half-integer hall response in topological insulators · this thesis studies the hall response of a...

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

mailto:[email protected]



http://www.uva.nl

http://www.uva.nl

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

[email protected]

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


4.3.2 Relativistic . . . . . . . . . . . . . . . . . . . . . . . . . . 120

4.3.2.1 Zero field . . . . . . . . . . . . . . . . . . . . . . 121

4.3.2.2 Constant field . . . . . . . . . . . . . . . . . . . . 123


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


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


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


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


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


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


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


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


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


2.2.2 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


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


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


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


2.2.3 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


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


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


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


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


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


2.2.4.3 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


↑

↑↑↑↑

↑

↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑

↑ ↑ ↑ ↑ ↑↑↑ ↑

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


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


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


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


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


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


〈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


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


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


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


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


2.4 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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


• (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


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


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


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


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


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


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


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


3.3.1 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


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


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


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


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


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


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


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


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


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


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


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


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


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


4.2.1.2 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


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


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


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


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


4.2.1.4 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


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


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


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


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


• 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


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


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


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


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)

æ æ æ æ æ æ æ ææ

æ æ æææ

æ

æ

æ

à à à à à à àà

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


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


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


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.


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


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


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


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


4.2.2.4 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


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


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


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


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


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


4.3.1 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


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.


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


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


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


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



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


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


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


ææ æ æ æ æ æ æ æ æ æ æ

æææææ æ æ æ

ææ

æ

æ

æ

àà à à à à à à à à à à à à à à à

ààà

à

à

à

à

à

ì

ìì ì ì ì ì

ììì

ì

ì

ìì ì ì ì

ìì

ì

ì

ì

ì

ì

ì

ò

òò ò ò ò ò ò

òòò ò ò ò ò

òò

ò

ò

ò

ò

ò

ò

ò

ò

-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.

æ

æ

æ

æ

æ

æ

æ

ææ æ

æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææææææ

ææææ

à

à

à

à

à

à

à

àà à

à à à à à à à à à à à à à à à àààààà

à

à

ì

ì

ì

ì

ì

ì

ì

ì

ìì ì

ì ì ì ì ì ììììì ì ì ì ì

ìììììì

ì

ì

ò

ò

ò

ò

ò

ò

ò

ò

òò ò

ò ò ò ò ò ò ò ò ò ò ò òòòòòòò

ò

ò

ò

ò

-30 -25 -20 -15 -10 -5 5m

100

200

300

E

Figure 4.14: The spectrum including the unphysical domain where m < 20.

119


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


4.3.2.1 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


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


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|.


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


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


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.


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


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


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).

æ æ æ æ æ ææ

ææ

ææ æ æ æ æ æ

ææ

ææ

ææ

æ

à à à à à à à à à à à à à à àà

àà

àà

àà

à

ì ì ì ì ì ì ì ì ì ì ìììììììììììì

ò ò ò ò ò òò

òò

òò ò ò ò ò ò ò

òò

òò

òò

ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ôô

ôô

ôô

ôô

ç ç ç ç ç ç ç ç ç ç çç

çç

çç

çç

çç

ç

ç

-15 -10 -5 5m

-15

-10

-5

5

10

15

E

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


æ

æ

æ

æ

æ

æ

æ

æ

æ

æ æææ æ æ æ æ æ æ æ

æææ æ æ æ æ æ æ æ

ææææææ

à

à

à

à

à

à

à

à

à

à ààà à à à à à à à à à à à à à à à à

ààààààà

ì

ì

ì

ì

ì

ì

ì

ì

ì

ì

ì

ì ì ì ì ì ì ì ì ì ì ì ì ì ììììììììììì

ò

ò

ò

ò

ò

ò

ò

ò

ò

ò òòò ò ò ò ò ò ò ò

òòò ò ò ò ò ò ò ò ò

òòòòò

ô

ô

ô

ô

ô

ô

ô

ô

ô

ô ôôô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô

ôôôôôô

ç

ç

ç

ç

ç

ç

ç

ç

ç

ç

ç

ç ç ç ç ç ç ç ç ç ç ç ç ç ççççççççççç

-30 -25 -20 -15 -10 -5 5m

-20

-10

10

20

E

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


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


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


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


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


5.1.2 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


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


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


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


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


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


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


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

Bibliography

Mathematica 7. www.wolfram.com/mathematica/. 99

M. Abramowitz and I.A. Stegun. Handbook of mathematical functions with for-

mulas, graphs, and mathematical tables. Number v. 55, no. 1972 in Applied

mathematics series. U.S. Govt. Print. Off., 1964. ISBN 9780486612720. URL

http://books.google.com/books?id=MtU8uP7XMvoC. 123

A. A. Abrikosov. Dirac operator on the riemann sphere, 2002. 121

Alexander Altland and Ben Simons. Condensed Matter Field Theory. Cambridge

University Press, June 2006. ISBN 0521845084. URL http://www.amazon.

com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/0521845084.

9, 95

Alexander Altland and Martin R. Zirnbauer. Nonstandard symmetry classes in

mesoscopic normal-superconducting hybrid structures. Phys. Rev. B, 55(2):

1142–1161, Jan 1997. doi: 10.1103/PhysRevB.55.1142. 9, 38

M.F. Atiyah and I.M. Singer. The index of elliptic operators: I. 87:484–530,

1968. 124

J E Avron and A. Pnueli. Ideas and methods in mathematical

analysis, stochastics, and applications, volume II: Ideas and meth-

ods in Quantum and Statistical physics, chapter Landau Hamiltoni-

ans on symmetric spaces, page 96. Cambridge University Press,

http://physics.technion.ac.il/ avron/files/pdf/pnueli.pdf, 1992. 82

148

http://books.google.com/books?id=MtU8uP7XMvoC

http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/0521845084


BIBLIOGRAPHY

Joseph. E. Avron. Adiabatic Quantum Transport. Course 1 Les Houches. 1995.

31

Joseph E. Avron and Ruedi Seiler. Quantization of the hall conductance for

general, multiparticle schrodinger hamiltonians. Phys. Rev. Lett., 54(4):259–

262, Jan 1985. doi: 10.1103/PhysRevLett.54.259. 21, 29

Xavier Bekaert and Nicolas Boulanger. The unitary representations of the

poincare group in any spacetime dimension, 2006. 61

B. Andrei Bernevig, Taylor L. Hughes, and Shou-Cheng Zhang. Quantum spin

hall effect and topological phase transition in hgte quantum wells. SCIENCE,

314:1757, 2006. URL http://www.citebase.org/abstract?id=oai:arXiv.

org:cond-mat/0611399. 42

Daniel Boyanovsky, Elbio Dagotto, and Eduardo Fradkin. Anomalous cur-

rents, induced charge and bound states on a domain wall of a semicon-

ductor. Nuclear Physics B, 285:340 – 362, 1987. ISSN 0550-3213. doi:

DOI:10.1016/0550-3213(87)90343-9. URL http://www.sciencedirect.com/

science/article/pii/0550321387903439. 142

Elie Cartan. Sur une classe remarquable d’espaces de riemann. Bulletin de la

Societe Mathematique de France, 54:214–216, 1926. 39

Peng Cheng, Canli Song, Tong Zhang, Yanyi Zhang, Yilin Wang, Jin-Feng Jia,

Jing Wang, Yayu Wang, Bang-Fen Zhu, Xi Chen, Xucun Ma, Ke He, Lili Wang,

Xi Dai, Zhong Fang, X. C. Xie, Xiao-Liang Qi, Chao-Xing Liu, Shou-Cheng

Zhang, and Qi-Kun Xue. Landau quantization of massless dirac fermions in

topological insulator. e-print arXiv:1001.3220, 2010. 55

Shing-Shen Chern. An elementary proof of the existence of isothermal parameters

on a surface. In Proceedings of the American Mathematical Society, volume 6,

pages 771–782. American Mathematical Society, 1955. 66

J. P. Dahlhaus, C.-Y. Hou, A. R. Akhmerov, and C. W. J. Beenakker. Geodesic

scattering by surface deformations of a topological insulator. Phys. Rev. B, 82

(8):085312, Aug 2010. doi: 10.1103/PhysRevB.82.085312. 62

149

http://www.citebase.org/abstract?id=oai:arXiv.org:cond-mat/0611399

http://www.citebase.org/abstract?id=oai:arXiv.org:cond-mat/0611399

http://www.sciencedirect.com/science/article/pii/0550321387903439


BIBLIOGRAPHY

P. A. M. Dirac. Quantised singularities in the electromagnetic field. Proceedings

of the Royal Society of London. Series A, 133(821):60–72, 1931. doi: 10.1098/

rspa.1931.0130. URL http://rspa.royalsocietypublishing.org/content/

133/821/60.short. 1, 112, 141

Benoit Estienne, Shanna Haaker, and Kareljan Schoutens. Particles in non-

abelian gauge potentials - landau problem and insertion of non-abelian flux.

New J.Phys.13:045012,2011, 2011. 144

Motohiko Ezawa. Supersymmetry and unconventional quantum hall effect in

monolayer, bilayer and trilayer graphene. Physica E: Low-dimensional Systems

and Nanostructures, 40(2):269 – 272, 2007. ISSN 1386-9477. doi: DOI:10.

1016/j.physe.2007.06.038. URL http://www.sciencedirect.com/science/

article/pii/S138694770700152X. International Symposium on Nanometer-

Scale Quantum Physics. 102

Z F Ezawa. Quantum Hall Effects: Field Theoretical Approach and Related Top-

ics. World Scientific, Singapore, 2000. 20

C. D. Fosco and A. Lopez. Dirac fermions and domain wall defects in 2+1

dimensions. Nucl.Phys. B538 (1999) 685-700, 1998. 142

Liang Fu and C. L. Kane. Time reversal polarization and a z2 adiabatic spin

pump. Phys. Rev. B, 74(19):195312, Nov 2006. doi: 10.1103/PhysRevB.74.

195312. vi, 26, 28

Liang Fu and C. L. Kane. Topological insulators with inversion symmetry. Phys.

Rev. B, 76(4):045302, Jul 2007. doi: 10.1103/PhysRevB.76.045302. 2, 42, 46,

47

Steven M. Girvin. The quantum hall effect: Novel excitations and broken sym-

metries. Topological Aspects of Low Dimensional Systems, ed. A. Comtet,

T. Jolicoeur, S. Ouvry, F. David (Springer-Verlag, Berlin and Les Editions de

Physique, Les Ulis, 2000), 1999. 20

150

http://rspa.royalsocietypublishing.org/content/133/821/60.short

http://rspa.royalsocietypublishing.org/content/133/821/60.short

http://www.sciencedirect.com/science/article/pii/S138694770700152X

http://www.sciencedirect.com/science/article/pii/S138694770700152X

BIBLIOGRAPHY

Jeffrey Goldstone and Frank Wilczek. Fractional quantum numbers on solitons.

Phys. Rev. Lett., 47(14):986–989, Oct 1981. doi: 10.1103/PhysRevLett.47.986.

142

M.B. Green, J.H. Schwarz, and E. Witten. Superstring theory: Introduction.

Cambridge monographs on mathematical physics. Cambridge University Press,

1988. ISBN 9780521357524. URL http://books.google.com/books?id=

ItVsHqjJo4gC. 63

F. D. M. Haldane. Fractional quantization of the hall effect: A hierarchy of

incompressible quantum fluid states. Phys. Rev. Lett., 51(7):605–608, Aug

1983. doi: 10.1103/PhysRevLett.51.605. 77

T. Hanaguri, K. Igarashi, M. Kawamura, H. Takagi, and T. Sasagawa.

Momentum-resolved landau-level spectroscopy of dirac surface state in bi2se3.

Phys. Rev. B 82, 081305(R) (2010), 2010. 55

A. Hatcher. Algebraic topology. Cambridge University Press, 2002. ISBN

9780521795401. URL http://books.google.com/books?id=BjKs86kosqgC.

10

P. Heinzner, A. Huckleberry, and M.R. Zirnbauer. Symmetry classes of disor-

dered fermions. Communications in Mathematical Physics, 257:725–771, 2005.

ISSN 0010-3616. URL http://dx.doi.org/10.1007/s00220-005-1330-9.

10.1007/s00220-005-1330-9. 10

D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z. Hasan. A

topological dirac insulator in a quantum spin hall phase. Nature, 452(7190):

970–974, 04 2008a. URL http://dx.doi.org/10.1038/nature06843. 42

D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava, and M. Z. Hasan.

A topological dirac insulator in a quantum spin hall phase : Experimental

observation of first strong topological insulator. NATURE, 452:970, 2008b.

URL doi:10.1038/nature06843. 2

151

http://books.google.com/books?id=ItVsHqjJo4gC

http://books.google.com/books?id=ItVsHqjJo4gC

http://books.google.com/books?id=BjKs86kosqgC

http://dx.doi.org/10.1007/s00220-005-1330-9

http://dx.doi.org/10.1038/nature06843

doi:10.1038/nature06843

BIBLIOGRAPHY

R. Jackiw and C. Rebbi. Solitons with fermion number 1/2. Phys. Rev. D, 13

(12):3398–3409, June 1976. doi: 10.1103/PhysRevD.13.3398. 4, 24, 131, 136,

142

C.G. Callan Jr. and J.A. Harvey. Anomalies and fermion zero modes on

strings and domain walls. Nuclear Physics B, 250(1-4):427 – 436, 1985.

ISSN 0550-3213. doi: DOI:10.1016/0550-3213(85)90489-4. URL http://www.

sciencedirect.com/science/article/pii/0550321385904894. 142

Tosio Kato. On the convergence of the perturbation method. i. Progress of

Theoretical Physics, 4(4):514–523, 1949. doi: 10.1143/PTP.4.514. URL http:

//ptp.ipap.jp/link?PTP/4/514/. 30

Alexei Kitaev. Periodic table for topological insulators and superconductors,

2009. 2, 41

Walter Kohn. Theory of the insulating state. Phys. Rev., 133(1A):A171–A181,

Jan 1964. doi: 10.1103/PhysRev.133.A171. 6

Markus Konig, Steffen Wiedmann, Christoph Brune, Andreas Roth, Hartmut

Buhmann, Laurens W. Molenkamp, Xiao-Liang Qi, and Shou-Cheng Zhang.

Quantum spin hall insulator state in hgte quantum wells. Science, 318(5851):

766–770, 2007. 42

Markus Konig, Hartmut Buhmann, Laurens W. Molenkamp, Taylor Hughes,

Chao-Xing Liu, Xiao-Liang Qi, and Shou-Cheng Zhang. The quantum spin

hall effect: Theory and experiment. Journal of the Physical Society of Japan,

77(3):031007, 2008. doi: 10.1143/JPSJ.77.031007. URL http://jpsj.ipap.

jp/link?JPSJ/77/031007/. 50

R. B. Laughlin. Quantized hall conductivity in two dimensions. Phys. Rev. B, 23

(10):5632–5633, May 1981. doi: 10.1103/PhysRevB.23.5632. 28

Dung-Hai Lee. Surface states of topological insulators: The dirac fermion in

curved two-dimensional spaces. Phys. Rev. Lett., 103(19):196804, Nov 2009.

doi: 10.1103/PhysRevLett.103.196804. viii, 3, 53, 62, 108, 116, 125, 128

152



http://ptp.ipap.jp/link?PTP/4/514/

http://ptp.ipap.jp/link?PTP/4/514/

http://jpsj.ipap.jp/link?JPSJ/77/031007/


BIBLIOGRAPHY

Chao-Xing Liu, Xiao-Liang Qi, HaiJun Zhang, Xi Dai, Zhong Fang, and Shou-

Cheng Zhang. Model hamiltonian for topological insulators. Phys. Rev. B 82,

045122 (2010), 2010. vi, 42, 44, 45, 49, 50, 55

Qin Liu, Chao-Xing Liu, Cenke Xu, Xiao-Liang Qi, and Shou-Cheng Zhang.

Magnetic impurities on the surface of a topological insulator. Phys. Rev. Lett.,

102(15):156603, Apr 2009. doi: 10.1103/PhysRevLett.102.156603. 54

Joseph D. Lykken, Jacob Sonnenschein, and Nathan Weiss. Anyonic supercon-

ductivity. Phys. Rev. D, 42(6):2161–2165, Sep 1990. doi: 10.1103/PhysRevD.

42.2161. 74

M. Nakahara. Geometry, Topology and Physics. Taylor & Francis, 2 edition,

June 2003. ISBN 0750306068. URL http://www.amazon.com/exec/obidos/

redirect?tag=citeulike07-20&path=ASIN/0750306068. 14

Kentaro Nomura, Shinsei Ryu, Mikito Koshino, Christopher Mudry, and Akira

Furusaki. Quantum hall effect of massless dirac fermions in a vanishing

magnetic field. Phys. Rev. Lett., 100(24):246806, Jun 2008. doi: 10.1103/

PhysRevLett.100.246806. vi, 75

V. Parente, P. Lucignano, P. Vitale, A. Tagliacozzo, and F. Guinea. Spin connec-

tion and boundary states in a topological insulator. Phys.Rev.B83:075424,2011,

2010. 62

A Pnueli. Spinors and scalars on riemann surfaces. Journal of Physics A:

Mathematical and General, 27(4):1345, 1994. URL http://stacks.iop.org/

0305-4470/27/i=4/a=028. 53, 79, 123

R.E. Prange and S.M. Girvin. The Quantum Hall effect:. Springer-Verlag,

1987. ISBN 9783540962861. URL http://books.google.com/books?id=

Y7XvAAAAMAAJ. 20

William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P.

Flannery. Numerical Recipes 3rd Edition: The Art of Scientific Computing.

Cambridge University Press, New York, NY, USA, 3 edition, 2007. ISBN

0521880688, 9780521880688. 98, 122

153



http://stacks.iop.org/0305-4470/27/i=4/a=028


http://books.google.com/books?id=Y7XvAAAAMAAJ

http://books.google.com/books?id=Y7XvAAAAMAAJ

BIBLIOGRAPHY

A. M. M. Pruisken, R. Shankar, and Naveen Surendran. General topological

features and instanton vacuum in quantum hall and spin liquids. Phys. Rev.

B 72, 035329 (2005), 2005. 26

Xiao-Liang Qi and Shou-Cheng Zhang. Topological insulators and superconduc-

tors, 2010. 47, 49

Xiao-Liang Qi, Taylor L. Hughes, and Shou-Cheng Zhang. Topological field the-

ory of time-reversal invariant insulators. Phys. Rev. B, 78(19):195424, Nov

2008. doi: 10.1103/PhysRevB.78.195424. 3

Nicolai Reshetikhin. Lectures on quantization of gauge systems. 14

Shinsei Ryu, Andreas P Schnyder, Akira Furusaki, and Andreas W W Lud-

wig. Topological insulators and superconductors: tenfold way and dimen-

sional hierarchy. New Journal of Physics, 12(6):065010, 2010. URL http:

//stacks.iop.org/1367-2630/12/i=6/a=065010. 2, 9, 39, 41, 53, 74

Adriaan M. J. Schakel. Relativistic quantum hall effect. Phys. Rev. D, 43(4):

1428–1431, Feb 1991. doi: 10.1103/PhysRevD.43.1428. 3, 74

Andreas P. Schnyder, Shinsei Ryu, Akira Furusaki, and Andreas W. W. Ludwig.

Classification of topological insulators and superconductors in three spatial

dimensions. Phys. Rev. B, 78(19):195125, Nov 2008. doi: 10.1103/PhysRevB.

78.195125. 10

Julian Schwinger. On gauge invariance and vacuum polarization. Phys. Rev., 82

(5):664–679, Jun 1951. doi: 10.1103/PhysRev.82.664. 70

Ryuichi Shindou. Quantum spin pump in s = 1/2 antiferromagnetic chains —

holonomy of phase operators in sine-gordon theory—. Journal of the Physical

Society of Japan, 74(4):1214–1223, 2005. doi: 10.1143/JPSJ.74.1214. URL

http://jpsj.ipap.jp/link?JPSJ/74/1214/. vi, 26, 28

William Shockley. On the surface states associated with a periodic potential.

Phys. Rev., 56(4):317–323, Aug 1939. doi: 10.1103/PhysRev.56.317. 26

154




BIBLIOGRAPHY

Michael Stone and Paul Goldbart. Mathematics for Physics: A Guided Tour

for Graduate Students. Cambridge University Press, 1 edition, August 2009.

ISBN 0521854032. URL http://www.amazon.com/exec/obidos/redirect?

tag=citeulike07-20&path=ASIN/0521854032. 14, 113

D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs. Quantized hall

conductance in a two-dimensional periodic potential. Phys. Rev. Lett., 49(6):

405–408, Aug 1982. doi: 10.1103/PhysRevLett.49.405. 28

John von Neumann and Eugene Wigner. Uber merkwurdige diskrete Eigenwerte.

Physikalische Zeitschrift, 30:465–467, 1929. 12, 29

J.T.M. Walraven. Elements of quantum gases: Thermodynamic and collisional

properties of trapped atomic gases. February 2010. 44

Jean Zinn-Justin. Path Integrals in Quantum Mechanics. Oxford University

Press, USA, illustrated edition edition, February 2005. ISBN 0198566743. URL

http://www.worldcat.org/isbn/0198566743. 95

155



http://www.worldcat.org/isbn/0198566743

half-integer hall response in topological insulators · this thesis studies the hall response of a...

Documents