand the relationship
between topology and physics.
If a voltage difference is produced in the transverse direction of a current
through a material, one speaks of a Hall effect. In the case of a material which
is penetrated by a perpendicular magnetic field one can understand this effect
on the classical level with the Lorentz force exerted on the charge carriers of the
current, as displayed in figure (2.5 a).
V
A⊗φ2
⊗φ1
⊗φ1⊗φ2
B BB
B
a) b) c)
Figure 2.5: Three equivalent diagrams of measuring the Hall effect.
The current is driven by the battery with voltage V and the Hall current,
induced by the Hall voltage, is measured with the Ampere-meter. In figure b)
we replaced the battery by a flux-tube, φ1, threading the loop. This flux drives
an emf1, equivalent to the current caused by the battery. Equally we replaced
the Ampere-meter with a flux-tube, φ2, threading the measurement loop. We
1electro-motive force
20
2. Topological phases of matter
see that the Hall effect essentially describes the physical response of a multiple
connected system: it relates the flux through one loop, with the flux through the
other loop. Using this insight, it turns out that the system can be studied in the
form of figure (2.5 c) Avron and Seiler [1985]. We consider the wires as part of
the system and rearranged them in such a way that the system became planar.
The system is now doubly-connected with the two flux-tubes penetrating the two
holes. This form is convenient for analyzing the topological properties of the qHe.
Figure 2.6: Schematic diagram of the iqHe system during the removal of one fluxquantum. The electrons are adiabatically lowered one orbit, and effectively theedges get polarized.
Before we show that the Hall conductance is robustly quantized, because it is a
topological invariant, the effect is schematically described to give the imagination
somewhat more flesh on the bones. The Hall conductance is:
σH = νe
Φ0
, with: Φ0 =h
e. (2.37)
This equation states that if one inserts an extra flux quantum, Φ0, through
the sample, the Hall response transports exactly ν electrons. Here ν is the filling
fraction and is for the integer quantum Hall effect, an integer. Later we shall see
that this filling fraction ν is a topological invariant. This number describes the
number of filled Landau levels. These well-known levels, or energy bands, emerge
21
2. Topological phases of matter
because of the magnetic field, as explained in chapter 4, and displayed in figure
(2.6). Here we displayed the spectrum of a Corbino disc with a perpendicular
field, in the Wannier state representation. If the system was an infinite plane we
would have the equidistant spectrum of the Harmonic oscillator, where each state
of the oscillator is a Landau level. The finite size of the Corbino disc induces a
confining potential, which causes the Landau Levels to curve up in energy near
the edge of the sample. We see from the figure that each level below the fermi
energy Ef , would contribute one edge state at the Fermi-level. Another crucial
ingredient of the qHe is disorder, remarkably it can be shown that without disor-
der the qHe would be non-exsistent, but this is now not part of this analysis. The
disorder causes the Landau levels to broaden as displayed in (2.6), but becomes
neglible near the edges.
As in our example of the ring a spectral flow is induced, if we adiabatically
insert an extra flux quantum Φ0. In this case the spectral flow causes all states
to move one orbit. From the Wannier-state representation of the states we can
see that this causes the edges of the Corbino disc to become polarized. To make
connection with the set-up of figure (2.5) we note that the driving flux is induced
through flux-tube, φ1, and the Hall response of the electron transport is recorded
through flux-tube φ2.
To conclude our remarks on the qHe we show the spectral flow of each state
in figure (2.7). Here we depicted the spectral flow of each state within the Lan-
dau level. All electrons are mapped one angular-momentum orbit lower. If one
considers this flux insertion in a Corbino disc, there arises a vacancy on the outer
edge and one electron surplus on the inner edge. This is the band-theoretical
picture of the iqHe. Important remark here is that these orbits are not gauge-
invariant and one could be mislead by it, but here it nicely illustrates the iqHe.
22
2. Topological phases of matter
Figure 2.7: Schematic diagram of the IQHE system during the removal of oneflux quantum. The electrons are adiabatically lowered one orbit, and effectivelythe edges get polarized. In a corbino disc the orbits with label m are of fixedradius r.
2.2.4 Z2-Thouless pump of the Shindou model
All topological phases of non-interacting fermions are either characterized by ei-
ther integer or binary topological quanta. In the previous section we saw an
explicit example of a topological phase with an integer order. In this section we
shall study an example of binary order: ν0 ∈ Z2.
We saw that in the iqHe we can pump an integer number of charges in the
system by adiabatically inserting flux. Now we discuss qualitatively a system
wherein one can pump a Z2-valued quantity, being a prototype of a topological
insulator characterized by a Z2-topological order.
Group theory of groups containing two elements is not so rich, since all groups
of two elements are isomorphic. Nonetheless there are two seperate classes of
Z2-valued topological insulators that we prefer to denote with different represen-
tations of the groups of two elements. The first class is represented by the cyclic
23
2. Topological phases of matter
group of two elements denoted as:
Z/2Z :=
ν0 ∈ ±1| under multiplication
(2.38)
and the second class is represented by the group of integers modulo two:
Z2 :=
ν0 ∈ 0, 1| under addition modulo 2
. (2.39)
The different representations are chosen such that the values of the order
parameters are the same as the value of the relevant Chern class.
2.2.4.1 Topological materials with ν0 ∈ Z2/Z
As we saw in the beginning of this chapter we encountered interfaces of topo-
logical materials in the time-domain and the spatial-domain. In the temporal
case the classical example of Z/2Z-order parameter stems from the system of a
double-well potential. One can consider each minimum of the well as a seperate
vacuum. There are two non-trivial solutions of a tunneling event, spanning in
time from t = −∞ to t = +∞, from one vacuum to the other and vice versa.
Those were dubbed by Gerardus ’t Hooft as instantons. The topological charge
is either ν0 ∈ ±1, defining a tunneling from A 7→ B or B 7→ A, with each event
having one of the topological charges.
An example in the spatial-domain, which we shall study extensively in chapter
5, is a system where the mass of relativistic particle flips sign along a rectilinear
domain wall. This wall separates the space into two domains, with each a mass
term of m = ±m0. In this case the domain wall contains a zero mode which is
topologically protected, which is known as a soliton and found by Jackiw and
Rebbi [1976]. The topological order is then defined as the sign of the mass term
as ν0 ∈ m/|m|.
Those are the archetypical examples of Z/2Z-valued topological phases, where
the solitons and instantons are the modes of interfaces between different topo-
logical orders. These classes of topological materials are characterized by the
24
2. Topological phases of matter
symmetry between the ν0 = +1 and ν0 = −1 phases. In the previous exam-
ples there are no inherent differences between both orders, and are thus assigned
somewhat arbitrarily. The profound consequences arise in considering interfaces
between them.
2.2.4.2 Topological materials with ν0 ∈ Z2
More recently a new type of Z2-valued systems emerged. In this class of materials
there ıs a distinction between both phases. This class, with ν0 ∈ 0, 1, has much
more resemblance with the example discussed in the previous section. There we
managed to adiabatically pump states, due to special properties of the adiabatic
curvature of the expanded Hilbert space (e.g. X × L2(M)). This topological
response is determined by topological invariants of the adiabatic curvature. The
class of Z2 materials is an analogous to this situation. If ν0 = 1 one can pump
adiabatically states, whereas with ν0 = 0 nothing is pumped at all, and one ends
up with the trivial insulator. So here the topological order parameter character-
izes whether one has a Thouless pump or not.
Another very distinctive feature is that the states, which are pumped, have
a Z2-character as well. During pumping cycles the system changes back- and
forth between two different states. The easiest analogous example is the wave-
function of two fermions. If we denote the operation of adiabatically permuting
the fermions with P we have the following chain:
Ψ(r1, r2)︸ ︷︷ ︸ P−−→ −Ψ(r2, r1)︸ ︷︷ ︸ P−−→ Ψ(r1, r2)︸ ︷︷ ︸ .∈ ∈ ∈
1 ∈ Z/2Z −1 ∈ Z/2Z 1 ∈ Z/2Z
(2.40)
For this oscillation between states we use again the multiplicative represen-
tation of Z/2Z, because of the symmetry between the two different states the
system can be in.
25
2. Topological phases of matter
2.2.4.3 An example of a Z2-system: The Shindou model
The Shindou model, introduced in Shindou [2005] and discussed in Fu and Kane
[2006], services the purpose of a simple prototype for the more complicated strong
topological insulator, which is discussed in the next chapter. A similar discussion
in a more general context can be found in Pruisken et al. [2005]. The Shindou
model defines a topological Thouless-pump protected by time-reversal symme-
try (TRS). Consider an anti-ferromagnetic spin-12
chain with two perturbations
added. The first perturbation is a staggered magnetic field (e.g. a magnetic field
opposite aligned at each adjacent site) which drives the system into a Neel state.
The second perturbation is an exchange-energy term, which drives the system
into a dimerized state, in which the electrons form singlets with one of the neigh-
bours. Now consider that we have a pumping parameter τ and the following
conditions on the Hamiltonian:
Spec(H(τ)) = Spec(H(τ + 2π)), (2.41)
and,
H(−τ) = ΘH(τ)Θ−1, (2.42)
where Θ is the time-reversal operator. On a ring-type geometry the pumping
parameter can be considered as a flux quantum τ = Φ0. It is possible to define a
cycle wherein the system undergoes a transition from the dimerized phase to the
Neel phase and vice versa every π/2, as depicted in the diagram (2.8). At τ = π
we have two unpaired spins at the edge of the sample, usually referred to as a
dangling edge spin. The dimers have moved one site, causing a mismatch such
that it is impossible to complete the dimerized phase to the end of the sample.
As one further tunes through the adiabatic cycle one has at the end of the cycle
two equally aligned spins, which are in the non-dimerized phase and opposite on
each edge. These scars, as Prof. C. Beenhakker calls them, of the adiabatic cycle
are close relatives of a Shockley state Shockley [1939].
In the course of an adiabatic cycle, wherein one tunes τ : 0 7→ 2π, time-
26
2. Topological phases of matter
↑
↑↑↑↑
↑
↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑
↑ ↑ ↑ ↑ ↑↑↑ ↑
τ = 0
τ = 2π
τ = π
τ = π/2
τ = 3/2π
x = 0 x = L
Figure 2.8: Diagram of the states of the Shindou model during a single adiabaticcycle.
reversal symmetry is broken, but because of (2.42) we at least have two points
of the adiabatic cycle wherein time-reversal symmetry is respected, relatively at
τ = 0 and τ = π. It is easy to prove that TRS implies that all states come
in pairs, known as Kramer’s doublets. Thus during a cycling the doublet can
separate but has to reunite at the points τ = 0 and τ = π. Now, one can define a
Z2 quantity that measures how the Kramer’s doublets are reunited, dubbed time-
reversal polarazation PΘ(τ). With the Berry-connection one can easily calculate
PΘ(τ), but is out of scope for this discussion. With this quantity one can easily
distinguish a trivial with a topological insulator as follows:
ν0 = 0 if PΘ(0) = PΘ(π) (2.43)
ν0 = 1 if PΘ(0) 6= PΘ(π). (2.44)
Thus if the PΘ(τ) changes during a cycle the system is topologically non-
trivial. Important remark is that at a time-reversal momentum, Λi, PΘ(Λi) itself
is not gauge-invariant, but its difference between different time-reversal momenta
ıs.
Considering the spectral flow of this system, the topological response of the
Shindou model becomes quite simple to understand. In a trivial material, both
partners of a Kramer’s doublet get reunited with eachother, whereas in a topo-
27
2. Topological phases of matter
logical non-trivial insulator the Kramer’s doublet switch partner, as depicted in
diagram (2.9). Now considering the states in a Wannier-state representation it is
clear that we end up with two dangling edge spins. Finally we have to remark
that on a system with periodic boundary conditions the Shockley scars are ab-
sent, such that we truly have the Z2-oscillating chain analogues to the fermion
permutation.
Figure 2.9: Schematic diagram of the Shindhou model Shindou [2005] Fu andKane [2006], we see that the Kramer’s doublets switch partner during the adia-batic cycle, thereby causing spin accumulation at the end of the sample.
2.3 Bundle Invariants
We saw that the spectral flow of states under an adiabatic cycle causes a topolog-
ical response. In this section we discuss the topological nature of this response.
We commence by proving that σH is the first Chern number of the line bundle
of the Schrodinger Hamiltonian over the flux torus. We conclude this section by
the introduction of the Chern classes and Chern numbers.
2.3.1 Z: The TKNN-integer as topological invariant
Now we move to the fact that the flux-averaged Hall response is a topological
invariant, and therefore robustly quantized, first found by Thouless et al. [1982].
The averaging over a flux-insertion, first considered by Laughlin [1981], has the
28
2. Topological phases of matter
advantage that persistent currents and the details of the flux-tubes vanish. Note-
worthy is that A.M.M. Pruisken strongly disputes this flux-averaging interpreta-
tion of the Laughlin-argument, but here we shall consider it to be valid. We shall
derive it from a geometric perspective in the form a theorem proved by Avron
and Seiler [1985].
Theorem 2.3.1 (Avron and Seiler) Let H(φ1, φ2) be a general multi-particle in-
teracting Schrodinger Hamiltonian threaded by two fluxes, φ1, φ2, with a non-
degenerate groundstate |Ω(φ1, φ2)〉, which is seperated by the rest of the spectrum
with a finite gap for ∀φi. Then the Hall response averaged over the fluxes, 〈σ12〉,is an integral over the first Chern class and therefore integer valued.
Proof Let H(φ1, φ2) be the Hamiltonian of the system on manifold M depicted in
figure (2.5 c), with Dirichlet boundary conditions on ∂M . Consider an adiabatic
flux insertion through φ1, while we keep the flux through the second hole constant:
φ2 = const.. This causes an electromotive force V around hole 1: φ1 = ω(t) =
−V t. The Hall response is then:
I2 = σ12V. (2.45)
Denote the physical state of the system as |Ψ(t, φ2)〉. At the start of the
process we are free to set: |Ω(0, φ2)〉 = |Ψ(t = 0, φ2〉). The ground state is
isolated and eigenvalue-crossings have co-dimension three, and are thus absent in
Hamiltonians with two adiabatic parameters von Neumann and Wigner [1929],
which guarantees the state can be chosen to be normalized and smooth in φ. We
rewrite the time-dependent Schrodinger equation:
i∂t|Ψ〉 = H(φ)|Ψ〉 (2.46)
idω
dt∂ω|Ψ〉 = H(φ)|Ψ〉 (2.47)
−iV ∂φ1|Ψ〉 = H(φ)|Ψ〉. (2.48)
Note that as V 7→ 0, the wavefunction adiabatically evolves as a function of
φ1. We can write the current as:
29
2. Topological phases of matter
I2 = 〈Ψ|∂H(φ)
∂φ2
|Ψ〉 (2.49)
= ∂φ2〈Ψ|H(φ)|Ψ〉 − 〈 ∂Ψ
∂φ2
|H(φ)|Ψ〉 − 〈Ψ|H(φ)∂Ψ
∂φ2
〉 (2.50)
= −iV(∂φ2〈Ψ|
∂Ψ
∂φ1
〉 − 〈 ∂Ψ
∂φ2
| ∂Ψ
∂φ1
〉+ 〈 ∂Ψ
∂φ1
| ∂Ψ
∂φ2
〉)
(2.51)
I2 = −iV(∂φ2〈Ψ|∂1Ψ〉 − 〈dΨ|dΨ〉
), (2.52)
where in the second last line we used 2.48, taking the Hermitian adjoint caused the
sign flip in the last term. With this expression we can find the Hall conductivity
averaged over the adiabatic insertion of flux φ1 denoted as 〈σ12〉. The integral is
performed over the Torus of area (2π)2 spanned by (φ1, φ2) denoted as D. The
physical interpretation of the integral over D is the induced charge averaged over
the insertion of one flux quantum and is found as:
〈σ12〉 =
∫D
I2
V(2.53)
=
∫ 2π
0
dφ1
∫ 2π
0
dφ2 − i(∂φ2〈Ψ|∂1Ψ〉 − 〈dΨ|dΨ〉) (2.54)
= −i(∣∣∣∣φ2=2π
φ2=0
∫ 2π
0
dφ1〈Ψ|∂1Ψ〉 −∫D
〈dΨ|dΨ〉), (2.55)
with Stokes theorem we have that∫∂D〈Ψ|dΨ〉 =
∫D〈dΨ|dΨ〉, we see that the
first term is cancelled and the Hall response can be written as:
〈σ12〉 = i
∣∣∣∣φ1=2π
φ1=0
∫ 2π
0
dφ2〈Ψ|∂φ2Ψ〉. (2.56)
Now we exploit a construction originally found by Kato [1949], which control-
lably allows one to adiabatically approximate a solution to (2.48), we write:
Ψad(φ) = limV 7→0
ei/V∫ φ1
0
dφ′1E(φ′1, φ2)Ψ(φ), (2.57)
30
2. Topological phases of matter
this is a solution to the Kato evolution equation1:
∂φ1|Ψad〉 = [∂φ1P, P ]|Ψad〉, (2.58)
with P the projection operator on the groundstate: P = |Ω〉〈Ω|. So we can
adiabatically transport |Ψad〉 along φ1, while it remains up to a Berry phase
equivalent to the groundstate: |Ψad〉 = eiϕ(φ)|Ω〉. Note that we can interpret
(2.58) as parallel transport in the Hilbert space:
iDΨ〉 = 0 (2.59)
i(d− [dP, P ]|Ψ〉 = 0 (2.60)
i(∂φ1 − [∂φ1P, P ])|Ψ〉 = 0, (2.61)
where [dP, P ] is the projective representation of the Berry connection. The
eigenvalues of H are periodic in both φ’s, in particular E(φ1) = E(φ1 + 2π).
From (2.57) we see |Ψad〉 is also periodic in φ1, so we can substitute |Ψad〉 into
the equation for the averaged Hall conductivity.
〈σ12〉 = i
∣∣∣∣φ1=2π
φ1=0
∫ 2π
0
dφ2〈Ψad|∂φ2Ψad〉 (2.62)
〈σ12〉 = i
∣∣∣∣φ1=2π
φ1=0
∫ 2π
0
dφ2〈e−iϕ(φ)Ω|∂φ2 eiϕ(φ)Ω〉 (2.63)
〈σ12〉 = i
(∣∣∣∣φ1=2π
φ1=0
∫ 2π
0
dφ2 i∂φ2ϕ(φ) + 〈Ω|∂φ2 Ω〉)
(2.64)
〈σ12〉 = i
(iϕ(2π, 2π)− iϕ(2π, 0) +
∣∣∣∣φ1=2π
φ1=0
∫ 2π
0
dφ2〈Ω|∂φ2 Ω〉), (2.65)
where we dropped all factors of ϕ(0, φ2), because of the setting of our initial
conditions. To calculate the Berry phase we use equation (2.58), but first we need
1For details see Avron [1995].
31
2. Topological phases of matter
to derive the identity:
〈dΩ|Ω〉+ 〈Ω|dΩ〉 = 0. (2.66)
This is derived by differentation the projection identity: P 2 = P as follows:
d(|Ω〉〈Ω|Ω〉〈Ω|
)= d
(|Ω〉〈Ω|
)(2.67)
|Ω〉〈dΩ|Ω〉〈Ω|+ |Ω〉〈Ω|dΩ〉〈Ω| = 0, (2.68)
which implies (2.66) by taking the trace. Now consider the inner product of
parallel transported adiabatic wavefunction with itself:
〈Ψad|Dφ1Ψad〉 = 0 (2.69)
〈Ψad|∂φ1Ψad〉 − 〈Ψad|[∂φ1P, P ]Ψad〉 = 0 (2.70)
Using the identity of (2.66) together with the fact that |Ψad〉 = eiϕ|Ω〉 and
that 〈Ω|eiϕΩ〉 = eiϕ, it is straightforward to show that the second term of (2.70)
is zero. From the first term we get the equality:
i∂φ1ϕ(φ) = −〈Ω|∂φ1Ω〉 (2.71)
With integration we obtain:
ϕ(2π, 2π)− ϕ(2π, 0) = i
∣∣∣∣φ2=2π
φ2=0
∫ 2π
0
dφ1〈Ω|∂φ1Ω〉. (2.72)
Going back to the flux-averaged Hall conductance we get:
32
2. Topological phases of matter
〈σ12〉 = i
(iϕ(2π, 2π)− iϕ(2π, 0) +
∣∣∣∣φ1=2π
φ1=0
∫ 2π
0
dφ2〈Ω|∂φ2 Ω〉)
(2.73)
〈σ12〉 = i
(∣∣∣∣φ2=2π
φ2=0
∫ 2π
0
dφ1〈Ω|∂φ1Ω〉+
∣∣∣∣φ1=2π
φ1=0
∫ 2π
0
dφ2〈Ω|∂φ2 Ω〉)
(2.74)
〈σ12〉 = i
∫∂D
〈Ω|dΩ〉 (2.75)
〈σ12〉 = i
∫∂D
A (2.76)
〈σ12〉 = i
∫D
〈dΩ| ∧ |dΩ〉 (2.77)
〈σ12〉 = i
∫D
F. (2.78)
In the last four lines we used Stokes’ theorem and the definition of the Berry
connection and Berry curvature. To show that the Hall response is integer-valued
we need to make our domain simply connected by applying two cuts with modified
boundary conditions as depicted in figure (2.10). Now the magnetic fields enter
the problem through the boundary conditions as:
Ψ(x) = eiφ1n1δ(x−c1)eiφ2n2δ(x−c2)Ψ(x), (2.79)
where x are the local coordinates on M . Now the magnetic field is removed
from the gauge transformed Hamiltonian H, which becomes φ independent.
⊗φ1⊗φ2
BB
c1c2 n2 n1
Figure 2.10: The original domain M is cutted along c1 and c2, which results inthe new domain M .
33
2. Topological phases of matter
In this new gauge the vector potential can be written as the gradient of a scalar
function χ(φ), satisfying A(φ) = dχ(φ). By Poincare’s lemma the magnetic field
is trivially zero on the domain M . χ(φ) is continuous within the domain M and
has discontinuities across the cuts c1, c2. The projection operation changes as
follows:
P = |Ω〉〈Ω| = U †QU = e−iχ(φ)|Ω〉〈Ω|eiχ(φ), (2.80)
where |Ω〉 is the ground state of H. After a tedious, nonetheless straightfor-
ward, calculation one obtains the following expression for the Hall conductance:
〈σ12〉 =
∫D
〈dΩ| ∧ |dΩ〉 (2.81)
〈σ12〉 =
∫D
〈dΩ| ∧ |dΩ〉+ idχ(φ) (2.82)
〈σ12〉 =
∫D
〈dΩ| ∧ |dΩ〉, (2.83)
where the total derivative of χ(φ) was dropped because the integration do-
main D, the torus, has no boundary. Equation (2.83) is manifestly periodic in
φ, and so is the Hall conductance. Now we prove that the flux-averaged Hall
conductance is quantized and the first Chern class of a line bundle over the torus,
which is the parameter space of the Hamiltonian H(φ) .
The wavefunction of the adiabatic process in the cutted picture, |Ψad(φ)〉 is
related to the ground state through the Berry phase:
|Ψad(φ)〉 = eϕ(φ)|Ω〉. (2.84)
Consider a closed loop in φ, denoted S1, our expression for the Berry phase
is:
ϕS1 = i
∫S1
〈Ω|dΩ〉. (2.85)
S1 partitions the parameter space D, over H(φ), into two segments: Din and
34
2. Topological phases of matter
Dout, sharing the same boundary S1. Because the Berry phase is gauge invariant
and |Ψad(φ)〉 is single valued we require:
∫Din
〈dΩ| ∧ |dΩ〉 −∫Dout
〈dΩ| ∧ |dΩ〉 = 2πν, with: ν ∈ Z. (2.86)
The Berry phase (in this case an element of U(1)) is single-valued, but the
angle itself can be multi-valued. Now we can consider shrinking S1 so that Din 'D and Dout = ∅, which is well defined because the integrand is smooth in changing
the path of integration. So we obtain the final result:
〈σ12〉 =
∫D
〈dΩ| ∧ |dΩ〉 (2.87)
〈σ12〉 = 2πν, (2.88)
with ν ∈ Z. Thus we conclude that the flux-averaged Hall conductance is a
topological invariant and an integer multiple of 2π.
This proves the theorem.
By dividing by the flux insertion of 2π we obtain the well-known result for
the Hall conductance:
σH = νe
Φ0
, with: ν ∈ Z. (2.89)
Here ν is the integer-valued topological-order parameter. It measures the
number of Landau levels below the Fermi level, which determines the number of
electrons pumped in during an adiabatic flux-insertion and also the number of
edge states.
2.3.2 Chern classes and numbers
To make the connection with the previous proof of the topological nature of the
Hall conductivity we cast the problem back to a system with non-interacting
35
2. Topological phases of matter
fermions and translation invariance. We get the following eigenvalue equation:
H(k)|ua(k)〉 = Ea(k)|ua(k)〉, a = 1, . . . , Ntot, (2.90)
where H(k) is a Ntot×Ntot single-particle Hamiltonian in k-space, and |ua(k)〉the Bloch wavefunctions of the band structure. Now consider the case where N−
bands are filled and N+ are empty, with N− + N+ = Ntot. We denote each type
of band as:
filled bands :=|u•a (k)〉
(2.91)
empty bands :=|vi (k)〉
. (2.92)
Because of the translational invariance we can consider the k-momenta as
mere parameters of the Hamiltonian. This is quite distinct from the previous
sections where we considered the Hilbert space H as follows:
H = X × L2(M). (2.93)
Here the temporal-spatial degrees of the system are parametrized by the man-
ifold M , with dim(M) = ds + 1, and the parameter space with X. In the pre-
vious cases we had to distinguish between the (slow)-parameter space (X) of
the Hilbert space, wherein one can adiabatically tune through, and the temporal-
spatial degrees of the manifold parameterizing the system itself (L2(M)). Thanks
to the translational invariance the Bloch wavefunctions become parametrized by
k-momenta, which we can adiabatically tune. This allows us to define a Berry
connection with regard to the momenta.
Define a non-abelian connection as:
Aab(k) = Aabµ (k)dkµ = 〈u•a (k)|du•b (k)〉, (2.94)
with: µ = 1, . . . , ds, (2.95)
and a, b = 1, . . . , N−. (2.96)
36
2. Topological phases of matter
The Berry curvature becomes:
F ab(k) = dAab + (A2)ab =1
2F abµν dk
µ ∧ dkν . (2.97)
The n-th Chern character, denoted as chn(F ), is:
chn(F ) =1
n!tr
[iF
2π
]n. (2.98)
The integral of the Chern character over d = 2n is called the n-th Chern
number, denoted as Chn(F ):
Chn(F ) =
∫B.Z.d=2n
chn(F ) =
∫B.Z.d=2n
1
n!tr
[iF
2π
]n. (2.99)
Here the integral is over the first Brillioun zone of a d = 2n k-space, which acts
as the parameter space of our Hamiltonian. Now our iqHe-system is described in
the case where n = 1:
Ch1(F ) =
∫ π
−πdkxdky
i
2πtr[F ] =
i
2π
∑a
∫ π
−πdkxdky〈du•a |du•a 〉. (2.100)
If we substitute the ground state |Ω〉, instead of our Bloch wavefunctions,
and replace the flux-torus by the Brillioun-zone, we see that we have the same
result as from the Avron & Seiler theorem. For completeness we state the second
Chern number, which is the basis of the topological protection of the d = 3 + 1
topological insulator:
Ch2(F ) =
∫B.Z.d=4
−1
8π2tr[F 2] =
−1
32π2
∫d4kεµνρσtr[FµνFρσ] (2.101)
For the second Chern number n = 2, so that the integral is over a four dimen-
sional Brillioun zone.
37
2. Topological phases of matter
2.4 Periodic system of Topological matter
2.4.1 Hamiltonian classes
In the article of Altland and Zirnbauer [1997] Altland and Zirnbauer classified
all random Hamiltonians according to their adherence to reality conditions. For
a long time this was regarded as mere an academic exercise, but now it turns
out to be of great importance. There are two basic symmetries single-particle
Hamiltonians, H, can adhere, which are anti-Hermitian. Recall the notation that
the full Hamiltonian (i.e. in second-quantized form) is defined as:
H =∑a,b
ψ†aHa,bψb. (2.102)
Then the first-quantized Ntot × Ntot-matrix H can be classified according to
its adherence to reality conditions. The first we have already seen is time reversal
symmetry (TRS) with the condition:
T : U †T H∗ UT = +H. (2.103)
The second condition is particle-hole symmetry (PHS), also known as charge
conjugation:
C : U †C H∗ UC = −H. (2.104)
Recall that for a unitary matrix we have the identity U∗U = ±I, then it is easy
to see that by operating both operators twice we have the possibilities: T2 = ±1
and P2 = ±1. Thus for each symmetry the Hamiltonian has three classes (i.e.
±1, 0). Since we have two of such operators we expect nine different classes of
Hamiltonians. There is however a special case where the Hamiltonian does not
adhere to both T and C. In this case the Hamiltonian can adhere to the product
operator S = T · C or not. This operator is called often chiral symmetry or sub-
lattice symmetry, because it reflects the interchange of two distinguishable sites
on a bipartite lattice. The S-operator is fixed for all classes except for the class
where T = 0 and C = 0, for these systems S = 0, 1. So we have 10 classes of
random Hamiltonians.
38
2. Topological phases of matter
Cartan label T C S Hamiltonian (eiHt) G/H (fermionic relica NLσM)
A (unitary) 0 0 0 U(N) U(2n)/(U(n)× U(n))AI (orthogonal) +1 0 0 U(N)/O(N) Sp(2n)/(Sp(n)× Sp(n))AII(sympletic) -1 0 0 U(2N)/Sp(2N) O(2n)/(O(n)×O(n)
AIII(chiral unitary) 0 0 1 U(N +M)/(U(N)× U(M)) U(n)BDI(chiral orthogonal) +1 +1 1 O(N +M)/(O(N)×O(M)) U(2n)/Sp(2n)CII (chiral symplectic) -1 -1 1 Sp(N +M)/(Sp(N)× Sp(M) U(2n)/O(2n)
D(BdG ) 0 +1 0 SO(2N) O(2n)/U(n)C(Bdg) 0 -1 0 Sp(2N) Sp(2n)/U(n)DIII(BdG) -1 +1 1 SO(2N)/U(N) O(2n)CI(BdG) +1 -1 1 Sp(2N)/U(N) Sp(2n)
Table 2.1: The Hamiltonian classes are listed by their symmetries. In the Hamil-tonian column we listed the symmetric space wherein the time-evolution operatoris element of. In the last column the target space T of the NLσM correspondingto the class is listed. The table is adapted from Ryu et al. [2010].
The most interesting part of this topic is that it connects a fundamental result
in mathematics to the time-evolution operator eiHt. It turns out that the time-
evolution operator of the above classes of random Hamiltonians runs over the 10
symmetric spaces found by Elie Cartan Cartan [1926]. Those spaces are finite-
dimensional manifolds with a constant curvature, parametrized by the radius of
curvature. They can be summarized in the following table, where we also list the
basic properties of the relevant classical Lie groups:
39
2. Topological phases of matter
A1
−1Time Reversal Symmetry
Particle Hole Symmetry
1
−1
AI
AII
BDI
CII
D
C
DIII
CI
AIII
1
−1Chiral symmetry
chiral unitary
symplectic
orthogonal
unitary
chiral orthogonal
chiral symplectic
BdG orthogonal
BdG symplectic
BdG
BdG
Figure 2.11: The ten symmetry classes according their basic symmetries.
Table 2.2: Classical Lie groups
Field Group definition
Real ROrthogonal O(n) = M ∈ GL(n,R)| MMT = I
Special orthogonal SO(n) = M ∈ GL(n,R)| detM = I,MMT = I
Complex CUnitary U(n) = M ∈ GL(n,C)| MM † = I
Special unitary SU(n) = M ∈ GL(n,C)| detM = I,MM † = I
Symplectic Sp(n) = M ∈ GL(2n,C)| MTωM = ω
40
2. Topological phases of matter
Cartan label/d 0 1 2 3 4 5 6 7 8
Real caseA (unitary) Z 0 Z 0 Z 0 Z 0 ZAIII (chiral unitary) 0 Z 0 Z 0 Z 0 Z 0
Complex caseAI (orthogonal) Z 0 0 0 2Z 0 Z2 Z2 ZBDI (chiral orthogonal) Z2 Z 0 0 0 2Z 0 Z2 Z2
D (BdG) Z2 Z2 Z 0 0 0 2Z 0 Z2
DIII (BdG) 0 Z2 Z2 Z 0 0 0 2Z 0AII (symplectic) 2Z Z2 Z2 Z2 Z 0 0 0 2ZCII (chiral symplectic) 0 2Z Z2 Z2 Z2 Z 0 0 0C (BdG) 0 0 2Z Z2 Z2 Z2 Z 0 0CI (BdG) 0 0 0 2Z Z2 Z2 Z2 Z 0
Table 2.3: The Hamiltonian classes are listed with their types of topologicalinsulators per dimension. The table is adapted from Ryu et al. [2010].
2.4.2 Homotopic classification
The study of Anderson localization of the boundary of a ds-dimensional system
described with a NLσM determines whether the long-range physics is governed by
a topological term, as explained in the beginning of this chapter. Upon completion
of this program one finds for which class of random Hamiltonians the system
is characterized by a topological order. This task has been accomplished very
recently by Ryu et al. [2010] and Kitaev [2009]. A complete thesis, or even
multiples of them, could be spent on it alone. Here we just display the table as it
has been found. In the next chapter we shall comment in which class the strong
topological insulator belongs and what consequences this has.
2.5 Conclusion
In this chapter we first explored the concept of the bulk-boundary correspondence.
Then we extensively studied of the physics of driving through adiabatic cycles
and the implications in gauge theories. Then we showed that the properties of
these cycles are topological in nature. We concluded with the complete list of
topological phases of non-interacting fermions.
41
Chapter 3
The strong topological insulator
In the previous chapter we introduced the general framework of topological ma-
terials. In this chapter we shall zoom in on the physics of the (3 + 1)d strong
topological insulator. The goal of this chapter is to show the microscopic origin
of the Dirac-type surface theory. Secondly we shall study some of the general
properties relevant to this study of the Dirac equation and introduce machinery
for the Dirac theory on curved spaces. We conclude the chapter by studying the
Hall response for the surface of a topological insulator.
3.1 Derivation of the surface theory
The (3+1)d topological insulator is a close relative to the famous theoretical pre-
diction of a topological phase in HgTe quantum wells by Bernevig et al. [2006],
and experimental verification by Konig et al. [2007]. This research triggered the
prediction of the strong TI in the materials: BixSb1−x by Fu and Kane [2007],
which was soon thereafter experimentally observed by using angle-resolved pho-
toemission spectroscopy (ARPES) Hsieh et al. [2008a]. Later this material was
joined by Bi2Te3, Sb2Te3 and Bi2Se3, as examples of 3d TI’s.
In this chapter we discuss the basic physics of these materials. These ma-
terials are found to be well described by one model Hamiltonian found by Liu
42
3. The strong topological insulator
et al. [2010], with different (fitting) parameters for the different materials. For
its simplicity we discuss Bi2Se3, and introduce its symmetries and some of its
microscopic details. Since this is really well documented in the literature, this
discussion shall by concise.
3.1.1 (3 + 1)d Topological insulator prototype: Bi2Se3
The crystal structure of Bi2Se3 is depicted in figure (3.1). The crystal consists
of a layered structure, chosen to be in the z-axis, and the unit cell consisting of
5 atoms: Bi1, Bi1′;Se1, Se1′;Se2, where the semicolon partitions the atoms
in equivalence classes under the space group. Thus both Bismuth-atoms are
equivalent and the Selenium-atoms are separated into two classes. The layers are
triangular and have a periodicity of 5-layers, known as quintuples. The structure
of the layers with a quintuple in the z-direction are: Se1, Bi1, Se2, Bi1, Se1, as
shown in figure (3.1). The quintuples are weakly coupled and can be considered
as disjunct in the low-energy sector and are thus treated separately. The system
has four basic symmetries:
• R3: a three fold rotation symmetry along the z-direction.
• R2: a twofold rotation symmetry along the x-direction.
• P : inversion symmetry, where the single Selenium-atom Se2 is mapped to
itself (e.g. Se2 is the inversion centre), and the other atoms in the unit cell
are interchanged.
• T : time-reversal symmetry
3.1.2 Band inversion of opposite parity
Now we shall consider some very general aspects of the band structure of Bi2Se3,
and see that at the Γ-point (e.g. k = 0) in k-space we have band inversion. The
43
3. The strong topological insulator
Figure 3.1: (a) Crystal structure of Bi2Se3. The quintuple layer is indicated withthe red box. (b) Top view along the z-direction. (c) Side-view of the quintuplelayer. The figure is from Liu et al. [2010].
atomic orbitals of Bismuth are: 6s26p3 and of Selenium are: 4s24p4, thus the out-
ermost shells are p-orbitals and we can neglect the inner s-shells. 1. In one quin-
tuple layer we have 5 atoms with each three p-orbitals. As can be seen from figure
(3.1), all the Se-layers are separated by Bi-layers. Therefore the most important
coupling is between the Se- and Bi-layers, which causes level-repulsion and the
hybridization of the original orbitals into a new set: |Bα〉, |B′α〉, |Sα〉, |S ′α〉, S0α〉,with α = px, py, pz.
Now we use the inversion symmetry to make the states eigenstates of the
parity operator, as follows:
|P1±, α〉 =1√2
(|Bα〉 ∓ |B′α〉
)(3.1)
|P2±, α〉 =1√2
(|Sα〉 ∓ |S ′α〉
). (3.2)
The antisymmetric states have a lower energy for the Coulomb repulsion, and
are therefore known as the bonding states. The symmetric states have a higher
1For good and concise lecture notes on atomic orbitals and interactions see Walraven [2010].
44
3. The strong topological insulator
Figure 3.2: Schematic figure of the origin of the band structure inversion. Thereare three steps taken into consideration: I) hybridization of Bi and Se orbitals, II)formation of bonding/anti-bonding, III) crystal field splitting and IV) Spin-orbitcoupling. The figure is from Liu et al. [2010].
energy cost related to the Coulomb repulsion, because they have a higher proba-
bility of being near to each other. In figure (3.2) we see that the states, |P2−, α〉and |P1+, α〉 have become the states nearest to the bandgap. The z-axis is special
because of the layered structure, this induces a so called crystal-field splitting,
where the pz-orbitals turn out to be the states closest to the gap Liu et al. [2010].
The spin-orbit coupling (SOC) term causes the band-inversion at the Γ-point.
The SOC Hamiltonian is of the form: HSOC = λL ·S, with λ the coupling param-
eter between the orbital angular- and spin-momentum. This term keeps the total
angular momentum J = L+S invariant, but mixes spin with orbital momentum.
It is quite easy to show that the SOC leads to a level repulsion between the states:
|Λ, pz, ↑〉 and |Λ, px + ipy, ↓〉 on the one hand and between its TR-counterpart:
|Λ, pz, ↓〉 and |Λ, px − ipy, ↑〉 on the other hand, with Λ = P1+, P2−. This
lowers energy of the states |P1+ ↑〉 and |P1+ ↓〉, whereas it increases the energy
of |P2− ↑〉 and |P2− ↓〉. Clearly for a critical strength of spin-orbit coupling (i.e.
45
3. The strong topological insulator
λ) the bands become inverted. Because both bands have an opposite parity we
shall see that this band inversion drives the material into a topological non-trivial
phase.
3.1.3 Topological non-trivial band inversion
In chapter 2 we saw that if in a one-dimensional system the time-reversal po-
larization PΘ(k) ∈ Z2 between two time-reversal invariant momenta (TRIM),
Λa,Λb, is opposite the system is in a topological non-trivial phase. In that case
the Kramer’s pair switch partners while tuning through k-space from Λa to Λb.
We now have to generalize this analyses into three dimensions where we have
Λa ∈ 2d = 23 = 8 points where the Kramer’s doublets are degenerate. In three
dimensions the Kramer’s degenerate band crossings at Λa lead to two-dimensional
Dirac cones in the dispersion of the surface of the material. This mechanism oc-
curs in any time-reversal invariant system with spin-orbit interaction, the topo-
logical non-trivial part is how the Dirac cones at Λa are connected to each other,
which is measured by the change in PΘ(k). Recall that the time-reversal polar-
ization itself is not gauge-invariant, but its change between two TRIM ıs.
A very convenient tool to find whether the system is topologically non-trivial
was presented in the article Fu and Kane [2007]. In a 2-dimensional plane, labelled
i, we have 4 TRIM, then one can construct a gauge invariant quantity identified
with that plane as:
(−1)νi =4∏
a=1
PΘ(Λa). (3.3)
This topological invariant measures weather the topology of the band struc-
ture is trivial or not: νi ∈ Z2. For a three-dimensional system this invariant es-
sentially indicates that the material is a stack of planar spin-Hall systems parallel
to plane i. This is not topologically protected because the states from different
levels in the stack can scatter on each other, therefore destroying the topological
46
3. The strong topological insulator
properties.
But in three dimensions one can also find another topological invariant which
is inherently three dimensional:
(−1)ν0 =8∏
a=0
PΘ(Λa). (3.4)
Materials which have ν0 = 1 are called the strong topological insulators, be-
cause they inherently have an odd number of Dirac cones in the surface dispersion
relation. From a simple glance at equation (3.4) one sees that one of the Λa needs
to be of opposite time-reversal polarization. In this case one can encircle this point
in the Brillioun zone, such a loop is commonly referred to as a Fermi-arc C. It is
easy to show that for any time-reversal invariant system with an electron encir-
cling a closed Fermi arc, the wavefunction is either mapped back to itself or to
minus itself, leading to a Berry phase of either ϕ = 0 or ϕ = π. If one traces an
adiabatic cycle which encloses the point of opposite sign, the Berry phase neces-
sarily has to be π. It is impossible to change this by continuously deforming the
Fermi-arc. This is the reason that the strong topological insulator is protected
by topology. For the weak topological insulator one can deform the arc in such
way that the point does not enclose a single Dirac point anymore. Thus the weak
topological insulator is not protected by topology, because it can continuously be
deformed to the trivial insulator.
To understand the fact that Bi2Se3 is a strong topological insulator we need
two more ingredients. In a system with inversion symmetry the problem of find-
ing the time reversal polarization, PΘ(k), is greatly simplified by the fact that it
is equal to the parity of the Bloch wavefunction at that particular momentum Fu
and Kane [2007]. In the previous section we saw that the band inversion at the
Γ-point had an opposite parity. The parity of the Bloch-wave function at all eight
Λa was found by ab initio methods by Qi and Zhang [2010]. The result of this
work is that without the band-inversion at the Γ-point all eight Λa have the same
parity. Thus if the spin-orbit coupling reverses the two-bands of opposite parity
at k = 0, we have driven the system through a topological phase transition, since
47
3. The strong topological insulator
now one of the eight TRIM has opposite parity.
To study the response of the system and the effective theory we shall derive
the model Hamiltonian for the Bi2Se3-system from symmetry principles in the
next section.
3.1.4 Model Hamiltonian
As we saw in the previous section the band inversion near the Γ-point determines
the topological nature of the material. It is possible to capture this behaviour
by considering the low-energy effective Hamiltonian around the Γ-point. At this
point in the Brillioun zone each state belongs to an irreducible representation of
the crystal symmetry group. Using this fact we construct the effective Hamilto-
nian. The four states closest to the bandgap form the basis of our Hamiltonian:
|P1+z , ↑〉, |P2−z , ↑〉, |P1+
z , ↓〉, |P2−z , ↓〉. Let σ be the regular Pauli-matrices act-
ing in the spin basis and τ in the basis of the P1+ and P2− sub-bands. Then the
symmetries of the crystal are translated in the following operators:
• R3 along the z-axis: R3 = eiΠ2θ, with: Π = σ3 ⊗ I2 and θ = 2π
3.
• R2 along the x-axis: R2 = eiσ1⊗τ3 .
• P inversion symmetry: I2 ⊗ τ3.
• T time-reversal symmetry: P = I2 ⊗ τ3.
Unfortunately using this four-band model, we have to abandon Eugene Wigner’s
saying that a system described by a matrix larger than 2×2 is probably not worth-
while studying. Our effective Hamiltonian is a 4×4-matrix that can be expanded
in the Dirac γ-matrices1as follows:
Heff (k) = ε(k) + di(k)γi + dij(k)γij. (3.5)
1For the γ-(vector) matrices we use as basis: γ1 = σ1 ⊗ τ1, γ2 = σ2 ⊗ τ1, γ3 = σ3 ⊗ τ1,γ4 = I2 ⊗ τ2, γ5 = I2 ⊗ τ3.
For the γ-(bivector) matrices we use as basis, where i, j ∈ 1, 2, 3: γij = εijkσk ⊗ I2, γi4 =σi ⊗ τ3, γi5 = −σi ⊗ τ2, γ45 = I2 ⊗ τ1.
48
3. The strong topological insulator
The above symmetry transformations allow us to make an irreducible rep-
resentation of each γi-vector and γij bi-vector matrix. That Heff is invariant
under these symmetries implies that di(k) has the same representation as γi and
equally for the bi-vectors. After some fiddling and puzzling, which is spelt out
in appendix B of Liu et al. [2010], one obtains the following Hamiltonian up to
O(k3):
Heff = ε(k) +M(k)γ5 + A1γ4kz + A2(γ1ky − γ2kx), (3.6)
with: (3.7)
ε(k) = C +D1k2z +D2(k2
x + k2y) (3.8)
M(k) = M0 −B1k2z −B2(k2
x + k2y). (3.9)
If one works out the matrices, and perform a unitary transformation with
U = diag(1,−i, 1, i) we get the well known Hamiltonian:
Heff (k) = ε(k)I4 +
M(k) A1kz 0 A2k−
A1kz −M(k) A2k− 0
0 A2k+ M(k) −A1kz
A2k+ 0 −A1kz −M(k)
, (3.10)
with: (3.11)
ε(k) = C +D1k2z +D2(k2
x + k2y) (3.12)
M(k) = M0 −B1k2z −B2(k2
x + k2y) (3.13)
k± = kx ± iky. (3.14)
The parameters A1, A2, B1, B2, C,D1, D2,M0 of the effective model are de-
pendent on the specific material and can for specific materials be found in Qi
and Zhang [2010]. The parameters of the mass-function are all positive (i.e.
M0, B1, B2 ≥ 0), so that we see that the band is indeed inverted at k = 0. Taking
into account higher orders causes the well known hexogonal warping effect in the
corners of the Brillioun zone in ARPES experiments, but do not alter the topo-
logical content of the theory.
49
3. The strong topological insulator
Now that we have obtained a general model for describing the long wave-
lengths dynamics of the strong topological insulator we are ready to study the
induced dynamics at the surface of the material in the next section.
3.1.5 Effective surface Hamiltonian
Consider a topological insulator described by (3.5) in the half-space z > 0. In
this discussion we follow Liu et al. [2010], but it was first derived by Konig et al.
[2008]. In the half-space kx- and ky remain good quantum numbers, and we need
to consider explicitely the momentum operator −i∂z for the z-axis. So we split
the Hamiltonian in a z-independent part:
Hxy = D2k+k− −B2k+k−γ5 + A2(γ1ky − γ2kx), (3.15)
and a part that depends on kz, that we replace with the operator −i∂z:
Hz = C −D1∂2z + (M0 +B1∂
2z )γ5 − iA1∂zγ4. (3.16)
Now we can solve the eigenvalue equation:
HzΨ = EΨ (3.17)
The off-diagonal parts of Hz are spanned by γ5 = I2⊗τ3 and γ4 = I2⊗τ2, so we
see that the Hamiltonian is block diagonal in spin-space such that the eigenstates
are of the form:
Ψ↑ =
χ
ξ
0
0
=
(ψ0
0
), Ψ↑ =
0
0
χ
ξ
=
(0
ψ0
), (3.18)
where both states are related under time-reversal symmetry. Now we can
solve the simplified eigenvalue equation. To simplify the discussion further we
set C = 0 and D1 = 0, because it was shown in Konig et al. [2008] to be of no
50
3. The strong topological insulator
significant importance for the surface theory. The eigenvalue equation becomes:(M0 +B1∂
2z −A1∂z
A1∂z −M0 −B1∂2z
)(χ
ξ
)= E
(χ
ξ
). (3.19)
We apply the ansatz that the spinor is of the form eλzψ0, where ψ0 is a two-
component spinor. Subsequently we use the fact that (3.19) has a particle-hole
symmetry, and are only interested in the E = 0 solution. The equation becomes:
σz(M0 +B1λ2)ψ0 − iσyA1λψ0 = 0 (3.20)
σx(M0 +B1λ2)ψ0 = A1λψ0 (3.21)
(M0 +B1λ2)± φ± = A1λφ±, (3.22)
where we defined:
φ+ = 1√2
(1
1
)φ− = 1√
2
(1
−1
). (3.23)
The eigenvalues are of the form:
λ± =−1
2B1
(−A1 ±√−4M0B1 + A2
1). (3.24)
The wavefunction is:
ψ0(z) = (aeλ+z + beλ−z)φ+ + (ce−λ+z + deλ−z)φ−. (3.25)
Our wavefunction was defined on half-space with z > 0, so we apply the
Dirichlet boundary conditions that ψ(0) = 0 and that the wavefunction is nor-
malizable. This implies that the wave-function is of the form:
ψ0(z) =
a(eλ+z − eλ−z)φ+ A1/B1 < 0
c(e−λ+z − e−λ−z)φ− A1/B1 > 0. (3.26)
Now we are in the position to derive the effective Hamiltonian on the surface
of the topological insulator. The surface Hamiltonian is found by sandwiching
51
3. The strong topological insulator
the former Hamiltonian between the explicit states just found:
Hab(kx, ky) = 〈Ψa|Hz +Hxy|Ψb〉, (3.27)
where we defined the basis: Ψ ∈ Ψ↑,Ψ↓. It suffices to study the result of the
γ-matrices under this inner-product, for example we show the procedure for the
γ1-matrix. We have γ1 = σ1⊗τ1, if we calculate: 〈Ψ↑|γ1|Ψ↑〉 we get the following:
〈Ψ↑|γ1|Ψ↑〉 =(ψ0 0
)( 0 σx
σx 0
)(ψ0
0
)= 0, (3.28)
and we note that by symmetry 〈Ψ↓|γ1|Ψ↓〉 = 0, now we calculate:
〈Ψ↑|γ1|Ψ↓〉 =(ψ0 0
)( 0 σx
σx 0
)(0
ψ0
)= 〈ψ0|σxψ0〉 = αx. (3.29)
Following the same procedure we obtain the following equalities:
〈Ψ|γ1|Ψ〉 = αxσx (3.30)
〈Ψ|γ2|Ψ〉 = αxσy (3.31)
〈Ψ|γ3|Ψ〉 = αxσz (3.32)
〈Ψ|γ4|Ψ〉 = 0 (3.33)
〈Ψ|γ5|Ψ〉 = αzI2, (3.34)
where αi = 〈ψ0|σiψ0〉 is a numerical factor depending on the specific boundary
conditions and system parameters. Now the surface Hamiltonian projected on
the edge-state becomes up to O(k3):
Hsurface = C +D2k+k− − αz(M0 +B2k+k−) + A2αx(σxky − σykx). (3.35)
Since we are interested in the low-energy, long-wavelength behavior of the
system, and know that the physics of the band-inversion is completely determined
52
3. The strong topological insulator
by the physics at the Γ-point it suffices to keep only terms up to order O(k2) giving
the (2+1)d massless Dirac equation:
Hsurface = σxkx + σyky. (3.36)
Here we set the constant in front of the Hamiltonian, the Fermi-velocity,
to unity. We also changed the basis of the Clifford basis and applied a space
inversion, to make contact with the convention in the rest of this study. We can
conclude that the surface of a TI defined in half-space is effectively described by
the massless Dirac Hamiltonian on the plane. In the next section we study its
basic properties.
3.2 Dirac Operator in (2+1)D
In this section we study the symmetries of the Dirac equation by the Lorentz
group in 2+1 dimensions. From there we depart into the Dirac equation on
curved spaces, because we need to study compact surfaces surrounding the TI.
There are two different perturbations one can add to the massless-Dirac Hamilto-
nian: a mass term and a magnetic field. First we derive the Hamiltonian from the
Dirac equation and then shortly discuss the physical origin of both perturbations.
3.2.1 Dirac equation on flat space
Within the veritable tower of Babylon of Dirac Hamiltonians one of the most
frequently used in TI-literature1 is of the form:
HD = σxkx + σyky + σzm (3.37)
HD = σx(−i∂x) + σy(−i∂y) + σzm (3.38)
To derive this form, one considers the Dirac equation in flat space, with metric
1We essentially follow the notations of Ryu et al. [2010] and with a reversion of the chargewith respect to the conventions of Pnueli [1994] and Lee [2009].
53
3. The strong topological insulator
signature (+ − −), with Clifford basis σz,−iσy, iσx. This basis statisfies the
Clifford algebra:
γi, γj = 2ηij I. (3.39)
Starting with the Dirac equation we can find the required Hamiltonian:
(iγµ∂µ −m)ψ = 0 (3.40)
(iσz∂t − σy∂x − σx∂y −m)ψ = 0 (3.41)
(i∂t + iσx∂x + iσy∂y − σzm)ψ = 0 (3.42)
i∂tψ = (−iσx∂x − iσy∂y + σzm)ψ (3.43)
i∂tψ = HDψ. (3.44)
Now if we couple it to a vector potential and note the dynamical momentum
as πi = −i∂i − Ai, we find the following form:
HD =
(m −i∂x − ∂y
−i∂x + ∂y −m
)(3.45)
HD =
(m πx − iπy
πx + iπy −m
). (3.46)
3.2.1.1 Perturbations
One possible perturbation is a mass-term, which can be induced by introducing
a ferromagnetic layer onto the TI Liu et al. [2009]. As we shall see later this
is a TRS-breaking perturbation that opens a gap. If one introduces magnetic
impurities on the surface of a TI, the Ruderman-Kittel-Kasuya-Yosida (RKKY)
interaction causes a mass gap where the sign of the mass and size of the mass are
determined by the direction perpendicular to the surface and magnitude of the
magnetization. Experimentally this proposal is plagued by the problem that the
ferromagnetic coating conducts current, which destroys the topological insulator.
At this moment there hasn’t been a solution to this problem.
54
3. The strong topological insulator
Another possible perturbation one could add is a perpendicular magnetic field.
In the next chapter we show the Landau level structure that arises from adding
a constant magnetic field. This has been experimentally studied by scanning
tunneling microscopy (STM) experiments by Cheng et al. [2010], Hanaguri et al.
[2010]. The results of these experiments are displayed in figure (3.3).
Figure 3.3: The STM tunneling spectra for the surface of Bi2Se3 in a magneticfield up to B = 11T . The resonance peaks are peaks in the density of states andtherefore the signal of a Landau level. The figure is from Liu et al. [2010].
In a STM-experiment one can measure the density of states at the surface
of the material. In figure (3.3) one sees that the density of states has resonance
peaks on the sequence of√n and a resonance-peak at E = 0, which we shall see
is typical for a relativistic quantum Hall system.
Since the surface of Bi2Se3 is relativistic it seems logical that we see the rel-
ativistic Landau levels, if we subject the system to a magnetic field. However it
immediately raises an unresolved question, because the magnetic field destroys
the time-reversal invariance inside the bulk of the TI. One would expect that
the topological nature of the surface states is destroyed by the magnetic field.
However the experimental evidence shows that this is not the case. A possible
argument could lay in the fact that due to the inversion symmetry of the system,
55
3. The strong topological insulator
the parity of the points at the time-reversal momenta still defines the topolog-
ical nature of the band structure, whereas the rest of the band-structure has a
broken time-reversal symmetry. Nonetheless experimental results show that the
topological properties of Bi2Se3 are not destroyed by applying a magnetic field.
The specific differences between both perturbations are mathematically clear:
the mass-term couples through σz, wheres the magnetic vector potential couples
through the σx and σy matrices. The physics is not so clear. The ferromagnetic
layer introduces the breaking of time-reversal invariance of the surface states,
while TRS remains intact within the bulk. The magnetic field penetrates through
the bulk of the TI and therefore necessarily breaks TRS on the surface and bulk
of the material. For now we are going to treat both perturbations as possible.
3.2.2 Lorentz- and Poincare group in 2+1D
As tradition dictates we commence studying the Dirac equation with some (Lie)-
group theory. The Lorentz group is a matrix-Lie group which preserves:
ds2 = −dt2 + dx2 + dy2, on R3, (3.47)
denoted as O(1, 2). This means the orthogonal group with signature (−+ +).
With requiring that the determinant is 1 one finds the sub-group SO(1, 2).
3.2.2.1 Lorentz transformations
The group transformations, denoted Λ ∈ O(1, 2), are the Lorentz-transformations.
Like in four dimensions the Lorentz-group is not connected, namely π0(O(1, 2)) =
Z2⊕Z2, known as the Klein-four group. The symmetry operations, time-reversal,
T , and partity, P , maps Λ’s from one connected part to the other. We denote
the topologically separate parts of O(1, 2) as: 1, P, T, PT. Some remarks on
the parity operator, P , are necessary. In four dimensions this operator is usu-
ally parametrized with space inversion, one sends: r 7→ −r. However this does
not work in O(1, 2), because we have two spatial degrees of freedom rendering
spatial inversion trivial. Here one needs to reverse the orientation of the plane,
56
3. The strong topological insulator
thus P : ex ∧ ey ←→ ey ∧ ex. We denote the parts of the Lorentz group that are
continuously deformable to the identity matrix, orthochronous proper Lorentz-
transformations as SO+(1, 2).
The Lorentz-group is three dimensional and has as generators in the basis
et, ex, ey:
Boost in the x-direction: Kx =
0 i 0
i 0 0
0 0 0
(3.48)
Boost in the y-direction: Ky =
0 0 i
0 0 0
i 0 0
(3.49)
Rotation in the z-direction: Jz =
0 0 0
0 0 i
0 −i 0
. (3.50)
3.2.2.2 The Lie algebra: so(1, 2)
Considering the commutators of the generators we acquire the Lie-algebra so(1, 2):
so(1, 2) =
[Kx, Ky] = −iJz[Jz, Kx] = iKy
[Ky, Jz] = iKx.
(3.51)
This algebra looks deceptively equivalent to a standard su(2), but is certainly
not, due to the minus sign in the upper-commutator. This algebra is isomorphic
to: so(1, 2) w su(1, 1) w sp(2,R). Both these groups are double covering groups
of SO+(1, 2):
SO+(1, 2) w SU(1, 1)/Z2 and: SU(1, 1) w Sp(2,R) (3.52)
57
3. The strong topological insulator
Another very important isomorphism is the following:
SO+(1, 2) w SL(2,R)/Z2. (3.53)
3.2.2.3 The Casimir operator & representations
As with the SU(2) case we switch to the basis K± = Kx ± iKy, and get the
commutation relations:
[Jz, K±] = ±K±, [K+, K−] = −2Jz. (3.54)
The Casimir operator, analogues to the angular momentum, becomes:
C = J2z −K2
x −K2y = J2
z −1
2(K+K− +K−K+). (3.55)
Consider the basis: j mj〉, with j > 0, j ∈ R and mj > 0,m ∈ Z, which are
eigenvectors of Jz and C as follows:
C|j mj〉 = j(j − 1)|j mj〉 (3.56)
Jz|l mj〉 = (j +mj)|l mj〉. (3.57)
Describing SO+(1, 2) we have j ∈ 1, 2, . . . , for SL(2,R) j ∈ 1/2, 1, 3/2, . . . ,and for the physically irrelevant universal covering group one has j > 0 and j ∈ R.
From the lowest state |j 0〉 one can construct all the states using the ’raising’ op-
erator K+ as following:
|j mj〉 =
√Γ(2j)
m!Γ(2j +m)(K+)m|j 0〉. (3.58)
In the case of d = 3 + 1 one has three boost operators and three rotation
operators, which one complexifies into a complexified Lorentz algebra. In that
case one obtains two sets of mutually commutating su(2) w sl(2,C) algebra’s,
the irreducible representations are then classified with the pair (a, b), with a, b ∈
58
3. The strong topological insulator
0, 1/2, 1, . . . . Or more formally:
so(3, 1)⊗ C w sl(2,C)⊕ sl(2,C) (3.59)
Interestingly by the space inversion operation, P, one has the mapping P :
(a, b) 7−→ (b, a) leading to left- and right handed representations. To list the
representations in d = 3 + 1 we get:
• (0, 0): the 0-dimensional scalar representation with spin zero.
• (12, 0): the 2-dimensional left-handed spinor representation. Left-handed
spinors transform trivial under one of the sl(2,C)-groups and according to
the spin-12-representation of the other group. The right-handed representa-
tion obviously has (0, 12).
• (12, 0)⊕(0, 1
2): this is the Dirac spinor representation, which has fixed parity.
Continuing one can construct all further spinor representations of for example
the vector-boson, or spin-2 graviton. In d = 2 + 1 we are in a simpler situation
here we can complexify the algebra as follows:
so(2, 1)⊗ C w sl(2,C). (3.60)
Explicitly we have the following complexification of the Lorentz algebra:
Kx = iSx, , Sy = −iSy, and Jz = Sz, (3.61)
with the commutation relations:
[Si, Sj] = iεijkSk. (3.62)
Clearly we now obtain the representations:
• (0): the 0-dimensional scalar representation with spin zero.
• (12): the 2-dimensional spinor representation according to the spin-1
2-representation
of sl(2,C).
59
3. The strong topological insulator
• (1): the 3-dimensional spinor representation with spin 1.
Here we see clearly why the Dirac spinor in d = 2 + 1 dimensions has only
two components and that left- and right handedness does not exist. Now we can
construct the Poincare algebra.
3.2.2.4 The Poincare algebra and its representations
The Poincare group is the Lorentz-group extended with translations. Its algebra
has six generators, three from the Lorentz group and three translations (one
temporal and two spatial). First we simplify the notation of the Lorentz group
and introduce the vector corresponding to the complexified Lorentzian algebra.
We can summarize the generators into the tensor Mµν :
Mµν =
0 Kx Ky
−Kx 0 Jz
−Ky −Jz 0
. (3.63)
From this tensor we can create a vector, which we denote as Lorentzian spin:
Sµ =1
2εµνρM
νρ. (3.64)
The commutation relations of the Poincare-algebra become:
[pµ, pν ] = 0 (3.65)
[pµ, Sν ] = iεµνρpρ (3.66)
[Sµ, Sν ] = iεµνρSρ. (3.67)
Obviously p2 = pµpµ is still a Casimir operator. The 2 + 1-dimesional equiva-
lent to the Pauli-Lubanksi operator is the scalar invariant W = pµSµ, a measure
of the helicity of a state. Now we can construct the states by considering the
Casimir operators of the Poincare- and its Lorentzian sub-group: p2, S2,Wand the regular momentum operator pi, with i = x, y. We denote the eigenvalues
as:
60
3. The strong topological insulator
p2|Ψ〉 = M2|Ψ〉 (3.68)
pi|Ψ〉 = pi|Ψ〉 (3.69)
pµSµ|Ψ〉 = q|Ψ〉 (3.70)
SµSµ|Ψ〉 = s(s+ 1)|Ψ〉, (3.71)
and thus can name a state as |M,p, s, q〉. In the massless case we have that
q has to be zero, unless the spin degree of freedom are infinity (i.e. lim s 7→ ∞)
Bekaert and Boulanger [2006]. We get the following eigenvalues:
p2|0,p, s, 0〉 = 0 (3.72)
p|0,p, s, 0〉 = p|0,p, s, 0〉 (3.73)
pµSµ|0,p, s, 0〉 = 0 (3.74)
SµSµ|0,p, s, 0〉 = s(s+ 1)|0,p, s, 0〉. (3.75)
Here we see that massless states have no internal helicity degree of freedom
as in d = 3 + 1. The state is solely determined by its spin-representation and mo-
mentum. We see that the system has a chiral symmetry (e.g. parity invariant),
there is no notion of left and right. Thus a massless fermionic particle is in the
AIII (chiral unitary) class.
In the massive case we can boost to the rest frame: p = (M, 0) and get the
following eigenvalues:
p2|M, 0, s,±〉 = M2|M, 0, s,±〉 (3.76)
p|M, 0, s,±〉 = 0 (3.77)
pµSµ|M, 0, s,±〉 = ±Ms|M, 0, s,±〉 (3.78)
SµSµ|M, 0, s,±〉 = s(s+ 1)|M, 0, s,±〉. (3.79)
61
3. The strong topological insulator
Here we see that the eigenvalue of the Pauli-Lubanksi operator q = ±1, which
is determined as follows:
pµSµ|Ψ〉 =
+Ms if: p0 > 0 (particles)
−Ms if: p0 < 0 (holes or anti-particles).(3.80)
Thus we see that the sign of the energy determines the helicity of the states.
This is an example of the relation between massive fields in d-dimensions and
massless fields in d + 1-dimensions. We conclude that for a relativistic massive
particle one has the notion of left and right and chiral symmetry is broken. Thus
a massive fermionic particle in (2 + 1)d is in the A (unitary) class.
In this section we have seen that the notion of spin is quite different in one-
dimension lower, and is essentially tied to the particle or hole nature of the solu-
tion. This causes the fact that the spin-12
fields have no ↑, ↓ degree of freedom
anymore, that we are so used to while considering these fields. The spin-degree
of freedom is tied to the momentum and hence the name: ’helical metal’, for the
surface of a topological insulator.
Now we are ready to introduce curvature into the Dirac equation to faithfully
study the surface of a topological insulator.
3.2.3 Dirac equation on curved space
In a finite universe the surface of a real-world topological insulator has to be
closed. The first article studying a topological insulator in a curved space (the
sphere) was published by Lee [2009]. For describing a relativistic fermion on a
curved space one needs to invoke a spin connection gauge field. The use of the
field in the describtion of the surface of a TI was conjectured and pioneered by
Lee. Later this conjecture was proven for the sphere by deriving analytically
the surface theory Parente et al. [2010]. Another article, Dahlhaus et al. [2010],
studied the scattering on surface deformations of a TI.
The Dirac equation on curved spaces is effectively a relativistic spin-1/2 field
62
3. The strong topological insulator
in a gravitational background. There are some complications in considering the
Dirac equation on curved spaces, originating from the fact that spinors transform
under Lorentz transformations and not under general coordinate transformations.
The construction of a spin connection to define the Dirac equation with a back-
ground field is now introduced.
3.2.3.1 Spin connection formalism
This part is mainly based on Green et al. [1988] (p.224 & p.271). Say we study
a system that is parametrized on a manifold M . Any theory defined on it should
be invariant under general coordinate transformations, formally called diffeomor-
phism invariance. For example if we have a vector V µ ∈ TxM , and consider a
general coordinate transformation xµ 7→ x′µ:
V µ 7→ V ′µ =∂x′µ
∂xνV ν = αV ν , α ∈ GLds+1(R). (3.81)
Thus we see that Bose fields transform under the general linear group. To
study the topological insulator surface we have to consider spinors on M , which
can be seen as a static spacetime background. The equivalence principle implies
that we have an inertial frame in x ∈ M . Locally M is flat so TxM is a normal
Minkowski space. We introduce an orthonormal basis in the local inertial frame
TxM :
e(x)mµ dsm=0, gµν = ηmnemµ e
nν , ηnm = gµνemµ e
nν , (3.82)
known as a vielbein or tetrad. From here on we shall set the spatial dimension:
ds = 2. The Latin index is the Lorentz index, belonging to the SO(1, 2) repre-
sentation and the Greek index is the spacetime index, belonging to the GL3(R)
representation. Now at each x ∈ M we can perform a local Lorentz transforma-
tion of the vielbein:
eaµ(x) = Λab (x) · ebµ(x). (3.83)
As with any local gauge invariance we need to introduce a gauge field ωabµ (x),
called the spin connection, which is the gauge field of the local Lorentz group.
63
3. The strong topological insulator
For completeness the spin connection can be found with:
ωabµ =1
2eνa(∂µe
bν − ∂νebµ)− 1
2eνb(∂µe
aν − ∂νeaµ)− 1
2eρaeσb(∂ρeσc − ∂σeρc)ecµ. (3.84)
The covariant derivative of the metric is zero, due to the Leibniz rule, and the
covariant derivative of the vielbein is required to be zero:
Dµeaν = 0 = ∂µe
aν − Γλµνe
aν + ωaµbe
bν . (3.85)
It can be shown that with these conditions the spin connection and vielbein
are fixed. Noteworthy is that the Riemann tensor can be found as:
Raµνb = ∂µω
aνb − ∂νωaµb + [ωµ, ων ]
ab . (3.86)
Now we can formulate spinors and the Dirac operator on curved space, let
ψ(x), be such a field in the spinor representation of the Lorentz group. Then the
covariant derivative of the field is:
Dµψ = ∂µψ +1
2ωabµ Σabψ, (3.87)
where Σab are the generators of the Lorentz group given as: Σab = −(i/2)[γa, γb].
Obviously the gamma matrices obey γa, γb = 2ηab, and the curved space gamma
matrices are then found via: Γµ(x) = eaµ(x)γa. It is easy to check that they obey:
Γµ(x),Γν(x) = 2gµν(x). (3.88)
The action of a spinor with a gravitational background is:
Sψ = − i2
∫dx ψΓµDµψ (3.89)
Example Dirac operator on a sphere
After all this formalism it is time for an example that we need in the next
chapter, which is the Dirac operator on the sphere. Consider the standard metric
64
3. The strong topological insulator
on a sphere with radius R = 1:
ds2 = (dθ)2 + sin2 θ(dφ)2, with: 0 ≤ θ < 2π, and: 0 ≤ φ < π. (3.90)
The natural diagonal zweibein connecting the Lorentz indices (in Latin) and the
coordinate indices (in Greek) is of the form:
eaµ = diag(1, sin θ), with: gµν = δabeaµebν . (3.91)
If one likes tedious exercises the components of the spin connection can be
found by equation (3.84), but an easier way is to vary the action (3.89) and solve
the spin connection from the geodesic equation for the free particle. Here we just
note that the non-zero components are:
ωxyφ = −ωyxφ = − cos θ, (3.92)
in the basis of Pauli matrices (σx, σy). The generator of the Lorentz group is:
Σxy = −Σxy = − i2
[σx, σy] = σz. (3.93)
With the covariant derivative of the field:
Dµψ = ∂µψ +1
2ωabµ Σabψ, (3.94)
we obtain:
Dθ = ∂θ and: Dφ = ∂φ −iσz2
cos θ. (3.95)
The Dirac operator contracted with the curved space gamma matrices, 6D, can
be calculated with:
6D = −iΓµDµ = −ieaµσaDµ. (3.96)
With noting that the only non-trivial zweibein inverse is eyφ = sin−1 θ one
65
3. The strong topological insulator
obtains the covariant Dirac operator on the sphere:
6D = −iσx(∂θ +
cos θ
2 sin θ
)− iσy
sin θ∂φ. (3.97)
3.2.3.2 Dirac equation in isothermal coordinates
In the spacetime backgrounds we need to consider only the spatial part of space
is curved, which makes life easier. In this case it is logical to split out the time-
dependence and write the Dirac Hamiltonian:
(6D + σzm)ψ = Eψ, (3.98)
where clearly the spinor ψ is time independent. Now the Dirac operator
operates in a two-dimensional space, which has the advantages that an isothermal
coordinate system exists Chern [1955]. In this system the metric has the following
convenient form:
gµν(x) = e2η(x)
(1 0
0 1
). (3.99)
The above metric is also known as the conformal metric. η(x) is a scalar field
defined over the manifold M . In this coordinate system the Dirac operator is of
the form:
6D = −iγa(eµa(x)∂µ +
1
2eµa(x)
(∂µ ln
√g
)+
1
2
(∂µe
µa(x)
))(3.100)
The natural diagonal vielbein of an isothermal coordinate system is: eµa(x) =
e−η(x)δµa .
66
3. The strong topological insulator
Plugging this in and working out the determinant of the metric gives:
6D = −iγa(eµa(x)∂µ + eµa(x)∂µη(x) +
1
2∂µe
µa(x)
)(3.101)
6D = −iγa(e−η(x)δµa∂µ + e−η(x)δµa∂µη(x)− 1
2e−η(x)δµa∂µη(x)
)(3.102)
6D = −ie−η(x)
(0 ∂x − i∂y + 1
2(∂x − i∂y)η(x)
∂x + i∂y + 12(∂x + i∂y)η(x) 0
)(3.103)
6DA = −ie−η(x)
(0 2∂ − 2iA+ ∂η(x)
2∂ − 2iA+ ∂η(x) 0
)(3.104)
6DA =
(0 K†
K 0
)(3.105)
In the last two equations we minimally coupled the Dirac operator, −i∂µ 7→−i∂µ − Aµ to a magnetic field with vector potential A = 1/2(Ax + iAy) and A
its conjugate. We also used the notation: ∂ = 12(∂x − i∂y) and ∂ = 1
2(∂x + i∂y).
From here on we shall drop the explicit coordinate dependence of the conformal
factor η(x). In this notation the magnetic field is found as follows:
B = dA = −2ie−2η(∂A− ∂A). (3.106)
3.3 Hall Response
Now that we have fully specified the theory of the surface of a topological insulator
it is time to consider its physical responses. The most important is the Hall
response as discussed in the first chapter. This response can be calculated in a
phenomenal number of ways. In this section we shall calculate the Hall response
for the infinite plane (R2) in two different ways. First we calculate it by using the
Berry curvature as discussed in chapter 1. Subsequently we calculate it by using
the propagator of the theory derived from the Swinger proper-time representation.
This has the advantage that we can explicitly consider a chemical potential, µ,
into the problem.
67
3. The strong topological insulator
3.3.1 Hall response from the Berry Curvature
Consider the Dirac Hamiltonian on R2, with translation invariance and a mass-
perturbation:
Heff = kxσx + kyσy +mσz (3.107)
The mass-term in the Hamiltonian has broken the chiral symmetry back to a
unitary symmetry (i.e. with Cartan class A). The energy of a state is:
E = ±√k2 +m2, (3.108)
with corresponding Bloch wavefunctions:
|vi (k)〉 =1√
2E(E −m)
(k−
E −m
)(3.109)
|u•a (k)〉 =1√
2E(E +m)
(−k−E +m
), (3.110)
where the lower Bloch functions are the occupied ones and the upper ones are
unoccupied. In the next chapter 4 we discuss these solutions in more detail and
here we just remark that the |u•a (k)〉 are well defined if m > 0. If m < 0 we can
obtain the correct Bloch wavefunction with a gauge transformation. Now we can
calculate the Berry connection of the occupied states:
Ax = 〈u•a (k)∂kx |u•a (k)〉 =iky
2E(E +m)(3.111)
Ay = 〈u•a (k)∂ky |u•a (k)〉 =−iky
2E(E +m), (3.112)
where it is important to keep in mind that E2 = m2 +k2. For the correspond-
ing Berry curvature we find:
Fxy = ∂kxAy − ∂kyAx =−im2E3
. (3.113)
68
3. The strong topological insulator
In the previous chapter we elaborately saw that the Hall conductance is found
as:
σH = Ch1[F ] =i
2π
∫d2kFxy =
i
2π
∫d2k−im2E3
. (3.114)
This integral is easily solved by using polar coordinates:
σH =−1
4π
∫ 2π
0
dφ
∫ ∞0
drrm
(r2 +m2)3/2(3.115)
=−1
2
∫du
2m
u3/2(3.116)
=1
2
m√m2
, (3.117)
where we used the substitution u = r2 + m2. We see that the Hall response
depends on the sign of m and that it is half of the normal integer Hall response:
σH =1
2sign(m). (3.118)
3.3.2 Hall response from the Schwinger proper-time rep-
resentation
In this section we derive the Hall conductance for arbitrary chemical potential µ.
Consider the Dirac Hamiltonian minimally coupled to a vector potential in the
Landau gauge A = Bxy:
Heff = −i∂xσx + (−i∂y +Bx)σy +mσz (3.119)
In the next chapter we shall study this Hamiltonian, and here we just state
the result of the eigenvalues. The lowest Landau level is of the following form:
E0 =
m if: B > 0
−m if: B < 0.(3.120)
It is interesting to ask whether it is possible to chose m such that the sign of m
69
3. The strong topological insulator
and B are opposite (e.g. m < 0, such that −m is positive). Both terms are time-
reversal symmetry breaking terms, and it is hard to imagine to break time reversal
in a non-binary manner. Our intuition, based on the Lorentz force, suggests that
TRS is either left- or right-handed broken. I tried to model a system wherein a
mass domain wall and a magnetic domain wall break against each other, but this
proved to be too difficult. So for now I shall assume that both have the same sign.
We see that depending on the sign of the magnetic field, the vacuum is either
below or above the E = 0 level. If m = 0 the zeroth Landau level is fixed at E = 0.
The Hamiltonian (3.119) has two symmetries, which allows to effectively reduce
the (2 + 1)-dimensional theory to a (0 + 1)-dimensional by Swinger’s proper-time
method Schwinger [1951]. The symmetries are:
• y 7−→ y + a
• x 7−→ x+ a with the gauge transformation: Ay 7−→ Ay +Bay.
We first consider the lowest Landau level, from which we can construct the
effective Lagrangian as:
L = ψ†i∂0ψ − E0ψ†ψ, (3.121)
with ψ an anti-commuting fermionic field. This Lagrangian has as Feynman
propagator SF (k):
SF (k) =1
k0 − E0 + ik0ε, (3.122)
where as usually ε > 0 is an infinitesimally regulator, for which we take
limε→+0 after the calculation. With this propagator we can construct the particle
density:
j0 = ig(B)
∫ ∞−∞
dk0
2π
1
k0 − E0 + ik0ε, (3.123)
where g(B) is the particle density of the Landau level: g(B) = |B|/2π 1. This
1The degeneracy of a Landau level of a planar system is the number of flux quanta nφ =|B/(2π)|+ 1, but here we drop this 1 since we are working with an infinite system.
70
3. The strong topological insulator
integral is solvable by contour integration, but we first need to regularize the
integral as follows:
j0 = i|B|2π
limε→+0
limΛ→+∞
∫ Λ
−Λ
dk0
2π
1
k0 − E0 + ik0ε. (3.124)
The pole is located at:
k0 = (1− iε)E0 + O(ε2), (3.125)
where we Taylor expanded the denominator of the propagator and kept only
linear terms in ε. Now consider the contour displayed in figure (3.4). The con-
tour, C, consists of two parts, including the original domain ΓI, which is on the
real axis from −Λ to Λ. The contour is closed with the semi-circle Γ∪ in the
lower-half plane of k0.
−Λ +Λ
E0
ǫ
k0
E0 > 0 E0 < 0
k0
−Λ +Λ
E0
ǫ
Γ∪Γ∪
Figure 3.4: Possible contours in the complex plane of k0 for the eigenvalues ±E0.
Let us first calculate the case where E0 > 0:∮C
SF (k0) =
∫ΓI
SF (k0) +
∫Γ∪
SF (k0) = −2πi, (3.126)
71
3. The strong topological insulator
since we encircle the single pole clockwise once. If one calculates a full circle
centered around the pole one equally finds −2πi, the contribution of the semi-
circle can easily be found to contribute half of the result leading to:∫Γ∪
SF (k0) = −iπ. (3.127)
Combining this result with equation (3.126) and taking both limits one ob-
tains: ∫ ∞−∞
dk0SF (k0) = −iπ. (3.128)
Note that if we had chosen our contour to be closed by a semi-circle in the
upper-half plane, the contour integral over C would yield zero. In that case the
semi-circle Γ∩ yields a contribution of iπ, because it now encircles the pole anti-
clockwise. This equally yields the result −iπ for the integral over ΓI, as it should.
From figure (3.4) one can see that if E0 < 0 we need that:∫ ∞−∞
dk0SF (k0) = iπ, (3.129)
to compensate for the contribution of the semi-circle. Now combining both
results we yield:
j0 = i|B|2π
∫ ∞−∞
dk0
2πSF (k0) (3.130)
j0 = i|B|4π2
−iπ if: E0 > 0
iπ if: E0 < 0(3.131)
j0 =|B|2π
1
2sign(m). (3.132)
If we realize that this result is the charge accumulated after one flux insertion
Φ0 = B/2π we see that the Hall conductance is of the form which we previously
72
3. The strong topological insulator
also found:
σH =1
2sign(m). (3.133)
Now it is time to implement the chemical potential µ. As we shall see in the
next chapter the energy eigenvalues different from the vacuum are:
E±n = ±√
2n|B|+m2, with: n ∈ 1, 2, . . . , (3.134)
and the induced current is changed into the form:
j0 = i|B|2π
∑n
∫ ∞−∞
dk0
2π
1
k0 − (En − µ) + ik0ε, (3.135)
where we assume that µ > 0, and thus effectively did the transformation
k0 7→ k0 + µ to incorporate the finite density of particles. If we denote:
In =
∫ ∞−∞
dk0
2π
1
k0 − (En − µ) + ik0ε, (3.136)
and follow the same logic of the vacuum case we obtain:
In =
−i/2 En − µ > 0
i/2 En − µ < 0. (3.137)
First we concentrate on all the energy levels except the E0 level. Its con-
tribution to the sum over In in (3.135) is simple. Consider the partial sum of
particle-hole symmetric terms: In + I−n, we then get:
In + I−n =
−i if: µ > |En|0 if: µ < |En|,
(3.138)
leading to the contribution to j0 of the form:
j0non-vacuum =
|B|2π
∞∑n=1
θ(µ− En), (3.139)
where θ(x) is the Heavyside step function. Now we consider the vacuum level.
If µ > |E0| the integral always yields i/2 and is thus independent of the sign of m.
73
3. The strong topological insulator
Whereas if µ < |E0| it is dependent on the sign and we yield the same expression
of the parity anomalous-Hall conductivity. We can summarize this as:
j0vacuum =
|B|2π
1
2
(sign(m)θ(|m| − µ) + θ(µ− |m|)
). (3.140)
Now we found the complete expression for the induced charge after one flux
insertion as:
j0 =|B|2π
1
2
(sign(m)θ(|m| − µ) + θ(µ− |m|) + 2
∞∑n=1
θ(µ− En)
). (3.141)
This result was first found by Lykken et al. [1990] by solving the associated
Green’s function and later by Schakel [1991] by using the same method as here.
Before we discuss this result we note that the Hall conductivity is now found as:
σH =
(1
2θ(µ− |m|) +
∞∑n=1
θ(µ− En)︸ ︷︷ ︸ +1
2sign(m)θ(|m| − µ)
)e
Φ0
.︸ ︷︷ ︸parity-normal term jnorm parity-anomalous term janomaly
(3.142)
We see that the Hall effect for the relativistic (2+1)d surface of the topological
insulators consists of two separate contributions. The first term is parity normal
and reflects the Landau level structure discussed in the next chapter. The key
property of this structure is that the lowest Landau level has half the number of
states than the other Landau levels, this is reflected by the fact that it is of the
form n+ 1/2, where n is the number of Landau levels below the Fermi energy.
The second term is the parity-anomalous term that can be found by the Pauli-
Villars regularization of the path integral and that we calculated with the first
Chern number of the Bloch wavefunctions. This term is the true topological term,
which can be described by the parity breaking Chern-Simons term. From Ryu
et al. [2010] we expect that this term shall evade Anderson localization and the
first term not, this has however, to our knowledge, not been shown in the litera-
74
3. The strong topological insulator
ture directly. But there is (circumstantial) evidence in the literature supporting
this view that deals with disordered graphene Nomura et al. [2008]. In the main
Figure 3.5: Disorder-averaged Hall conductivity σxy as a function of the fillingfraction ν for the dimensionless disorder strength: a) 0.4, b) 0.7 c) 1.1 The figureis from Nomura et al. [2008].
diagram of figure (3.5) the Hall conductivity is displayed for three strengths of
disorder. Here one sees clearly the vanishing of the Hall plateaus for stronger
disorder except for the plateaus at ν = ±1/2. This research is relevant for the
topological insulator because the spin channel and inter-valley scattering chan-
nels are neglected. Basically this implies that this result applies to the topological
insulator, since it has no valley degeneracy and spin channels as well.
The experimental result we displayed earlier in figure (3.3) does not support
the argument. Here it was shown that with STM-experiments one can measure
up to the 10-th Landau level. This suggests strongly that the parity normal term
ıs robust against disorder.
Finally there is a interesting question that arises from equation (3.142). An
75
3. The strong topological insulator
anomaly that arises in a theory is induced by the necessary regularization of the
path integral. In this sense it is reasonable to expect that a massless femion in
(2 + 1)d acquires a (small) mass term due to the parity anomaly and thus is the
theory always equipped with the anomalous Hall conductance, which is robust
against disorder. However from equation (3.142) we see that to observe this Hall
conductance one needs to be able to drive the chemical potential to |m| > µ. As
Pruisken puts it: the chemical potential, or Fermi energy at T = 0, is merely the
energy of the last electron added to the system. In that perspective is placing the
Femi level in an energy gap a very academic consideration since there are no states
available for electrons. Combining these insights, the experimental accessibility
of the parity anomalous Hall conductivity depends on the size of the induced
mass gap, either by the RKKY -interaction or by the anomaly itself, versus the
ability to drive the chemical potential to E = 0, which is experimentally hard.
3.4 Conclusion
With the theoretical framework of the first chapter we embarked on the physics
of the (3+1)d topological insulator. We discussed its origin in material science by
discussing the prototype Bi2Se3 and derived its Dirac-type surface theory. Then
we studied some of the interesting characteristics of the (2 + 1)d Dirac theory
and finally we studied its Hall response. In the next chapters we focus on this
Hall conductance by studying different configurations as the sphere, plane and
cylinder in different magnetic fields and mass configurations.
76
Chapter 4
The toy model collection of
Landau problems
4.1 Overview of the models and field configura-
tions
In this chapter we systematically study a collection of systems with different field
configurations. The central goal is to see the structure between the Laplacian
and the Dirac operator in these different settings and the role of the vector po-
tential in combination with non-trivial gauge transformations. The Laplacian,
∆, describes a standard non-critical (2 + 1)d electron-gas, corresponding to the
iqHe-effect and integer topological quanta. The Dirac operator, 6D, describes a
relativistic massless fermionic degree of freedom, often referred to as a helical
metal, with binary topological quanta.
We study two topologically trivial geometries: the flat geometry and the spher-
ical geometry. The field configurations we consider are depicted in figure (4.1).
The right figures correspond to a constant field configuration with respect to the
area two-form of the surface itself. This results in the standard Landau problem
for the flat geometry and the spherical Landau problem on a magnetic monopole
solved by Haldane [1983]. The second configuration we consider, depicted on the
left side in (4.1), is a sphere placed in a constant magnetic field. The magnetic
77
4. The toy model collection of Landau problems
field is constant with respect to the metric of the space wherein our surface is
embedded. This leads to a magnetic field with a domain wall in it.
Figure 4.1: An overview of the two field configurations of the magnetic field.The left column is the field corresponding to a sphere immersed in a constantmagnetic field, leading to a kinked field configuration with respect to the areatwo-form of the immersed surface. The right column corresponds to a constantfield configuration with respect to the area two-form itself, leading to a monopoleconfiguration on the sphere.
Of the constant field configuration the exact spectrum and wavefunctions
are known and listed in figure (4.2), which we shall derive. Of the domain wall
configurations hardly anything is known, as far as we have found in the literature,
and we rely on an instanton analyses. We also solve the spectra numerically.
78
4. The toy model collection of Landau problems
Figure 4.2: An overview of all the toy models with constant magnetic fields,including spectra, wavefunctions of the lowest Landau level and ground statedegeneracy.
4.1.1 Relation between the Dirac and Schrodinger opera-
tor
In this section we shall derive some powerful results that relate the Dirac operator
with the Schrodinger operator, which we need in this chapter. Normally one is
used to the fact that the square of the Dirac operator yields the Klein-Gordon
equation. In our Hamiltonian setup this translates to the fact that the square
of the Dirac operator is equal to minus the Laplacian, which is the Schrodinger
Hamiltonian. Coupling spatial dependent connections to the Dirac operator how-
ever causes the derivative operator to fall on the connections so that 6D2 is not
longer proportional to the Laplacian. Happily not all structure is lost so that
the tools we have for solving the Laplacian remain available for solving Dirac
equations with magnetic fields and curvature. Here we shall study this remaining
structure in some detail, found by Pnueli [1994].
In the previous chapter we showed the form of the Dirac operator in isothermal
79
4. The toy model collection of Landau problems
coordinates. Consider the square of the Dirac operator:
6D2A =
(K†K 0
0 KK†
), (4.1)
with:
K† = −ie−η(
2∂ − 2iA+ ∂η
)(4.2)
K = −ie−η(
2∂ − 2iA+ ∂η
), (4.3)
with all the same definitions of the last chapter. Let us work out the first
term of the 6D2, it is unfortunately somewhat tedious, but sure worthwhile the
effort:
K†K = −e−η(
2∂ − 2iA+ ∂η
)e−η(
2∂ − 2iA+ ∂η
)(4.4)
= −e−2η
(− 2∂η
(2∂ − 2iA+ ∂η
)+ (4.5)
+
(2∂ − 2iA+ ∂η
)(2∂ − 2iA+ ∂η
)). (4.6)
Collecting all the terms gives:
K†K = −e−2η
(4∂∂ − (∂η)(∂η) + 2∂∂η + 2
((∂η)∂ − (∂η)∂
)(4.7)
+2i
(A(∂η)− A(∂η)
)− 4i
(A∂ + A∂
)− 4i∂A− 4AA
). (4.8)
Now we can compare this with the Schrodinger equation with a different
magnetic field B′ and vector potential A′ in isothermal coordinates:
80
4. The toy model collection of Landau problems
Hs(B′) = −e−2η
((∂x − iAx)2 + (∂y − iAy)2
)(4.9)
= −e−2η
(4∂∂ − 4i(A′∂ + A′∂)− 2i(∂A′ + ∂A′)− 4A′A′
).(4.10)
It is well known that curvature manifests itself as an effective magnetic field,
which can be understood from the Berry phase. The changing normal of the
surface due to the curvature induces a Berry-connection one-form that can be
absorbed into the magnetic field one-form.
To find the form of the curvature term added to the magnetic field we first
denote that the Gaussian curvature in an isothermal coordinate system is given
as:
k = −4e−2η∂∂η. (4.11)
If we now change the vector potential as:
A′ 7→ A+ iα∂η, A′ 7→ A− iα∂η, (4.12)
then the magnetic field changes as:
B′ = dA′ (4.13)
= −2ie−2η(∂A′ − ∂A′) (4.14)
= −2ie−2η(∂A− ∂A) + 2αe−eta∂∂η (4.15)
= B − αk. (4.16)
Now looking at the form of K†K, we see that in any case the added term
needs to cancel the term (∂η)(∂η) with the term −4A′A′. This suggests we need
to set α = 12. Let’s look how close we got by evaluating the following difference:
81
4. The toy model collection of Landau problems
K†K(B)−Hs(B −1
2k) = −e−2η
(2∂∂η − 2i
(∂A− ∂A
))(4.17)
= −B +1
2k. (4.18)
We see that it is possible to completely express the square of the Dirac operator
in terms of the Schrodinger Hamiltonian. Following the same procedure one finds
for the other spinor component:
KK†(B) = Hs(B +1
2k) +B +
k
2. (4.19)
These results can be summarized as:
6D2 =
(Hs(B − 1
2k)− (B − 1
2k) 0
0 Hs(B + 12) + (B + 1
2k)
)(4.20)
This is the generalization of the ’rule’ that squaring the Dirac operator leads
to the Klein-Gordon equation. We shall exploit this result to find the energy
eigenvalues of the different systems on the sphere.
4.2 Landau problem on the plane
The Landau problem has approximately been discussed by 2n authors in (2n +
1) different ways Avron and Pnueli [1992]. Despite this ’fact’ we like to start
our collection with discussing this basic case, for the sake of completeness and
introduction of notation and apparathus. The characteristics of the infinite plane
of course does not require much introduction. It can be parametrized with R2 'C. And its fundamental group is trivial, π1(C) = 1.
4.2.1 Non-relativistic
In quantum mechanics the partial derivative is changed as: ∂i 7→ −i∂i, due to
the deformation of the Poisson bracket. Therefore the Schrodinger equation for
82
4. The toy model collection of Landau problems
a free particle, with mass m = 2, on the plane is minus the Laplacian:
i∂tφ = Hsφ = −∆φ (4.21)
If we minimally couple a magnetic field to our system, we change the partial
derivative into a covariant one, which is often dubbed the dynamical momentum
operator:
πi = −i∂i − Ai. (4.22)
The Hamiltonian becomes:
Hs(B)φ = π2φ = π · (πφ) = Eφ. (4.23)
Verifying the commutation relation of the two orthogonal components of the
dynamical momentum one finds:
[πx, πy]φ = −i(∂xAy − ∂yAx)φ = iBz(x, y)φ, (4.24)
where Bz denotes the magnetic field perpendicular to the plane (z = ex ∧ ey),and we often abbreviate it to B.
4.2.1.1 Zero field
In zero field both components commute and the Schrodinger equation reduces to:
Hs(0)φ = π2φ = −(∂2x + ∂2
y)φ = Eφ. (4.25)
The eigenstates can be labelled with (kx, ky) ∈ R2 and we obtain the spectrum
and eigenstates as:
E = k2 φ(r) = φ0eik·r. (4.26)
The degeneracy of states are circles of equal radius in k-space : rk =√k2x + k2
y.
83
4. The toy model collection of Landau problems
4.2.1.2 Constant field
Now we apply a perpendicular magnetic field to the plane with a constant flux
density: B = B0z, with B > 0. Both components cease to commute and we
rewrite the Hamiltonian as following:
Hs(B)φ = Eφ (4.27)
(π2x + π2
y)φ = Eφ (4.28)
1
2
πx + iπy), (πx − iπy)
φ = Eφ (4.29)
B
a†, a
φ = Eφ. (4.30)
In the third row we factorized the Hamiltonian into two components and sym-
metrized it with the anti-commutator. In the last line we defined the operators:
a = 1√2B
(πx − iπy) a = −√
2B4
(z + 4B∂)
a† = 1√2B
(πx + iπy) a† =√
2B4
(z − 4B∂).
(4.31)
As the notation suggest these operators obey bosonic harmonic oscillator com-
mutation relations: [a, a†] = 1, and for later reference we gave its complex co-
ordinate representation on the right hand side. Using this commutation relation
the Hamiltonian and energy spectrum become:
Hs(B)φ = Eφ = B(2a†a+ 1)φ = B(2n+ 1)φ. (4.32)
One of the continuous parameters labelling the states of the spectrum of the free
particle, (kx, ky), is transformed into a bosonic occupation number n = a†a ∈ Z+
of the harmonic oscillator. The energy levels became degenerate.
To find the degeneracy we unfortunately have to leave gauge-invariant grounds.
Basically there are two ways of doing this, which have necessarily the same phys-
ical observables.
84
4. The toy model collection of Landau problems
Symmetric Gauge
In the symmetric gauge, we solve the problem by using the rotational sym-
metry in the Hamiltonian. In this way the angular momentum becomes a good
quantum number and to find it we first introduce guiding-center coordinate op-
erators: rx, ry. To appreciate these operators, recall that classically the electrons
trace out circular orbits, known as Landau orbits, with a trajectory:
r =(rx +
πyB, ry −
πxB
). (4.33)
Which leads us to adopt the following definition for the guiding coordinates:
rx = x− πyB
ry = y +πxB. (4.34)
The commutation relations of these guiding coordinates are easily found as:
[rx, ry] =i
B, [ri, πj] = 0. (4.35)
Now we have two commuting pairs of harmonic oscillator operators. We define
in the same spirit as for the π-operator, the new operators:
b =√
B2
(rx + iry) b =√
2B4
(z + 4B∂)
b† =√
B2
(rx − iry) b† =√
2B4
(z − 4B∂)
. (4.36)
To find the angular momentum operator perpendicular to the plane, Lz =
xpy − ypx, we have to chose a gauge:
A =B
2(−yx+ xy) (4.37)
=B
2
((ry −
πxB
)x+ (rx +πyB
)y. (4.38)
In this gauge the Lz operator becomes:
85
4. The toy model collection of Landau problems
Lz =1
2B(π2
x + π2y)−B(r2
x + r2y) (4.39)
= a†a− b†b (4.40)
= z∂ − z∂. (4.41)
The problem is now written in the form of two bosonic harmonic oscillators, a
and b, satisfying [a, a†] = 1, [b, b†] = 1, and the a- and b-operators are commuting
amongst each other. If we denote the states with |n m〉, we get the following
relations:
Hs(B)|n m〉 = B(2n+ 1)|n m〉, (4.42)
Lz|n m〉 = (n−m)|n m〉. (4.43)
Now we can focus on the wavefunctions, and for simplicity let us consider
the Landau level with n = 0, known as the lowest Landau level (LLL). These
wavefunctions are annihilated by a, and the solutions to the differential equation
aφ = 0 are:
φ0 = h(z)e−Bzz4 . (4.44)
The analytic function can be any polynomial in the orthogonal basis zmm=∞m=0 ,
which are all linear independent, and are eigenstates of the angular momentum
operator Lz.
By using the Schwinger construction of bosonic oscillators we were able to
construct the complete spectrum and its eigenfunctions in the symmetric gauge.
Landau Gauge
Although this treatment is beaten to death, for many of us, I’m inclined to
discuss it briefly for later reference. In the original Landau gauge: A = Bx y one
diagonalizes not the angular momentum within the Landau level, but the linear
momentum. Hence one uses the translational symmetry of the Hamiltonian as a
86
4. The toy model collection of Landau problems
method to solve the problem. In this case the Hamiltonian is translational invari-
ant in the y-direction, so we can substitute (−i∂y) 7→ ky. In all of the following
problems where we use this translational invariance we drop the subscript and
denote ky as k.
Using this gauge the Hamiltonian is of the form:
Hs(B)φ = (−i∂x)2φ+ (−i∂y +Bx)2φ (4.45)
= −∂2xφ+B2(k/B + x)2φ, (4.46)
which is just a one-parameter, k, family of harmonic oscillators with normal-
izable solutions:
φnk(r) = eik yHn
(√B(k/B + x)
)e−
B2
(k/B+x)2
. (4.47)
We see that in another gauge the problem has again a harmonic oscillator
spectrum, the Landau levels, but now the states within the levels do not label
the angular momentum, m but rather linear momentum k.
The mind boggling part is that k ∈ R, which is an uncountable infinite set,
and m ∈ Z, which is a countable infinite set, give the same number of states once
the system is regularized. This property is needed because the number of states
is a gauge invariant property. This feature of uncountable infinite sets being in
conjunction with countable sets is an interesting feature that is more common in
physics.
4.2.1.3 Constant field of compact support
We saw that the degeneracy was infinite for each Landau level. In this section we
consider the degeneracy of the lowest Landau level for a finite system. The result
is well-known and the derivation here works, but is not intended to be rigorous.
Suppose that our system is a bounded disc with area of A = πR2. The m-th
87
4. The toy model collection of Landau problems
wavefunction in the symmetric gauge in polar coordinates is:
φ0m(ρ, θ) = ρmeimθe−Bρ2
4 . (4.48)
Clearly this function has a Gaussian character in the ρ-direction, by differen-
tiation with respect to ρ we find the mean at:
ρ =
√2m
B, (4.49)
with Gaussian-spread of 4/B. Now we can turn it around as follows:
m =Bρ2
2. (4.50)
It is quite reasonable to demand that in the finite system with area A, there
are no states that have a mean radius larger than the system. This suggests that
we can set m to be the largest eigenvalue for m. Then if we equate the mean
radius of the most outer state ρ with the radius of the system we get the condition
that the largest eigenstate m obeys:
m =BA
2π. (4.51)
This formula has a nice interpretation. Recall that in our units the flux
quantum is Φ0 = 2π and that B is the flux density, this implies that:
nφ =BA
2π, (4.52)
is the total number of flux quanta penetrating through the surface. Thus we
can conclude that the degeneracy of the lowest Landau level d0[Hs(B)] is given
as:
d0[Hs(B)] = nφ + 1, (4.53)
because the basis of eigenfunctions of the ground state are labelled m ∈0, 1, . . . , nφ. For a rectangular domain in the Landau gauge one gets the same
result, but this is left to the reader. The ground state degeneracy has some very
nice geometrical aspects to it, where we shall touch upon later in this chapter.
88
4. The toy model collection of Landau problems
4.2.1.4 Field with a single magnetic domain wall
This field configuration is naturally interpretted as a (2+1)d non-relativistic elec-
tron gas living on a sphere embedded in an environment with a constant mag-
netic field. Through the stereographical projection this naturally leads to the
picture of the magnetic island of figure (4.1). First we note that now we have
[πx(r), πy(r)] = iB(r), where the field can vary over the plane. We still can write
the Hamiltonian as:
Hs(B(r)) =1
2
πx(r) + iπy(r), πx(r)− iπy(r)
. (4.54)
We now define the operators:
a(r) = 1√2B(r)
(πx(r)− iπy(r))
a†(r) = 1√2B(r)
(πx(r) + iπy(r)) .(4.55)
Now we are going to find the Hamiltonian for an arbitrary perpendicular field.
In the following we suppress the variables and assume the field only depends on
x:
Hs(B(r)) =1
2
πx + iπy, πx − iπy
(4.56)
Hs(B(r)) =1
2
√2Ba†,
√2Ba
(4.57)
Hs(B(r)) =1
2
(|2B|a†, a+
√2B([a†,
√2B]a+ [a,
√2B]a†)
). (4.58)
To continue we calculate the relevant commutators, where in the last equality
89
4. The toy model collection of Landau problems
we assumed that the field only depends on x:
[a†(r),
√2B(r′)
]= −i
√2
B(r)
(∂√
2B(r)
)δ(r− r′) = −i
|2B(x))|
(∂xB(x)
)δ(r− r′)
[a(r),
√2B(r′)
]= −i
√2
B(r)
(∂√
2B(r)
)δ(r− r′) = −i
|2B(x)|
(∂xB(x)
)δ(r − r′)
[a(r), a†(r)
]= 1 + 2i
(a(r)∂ 1√
2B(r)− a†(r)∂ 1√
2B(r)
)= 1 + i
(2B(x))3/2
(∂xB(x)
)×
×(a(x)− a†(x)
)(4.59)
.
The last commutator is exactly what we would expect; for a constant field it
is equal to unity and it induces Landau level mixing proportional to the gradient.
Using these commutators we find for the general Hamiltonian the following
form:
Hs(B(r)) =1
2
(|2B|a†, a+
√2B([a†,
√2B]a+ [a,
√2B]a†)
)(4.60)
Hs(B(r)) = |B|(2a†a+ 1)− i√2B
(∂xB)a† (4.61)
We find the regular Hamiltonian with an additional mixing term proportional
to the gradient of B. Now we can make this more specific we consider a field that
has a straight domain wall at x = 0 for simplicity. If we chose the magnetic island,
or the radius of the sphere, large enough we can always do this approximation.
We parametrize the magnetic field as:
Bz(x) = B0 tanh(x/w), (4.62)
where B0 is the magnitude of the field and w the width of the domain wall.
With the Maxwell equation ∇ × B = 4πj, where we tossed the Maxwell term,
90
4. The toy model collection of Landau problems
due to the non-relativistic limit, we find a current of the form:
Jy(x) =B0
w(cosh(x/w)2= ∇B. (4.63)
Thus on the domain wall we expect a one-way propagating current, with
a gaussian type of spread. Away from the domain wall ∇B is exponentially
suppressed and we retrieve our normal Hall-liquids. Now we focus on the physics
happening at the domain wall. In figure (4.3) the wall is depicted with the relevant
diagrams.
Figure 4.3: Diagram of the domain wall field configuration and relevant functions.
B⊙ B⊗
Figure 4.4: Illustration of the three types of classical orbits.
91
4. The toy model collection of Landau problems
Before we dive into quantum mechanical no-mans-land 1 we focus on a classical
non-quantum mechanical particle in such a field. There are basically three types
of orbits a particle can follow:
• A particle that starts at the domain wall with no momentum perpendicular
to the domain wall shall continue to follow its straight trajectory along the
domain wall.
• Particles far from the domain wall shall circulate in their standard cyclotron
orbits.
• Finally particles that are close to the domain wall shall try to complete their
cyclotron orbit but encounter a sign flip of the Lorentz-force when they
cross the domain wall. Therefore they oscillate back- and forth between
both domains while having a net momentum in the direction along the
domain-wall.
These types of orbits are schematically drawn in figure (4.4). The goal of
this thesis is to study the quantum Hall effect of the topological insulator, thus
unfortunately we have to leave this classical picture, while there are still a lot of
open questions such as: what are the fixed points, attractors, basins of attraction,
etc.
The Schrodinger equation for a particle in the tangent hyperbolic magnetic
field in Landau gauge is:
Hs(B(x)) = (−i∂x)2 + (−i∂y + Ay)2. (4.64)
The vector potential for the magnetic field B0tanh(x/w) in Landau gauge is:
Ay = B0w ln(cosh(x/w)). (4.65)
1Remarkably no literature was found that dealt with this problem, but perhaps that saysmore about this author than the literature.
92
4. The toy model collection of Landau problems
When we split off the y-dependency by using φ = fk(x)eik·y we find the fol-
lowing Hamiltonian:
Hs(B(x)) = −∂2x +
(k +B0w ln(cosh(x/w))
)2
(4.66)
From figure one can see that we essentially got rid of the magnetic field and
found just the problem of a double-well potential depending on the y-momentum!
-4
-2
0
2
4
x
-10
-5
0
5
10
k
0
50
100
VHx,k
Figure 4.5: The effective potential for a particle in Landau gauge, V (x, k) =(k+B0w ln(cosh(x/w))
)2
At k = 0 there is a minimum, that for k ≤ 0 branches
in two parts.
From this picture alone, we can already draw the following conclusions:
• We see that the y-momentum continuously deforms the potential between
a single-well and a double-well.
• When k << 0 the minima are very separated and the states in both wells
become degenerate. In this case one is deep in one of the Landau levels.
• As k 7→ 0 the minima approach each other, and tunneling becomes possible.
Instantons emerge and this induces a band splitting between the even and
odd states.
93
4. The toy model collection of Landau problems
• As k >> 0 we have a harmonic-oscillator like system with a splitting be-
tween even and odd states.
Before we continue, these properties can be modelled with a very simple toy-
model suggested by my supervisor prof.dr. K. Schoutens.
Example Band splitting in the infinite well
Consider a particle in a infinite square well of size −π ≤ x ≤ π, with a
deltafunction gδ(x) in the middle of the well:
H(g)φ = −∂2xφ+ V (x, g)φ = Eφ (4.67)
V (x, g) =
gδ(x) if −π ≤ x ≤ π
∞ everywhere else.(4.68)
At g = 0, we have two types of solutions that are ordered according to the
reflection operator P : x 7→ −x, into even and odd solutions as:
φn,even(x) = 1√2π
cos((n+ 1/2)x), En = (n+ 1/2)2, n ∈ 0, 1, 2, · · ·φn,odd(x) = 1√
2πsin((n+ 1)x), En = (n+ 1)2, n ∈ 0, 1, 2, · · ·
(4.69)
−π π0
φ1,evenφ1,odd
0 ∞
V (x) with g = 0 V (x) with g = ∞
g !→x !→ x !→
1/4
4/4
9/4
16/4
φ1,even
φ1,odd
φ1,even
φ1,odd
φ2,even
φ2,odd
−π π0
Spec(H(g))
Figure 4.6: Wavefunctions and spectrum of the infinite well with a delta functiongδ(x) in the middle.
94
4. The toy model collection of Landau problems
We nicely see that we get an even sequence of eigenstates with a lower energy
than the odd sequence. If we switch on the delta potential, we suppress the
solutions to have a small value at x = 0. The odd sequence has already a
zero-modulus at x = 0, but the even function changes. If we consider the case
where g 7→ ∞, the even state and odd state become physically degenerate, while
remaining eigenfunction of the reflection operator, as displayed in figure (4.6).
From the example we expect a Landau-band spectrum which is doubly de-
generate at k << 0, and shows a splitting as k approaches zero. For an analyses
of this double-well potential problem we can proceed with a standard routine of
finding the classical solutions in the wells and the associated instanton solutions,
describing tunneling events between the minima, which can be found in detail in
Zinn-Justin [2005] and Altland and Simons [2006].
Close to x = 0 we can make the linear approximation and write the Hamilto-
nian in Landau gauge as:
Hs(B(r)) = −∂2x + (k +
B0
2wx2)2 = −∂2
x + (αx2 + k)2, (4.70)
where we renamed B0/2w = α. The action corresponding to (4.70) is:
S[φ] =
∫M
dtdx ∂xφ∗∂xφ− (αx2 + k)2φ∗φ. (4.71)
With variation of this action one finds that the classical solutions solve the
saddle-point equations:
−mx+ V ′(x) = 0, (4.72)
with corresponding classical solutions:
qcl = ±√−kα
= ±√πm
αL, m ∈ N, (4.73)
where we regularized k in the second equation. Remark that we see that for
a classical solution to exists we require that k ≤ 0 or that m ∈ N, reflecting the
one-way propagating nature of the domain wall.
95
4. The toy model collection of Landau problems
In both minima we can build towers of harmonic oscillator states, but in-
evitably the levels shall split in bonding (symmetric) and anti-bonding (anti-
symmetric) states by the tunnelling processes back and forth. So effectively both
harmonic oscillators hybridize. To find the size of splitting we consider the action
of one instanton (a tunnelling event):
Sinstanton =
∫ τ
0
dτ ′m
2q2cl + V (qcl) =
∫dτ ′
dqcldτ ′
mqcl =
∫ a
−adq√
2mV (q), (4.74)
where we denoted the solutions in the potential minima as ±a. Performing
the integral over the action we find:
Sinstanton =8
3
√−k3
2α, with: k ≤ 0. (4.75)
Now by using the dilute instanton gas assumption, one finds the following
relation to the action of the instanton and the energy-scale of the band splitting:
δε = ηe−Sinstanton = ηe−83
√−k3
α . (4.76)
Figure 4.7: Diagram of the bandsplitting δε as a function of momentum k in threeorders of magnitude of α = B0/w, α ∈ 0.1, 1.0, 10.
We see that if the width of the domain wall is large compared to the field
magnitude, parametrized with α = B0/w, that the tunnelling between the clas-
sical solutions on both sides of the domain wall is quickly supressed with larger
y-momenta. What we now want to check is the proportionality between the
96
4. The toy model collection of Landau problems
oscillator spacing ω and bandsplitting δε. The constant can be found to be as1:
η = ω
√Sinstanton
2π
√det(J) (4.77)
η =8
2π
(−k5B0
w
)1/4
. (4.78)
Thus we find that the bandsplitting term due to the tunnel is as:
δε = ηe−Sinstanton =8
2π
(−2k5B0
w
)1/4
e− 8
3
√−wk3
B0 , (4.79)
The oscillator spacing is mω2 = V ′′(qcl) and becomes:
ω =√
8kα = 2
√B0k
w. (4.80)
Thus the spectrum around the domain wall, for k ≤ 0 is found as:
En,k = ω(k)(n+1
2)± δε(k). (4.81)
This equation has a limited scope. First of all as k 7→ 0 it goes to zero, the
analyses becomes invalid because we loose our double-well potential and should
analyze the problem with a WKB-method. Also if k becomes large the linear
approximation breaks down. However for small k this formula gives the correct
behaviour.
Let us now focus on the numerical results of the spectrum. We obtained the
results by using the so called ’shooting method’. This method is a way to obtain
solutions of a boundary value problem, which is a differential equation with values
specified at the boundary. In such a problem you always have at least one free
parameter to chose. With a smart guess you chose a set of initial conditions and
integrate your differential equation to the other end of the boundary. In most
situations the miss-match at the boundary with the required boundary values
1For more details see the earlier mentioned references.
97
4. The toy model collection of Landau problems
defines a system of equations in conjunction with the guessed initial values. In
that way one can find the correct initial conditions that solve the boundary value
problem. For more details we refer to Press et al. [2007].
Our eigenvalue problem has to be translated into a boundary value problem.
To find the eigenvalues we need to find the eigenfunctions, which are normalizable.
For this we use the fact that the potential obeys the symmetry:
V (x) = V (−x). (4.82)
Therefore we can separate our solutions into odd and even functions. Note
also that the potential goes to infinity as |x| becomes large. Then we can always
choose an L such that the eigenfunctions obey φ(|L|) = 0 and φ′(|L|) = δ, with
δ > 0 as initial conditions. Then we have for the boundary conditions at x = 0:
φodd(0) = 0 (4.83)
φ′even(0) = 0. (4.84)
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æ æ æææ
æ
æ
æ
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ààà
à
à
à
à
à
à
ì ì ì ì ì ì ì ì ì ì ììì
ì
ì
ì
ì
ò ò ò ò ò ò ò ò ò
òò
ò
ò
ò
ò
ò
ò
-5 -4 -3 -2 -1 1k
2
4
6
8
E
Figure 4.8: Diagram of the spectrum of the two lowest Landau levels for B = 1and w = 1 and as a function of k.
98
4. The toy model collection of Landau problems
We solved the spectra using Mathematica 7 mat. We see that as k << 0
the energy levels are degenerate and the particle is ’deep’ in a Landau level. As
k approaches zero the symmetric eigenfunction lowers its energy due to the hy-
bridization. At k > 0 one has the fourth-order potential problem where the even
and the odd states are non-degenerate.
Next we consider the case of compactifying the y-direction into a circle with
a circumference of 2π. The plane is transformed into a cylinder and set m = k/π
with m ∈ Z. Like with the particle in a ring example we can now think of
adiabatically driving one flux quantum through the cylinder. Then all the states
are mapped m 7→ m + 1. The eigenstates in state m have a large expectation
value at both ±〈(qcl)m〉. Under this adiabatic flux insertion this is mapped to
±〈(qcl)m+1〉, thus these states effectively move towards to the domain wall. If one
would thread the flux in the opposite direction, the states flow away from the
domain wall.
+1
2Q−Q+
1
2Q
−2Q+Q +Q
+Q
−Q = −2Q+Q
Ef
+1
2Q +
1
2Q−Q
a) b) c)
Figure 4.9: a) For every Landau level the system has two edge modes. b) Underthreading a flux through the cylinder two electrons are transported to the domainwall. The ends of the cylinder have a charge of +Q, which are a superpositions ofthe odd and even states. c) Under the mapping w = ln(z) the cylinder becomesa plane, if one shrinks the domain wall to z = 0 we recover the standard pictureof the iqHe.
In figure (4.9) we displayed the effect of the adiabatic flux insertion. We can
conclude that for every Landau level underneath the Fermi energy, two electrons
are transported to the domain wall, signaled by the two edge modes in the spec-
trum. Interesting to see is that the excitations at the end of the cylinder are
actually superpositions of the odd and even states. By transforming the system
back to the plane, and shrinking the domain wall to the origin, we arrived at the
99
4. The toy model collection of Landau problems
standard iqHe response of a flux insertion. The accumulated charge at the origin
becomes: −2Q + Q = −Q and at the edge of the disc we have the charge +Q.
We see that we have recovered the Hall response:
σH = νe
Φ0
, with: ν ∈ Z. (4.85)
4.2.2 Relativistic Landau problem
In this section we study the same magnetic configurations for the Dirac equation.
Recall that we found the Dirac Hamiltonian to be of the form:
HD =
(m −i∂x − ∂y
−i∂x + ∂y −m
)(4.86)
HD =
(m πx − iπy
πx + iπy −m
)(4.87)
HD =
(m
√2Ba†√
2Ba −m
). (4.88)
Where we used the dynamical momentum operators i∂i − Ai = πi and the
Schwinger bosonic operators.
4.2.2.1 Zero field
In this section we consider a zero magnetic field and a constant mass term. The
easiest way is to consider the square of the dirac operator:
H2Dψ = E2ψ (4.89)
H2Dψ =
(m −i∂x − ∂y
−i∂x + ∂y −m
)2
ψ (4.90)
H2Dψ =
(m2 −∇2 0
0 m2 −∇2
)ψ. (4.91)
(4.92)
100
4. The toy model collection of Landau problems
In zero field we have the energy eigenvalues: E± = ±√k2 +m2, which cor-
responds to conduction-band electrons (+), or valence-band holes (−) and this
spectrum is usually called the Dirac cone. We can separate variables as ψ±(x) =
ξ±(k)e∓ik·x1. Substituting this into the Dirac equation and using light-cone coor-
dinates k+ = kx + iky, k− = kx − iky,
HDψ =
(m kx − iky
kx + iky −m
)ψ =
(m k−
k+ −m
)ψ, (4.93)
gives the eigenspinors:
ψ+(x) =e−ik·x
N
(kx − iky√k2 +m2 −m
)=
e−ik·x√2E(E −m)
(k−
E −m
), (4.94)
where it is understood that the inner product is Lorentzian and E > 0. The
spinor belonging to the valence band is found to be:
ψ−(x, k) =e−ik·x√
2E(E +m)
(−k−E +m
). (4.95)
The degeneracy of states are like in the non-relativistic case circles of equal
radius in k-space: rk =√k2x + k2
y. There are no extra internal (spin) degrees of
freedom.
4.2.2.2 Constant field
In order to see some subtleties in solving the relativistic Landau problem we shall
first do it explicitly in terms of Hermite polynomials, not to obscure the structure
behind the oscillator algebra. As we did before we can find the spectrum by
squaring the Dirac Hamiltonian:
1Here we use the covariant notation: ψ±(x) = ψ±(t,x) = ξ±(ω,k)e∓i(−Et+x·k).
101
4. The toy model collection of Landau problems
H2D =
(m2 + (πx + iπy)(πx − iπy) 0
0 m2 + (πx + πy)(πx + πy)
)(4.96)
=
(m2 + π2
x + π2y − i[πx, πy] 0
0 m2 + π2x + π2
y + i[πx, πy]
)(4.97)
= (m2 +Hs(B))I−Bσz. (4.98)
where we again find the relation between the Schrodinger Hamiltonian Hs(B)
and H2D(B) as derived in the beginning of this chapter. Although one should
be careful in interpreting the physics of the square of the Dirac operator, this
expression has a nice interpretation, due to Ezawa in the context of graphene
Ezawa [2007]. Recall that this Hamiltonian does not contain a potential term, so
except for the mass term all energy is kinetic. The term Hs(B) accounts for the
kinetic energy of the cyclotron orbits, normally constituting the Landau levels.
The term Bσz can be seen as the intrinsic Zeeman effect of the spin-1/2 particle.
The cyclotron energy spacing is in the non-relativistic case 2B, so we see here
that the offset due to the Zeeman term is exactly half of the cyclotron energy.
We expect that the Landau levels become mixtures of the two types of spinors
from different cyclotron levels. Because it is the square of the Dirac operator
we get two branches of solutions due to the square-root. We shall pay special
attention at the zeroth-level, where the two branches meet, and we will encounter
orthogonality issues.
Again we split off the y-dependence and write our spinor as:
ψ(x) = eiky
(χ(x)
ξ(x)
). (4.99)
The differential equations of the components of the spinor become:
−χ′′(x) +B2(k/B + x)2χ(x) = (E2 +B)χ(x), (4.100)
−ξ′′(x) +B2(k/B + x)2ξ(x) = (E2 −B)ξ(x). (4.101)
102
4. The toy model collection of Landau problems
There are two sets of solutions, which are square normalizable (i.e. ∈ L2(R)).
The solutions for χ(x) have the familiar relativistic eigenvalues E2 = 2Bn, with
solutions:
χnk(r) = eikyHn
(√B(k/B + x)
)e−
B2
(k/B+x)2
. (4.102)
The equation for ξ(x) has normalizable solutions with spectrum as E2 =
2B(n+ 1):
ξnk(r) = eikyHn
(√B(k/B + x)
)e−
B2
(k/B+x)2
. (4.103)
The non-relativistic spectrum is Zeeman-splitted in exactly such a way that
we transit from an odd sequence in B to an even sequence. Every Landau level is
populated with up- and down-spinors from different cyclotron orbits (e.g. order
of the Hermite-polynomial). At E = 0 we get the solution |χ〉0. Solving the
eigenfunction equation for a particle in the n-th Landau level, n > 0, with energy
+√
2Bn above the fermi-energy gives:
ψ+nk(r) = e−ikye−B2
(k/B+x)2
(−Hn(
√B(k/B + x))
Hn−1(√B(k/B + x))
). (4.104)
In solving the equation for the same Landau level, but now for a hole with
energy −√
2Bn one has the result:
ψ−nk(r) = e−ikye−B2
(k/B+x)2
(Hn(√B(k/B + x))
Hn−1(√B(k/B + x))
). (4.105)
If we also consider the solutions in symmetric gauge and concentrate on the
lowest Landau level (LLL) with E = 0 we can summarize the results if B > 0:
Landau Gauge Symmetric Gauge
ψ0m = eiky
(e−kx−
Bx2
2
0
)ψ0m =
(zme−
Bzz4
0
).
(4.106)
103
4. The toy model collection of Landau problems
And if B < 0 we get:
Landau Gauge Symmetric Gauge
ψ0m = e−iky
(0
ekx+Bx2
2
)ψ0m =
(0
zmeBzz
4
).
(4.107)
If we consider the symmetric gauge that diagonalizes the angular momentum,
we realize that due to the Lorentz invariance of the Dirac equation Lz ceases to
be a good quantum number and it is modified to the total angular momentum:
Jz = LzI + Sz =
(a†a− b†b+ 1
20
0 a†a− b†b− 12
), (4.108)
which commutes with the Hamiltonian. We now get the following infinitely
degenerate spectrum for |n m〉:
H|n m〉 = sign(n)√
2B|n||n m〉, (4.109)
Jz|n m〉 = (n−m)|n m〉, (4.110)
with n ∈ Z and m ∈ Z+.
4.2.2.3 Constant field of compact support
The analyses for the degeneracy is completely analogues to the non-relativistic
case and yields:
d0[HD(B)] = nφ + 1. (4.111)
104
4. The toy model collection of Landau problems
4.2.2.4 Field with magnetic domain wall
First we check the form of the operator in a varying magnetic field, the square of
the Dirac operator with a varying field is:
6D2 =
((iπx + πy)(−iπx + πy) 0
0 (−iπx + πy)(iπx + πy)
)(4.112)
=
(π2x + π2
y + i[πx, πy] 0
0 π2x + π2
y − i[πx, πy]
)(4.113)
= Hs(B(r))I−Bz(r)σz. (4.114)
We see that we recover for both spinor components the Laplacian and a linear
term in the magnetic field opposite for each component, as derived in the be-
ginning. Written in the bosonic oscillators we can write the square of the Dirac
operator as:
6D2 =
( √2Ba†
√2Ba 0
0√
2Ba√
2Ba†
)(4.115)
= 2|B|(a†a 0
0 aa†
)− i∇B|B|
(a 0
0 a†
). (4.116)
Here we see the structure of the normal operator added with a mixing term
proportional to the gradient of the field normalized with its magnitude. 1
Now it is time to consider our domain wall configuration again of the form:
Bz(x) = B0 tanh(x/w), (4.117)
1This equation may permit a solution of the form of coherent states because of particle-hole symmetry. Both operators in the Landau-mixing term are annihilation operators with the
coherent states as eigenstate: ψϕ =
(exp
(∑i ϕia
†i
)exp
(∑i ϕiai
) ) |0〉. But for now we leave this as an
aside.
105
4. The toy model collection of Landau problems
Compared to the non-relativistic case we now have a term extra:
6D2σ = ∂2
x + (−i∂y +B0w ln(cosh(x/w)))2 + ασB0 tanh(x/w), (4.118)
where ασ = ±1 for the upper and lower spinor. This equation has the com-
bined symmetry x 7→ −x and ↑〉 7→ ↓〉.In the linearized version we get:
6D2σ = ∂2
x + (−k + αx2)2 + ασx. (4.119)
We see that the degeneracy of the double-well minima is broken by the spin
term. Another big difference is that we have two opposite spin states per harmonic
oscillator.
Figure 4.10: Both diagrams of the potential of the relativistic case (right) and itsnon-relativistic counter-part (left). The parameters are: k = 6. and α = 1. .
Note that the classical solutions are −qcl for | ↑〉 and +qcl for | ↓〉. We see
that we get a sort of two-site Hubbard model, with an opposite field on each site.
As with the Hubbard model the chemical potential, occupancy of the states will
influence the spectrum and behavior of the model.
We shall investigate the half-filled case, because it is the most relevant, since
in the relativistic Landau Level at n = 0 is half-filled as well. First note that Jz
is conserved. Thus a tunneling event is accompanied with a spin-flip event. This
106
4. The toy model collection of Landau problems
separates the problem into two channels. First we have the low-energy channel
with | ↑−qcl〉, ↓+qcl〉 and a higher energy level: | ↓−qcl〉, ↑+qcl〉. It would be interest-
ing to deploy an instanton analyses on this problem, but since a similar problem
including spin has not been dealt with in the literature, it takes us to much astray
of the goal of this project and we move directly to the numerical solved spectrum.
Following the same procedure as described in the non-relativistic case we
obtain the spectrum as displayed in figure (4.11).
æ æ æ æ æ æ ææ
æ
æ
æ
æ
æ
æ
æ
æ
æ
à à à à à à àà
à
à
à
à
à
à
à
à
à
ì ì ì ì ì ì ì ìì
ìì
ì
ì
ì
ì
ì
ì
ò ò ò ò ò ò ò òò
òò
ò
ò
ò
ò
ò
ò
ô ô ô ô ô ô ô ôô ô
ôô
ô
ô
ô
ô
ô
ç ç ç ç ç ç ç çç ç
çç
ç
ç
ç
ç
ç
-5 -4 -3 -2 -1 1k
-2
-1
1
2
E
Figure 4.11: Spectrum of the n ∈ −1, 0, 1 Landau level of the massless Diracsystem on a plane.
First we note that the spectrum has a particle hole symmetry. The branches
of each Landau level are the eigenstates of the symmetry operator, which flipped
parity and spin. If we look at the Landau levels with E 6= 0 we see the same struc-
ture as in the non-relativistic case. The Landau level at E = 0 is special. This
Landau level splits into two parts, where one branch is going to E 7→ +∞ and the
other is going to E 7→ −∞. Another difference with the non-relativistic case is
that asymptotically the branches with k 7→ +∞ rise linearly instead of quadrati-
cally, which is reminiscent of the massless dispersion relation of the Dirac particle.
107
4. The toy model collection of Landau problems
If we employ a compactification in the y-direction and adiabatically ramp m
to m + 1, we see that we always have a odd-number of edge modes. The lowest
Landau level contributes a single edge mode making the number of edge modes
odd. Imagine we place the fermi energy Ef just above E = 0 as displayed in
figure (4.12). If we now consider the adiabatic flux insertion a single electron is
pulled towards the domain wall. Now the edges become excited with +12Q. As
in the non-relativistic case the state that was pulled to the domain wall was a
super postion of both domain walls. The excitation at the end of the cylinder
is an entangled pair of both domains. Finally if we project the cylinder back to
plane and shrink the domain wall to the origin, the total accumulated charge at
the origin is: −Q + 12Q = −1
2Q, and the accumulated charge at the edge of the
disc is +12Q. We see that we have recovered the half-integer quantum Hall effect:
σH = (ν +1
2)e
Φ0
, with: ν ∈ Z. (4.120)
This analyses was performed at a mass of m = 0. Thus if we consider the
Hall response that we found in the previous chapter:
σH =
(1
2θ(µ− |m|) +
∞∑n=1
θ(µ− En)︸ ︷︷ ︸ +1
2sign(m)θ(|m| − µ)
)e
Φ0︸ ︷︷ ︸parity normal-term parity anomalous-term ,
(4.121)
we have explicitly found the parity normal-term of the Hall conductance. Now
it is time to tackle the same system on the sphere to make connection with the
results of Lee [2009]. After that we shall try to hunt at the partity anomalous
part of the Hall conductance.
4.3 Landau problem on the sphere
In this case we consider a two-dimensional electron gas that lives on the surface of
the sphere of radius R = 1. We consider the same systems as last section but now
in a different geometry. For the sphere we use two types of coordinate systems:
108
4. The toy model collection of Landau problems
Ef
+1
2Q −Q +
1
2Q
+1
2Q
−1
2Q = −Q+
1
2Q
+1
2Q +
1
2Q−Q
Figure 4.12: a) The system has an odd number of edge modes. b) Threading aflux attracts a single electron to the domain wall. c) For the plane this impliesthe half-integer qHe.
the standard spherical coordinates in R3 with fixed radius and the isothermal
coordinates, in which the metric of the surface itself is diagonal. We use each
coordinate system, when it is useful for the specific problem. The standard coor-
dinate system for a fixed radius R = 1 expressed from the Cartesian coordinates
in R3 is given as: x
y
z
=
sin θ cosφ
sin θ sinφ
cos θ
. (4.122)
The curl, divergence and gradient in this coordinate system are well docu-
mented and understood as given. We also use the isothermal coordinate system,
where we shall explicitly use a radius of R to later set it back to 1. The conformal
metric tensor is:
ds2 = e−2η(x)
((dx1)2 + (dx2)2
). (4.123)
In Cartesian coordinates we have the representation: x
y
z
= R
sin(α(u) cosφ
sin(α(u)) sinφ
cos(α(u))
, (4.124)
with: α(u) = 2arctan(eu), u ∈ R, and 0 ≤ φ < 2π. (4.125)
109
4. The toy model collection of Landau problems
In this coordinate system the conformal factor is of the form:
e−2η(u) =R2
cosh2(u), or η(u) = ln (cosh(u))− lnR. (4.126)
It is convenient to change this coordinate system to the complex coordi-
nate representation we used before to derive the relation between the Dirac and
Schrodinger operator:
z = u+ iφ ∂ = 12(∂u − i∂φ) A = 1
2(Au − iAφ)
z = u− iφ ∂ = 12(∂u + i∂φ) A = 1
2(Au + iAφ).
(4.127)
In this coordinate system the magnetic field was of the form:
B = −2ie2η(∂A− ∂A). (4.128)
The Gaussian curvature in this coordinate system is of the form:
k(z, z) = 4e2η∂∂η (4.129)
=cosh(u)2
R2∂2uln(cosh(u)) (4.130)
=1
R2, (4.131)
which is a constant. The total curvature for a curved space with a constant
Gaussian curvature is equal to the curvature times the total area:
κ = k · A (4.132)
κ = 4π, (4.133)
which is the total curvature of a sphere.
110
4. The toy model collection of Landau problems
4.3.1 Non-relativistic
As in the previous section we commence with studying the non-relativistic elec-
tron gas living on the sphere. From here we shall depart into the relativistic
case.
4.3.1.1 Zero field
This very well-known problem which is easiest considered using spherical coordi-
nates in the embedding R3 space, and restricting the radius to be R. It essentially
boils down to solving the spherical part of the Schrodinger equation in a central
potential. The Schrodinger equation on the surface of a sphere with radius R and
mass of m = 2:
Hs(0) = L2 (4.134)
with L2 the square of the angular momentum L = −ir×∇. If one explicitly
works out the square of the angular momentum operator in spherical coordinates
and use the substitution u = cos(x), the eigenvalue equation (4.134) reduces to
the Legendre differential equation. Therefore the solutions of this system are the
spherical harmonics Y mll (φ, θ), which have also Lz diagonal:
Hs(0)Y mll (φ, θ) = l(l + 1)Y ml
l (φ, θ), with: l ∈ 0, 1, . . . (4.135)
LzYmll (φ, θ) = mY ml
l (φ, θ), with: ml ∈ −l,−l + 1, . . . , l. (4.136)
The degeneracy of the spectrum is:
dl[Hs(0)] = 2l + 1, (4.137)
because that are the allowed possibilities for the eigenvalues of Lz such that the
associated Legendre polynomials of the spherical harmonics remain normalizable.
Implicitly we made use of the SO(3) symmetry of the sphere and used the
111
4. The toy model collection of Landau problems
fact that its Casimir operator is L2, with eigenvalues of l(l + 1). In this way
we circumvented the intrinsic curvature of the sphere by using the metric of the
embedding space.
4.3.1.2 Constant field
A constant magnetic field perpendicular to the surface of a sphere is a monopole
field. This has well-known topological aspects first found by Dirac [1931] that we
need to discuss first. The magnetic field on the sphere defines a U(1)-bundle on
the sphere S2. The first Chern number of this bundle is integral, which implies
that the flux is quantized: nφ ∈ N. Let us look at it specifically:
The total flux, Φ, of the monopole through the surface of the sphere is:
Φ = 4πB, (4.138)
where like previously B is the local flux density. In our units the flux quantum
is Φ0 = 2π which implies the number of flux quanta nφ is equal to twice the flux
density: nφ = 2B. For the total flux we also have the equation:
Φ =
∫S2
B d(area) =
∫S2
dA d(area) =
∫S2
F d(area). (4.139)
We know that this is the first Chern number of the U(1)-bundle over S2,
but let us show explicitly that it is integral valued. Denote the northern and
southern hemisphere of the sphere as D± and the gauge fields as A±. The gauge
field transforms as:
A− = A+ + ndφ, with: n ∈ Z. (4.140)
The fact that n ∈ Z can easily be seen that after the gauge transformation
the angle of U(1) can be multi-valued but the element of U(1) itself has to be
single-valued: einφ = 1. Now we can calculate the flux as follows:
112
4. The toy model collection of Landau problems
Φ =
∫S2
dA (4.141)
Φ =
∫D+
dA+ +
∫D−
dA− (4.142)
Φ =
∫D+∩D−
A+ − A− (4.143)
Φ =
∫S1
ndφ (4.144)
Φ = 2πn, (4.145)
where in the second last line applied Stokes’ theorem and an integral over the
equator remained. From both these results we see that:
nφ = n, with: n ∈ Z. (4.146)
We see thus that the number of flux quanta penetrating the sphere is a topo-
logical invariant and is thus quantized. nφ is known as the monopole charge
of the field configuration. There are many ways to derive the solutions of the
Schrodinger particle on the sphere with a monopole inside, known as the monopole
harmonics. The most eloquent way, at least in our opinion, uses the Hopf map
and is for example discussed in Stone and Goldbart [2009].
In this derivation we use the fibre-bundle framework discussed in chapter 2.
Consider the bundle with total space G = S3 and as base space G/H = S2. The
projection π of the bundle is then the Hopf map:
π : S3 7−→ S2. (4.147)
Note that for G = S3 = SU(2), with quotient space G/H = S2 we have that
the fibre is π−1 = H = U(1), if H is generated by J3. The Hopf map, which we
shall not discuss here explicitly but can be found in Stone and Goldbart [2009],
has the remarkable property that:
π3(S2) = Z. (4.148)
113
4. The toy model collection of Landau problems
This implies that the bundle G can be twisted, which can also be seen from the
fact that S3 is not a product S2×S1. Now consider the representation DJ(θ, φ, ψ)
of the SU(2) group:
DmnJ (θ, φ, ψ) = 〈J,m|e−iφJ1e−iθJ2e−iψJ3|J, n〉, (4.149)
= e−imφdmnJ (θ)e−inψ (4.150)
where the representation matrices DmnJ form a complete orthonormal set of
functions on S3. Note that if n = 0 the functions become ψ independent, with
some puzzling one can find the identification with the spherical harmonics:
Y mll (θ, φ) =
√2l + 1
4π
(Dm0l (θ, φ, 0)
)∗, (4.151)
where the complex conjugation ∗ is necessary to convert e−imφ into eimφ. If
one considers the spherical harmonics as the set of orthonormal functions on the
sphere for a monopole charge with nφ = 0, it looks promising to attempt the
ansatz that for DmnJ the n should be proportional to the monopole charge nφ. We
then obtain the sections:
YmnJ (θ, φ, ψ) =
√2J + 1
4π
(DmnJ (θ, φ, ψ)
)∗, (4.152)
known as the monopole harmonics. If one computes the first Chern number
of these section one discovers that ch1 = 2n, and we see that from (4.146) we
can indeed identify nφ = 2n. The Hopf map (i.e. projection π) is now simply by
forgetting ψ as follows:
Hopf = π : [(θ, φ, ψ) ∈ S3] 7−→ [(θ, φ) ∈ S2], (4.153)
and where nφ ∈ Z denotes the element of the homotopy group of the Hopf
map.
Now we have an orthonormal set of functions of our problem, we need to
construct the correct operators and find their eigenvalues. DJ(θ, φ, ψ) satisfies
114
4. The toy model collection of Landau problems
the eigenvalue equation:
J2DmnJ (θ, φ, ψ) = J(J + 1)Dmn
J (θ, φ, ψ), (4.154)
since J2 = J21 + J2
2 + J23 is the Casimir operator of the SU(2) group. Now the
question is how does the Schrodinger equation on S2 acts on DJ . This Casimir
operator operates on the total space of the bundle, whereas the Schrodinger
operator only operates on the base space S2 including the connection in the
U(1)-fibre. We saw the J3 operator generated the sub-group U(1), which was the
fiber over the base space. Thus the J3 moves the section horizontally in the fiber
and we can conclude that the Schrodinger equation is:
Hs(nφ) = J21 + J2
2 = J2 − J23 , (4.155)
which are the generators in the base space S2. Now we have that DJ satisfies:
J3DmnJ = nDmn
J =nφ2DmnJ , (4.156)
which shows that the eigenvalues of J3 are half-integral. Combining all this
steps we see that the monopole harmonics are the solutions to the Schrodiner
equation with a monopole field:
Hs(nφ)YmnJ =
(J(J + 1)− (
nφ2
)2
)YmnJ . (4.157)
Recall that the z-projection in the SU(2) can never exceed the total spin in
the representation: J ≥ |n| so that we get the following spectrum:
EJ,m = J(J + 1)− (nφ2
)2, J ≥ |nφ2|, −J ≤ m ≤ J, (4.158)
and thus have the degeneracy:
dJ [Hs(B)] = 2J + 1 = nφ + 1. (4.159)
Note that this is the same degeneracy for the Landau problem on the plane
with a finite radius.
115
4. The toy model collection of Landau problems
4.3.1.3 Field with magnetic domain wall
In this section we study a sphere emerged in a constant magnetic field. At the
equator there is a zero-flux band, and the flux on both hemispheres is opposite.
So in total there is zero-flux through the sphere, thereby obeying the Maxwell
equation: ∇ ·B = 0.
This problem is most convenient formulated in the isothermal coordinate sys-
tem introduced earlier. Recall that the Schrodinger Hamiltonian is of the form:
Hs(B) = −e2η
(4∂∂ − 4i
(A∂ + A∂
)− 2i
((∂A) + (∂A)
)− 4AA
). (4.160)
The conformal factor for a sphere with R = 1 was:
e2η = cosh2(u) and η = ln(cosh(u)). (4.161)
Now we need to construct our magnetic field where we follow Lee [2009].
First we construct the monopole field on a sphere. We found that nφ = 2B for
the monopole. Both the magnetic field and the Gaussian curvature, k(z, z), are
constant thus we have the relation:
B =nφ2k(z, z), (4.162)
by plugging back in the radius one explicitly sees that this relation works.
Now we pick the gauge wherein:
∂A+ ∂A = 0, (4.163)
this enables us to find a potential Λ(z, z), such that:
A = −i∂Λ, and: A = i∂Λ. (4.164)
Using this potential we find for the magnetic field:
116
4. The toy model collection of Landau problems
B = −2ie2η(∂A− ∂A) (4.165)
= 4e2η∂∂Λ. (4.166)
The expression for the gaussian curvature was:
k(z, z) = 4e2η∂∂η. (4.167)
Combining this with the relation between the magnetic field and curvature
(4.162) we can set:
Λ =nφ2η, with: η = ln(cosh(u)) (4.168)
Now we have an expression for the constant field it is easy to construct the
domain wall configuration, by simply multiplying with the factor tanh(u/w):
Λ =nφ2
tanh(u/w)ln(cosh(u)). (4.169)
For the vector potential we need the derivative of Λ and for simplicity we set
w = 1, so we define:
f(u) = ∂u(tanh(u)ln(cosh(u))
)(4.170)
f(u) =ln(cosh(u))
cosh2(u)+ tanh2(u). (4.171)
If we note that A = −A it is easy to find that the Hamiltonian is of the form:
Hs(B)φ = −cosh2(u)
(∂2u + ∂phi
2 − 4A∂φ − 4AA
)φ (4.172)
= −cosh2(u)
(∂2u −m2 −mnφf(u)−
n2φ
4f(u)2
)χ, (4.173)
where we used translation invariance in φ to define φ = eimφχ, with m ∈ Z.
117
4. The toy model collection of Landau problems
Finally we obtain the eigenvalue equation:
−χ′′ +(m2 +mnφf(u) +
n2φ
2f(u)2 − E
cosh2(u)
)= 0. (4.174)
For a mathematician this eigenvalue equation looks worrisome, and it is in-
deed, despite many substitution attempts it appears to be analytically intractable
so we analyze it numerically. First we note that the equation is invariant under
the substitution u 7→ −u so we can use our standard approach of the previous
section.
The spectrum that is displayed in figure 4.14 had as values R = 1, w = 1 and
nφ = 40. As expected the spectrum is very similar to the one for the plane. There
are some interesting features that are different. From the instanton analyses we
learnt that for higher Landau levels the bandsplitting emerges by lower k-values.
On the plane this feature is not so profound visible as on the sphere.
Another interesting feature is the fact that the Landau levels do not become
flat in the (E,m)-diagram. In the diagram one clearly sees that the levels peak
at around m = −15, and then they decay towards zero. At around m = −20 the
results showed a kink and from there on a sharp rise. These results were stable of
changing the size of the integration domain L, thus the algorithm was not instable.
We speculate that this dip at m = −20 in the results mark the degeneracy of
each Landau level. The total number of flux quanta was set to be nφ = 40. Thus
if the configuration was a monopole, 40 quanta would penetrate the surface.
The tanh(u)-term reversed the orientation of the flux quanta at the southern
hemisphere leading to the situation of roughly 20 flux quanta penetrating both
hemispheres in opposite direction. This suggest that the degeneracy of each mini-
Landau level is about 20. This suggest that there is a lower bound on m:
m ∈ −nφ2,−nφ
2+ 1, . . . . (4.175)
This has some interesting consequences. Formally all states are equal super-
positions, odd and even, from both hemispheres. This was also true in the plane.
Physically however it is very likely to expect that for particles that are very far
118
4. The toy model collection of Landau problems
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-15 -10 -5 0 5m
50
100
150
200
E
Figure 4.13: Spectrum of the Schrodinger Hamiltonian on the sphere with amagnetic field with a domain wall. The parameters are R = 1, w = 1 andnφ = 40.
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-30 -25 -20 -15 -10 -5 5m
100
200
300
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Figure 4.14: The spectrum including the unphysical domain where m < 20.
119
4. The toy model collection of Landau problems
from the domain wall, this entanglement is mere an academic result. It is likely
that the entanglement decreases if the particle is far from the domain wall, in
units of the magnetic length. A reasonable measure for this entanglement is the
size of the band-splitting. As soon as the band-splitting sets in, the particles
are entangled from both Landau domains. Since the band-splitting sets in for
higher Landau levels at lower values of m more particles are entangled in higher
Landau levels. Finally we note that if we adiabatically insert a flux quantum in
Ef
+1
2Q +
1
2Q
+1
2Q +
1
2Q
−Q−Q −2Q
+Q
+Q
Figure 4.15: Illustration of the process of an adiabatic flux insertion.
a system that has ν = 1, this means that we ramp nφ 7→ nφ + 1. 1 This causes
that m 7→ m+ 1. Like in the cylindrical case two electrons march to the equator
and the poles become polarized with charge +Q, because we necessarily have two
edge modes per ν.
4.3.2 Relativistic
In this section we shall study the Dirac equation on the sphere. Here we shall
use the Dirac operator on the sphere as derived in the previous chapter, and we
shall use its form in isothermal coordinates.
1Because our construction of the magnetic field was based on the monopole, which wedeformed with a tangent hyperbolic, it is likely that the number of flux quanta is a topologicalinvariant. If that is the case the process of adiabatically inserting a flux quantum is of coursenonsense. For now we shall ignore this subtlety and just assume that we have a process thatcan drive: m 7→ m+ 1.
120
4. The toy model collection of Landau problems
4.3.2.1 Zero field
Here we follow Abrikosov [2002]. In the previous chapter we found the form of
the Dirac operator as:
6D0 = −iσx(∂θ +
cot θ
2
)− iσy
sin θ∂φ. (4.176)
Where we denoted the zero field by appending a zero to the operator. The
Dirac operator on the sphere is invariant under SU(2)-transformations, which
implies:
[6D0, L2] = 0. (4.177)
The Weyl-Cartan set of generators for S2 in the spin-1/2 representation are:
Lz = −i∂φ, (4.178)
L+ = eiφ(I2(∂θ + i cot θ∂φ) +
σz2 sin θ
), (4.179)
L− = e−iφ(I2(−∂θ + i cot θ∂φ) +
σz2 sin θ
). (4.180)
These generators satisfy the SU(2) algebra. Plugging these generators into
the Casimir operator L2 = 12(L+L− + L−L+) + L2
z we obtain:
L2 = −(I2
(1
sin θ∂θ(sin θ∂θ) +
1
sin2 θ∂2φ −
1
sin2 θ
)− iσz
cos θ
sin2 θ∂φ
), (4.181)
we recognize the first two terms from the spherical harmonics and the last
terms are new. If we take a look at the Dirac operator and square it we obtain:
(−i6D0)2 = −[I2
( 1
sin θ∂θ(sin θ∂θ)+
1
sin2 θ∂2φ−
1
sin2 θ− 1
4
)− iσz
cos θ
sin2 θ∂φ
]. (4.182)
121
4. The toy model collection of Landau problems
Comparing both expressions we find that:
−6D20 = L2 +
1
4. (4.183)
However this is not enough to solve the spectrum of the Dirac operator com-
pletely, because we need two quantum numbers. Unfortunately there exist no-
short cut that can solve the spectrum straight away, so we need to solve the
eigenvalue equation of the square of the Dirac operator. First we define the
spinor, ψ, with Fourier decomposition:
ψ =
(χ(θ, φ)
ξ(θ, φ)
)=∑m
eimφ√2π
(χm(θ)
ξm(θ)
), m = ±1
2,±3
2, · · · (4.184)
If we let −6D20 operate on this spinor, and use the substitution u = cos θ, with
x ∈ [−1, 1] we get:
(∂u(1− u2)∂u −
m2 − σzmu+ 14
1− u2
)(χm(u)
ξm(u)
)= −
(E2 − 1
4
)(χm(u)
ξm(u)
).
(4.185)
This is a generalized hypergeometric equation. Note that if we map m 7→ −mor u 7→ −u we exchange the spinor components. The pesky property of these
type of equations is that they are singular at the boundary of the domain of
u = ±1. For example this makes them very intractable for numerical analyses,
because one needs to integrate into a singularity. One solution is to integrate
out of the singularities and the eigenvalue can be found by matching the two
pieces at u = 0. This is the double shooting method that can be found in Press
et al. [2007]. Here we can circumvent this problem by the following, very clever,
substitution: (χm(u)
ξm(u)
)=
((1− u)
12|m− 1
2|(1 + x)
12|m+ 1
2|χ′m
(1− u)12|m+ 1
2|(1 + x)
12|m− 1
2|ξ′m
). (4.186)
122
4. The toy model collection of Landau problems
Now the eigenvalue equation becomes:
((1−u2)∂2
u+
(sign(m)σz− (2|m|+2)u
)∂u−m(m+1)+(E2− 1
2)
)(χ′m(u)
ξ′m(u)
).
(4.187)
This is the differential equation of the Jacobi polynomials with α = β± 1, see
for example Abramowitz and Stegun [1964]. The Jacobi polynomials, P(α,β)n (u),
apparently are the spinorial brothers (spin-1/2 representation) of the associated
Legendre polynomials (spin-0 representation). The Jacobi polynomials are square
normalizable on the domain u ∈ [−1, 1] if E satisfies:
E = ±√n+ |m|+ 1
2n ≥ 0, m = ±1
2,±3
2, · · · (4.188)
The groundstate is for n = 0 and m = ±1/2 and is thus doubly degenerate
(spin up and down). For completeness we state that the eigenspinors have the
form: (χm(θ)
ξm(θ)
)=
(c1P
|m− 12|,|m+ 1
2|
n (cos(θ))
c2P|m+ 1
2|,|m− 1
2|
n (cos(θ))
). (4.189)
Now it is easy to check that for the Casimir operator eigenvalue l we get
l = n+ |m|.
4.3.2.2 Constant field
It seems logical to proceed the way we did for the non-relativistic case, to con-
sider the Jacobian polynomials as the monopole charge n = 0 solution, and by
using the Hopf map construct solutions for other monopole charges. Since this
particular problem is not at the heart of this thesis, and this has probably been
done somewhere else, although we have not encountered it so far, we move on,
and this pebble remains dormant.
In this section we employ a trick to solve this spectrum found by Pnueli [1994],
which only works for surfaces of constant curvature. For this class of surfaces all
123
4. The toy model collection of Landau problems
Landau levels are degenerate, so we can use the Atiyah-Index theorem. First we
write the Dirac operator in the form:
6D =
(0 K†
K 0
). (4.190)
Recall that its square obeys:
6D2 =
(K†K 0
0 KK†
)=
(Hs(B − 1
2k)− (B − 1
2k) 0
0 Hs(B + 12k) + (B + 1
2k)
).
(4.191)
Since this eigenvalue equation has a single-valued eigenvalue for all spinors of
the spectrum we have the condition:
Spec(K†K)/Ker(K†K) = Spec(KK†)/Ker(KK†). (4.192)
Thus these operators share all eigenvalues (even multiplicities) except for pos-
sible zero-modes (i.e. Kf = 0 or K†f = 0, where f is a spinor component). If
we denote the n-th eigenvalue of the operator K†K(B) as: λK†K
n (B) we have:
λK†K
n = En(B − 1
2k)− (B − 1
2k), (4.193)
λKK†
n = En(B +1
2k) + (B +
1
2k). (4.194)
For a compact, closed, two-dimensional surface the Atiya-Singer index theo-
rem Atiyah and Singer [1968] states:
Index6D = Dim(Ker(K))−Dim(Ker(K†)) =1
2π
∫B. (4.195)
This guarantees that there are zero-modes. For the spherical monopole both
k = 1 and B are constant and without loss of generality we can assume that
B > 0. The eigenvalues of a Schrodinger Hamiltonian are positive definite so that
Ker(KK†) = 0. Because both operators share the same non-zero eigenvalues we
have that the ground state of KK† must be the same as the first excited state of
124
4. The toy model collection of Landau problems
K†K:
λKK†
n (B) = λK†K
n+1 . (4.196)
Using this together with equations (4.193) and (4.194) we obtain the recur-
rence relation:
λKK†
n (B) = λK†K
n+1 = λK†Kn + (2B + k). (4.197)
Because λK†K0 (B) = 0 we get:
λK†K
n (B) = 2nB + n2k. (4.198)
For the sphere we had k = 1 and we acquire the spectrum by simply taking
the squareroot:
En = ±√n(2B + n), (4.199)
or in terms of the flux quanta:
En = ±√n(nφ + n). (4.200)
We see with this ’simple’ calculation that the Atiyah-Singer index theorem can
be very powerful in solving problems. Now we embark on our magnetic domain
wall problem, where this theorem is not applicable.
4.3.2.3 Field with magnetic domain wall
In this section we shall check the results reported by Lee [2009]. We study
the Dirac equation on the sphere using the isothermal coordinate system and
the magnetic field, which we introduced in section 4.3.1.3. Recall the relation
between the Schrodiner and Dirac operator derived in section 4.1.1:
6D2 =
(Hs(B − 1
2k)− (B − 1
2k) 0
0 Hs(B + 12) + (B + 1
2k)
). (4.201)
125
4. The toy model collection of Landau problems
First we calculate Hs(B − 12k) for the upper-spinor component, the adjusted
vector potentials are of the form:
A 7→ A+i
2∂η, and: A 7→ A− i
2∂η. (4.202)
Using the definitions of section 4.3.1.3 the vector potential becomes:
A =i
4
(tanh(u)− nφ
(ln(cosh(u))
cosh2(u)+ tanh2(u)
))︸ ︷︷ ︸ . (4.203)
f(u)
Plugging this in the Hamiltonian and use the translation invariance in φ, to
write ψ = eimφχ we get:
Hs(B −1
2k) = −cosh2(u)
(∂2u −m2 +mf(u)− f(u)2
4
). (4.204)
Recall that the curvature was k = 1, thus we are left with determining the
magnetic field B. If we use the definition of the field in this coordinate system
and plug our vector potential in, and simplify the result a little bit we get:
B =nφ2
tanh(u)
(3− 2ln(cosh(u))
). (4.205)
Collecting all the terms we are left with the eigenvalue equation for the upper
spinor component:
−χ′′+(m2−mf(u)+
f(u)2
4− nφ
2
tanh(u)
cosh2(u)
(3−2ln(cosh(u))
)+
12− E2
cosh2(u)
)χ = 0.
(4.206)
Following the same procedure we obtain for the other spin component, which
126
4. The toy model collection of Landau problems
we denote with ξ:
−ξ′′+(m2 +mg(u)+
g(u)2
4+nφ2
tanh(u)
cosh2(u)
(3−2ln(cosh(u))
)+
12− E2
cosh2(u)
)ξ = 0,
(4.207)
where we defined:
g(u) = tanh(u) + nφ
(ln(cosh(u))
cosh2(u)+ tanh2(u)
). (4.208)
Of course we do not attempt to solve this analytically, and plug it right away
into our shooting algorithm, which results are displayed in figure (4.16).
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15
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Figure 4.16: Spectrum of the n ∈ −1, 0, 1 Landau level of the massless Diracsystem on a sphere. We used nφ = 40, R = 1 and w = 1.
Combining the insights of the massless Dirac spectrum on the plane together
with the Schrodinger system on the sphere, this spectrum has no big surprises.
For this problem we see again that near the momenta m = nφ/2 the Landau level
starts to drop and start a steep ascend afterwards, as displayed in figure (4.17).
We checked this property with a different number of quanta and this point
indeed shifted along with nφ. On closer inspection the lowest Landau level starts
127
4. The toy model collection of Landau problems
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Figure 4.17: Dirac spectrum on the sphere including the unphysical domain.
its ascend already at m = −19, which might indicate that the degeneracy of
the Landau domain is somewhat smaller than nφ/2. Noteworthy is that the first
Landau level remains regular somewhat longer.
Figure 4.18: Eigenspectrum of the same system, with unmentioned parameters.This figure is from Lee [2009]
In figure (4.18) we show the results that were obtained in the article Lee
[2009]. We clearly see that both spectra are of the same form. Nice is the plot
of the modulus of the eigenfunctions |ψ|2, here one sees clearly that the state is
in a superposition in both Landau domains, depending on the m-momentum. In
figure 4.19 we illustrated the result of a flux insertion.
128
4. The toy model collection of Landau problems
Ef
+1
2Q
+1
2Q
−Q −2Q
+Q
+Q
Figure 4.19: Illustration of a flux insertion of the relativistic system on the sphere.
4.4 Summary
In this chapter we saw explicitly the parity normal part of the Hall conductance
of the topological insulator: σH = (n+ 1/2)e/Φ0. We compared it with the non-
relativistic system of the integer quantum Hall effect. By considering a magnetic
domain wall we concluded that the relativistic system always has an odd number
of edge modes and the non-relativistic case an even number. Summarized in the
figure below:
Ef
+1
2Q −Q +
1
2Q
+1
2Q−Q+
1
2Q
−2Q+Q +Q
+1
2Q
−1
2Q = −Q+
1
2Q
+Q
−Q = −2Q+Q
Ef
+1
2Q +
1
2Q−Q
+1
2Q +
1
2Q−Q
Figure 4.20: Illustration comparing the relativistic Hall effect versus its non-relativistic counterpart.
The half-integral excitations are part of an entangled pair, and seem to be
vulnerable to disorder. In principle it should be possible to measure the half-
integral excitation, if one considers connecting a lead between the north-pole and
129
4. The toy model collection of Landau problems
the equator of the sphere as depicted in figure 4.19. If one wants to study this
Corbino type of setting faithfully the leads should be taken into consideration. A
version of the Avron & Seiler theorem is imaginable for a TI surface with genus
g = 2.
In the next chapter we shall try to shine some light one the parity anomalous-
term in the Hall conductance.
130
Chapter 5
Mass domain walls in topological
insulators
5.1 Mass domain wall on the plane
In this short chapter we shall discuss the problem of a one-dimensional mass do-
main wall for the Klein-Gordon- and Dirac-equation. We chose the mass term
to be of the form a tanh(x), which is solved exactly with the (generalized) Leg-
endre functions in 2+1 dimensions. For the Dirac-equation we find the massless
Jacki-Rebbi state Jackiw and Rebbi [1976] and ba − 1c bound states at the do-
main wall. By considering the spectral flow of this system on a cylinder we find
the parity-anomalous Hall effect. The spatial degree of freedom in the direction
perpendicular to the domain wall is eliminated from the spectrum and manifests
itself as an effective mass term of the dimensional-reduced theory. Perhaps this
mechanism could be a new way of ’compactifying’ dimensions and could be a can-
didate for explaining the three fermionic-mass generations in the standard model.
We consider a Minkowskian space time of (2+1)-dimension with a mass field
of the form:
m(x) = a tanh(x). (5.1)
The mass is regarded as static background field. To find the solution we start
with the Klein-Gordon equation.
131
5. Mass domain walls in topological insulators
5.1.1 Klein-Gordon solution
The solution of the Klein-Gordon equation is remarkably simple. First we reduce
the Klein-Gordon equation into a second-order differential equation for x in the
following way:
(2 +m(x)2)φ = 0 (5.2)
(∂2t −∇2 + a2 tanh2(x))φ = 0 (5.3)
(−E2 + k2 − ∂2x + a2 tanh2(x))φ = 0 (5.4)
−φ′′ + (a2 tanh2(x)− E2 + k2)φ = 0, (5.5)
where we used translational invariance in the y-direction and time. To simplify
this equation we can use the substitution u = tanh(x), and note that −1 ≤ u ≤ 1.
The second derivative on a function f(u(x)) transforms into:
∂2xf(u(x)) = ∂x(
df
du
du
dx) (5.6)
=d2f
du2
(du
dx
)2
+df
du
d2u
dx2(5.7)
=d2f
du2
1
cosh4(x)− 2
df
du
tanh(x)
sinh(x)(5.8)
=d2f
du2(1− u2)2 − 2u(1− u2)
df
du, (5.9)
where we used in the last line the identities: cosh(arctanh(x)) = 1/(√
1− x2)
and sinh(arctanh(x)) = u/(√
1− x2). The Klein-Gordon equation is now brought
into the form:
(1− u2)φ′′ − 2uφ′ +
(−a2u2 + E2 − k2
1− u2
)φ = 0 (5.10)
(1− u2)φ′′ − 2uφ′ +
(a2 +
−a2 + E2 − k2
1− u2
)φ = 0, (5.11)
132
5. Mass domain walls in topological insulators
Now this differential equation is of the form of the Legendre differential equa-
tion:
(1− u2)Pm′′
l (u)− 2uPm′
l (u) +
(l(l + 1)− m2
l
1− u2
)Pml (u) = 0. (5.12)
The Legendre polynomials, the basis of the spherical harmonics, are normal-
izable on the domain −1 ≤ u ≤ 1 if l ∈ N and |ml| ∈ 0, 1, 2, . . . , l. So if the
mass satisfies the condition:
±a =√l(l + 1), (5.13)
The spectrum is of the simple form:
E = ±√k2 + a2 −m2
l (5.14)
E = ±√k2 + l(l + 1)−m2
l , (5.15)
with: k ∈ R, l ∈ 1, 2, . . . , |ml| ∈ 1, 2, . . . , l. (5.16)
Now an interesting question arises. Are these all the possible solutions that
are square integrable on R2, which implies that the mass-amplitude should be
quantized? Physically this would be remarkable. It would mean that the effec-
tive mass gap induced by the RKKY -interaction has to be certain values for
states to be allowed at all. However if one relies on the overwhelming majority
of literature on Legendre polynomials one is lured into this conclusion. Thanks
to my supervisor K. Schoutens we found that these are a special set of solutions
and the mass amplitude can be any real number. We show this for the case of
the Dirac equation since we are interested in those solutions.
To finish this section on this special set of solutions we pay some attention
to the normalization. For the spherical harmonics the argument of the Legendre
polynomial is cos(x), whereas we have the hyperbolic tangent. Hence the normal-
ization of the wavefunction requires that the m = 0 is discarded because those
have a constant term in the Legendre polynomial, which are not convergent on
133
5. Mass domain walls in topological insulators
R. The ’s-wave’ solution (e.g. l = 0) is special because it requires a = 0, which
implies a massless particle, and we have lost our mass defect. The corresponding
solutions are of the form:
φklm(x) =1
NeikyPm
l (tanh(x)). (5.17)
l = 1, a =√2 l = 2, a =
√6 l = 3, a = 2
√3
m = 1
m = 2
m = 1
m = 2
m = 3
k !→k !→k !→
x !→ x !→ x !→
|φlm(x)|2
Elm(k)
Figure 5.1: Density plots and spectra for the solutions of the Klein-Gordon equa-tion for mass amplitudes a ∈ ±
√2,±√
6,±2√
3.
We see that depending on the mass-level l we have l number of states around
the domain wall with the quantum number m. The lowest state has |m| = l as
quantum number and there are no gapless states in the problem.
134
5. Mass domain walls in topological insulators
5.1.2 Dirac solution
The Dirac Hamiltonian in (2+1)-dimension, with m = a tanh(y), where we chose
y for later convenience, is of the form:
HD =
(m −i∂x − ∂y
−i∂x + ∂y −m
)(5.18)
HD =
(au k − (1− u2)∂u
k + (1− u2)∂u −au
), (5.19)
where we expanded our wave function in plane-waves in k for the y-component.
If the Dirac operator is translational invariant, each component of the spinor
satisfies the Klein-Gordon equation, by applying the Dirac operator twice, as in
the relativistic Landau problem the derivative falls on the second operator and
the equations are slightly modified. Considering the equation:
H2Dψ = E2ψ, with : ψ = eikx
(χ
ξ
), (5.20)
we get the following coupled system of differential equations:
(1− u2)χ′′ − 2uχ′ − −a2u2 + k2 − E2
1− u2χ− aξ = 0, (5.21)
(1− u2)ξ′′ − 2uξ′ − −a2u2 + k2 − E2
1− u2ξ − aχ = 0. (5.22)
Note that both equations are identical, so it seems to be justified to do the
ansatz: ξ = αχ, with α a numerical constant. Filling in the ansatz and doing the
partial fraction we obtain:
(1− u2)χ′′ − 2uχ′ + (a2 − a)χ+−a2 + k2 − E2
1− u2χ = 0, (5.23)
(1− u2)ξ′′ − 2uξ′ + (a2 − a
α)ξ +
−a2 + k2 − E2
1− u2ξ = 0. (5.24)
135
5. Mass domain walls in topological insulators
For equation (5.23) we have the Legendre polynomials as solution if:
l(l + 1) = a(a− α) and m2 = −a2 + k2 − E2, (5.25)
where for equation (5.24) we have:
l(l + 1) = a(a− α−1) and m2 = −a2 + k2 − E2, (5.26)
we see that for both solutions to match we have the condition that α = ±1.
If we denote α = (−1)ν , then we solve the mass amplitude with:
(−1)νa = l (−1)ν+1a = (l + 1) (5.27)
The spectrum of the special set of solutions is found as:
E = ±√k2 + l2 −m2, (5.28)
where now we do have massless modes as expected from the Jackiw-Rebbi
result Jackiw and Rebbi [1976]. Note that equation (5.27) suggests that for the
higher bound states we get different Legendre polynomials for each spinor com-
ponent, like in the relativistic Landau problem where the Landau levels were
populated with different Hermite polynomials for each spinor component.
k !→k !→k !→
Elm(k)
l = 1, a = ±1 l = 2, a = ±2 l = 3, a = ±3
Figure 5.2: The spectra of the Dirac equation for mass amplitudes a ∈±1,±2,±3.
Now that we have the spectrum, we can try to find the eigenspinors of the
Dirac Hamitonian, thus far I have only found the ’diagonal’ states, first we take
136
5. Mass domain walls in topological insulators
a = −l, we get the following unnormalized solutions:
HD
(P ll
P ll
)= −k
(P ll
P ll
)(5.29)
HD
(P−llP−ll
)= k
(P−llP−ll
). (5.30)
For l = a we get the other set of solutions:
HD
(P ll
−P ll
)= −k
(P ll
−P ll
)(5.31)
HD
(P−ll−P−ll
)= k
(P−ll−P−ll
). (5.32)
To find whether these states are normalizable and if we can find solutions for
general, non-integer values, of a it is instructing to tabulate a couple of Legendre
polynomials:
P 11 (u) = c1(1− u2)
12 = c1cosh(y)−1 (5.33)
P 22 (u) = c2(1− u2)
22 = c2cosh(y)−2 (5.34)
P 33 (u) = c3(1− u2)
32 = c3cosh(y)−3, (5.35)
where in the second equality we used that u = tanh(y). The fact that the
solution of the Jackiw-Rebbi state does not require a mass quantization condition,
lead K. Schoutens, my supervisor, to try the following ansatz:
P aa (u) = ca(1− u2)
a2 = cacosh(y)−a. (5.36)
Plugging this into the Dirac equation one gets the following solutions:
137
5. Mass domain walls in topological insulators
ψ+(y) = cacosh(y)−a
(1
−1
), with: HDψ+ = −kψ+ (5.37)
ψ−(y) = cacosh(y)a
(1
1
), with: HDψ− = +kψ− (5.38)
Thus if a > 0 only ψ+ is normalizable and if a < 0 then ψ− is normalizable.
And if a is integer-valued it corresponds to a Legendre polynomial. As with
the magnetic domain wall we see that the mass domain wall causes a single-way
propagating current along the wall as depicted in figure (5.3).
!4 !2 2 4
!4
!2
2
4
!4 !2 2 4
!4
!2
2
4
k !→ k !→
a < 0 a > 0
E E
Figure 5.3: The spectra of the Jackiw-Rebbi states for a < 0 and a > 0.
In chapter 2 we saw that a mass term breaks time-reversal invariance. The
fact that a mass-domain has a single-way propagating edge state suggests that
the sign of the mass-term determines the orbital chirality. In figure (5.4) one
sees that if the mass has a positive sign the orbits are clock-wise, and for a nega-
tive mass anti-clockwise. This raises some questions: for example in the Landau
problem one has the notion of the magnetic length determining the length-scale
of a Landau-orbit. Also one has the cyclotron-frequency determining the orbital
frequency. Could this concepts be translated to this mass setup?
138
5. Mass domain walls in topological insulators
a < 0a > 0
!→
y
!→
y
Figure 5.4: Diagram of the mass-domain wall and the single-way propagatingcurrent. It turns out that the sign of the mass determines the chirality of theorbits of the particles.
The eigenspinors for the bound states are still to be found, however we are
quite close. Schoutens suggested the following logic, consider the following table
with the same l but different m:
P 33 (u) = c3(1− u2)
32 (5.39)
P 23 (u) = c2u(1− u2)
22 (5.40)
P 13 (u) = c1(−1 + u2)(1− u2)
12 . (5.41)
Then it seems likely to find a solution with the ansatz:
P bac−ma (u) = f(u)(1− u2)a−m
2 . (5.42)
Annoyingly I haven’t been able to find the exact spinorial form of the bound
states. But we know, or at least expect, from the set of special solutions of the
Legendre polynomials, that there are ba−1c massive bound states at the domain
139
5. Mass domain walls in topological insulators
wall. For the exact form of the spinors I expect that it requires different orders
of the Legendre polynomials like in the case of the magnetic field. This would
make the analogy with the magnetic field complete! Since these bound states are
not of our primary interest we summarize the obtained results and move on to
the Hall response of this system.
In this section we have shown that at a mass domain wall we have a single-
way propagating edge current. From the direction of the current we can assign a
helicity to the sign of the mass term. Next to the massless state we have ba− 1cmassive states localized at the domain wall.
5.1.3 Adiabatic Cycles
The goal of researching the mass domain wall was to find the parity-anomalous
Hall effect, and to construct a topological Thouless-pump from it. By compact-
ifying the plane into a cylinder in the y-direction and sending a flux quantum
through the cylinder we induce the involution: k 7→ k+1. Depending on the sign
of a we get either an electron above the fermi level or a hole at the fermi level.
Because all the massless states are bound at the domain wall we either attract an
electron towards the domain wall or repel one from it as depicted in figure (5.5).
Ef
+1
2Q −Q +
1
2Q
Ef
+Q −1
2Q−1
2Q
a < 0
a > 0
Figure 5.5: Spectral flow of a cylinder with a mass domain wall on it. Dependingon the sign of a we get an opposite Hall response.
140
5. Mass domain walls in topological insulators
Now if we pull the domain wall to the right of the cylinder the upper figure
with a < 0 will have a constant mass of m = −a and the lower figure with a > 0
will have a constant mass of m = +a, because of the tangent hyperbolic. Then
if we map the right end of the cylinder to the origin of the plane and the left end
to infinity we get the picture as displayed in figure (5.6).
+1
2Q
−1
2Q = +
1
2Q−Q +
1
2Q = −1
2Q+Q
−1
2Q
m < 0m > 0
Figure 5.6: Spectral flow of the a relativistic system on the plane derived fromthe domain wall.
We see that we have explicitly constructed the spectral flow of the anomalous
Hall effect:
σH =1
2sign(m)
e
Φ0
. (5.43)
5.1.4 Further ideas
Another wild speculation resulting from this chapter is that this mass domain-
wall model could be interesting for the standard model in regards that it could be
a way for understanding the fermionic mass generations. These masses currently
enter in the standard model as parameters and here the mass is determined
through quantum numbers in the spectrum as follows:
E = ±√k2 +M2 with: M2 = l2 −m2, (5.44)
where: l ∈ R, and m ∈ 0, 1, . . . , bl − 1c.
As in the seminal article of Dirac Dirac [1931] where the existence of a mag-
141
5. Mass domain walls in topological insulators
netic monopole leads to the quantization of the electric charge e, we see that a
stringlike mass defect quantizes the effective mass. However here the mass defect
is parametrized in the internal space of the system. An interesting consequence
is that by introducing a mass defect in a particular spatial degree of freedom,
the degree manifests itself as a mass term in the remaining degrees of freedom.
Hence we have found a new way of ’compactifying’, or a mechanism that effec-
tively compactifies, the degree of freedom containing the mass defect. Perhaps
a measurable consequence of us living on the domain wall of a fifth-dimension
are the massless Jackiw-Rebbi states. Those could perhaps be interpretted as
the fermionic versions of Goldstone bosons. Another interesting perspective re-
sulting from this scheme is that it could explain the fact that parity is broken in
our universe. The broken parity is the sign of the single-way propagating edge
current at the domain wall. Some relevents earlier articles on mass-domain walls
are: Jackiw and Rebbi [1976] Boyanovsky et al. [1987], Jr. and Harvey [1985]
Goldstone and Wilczek [1981] Fosco and Lopez [1998].
.
142
Chapter 6
Conclusions & Outlook
6.1 Conclusions
After this long journey we have seen that the topological insulator is equipped
with two different Hall responses. With the use of domain walls we have con-
structed a topological Thouless pump for the parity-normal and parity-anomalous
Hall response.
Afterwards the big remaining question is: how are the mass term and a mag-
netic field (physically) related? Although this thesis has not really answered this
question, I want to share some thoughts on it.
One way of deriving the parity-anomalous Hall response is by a Pauli-Villars
regularization of the action. An anomaly breaks a symmetry that the action
classically has, which is in our case chiral symmetry. Like the chiral anomaly
in d = 3 + 1, this is unavoidable and has measurable consequences. If we think
back to the relativistic spectra, where the zeroth Landau level splitted into two
branches going up to infinity, we can interpret that as the anomaly. No matter
how close we position our Fermi energy to zero, we always have a single-edge
mode.
The fact that both terms, the mass term and the magnetic field break time
reversal invariance, indicates that they are related. Another very puzzling fact is
143
the experimental evidence that the topological phase of a TI sustains magnetic
fields of about 11T . This is remarkable because in the derivation one explicitly
uses TRS.
Nonetheless all these remaining questions I have had a great journey through
a variety of fields within theoretical physics. The point that I liked the most is
that a simple question can lead you from Chern classes, to particles in a box and
from adiabatic curvature to instantons. The broad scope was the part of this
project that I enjoyed the most. Now I take a short holiday and the first thing
I do when I’m back is to try to solve the Dirac equation using the Hopf map!
Followed by studying the mass domain wall in (3 + 1 + 1)-dimensions and its
experimental consequences!!
6.2 Outlook
There are many directions and leads that could be continued from here on, let us
name a few:
• Entanglement in meso-scopic Landau systems by studying the band split-
ting.
• Study domain wall configurations, adiabatic cycles with another gauge
group than U(1), like in Estienne et al. [2011]
• Laughlin gauge argument for the TI, by taking the leads into account, which
leads to an analyses of a g = 2 surface in the style of Avron & Seiler.
• TI’s on genus g surfaces, what about the multiplicity of spin connections
22g, is it possible to measure spin (connection) multiplicity? Label genera
with gi, what is the topological response if it is threaded with (non-abelian)
fluxes Φi?
• Mass domain wall compactification for the standard model. Would there
be ways to detect that we live in a 5d world? How do we know if we are
144
not living on the domain wall of the fifth dimension? Could it explain
the broken parity in our universe? And could the Jackiw-Rebbi states be
understood as the fermionic brothers of the Goldstone bosons?
• Why is the relativistic nature of the physics at the surface of a topological
insulator maintained in a magnetic field?
• Is it possible to disentangle (also experimentally) the two contributions to
the Hall conductance?
• In the Fock space the quantum numbers are Z2 for fermions and Z for
bosons, could one consider TI’s in the topological classification as bosonic
and fermionic as well?
145
Acknowledgements
First I want thank my supervisor Kareljan Schoutens for giving me
the opportunity to conduct this challenging, open-ended, free, and at
the frontier of theoretical physics, project. I really appreciated the
contributions and comments sent from Aspen! That was a great help
and stimulus. To conclude I really have learnt a lot. Glancing at my
bibliography it has become a project that is based in 21-century theo-
retical physics. However all triggered by Paul Adrien Maurice Dirac.
Another one who I really need to thank is my girlfriend Reka Fekete,
who suffered quite a bit because of this project. First she learnt ev-
erything about dimers and now she knows everything about the dif-
ference between donuts and oranges. Without any jokes, conducting
this project without her would have been much harder. Thank you
very much!
Of course I need to thank my buddy’s of the Nieuw Amsterdams
Genootschap voor de Theoretische Fysica , noteworthy Willem-Victor,
Paul de Lange, Gijs Leegwater and Philip van Reeuwijk. With whom
it has always been a pleasure to brainstorm about physics and the
nature of nature. I would like to thank Willem-Victor van Gerven
Oei separatly, who did a great job in proof reading my manuscript.
(hyphens, hyphens and more hyphens...;-)
A lot of great people at the ITFA were of great help during my project,
especially I would like to thank Balazs Pozsgay, Benoit Estienne, Jes-
per Romers, Shanna Haaker, A.M.M. Pruisken, J.-S. Caux and Erik
Verlinde. And of course I need to thank the experimentalists from the
van der Waals-Zeeman institute, with whom we could have heated
discussions about Ti’s: Erik van Heumen, Jeroen Goedkoop, Jook
Walraven. What a passion for physics!
Very important are my parents, who made it possible for me to do
second study. It is indescribable how grateful I am for that gift.
Then I want to thank my house mates of Langewagt (www.langewagt.nl),
who were so kind to give me a great refuge these years, in a great artis-
tic inspiring environment in the centre-centre of Amsterdam.
And last but not least the children of 3 HAVO and 4 VWO: for the
honor I had to teach them classical Newtonian mechanics this year.
Among many great moments I would like to mention the concept of
Mees Bartz, who came with the interesting concept of anti-time, or
its covariant version anti-spacetime. Very intruiging!
Amsterdam, 22nd of August 2011,
Sal Jua Bosman
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