Axion Electrodynamics and Measurable Effects in Topological Insulators

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Axion Electrodynamics and

Measurable Effects in Topological

Insulators

Axion Elektrodynamik och Mätbara Effekter i Topologiska Isolatorer

Andreas Asker

Faculty of Health, Science, and Technology Master thesis

30 hp (ECTS)

Supervisor: Jürgen Fuchs Examiner: Lars Johansson Date 2018-06-08

Serial number N/A

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AXION ELECTRODYNAMICS AND MEASURABLE

EFFECTS IN TOPOLOGICAL INSULATORS

Sweden, 2017-2018

by

ANDREAS ASKER

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Abstract

Topological insulators are materials with their electronic band structure in bulk resembling that of an ordinary insulator, but the surface states are metallic. These surface states are topologically protected, meaning that they are robust against impurities. The topological phenomena of three dimensional topological insulators can be expressed within topological field theories, predicting axion electrodynamics and the topological magnetoelectric effect. An experiment have been sug-gested to measure the topological phenomena. In this thesis, the underlying theory and details around the experiment are explained and more detailed derivations and expressions are provided.

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Acknowledgment

I would like to thank my supervisor Prof. Jürgen Fuchs for introducing me to the subject of topological field theory applied to topological insulators, knowing that it would suit me well. Our discussions, his recommendations and that he takes time for questions has been greatly appreciated and will always be remembered.

A special thank you to my friends Rickard Barkman, Gabriel Khajo, Simon Kronberg and Rasmus Lavén for helping me with MATLAB-plots and Mathematica calculations (and/or other computer related problems), due to my limited competence with these programs.

Lastly, I would like to thank my friends and family for all of their support that they have given me.

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List of Figures

3.1 A sphere (g = 0), a torus (g = 1) and a genus-2 surface (g = 2). . . 36 3.2 The plane P , spanned by T and N at point p, cuts out the normal section C of

M. The curvature vector κ points from p towards the centre of curvature (at a distance of κ−1_{). . . .} ₃₇ 4.1 The shear and squeezing deformation of a torus generates a non-Abelian geometric

phase T and S, respectively. The last shear deformed torus is the same as the original torus after the coordinate transformation (x, y) 7→ (x+y, y). Similarly, the last squeeze-deformed torus is the same as the original torus after the coordinate transformation (x, y) 7→ (y, −x). . . 42 4.2 Schematic representation of the surface energy levels of a crystal in either 2D or

3D, as a function of surface crystal momentum. (a) shows a conventional insulator, whereas (b) shows a topological insulator. The shaded regions show the bulk con-tinuum states, and the lines show discrete surface (or edge) bands localized near one of the surfaces. In topological insulators, (b), the surface bands are guaranteed to cross the Fermi level inside the bulk, allowing metallic conduction on the surface. 44 5.1 Illustration of the QH effect in ferromagnetic-topological insulator

heterostruc-ture. The electric field Ex, with the direction into the paper represented by ”⊗”, induces Hall currents jtand jbfor (a) parallel and (b) anti-parallel magnetization, respectively. . . 63 5.2 Illustration of the magnetization induced by an electric field E in a cylindrical

geometry. The magnetization of the FM layer points outward from the side surface of the TI, and a circulating current is induced by the electric field. . . 64 5.3 Illustration of the charge polarization induced by a perpendicular magnetic field

B in a cylindrical geometry. The positive and negative charges induced by the magnetic field on the top and bottom surfaces are denoted ”⊕” and ” ”, respectively. 65 6.1 Faraday-Kerr effect: the polarization (black arrow) of the incident light becomes

rotated (blue arrow) due to the magnetized material, with magnetization vector M. The rotated polarization of the reflected and transmitted light defines the angle θK(Kerr) and θF (Faraday), respectively. . . 69 6.2 Setup of the proposed optical experiment, where a topological insulator film of

thickness ` and optical constants ε2, µ2 is deposited on a topologically trivial substrate with optical constants ε3, µ3. The objective is to shine monochromatic light with frequency ω onto the topological insulator film and measure the Kerr angle θKand Faraday angle θFof the reflected and transmitted light, respectively, in the presence of an external magnetic field B. . . 70

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6.3 The universal function h

e2f (θ) for various material constants ε2, ε3, p (setting µ2 = µ3 = 1 without loss of generality). The zero crossing of f(θ) provides the bulk axion angle θ. Here, we are using atomic units such that e2

h = 1. Furthermore, a scale factor of 0.425 has been multiplied to f(θ) in order to compare with the obtained plots of Ref. [1]. . . 74 6.4 The reflectivity R as a function of ω

ω`. The optical constants ε2 = 100, ε3 = 13

and µ2 = µ3 = 1, which are appropriate for topological Bi2Se3 thin films on a Si substrate [1, page 2]. The reflective minima and maxima occur at ω/ω` = n and ω/ω` = (n + 1₂), n ∈ Z respectively, which agrees with the results of Ref. [1]. However, the values for the reflectivity at these points do not agree with the results of Ref. [1]. . . 77 6.5 The reflectivity R as a function of ω

ω`, where we have taken into account for

transmission from medium 2 into 1. The optical constants ε2 = 100, ε3 = 13 and µ2 = µ3= 1, which are appropriate for topological Bi2Se3 thin films on a Si substrate [1, page 2]. The reflective minima and maxima occur at ω/ω`= n and ω/ω` = (n + 1₂), n ∈ Z, respectively, which agrees with the results of Ref. [1]. However, similar to Fig. 6.4, the values for the reflectivity at these points do not agree with the results of Ref. [1]. . . 81 6.6 Comparison of the reflectivity R in Fig. 6.4 (curve R1) and 6.5 (curve R2) as a

function of ω

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Chapter 1 Introduction

In recent decades, extensive research and developments have taken place in science and techno-logy, especially in the field of condensed matter and nanotechnology. One reason for the amount of interest in these fields is the discovery of the integer quantum Hall effect (1980) and frac-tional quantum Hall effect (1982). At the time, the latter demonstrated the limits of theoretical models and, in turn, motivated for a new type of classification of phases and phase transitions. Furthermore, from a technological perspective, a quantized Hall conductance and dissipationless currents have exciting prospects. However, the integer quantum Hall effect only manifests at strong magnetic fields and extremely low temperatures. This limited any further developments, in particular the possibility for new technology in both industrial and daily applications. There-fore, it became of great interest to find a quantum Hall effect without these specific restraints. The discovery of the integer quantum Hall effect and the fractional quantum Hall effect formed the theoretical basis for the topological insulators. Furthermore, the theoretical prediction and experimental validation of the quantum spin Hall effect in 2005 marked the beginning of topolo-gical insulator exploration [2, page 1-3].

Within the last decade, many specific types of topological insulators have been theoretically pre-dicted and found. For example, experiments have shown Hg1−xCdxTe to be a two dimensional topological insulator system [3], while the materials Bi2Se3, Bi2Te3 and Sb2Te3 show three di-mensional topological insulator characteristics [4]. At present time, main applications for topolo-gical insulators are electronic and semiconductor devices; e.g. photodetectors, magnetic devices, field-effective-transistors and lasers. However, due to the constraints of preparation methods and processes the topological insulator material inevitably will have defects, making the material properties differ significantly from the desired ideal ones. In turn, the practical application of topological insulators are, for now, limited and still at their initial stage [2, page 19-20]. The electromagnetic features of topological insulators can be defined within the theoretical frame-work of topological field theory. The topological responses support many novel topological phe-nomena, some of which theoretically predict magnetic monopole charges [5]. We will mainly consider the topological field theory of three dimensional topological insulators and how topolo-gical phenomena can be experimentally tested.

In this thesis, we start in chapter 2 by going through some general background; e.g. notations, conventions, field theory, and so on. Next, we spend some time deriving the quantized Hall conductance and discussing its connection to topology in chapter 3. We also discuss some clas-sifications of phases of matter in chapter 4 and what classification topological insulators fall into. In chapter 5, we introduce the field theory aspect of topological insulators and how axion

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12 CHAPTER 1. INTRODUCTION electrodynamics emerges from the theory. We also emphasize how consequences of axion electro-dynamics can be measured. In chapter 6 we discuss thoroughly a specific experiment, which has been proposed by J. Maciejko and et al (Ref. [1]). In this thesis, we perform additional calcula-tions and obtain explicit expressions, which are not provided in Ref. [1]. These expressions are plotted and are in good agreement with the results in Ref. [1].

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Chapter 2 General background

In this chapter we consider some of the preliminaries required for this paper in more detail, which should aid the reader. The topics mentioned here includes notations, conventions and units used in this paper and more in depths about tensors and field theory. The following concepts will not be mentioned in greater, if any, detail: theory of quantum mechanics, basics of electromagnetic field theory or solid state theory. The author anticipates that the reader has some background in these fields.

2.1 Notations and conventions

2.1.1 Tensors, index and units

In this paper we denote 4-vectors xµ_{= (t, x, y, z)}T with Greek indices, running over 0,1,2,3, and 3-vectors xi_{= (x, y, z)}T with Latin indices, running over 1,2,3. From now on, we write x0for the time component of a 4-vector xµ. In addition, 4-vector are denoted by light italic type; 3-vectors are denoted by boldface type; e.g. xµ _{= (x}0_{, x)}T, with unit 3-vectors denoted by boldface with a hat over it, e.g. ˆx. Lastly, in order to avoid confusion we will be represented matrices in parentheses-form, e.g. a b

c d

, whereas rank 2 tensors will be represented in bracket-form, e.g. _{a b}

c d

. We use the metric

gµν=     1 0 0 0 0 −1 0 0 0 0 −1 0 0 0 0 −1     , (2.1.1)

where gµν is known as the Minkowski metric and gµν = diag [ 1, −1, −1, −1 ] is the inverse. With this metric we can construct a covariant 4-vector xµ from the contravariant 4-vector xµ: xµ= 3 X ν=0 gµνxν≡ gµνxν= (x0, −x) . (2.1.2) 13

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14 CHAPTER 2. GENERAL BACKGROUND In Eq. (2.1.2) we have introduced another convention, i.e. ignore writing out the sum. In fact, the reason we have written upper and lower indices is that it enables us to introduce the Einstein

summation convention; whenever an expression contains indices as superscripts and the same

indices as subscript, a summation is implied over all values that those indices can take. In Eq. (2.1.2), µ is known as a free index, whereas ν is called a dummy index. If a free index is relabelled, it must be so everywhere in the expression. In contrast, since a dummy index merely denotes the summation from 0 to 3, it does not matter what Greek letter is used, e.g. gµνxν = gµαxα= gµζxζ and so on. Furthermore, note that the index structure in Eq. (2.1.2) is balanced; i.e. 1 = 2 − 1 or 1 ”lower index” = 2 ”lower indices” − 1 ”upper index”. The number of indices involved in a tensor is called the rank, so what we mean by ”balanced index structure” is that the rank are equal on both sides of the equation. We require that the rank of any equation is conserved since if an expression states that v = Vµ, with v a scalar and Vµ a 4-vector, it is clearly wrong. Therefore, we will keep the index structure intact throughout this paper.

From Eq. (2.1.2) we note that the connection between any covariant vector (or one-form) Vµ and the contravariant Vµ is the Minkowski metric g

µν. In fact, for an _MN

tensor, a linear function of N vectors and M one-forms, the metric maps it into an N −1

M +1

tensor. Similarly, the inverse metric maps an N

M

tensor into an N +1 M −1

tensor [6, page 73-74]. Suppose Tαβ

γ are the components of a 2

1

tensor, then

Tαβµ= gµγTαβ_γ (2.1.3)

are the components of a 3 0

tensor, while

Tα_µγ= gµβTαβγ (2.1.4)

are the components of a 1 2

tensor. These operations are called index ”raising” and ”lowering”, respectively.

The Minkowski inner product of xµ is written as

x · x = gµνxνxµ = (x0)2− (x1)2− (x2)2− (x3)2≡ t2− x2− y2− z2= t2− x · x , (2.1.5) or

gµνxνxµ= xµxµ= x0x0+ x1x1+ x2x2+ x3x3. (2.1.6) The operation xµxµ in Eq. (2.1.6) is formally known as a contraction. A contraction is more fundamental in tensor analysis than the scalar product because it can be performed between any one-form and vector without reference to other tensors [6, page 59]. Using index raising and lowering the following contractions are equivalent:

VµWµ= gµνVνWµ= gµνgµρVνWρ= VµWµ. (2.1.7) Note that in last equality of Eq. (2.1.7) we have used the following; the metric and its inverse satisfy gµνgµρ= δρν, with δ_νρ= ( 1, for µ = ν 0, for µ 6= ν , (2.1.8) such that δρ

νVνWρ= VρWρand lastly we relabelled the dummy index ρ 7→ µ. When considering a transformation Λµ

ν, 4-vectors Vµ transform as Vµ 7→ Λµ

νV

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2.1. NOTATIONS AND CONVENTIONS 15 and is a defining property of a 4-vector. If Vµis not a number but depends on x, then Vµ_{= V}µ_(x) is called a vector field with similar transformation property as 4-vectors. Furthermore, a tensor Tµν, say, transforms under Λµ

ν as Tµν 7→ ΛµαΛ ν βT αβ , (2.1.10)

and tensor fields Tµν _{= T}µν_(x)have similar transformation property.

Let us consider some important objects which will appear in this paper. For the position 4-vector xµ the derivatives with respect to xµ is the following one-form:

∂µ≡ ∂

∂xµ = (∂t, ∂x, ∂y, ∂z) . (2.1.11) It should be mentioned that when we write ∂µ, then, throughout this paper, we mean the derivatives with respect to xµ. If we differentiate with respect to the wave number kµ, say, then I write, for example, ∂kµ to clarify this. The operator that involves second-order derivatives is

the d’Alembertian = ∂µ∂µ= ∂2t− ∂ 2 x− ∂ 2 y− ∂ 2 z= ∂ 2 t − 4 , (2.1.12)

which is the relativistic generalization of the Laplacian 4 = ∇2_{= ∂}2 x+ ∂ 2 y+ ∂ 2 z. (2.1.13)

We define the totally antisymmetric tensor εµνρσ so that

ε0123= + 1 . (2.1.14)

This antisymmetric tensor is the 4 dimensional analogue of the Levi-Civita symbol εijk =      + 1 for (123), (231) or (312) − 1 for (321), (132) or (213) 0 for i = j, j = k, or k = i . (2.1.15)

However, we have to be careful using the definition Eq. (2.1.14) since it implies

ε0123= g0µg1νg2ρg3σεµνρσ= g00g11g22g33ε0123= (+1)(−1)(−1)(−1)ε0123= −ε0123, (2.1.16) and

ε0123= −ε1023= ε1203= −ε1230. (2.1.17) The antisymmetric tensor is useful when writing cross-products in index notation. Recall that the cross-product operation between two vectors yields a vector. As a consequence of keeping the index structure intact, we will write C = A × B in index notation in the following way:

Ck = ε_ijkAiBj = εijkAiBj. (2.1.18) Lastly, let us mention the convention of units. In most cases, we will work in units where

~ = c = 1 . (2.1.19)

In this system,

[length] = [time] = [energy]−1_{= [mass]}−1_. _(2.1.20) For illustrative purposes, we will also restore the units for some final results.

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16 CHAPTER 2. GENERAL BACKGROUND

2.2 Field theory and gauge theory

2.2.1 Lagrangian and Hamiltonian field theory

Two possible ways to define a field theory are in terms of either a Lagrangian or a Hamiltonian. We will almost exclusively use Lagrangians. For the moment, we will consider 3+1 dimensional space (that can easily be generalized). The Lagrangian, L, can be written as an integral over all space of a Lagrangian density, L, which is a functional of one or more fields φ(x) and their derivatives, i.e. , L = L(φ, ∂µφ). The dynamics of a Lagrangian system are determined by the principle of least action, where the action S is the time integral of the Lagrangian [7, page 15-16];

S = Z

dt L =Z d4_{x L(φ, ∂}

µφ) . (2.2.1)

Since it is L that is of most interest I will refer to it just as the Lagrangian. The principle of least action states that a system evolves from a given configuration, at time t1, to another, at time t2, along a path in configuration space such that S is stationary to first order, i.e., δSδφ = 0. This condition can be written as

δS = Z d4_x ∂L ∂φδφ + ∂L ∂(∂µφ) δ(∂µφ) = Z d4_x ∂L ∂φ− ∂µ ∂L ∂(∂µφ) δφ + ∂µ _∂L ∂(∂µφ) δφ = 0 . (2.2.2)

The last term can be turned into a surface integral over the boundary of the 4 dimensional spacetime. We will always make the assumption that the fields vanish sufficiently fast at the boundary, which lets us drop total derivative terms from Lagrangians. This will in turn allow for integration by parts within Lagrangians without any boundary term; i.e., A∂νB = −(∂µA)B. From this we have

∂L ∂φ− ∂µ

∂L ∂(∂µφ)

= 0 , (2.2.3)

which are known as the Euler-Lagrange equations. They give the equations of motion following from a Lagrangian.

Although Lagrangians will be of considerable interest in this paper we should still mention how the Hamiltonian can be obtained. The Hamiltonian H can be written as an integral over all space of a Hamiltonian density, H. The Hamiltonian density is a functional of fields and their conjugate momenta; i.e., H = H(φ, π). The Lagrangian (density) and Hamiltonian (density) are associated to each other via Legendre transformations [8, page 29]. Formally, the Lagrangian is obtained from the Hamiltonian as

L(φ, ˙φ) = π(φ, ˙φ) ˙φ − H(φ, π(φ, ˙φ)) , (2.2.4) where ˙φ = ∂tφand π(φ, ˙φ) is implicitly defined by ∂H(φ,π)_∂π = ˙φ. The inverse transformation is

H(φ, π) = π ˙φ(φ, π) − L(φ, ˙φ(φ, π)) , (2.2.5) where ˙φ(φ, π) is implicitly defined by ∂L(φ, ˙φ)

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2.2. FIELD THEORY AND GAUGE THEORY 17

2.2.2 Noether’s theorem and currents

Let us study the effect of a continuous transformation on the fields φ of the form φ 7→ φ + αδφ δα, where α is an infinitesimal parameter and δφ

δα is some deformation of the field configuration. If this transformation leaves the equations of motion invariant we call it a symmetry of the Lagrangian. The invariance is insured if the action changes at most by a surface term. Therefore, the Lagrangian must be invariant under the transformation up to a 4-divergence; i.e., L 7→ L + α∂µJµ, for some Jµ. If we let φn represent the set of fields the Lagrangian depends on, we find ∂µJµ= δL δα = X n ∂L ∂φn − ∂µ ∂L ∂(∂µφn) δφn δα + ∂µ ∂L ∂(∂µφn) δφn δα . (2.2.6) The first term vanishes due to the Euler-Lagrange equations and we find

∂µjµ= 0 , (2.2.7) where jµ=X n ∂L ∂(∂µφn) δφn δα − J µ_. _(2.2.8)

The vector field jµ is known as a Noether current and when such a vector field satisfies ∂µjµ = 0 it is called a conserved current. We are now ready to state Noether’s theorem: For each continuous symmetry of L there exist conserved currents associated to the symmetries and we have such a conservation law. The conservation law can be expressed by saying that the total charge Q, defined as

Q ≡ Z d3 x j0, (2.2.9) satisfies ∂tQ = Z d3_{x ∂} tj0= Z d3_{x ∇ · J = 0 .} _(2.2.10)

We have used the divergence theorem in the last step such that integral is zero, since, by assump-tion, nothing is leaving the system. The total charge does not change with time and is, just as jµ,

conserved. The concept of currents is useful in field theory since they can be used in many ways.

For example, currents can be Noether currents or external currents (such as electrons flowing through a wire). Currents can be used as sources for fields appearing in a Lagrangian as, say, L(x) = · · · − Aµ(x)Jµ(x). We will encounter currents as sources later in Sec. 2.3.2. Currents can simply be a formal place-holder. For example, if L = · · · − ieAµ(φ∗∂µφ − φ∂µφ∗), we could write it as L = · · · − AµJµ with Jµ=ie(φ∗∂µφ − φ∂µφ∗). It should be noted that currents are not dynamical fields; that is, they never have their own kinetic terms. Therefore, to solve the dynamics of Jµ are not of interest.

2.2.3 Covariant formulation of electromagnetism

Classical electromagnetic field theory is summed up by Maxwell’s equations, so we would like to express them in a covariant form. This can be achieved by introducing the following

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antisym-18 CHAPTER 2. GENERAL BACKGROUND metric field tensor (Eq. (5.1) in [9]):

Fµν =     0 Ex Ey Ez −Ex 0 −Bz By −Ey Bz 0 −Bx −Ez −By Bx 0     . (2.2.11)

In terms of Fµν , Maxwell’s equations takes on the simple forms

∂µFµν = Jν, (2.2.12)

εµνρσ∂νFρσ = 0 , (2.2.13)

where Jν _{= (ρ, J )}is a 4-vector current density. In Appendix 8.1.1 we justify Eqs. (2.2.12) and Eqs. (2.2.13) by explicit calculation to regain the usual forms of Maxwell’s equations. Fields Fµν that satisfy Eqs. (2.2.12)-(2.2.13) can be expressed in terms of a 4-vector potential Aµ_{= (Φ, A)} as

Fµν = ∂µAν− ∂νAµ. (2.2.14)

In terms of the potential Eqs. (2.2.13) are satisfied identically (see Appendix 8.1.1), and Eqs. (2.2.12) become

∂µFµν = ∂µ∂µAν− ∂µ∂νAµ= Aν− ∂ν(∂µAµ) = Jν. (2.2.15) The Eqs. (2.2.15) can also be derived from the Euler-Lagrange equations. Consider the Lag-rangian

L = −1 4FµνF

µν_{− A}

µJµ, (2.2.16)

where we treat the four components Aµas the independent fields. The Lagrangian in Eq. (2.2.16) is known as the Maxwell Lagrangian with sources and the name will be justified in the following calculations. We expand L by inserting Eq. (2.2.14) and simplify by exchanging dummy indices:

L = −1 4 ∂µAν− ∂νAµ ∂µAν− ∂ν_Aµ_{− A} µJµ = −1 2∂µAν∂ µ_Aν₊1 2∂µAν∂ ν_Aµ_{− A} µJµ. (2.2.17)

Next, by lowering indices, ∂ν_Aµ _{= g}να_gµβ_∂

αAβ, and using ∂(∂µAν)/∂(∂αAβ) = δαµδ β ν we note that ∂µ ∂(∂ρAσ∂ρAσ) ∂(∂µAν) = ∂µ ∂(∂ρAσ∂αAβ) ∂(∂µAν) gαρgβσ= ∂µ 2 δρµδ ν σ∂αAβgαρgβσ = 2 ∂µ∂µAν. (2.2.18) Then, the Euler-Lagrange equations implies

∂L ∂Aν − ∂µ ∂L ∂(∂µAν) = −Jν+ ∂µ∂µAν− ∂µ∂νAµ= 0 . (2.2.19) Hence, with the Lagrangian Eq. (2.2.16) we obtain Maxwell’s equations from the Euler-Lagrange equations. It seems, however, that we only have regained Eqs. (2.2.12), but we have, in fact, also obtained Eqs. (2.2.13). This will be demonstrated in Appendix 8.1.1. This covariant formulation of electromagnetic field theory and incorporating it in a Lagrangian description will be useful for us throughout this thesis.

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2.2. FIELD THEORY AND GAUGE THEORY 19

2.2.4 Gauge theory

Let us consider the Lagrangian in Eq. (2.2.16) for Jµ = 0, known as the Maxwell Lagrangian

without sources:

L = −1 4FµνF

µν_. (2.2.20)

This Lagrangian has an important property. Namely, under the gauge transformation

Aµ7→ Aµ+ ∂µα(x) , (2.2.21)

for some function α(x), the field strength tensor becomes

Fµν 7→ Fµν? = ∂µ Aν+ ∂να(x) − ∂ν Aµ+ ∂µα(x)

= ∂µAν− ∂νAµ+ ∂µ∂να(x) − ∂ν∂µα(x) = Fµν. (2.2.22) In the last equality we used that ∂µ∂νis a symmetric tensor operator in the index µ and ν, so the two ∂∂α-terms cancel each other. Using Eq. (2.2.22), the Lagrangian Eq. (2.2.20) under gauge transformation yields L 7→ L?= −1 4F ? µνF ? µν _{= −}1 4FµνF µν _{= L .} _(2.2.23)

Any Lagrangian that is unaltered by a gauge transformation, such as Eq. (2.2.20) demonstrated in Eq. (2.2.23), is known to be gauge invariant [8, page 118]. Hence, two fields Aµthat differ by the derivative of a scalar field are physically equivalent. Symmetries parametrized by a function α(x)are called gauge or local symmetries, while if they are only symmetries for constant α they are called global symmetries. A gauge symmetry automatically implies a global symmetry [8, page 122], which in turn imply a conserved current by Noether’s theorem.

Having gauge invariance, we have the freedom to transform the fields in Eq. (2.2.21) to impose constraints on Aµ, a procedure known as gauge-fixing, in order to reduce the degrees of freedom in the equation of motion following from the given Lagrangian [8, page 119]. For example, the equations of motion from Eq. (2.2.20) are (compare to Eq. (2.2.15))

Aν− ∂ν_(∂

µAµ) = 0 . (2.2.24)

Since ∂iAi 7→ ∂iAi+ 4α and A0 7→ A0+ ∂tαunder gauge transformation we can choose α so that ∂iAi = 0, known as Coulomb gauge, and allow us to set A0 = 0. Furthermore, in this gauge the equations of motion reduce to Ai= 0. In Fourier space

Aµ(x) =

Z d4_p (2π)4µ(p)e

ipx _(2.2.25)

and the equations Ai = 0, ∂iAi = 0 and A0 = 0 become pµpµ = 0, pii = 0 and 0 = 0. Choosing a frame were we can write the momentum as pµ = (E, 0, 0, E), the equations above have two solutions

1

µ= (0, 1, 0, 0) , (2.2.26)

2

µ= (0, 0, 1, 0) , (2.2.27)

which represents the linear polarization of light.

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20 CHAPTER 2. GENERAL BACKGROUND not observable and is not a symmetry of nature. In contrast, global symmetries are physical, since they have physical consequences: conservation of charge. If the total charge of a region is measured at separate times (t1, t2)and nothing leaves this region, then the two measurements will be equal. Gauge invariance has no associated measurement to it. An easy way of seeing the non-physicality of gauge invariance is that we can choose any gauge, and the physics is going to be identical. In fact, we have to choose a gauge to do any computations. Therefore, there cannot be any physics associated with this gauge-fixing. Gauge invariance is merely a redundancy introduced to be able to describe a theory that is associated with a local Lagrangian [8, page 130-131].

2.3 Other preliminaries

2.3.1 Time-reversal symmetry

One discrete symmetry which will appear the most in this paper is time-reversal. Denoting the time-reversal operator by T , it transforms as

T : (t, x) 7→ (−t, x) . (2.3.1)

Furthermore, the time-reversal operator T is defined to be an antiunitary operator [10, page 291]. That is, the transformation |αi 7→ |˜αi = T |αi and |βi 7→ | ˜βi = T |βi satisfies [10, page 287]

h ˜β | ˜αi = hβ | αi∗ , (2.3.2)

T c1|αi + c2|βi = c∗1T |αi + c ∗

2T |βi . (2.3.3)

Relation Eq. (2.3.2) imply that T is unitary, i.e. T†T = I, and relation Eq. (2.3.3) defines an

anti-linear operator.

Let us consider some important cases of time-reversal. In classical electrodynamics, Maxwell’s equations Eq. (2.2.12)-(2.2.13) and the Lorentz force F = e E +1

cv × B

are invariant under time-reversal provided (Eq. (4.4.3) in [10])

E 7→ E , (2.3.4)

B 7→ −B , (2.3.5)

ρ 7→ ρ , (2.3.6)

J 7→ −J , (2.3.7)

v 7→ −v . (2.3.8)

From this, we find that the 4-vector potential Aµ = Aµ(t, x) transforms under time-reversal as (Eq. (11.89) in [8])

( A0(t, x), Ai(t, x) ) 7→ ( A0(−t, x), −Ai(−t, x) ) . (2.3.9) In quantum mechanics, given a position eigenstate |xi and momentum eigenstate |pi the time-reversal operator transforms these states as [10, page 293]

T |xi = eiδ1 _{|xi ,} (2.3.10)

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2.3. OTHER PRELIMINARIES 21 From now on, we choose the phase convention δ1= δ2= 0. Considering a spinless single-particle system represented by the state |αi. The wave functions ψα(x) = hx | αi and Φα(p) = hp | αi appears as the expansion coefficient in the position representation and momentum representation, respectively, of |αi: |αi = Z d3_{x |xi hx | αi ,} (2.3.12) |αi = Z d3_{p |pi hp | αi ,} _(2.3.13)

Applying the time-reversal operator to Eq. (2.3.12)-(2.3.13) and using Eq. (2.3.3) and (2.3.10)-(2.3.11) yields T |αi = Z d3_{x T}_{|xi hx | αi =} Z d3_{x |xi hx | αi}∗ , (2.3.14) T |αi = Z

d3_{p T}_{|pi hp | αi =}Z _d3_{p |−pi hp | αi}∗ =

Z

d3_{p |pi h−p | αi}∗

. (2.3.15) The position-space and momentum-space wave functions of the time-reversed state |˜αi = T |αi are identified from Eq. (2.3.14)-(2.3.15), implying that under time-reversal transformation T

ψα(x) 7→ ψ∗α(x) , (2.3.16)

Φα(p) 7→ Φ∗α(−p) , (2.3.17)

Lastly, let us consider matrix elements hβ | A | αi of a linear operator A and discuss the behaviour under time reversal. We start by define a state |γi ≡ A†|βi. Then, we also have that hγ| ≡ hβ| A. Using Eq. (2.3.2), with |˜αi = T |αi and | ˜βi = T |βi we have

hβ | A | αi = hγ | αi = hα | γi∗= h ˜α | ˜γi = h ˜α | T | γi = h ˜α | T A†| βi

= h ˜α | T A†T†T | βi = h ˜α | T A†T†| ˜βi . (2.3.18) The identity Eq. (2.3.18) is useful when considering operators that are even or odd under time-reversal. In particular, for hermitian operators, such that T A†_T† _{= ±A}_{, we can, together with} Eq. (2.3.18), determine phase constrictions on matrix elements of A when taken with respect to time-reversed states.

2.3.2 Generating functional

The topics discussed throughout Sec. 2.2.1 can be considered as classical field theory. That is, there is no quantization involved and no connection to quantum systems; no quantum field theory. To get from classical field theory to quantum field theory require some form of quantization procedure. One such procedure is called the path integral formulation and the object we are going to introduce originates from this formulation. Only a brief description will be given in this thesis†_.

First, we define the functional derivative, δ

δJ (x), as follows. Considering 3+1 dimensional space

†_{For more details about path integrals and the quantization of fields through path integrals, I recommend to}

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22 CHAPTER 2. GENERAL BACKGROUND (that can easily be generalized) the functional derivative obeys the following axiom (Eq. (9.31) in [7]): δ δJ (x)J (y) = δ (4)_{(x − y) ,} _(2.3.19) or δ δJ (x) Z d4_{y J (y)φ(y) = φ(x) .} _(2.3.20)

Taking the functional derivative of more complicated functionals simply amounts to the ordinary rules for derivatives of composite functions. For example,

δ

δJ (x)exph i Z

d4_{y J (y)φ(y)}i₌iφ(x)exph iZ _d4_{y J (y)φ(y)}i_. _(2.3.21) If the functional involves derivatives of J we integrate by parts before applying the functional derivative [7, page 289]: δ δJ (x) Z d4_{y ∂} µJ (y)Vµ(y) = −∂µVµ(x) . (2.3.22)

Consider a system with the Lagrangian L = L(φ, ∂µφ), where φ(x) is a scalar field. Then, the

generating functional, which we denote Z[φ, J], is defined as (Eq. (9.34) in [7])

Z[φ, J ] ≡ Z Dφexph i Z d4_x_{L(φ, ∂} µφ) + J (x)φ(x) i . (2.3.23)

The functional measure Dφ represents integration over all possible classical field configurations φ(t, x). Furthermore, for J = 0 the functional integral Z0[φ] = Z[φ, J = 0] includes a phase eiS given by the classical action evaluated in that field configuration. The added term to L in the exponent, J(x)φ(x), is called a source term. The function J(x) is a classical current, similar to our discussion about currents in Sec. 2.2.2. Note that the expression Eq. (2.3.23) is in units where ~ = c = 1. When restoring units we require i 7→ i

~ in the exponential term.

At this point it is not clear what the generating functional is nor why it is useful. To get some understanding of it we consider the following: We transform the 3+1 dimensional Minkowski space into the Euclidean 3+1 dimensional space xE by a Wick rotation:

t 7→ −ix0, x 7→ x_E, (2.3.24)

which produces the Euclidean 4-vector product

x2= t2− |x|2_{7→ −(x}0₎2_{− |x}

E|2_{= |x}

E|2_. (2.3.25)

The Wick rotated generating functional becomes (Eq. (9.47) in [7]) Z[φ, J ] ≡ Z Dφexph− Z d4_x E LE− J φ i . (2.3.26)

The functional LE[φ] has the form of an energy and the exponential is a reasonable statistical weight for fluctuations of φ. To see this, we set J = 0 in Eq. (2.3.26) and require the following

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2.3. OTHER PRELIMINARIES 23 relations. For a particle propagating from a Euclidean time tE = 0 to t_E = β the quantum amplitude is given by (Eq. (A.1.30) in [11])

Z φ(β)=φf

φ(0)=φi

Dφ e−SE _{= hφ}

f| e−βH| φii , (2.3.27) where H is the Hamiltonian. In addition, the trace Tr(A) of an operator A in some continuous basis ξ is written as [10, page 37-38, 40-41]

Tr(A) =Z dξ hξ | A | ξi . (2.3.28)

Considering the path of the particle to be closed, φ(0) = φ(β) = φi, then Eq. (2.3.26) (for J = 0) can be rewritten using Eq. (2.3.27)-(2.3.28):

Z0[φ] = Z φ(0)=φ(β) Dφ e−SE ₌ Z dφi Z φ(β)=φi φ(0)=φi Dφ e−SE = Z dφihφi| e−βH| φii =Tr(e−βH) . (2.3.29) In this form, Z0[φ] is precisely the partition function describing the statistical mechanics of a macroscopic system by treating the fluctuating variable as a continuous field [7, page 293]. In other words, the generating functional Eq. (2.3.23) is the quantum field theory analogue of the partition function in statistical mechanics: it incorporates all information about the system [8, page 262].

2.3.3 Entangled states

We will mention states of many-body systems in this thesis, so let us formulate the vector space and basis of such states.

We begin with Kronecker product V1× V2 of two vector spaces V1 and V2, which is the set of all ordered pairs (v1; v2)of elements v1∈ V1, v2∈ V2. This product is itself not a vector space. However, if V1and V2are vector spaces over the same field F (say, the field of complex number C), then one obtains a new vector space if one identifies the pairs (ξv1; v2)and (v1; ξv2)for ξ ∈ F , as well as (v1 + w1; v2 + w2)and (v1; v2) + (v1; w2) + (w1; v2) + (w1; w2). The set obtained this way from V1× V2is called the tensor product V1⊗ V2of V1and V2. We denote the element of the tensor product that is the class of the pair (v1; v2)as v1⊗ v2. The scalar multiplication then acts as ξ(v1⊗ v2) = (ξv1⊗ v2) = (v1⊗ ξv2). Given the bases Bj= { v

[j]

(i)| i = 1, 2, · · · ,dim(Vj) } of Vj for j = 1, 2, the set

B = { v_(i)[1]⊗ v_(j)[2]| i = 1, 2, · · · ,dim(V1), j = 1, 2, · · · ,dim(V2) } (2.3.30) is a basis of the tensor product V1⊗V2. Thus the dimension of the tensor product of vector spaces is given by the product of dimensions of each vector space; that is, dim(V1⊗V2) =dim(V1)dim(V2) [12, page 49-50].

The vector space considered for a one particle quantum system is a Hilbert space H [10, page 11] with basis { ψα}. For a N-body quantum system we have N tensor products of Hilbert spaces:

H_tot= N O

i=1

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24 CHAPTER 2. GENERAL BACKGROUND with the basis Ψ = { ψ(1)

α1 ⊗ ψ

(2)

α2 ⊗ · · · ⊗ ψ

(N )

αN }. Lastly, if a state |Φi ∈ Htot can be written as a

product of states |φii ∈ H(i), say

|Φi = |φ1i ⊗ |φ2i ⊗ · · · ⊗ |φNi = ⊗Ni=1|φii , (2.3.32) then the state is an unentangled state. If the state cannot be put into product-form, such as Eq. (2.3.32), it is an entangled state.

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Chapter 3 Topological nature of the Hall

Conductance

In this chapter we start from basic quantum mechanics and work through concepts in condensed matter physics in order to derive the quantized Hall conductance. We will also emphasize its topological nature and symmetry features.

3.1 Berry connection, Berry curvature

3.1.1 Adiabatic approximation

Let us study a quantum-mechanical system for which the Hamiltonian H(t) depends on a set of parameters, which vary ”slowly” with time. Here by slowly we mean that the parameters change on a time scale T that is much larger than 2π~/Eabfor some typical difference Eab= Ea− Eb in energy eigenvalues. This assumption is known as the adiabatic approximation [10, page 346]. The eigenstates of H are labelled by n, suppressing any possible degeneracy (which can easily be restored), and we will take the initial time t0= 0. We have the eigenvalue problem

H(t) | n; ti = En(t) | n; ti , (3.1.1) where | n; ti is an energy eigenvector with corresponding eigenvalue En(t)at time t. Looking at the general solution to the Schrödinger equation

i~_∂t∂ | α; ti = H(t) | α; ti , (3.1.2) for some state vector | α; ti. Assuming that the energy eigenvalues are discrete we can write the state vector as (Eq. (5.6.5) and (5.6.6) in [10])

| α; ti =X n

cn(t)eiθn(t)| n; ti , (3.1.3)

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26 CHAPTER 3. TOPOLOGICAL NATURE OF THE HALL CONDUCTANCE where θn(t) = − 1 ~ Z t 0 dt0_E n(t0) . (3.1.4)

For convenience, we have separated the time-dependence into two parts in Eq. (3.1.3) (cn(t) and eiθn(t)), which will result in nicer equations below. Inserting Eq. (3.1.3)-(3.1.4) into the

Schrödinger equation and using Eq. (3.1.1) we obtain X n eiθn(t) h ˙cn(t) | n; ti + cn(t) ∂ ∂t| n; ti i = 0 . (3.1.5)

Taking the inner product of Eq. (3.1.5) with hm; t | and invoking hm; t | n; ti = δmn we obtain 0 = eiθm(t)_˙c m(t) + X n eiθn(t)_c n(t) hm; t | ∂ ∂t| n; ti , (3.1.6) or equivalently ˙cm(t) = − X n ei[θn(t)−θm(t)]_c n(t) hm; t | ∂ ∂t| n; ti . (3.1.7)

We can arrive at an expression for the matrix element hm; t | ∂

∂t| n; tiby studying Eq. (3.1.1). By differentiating Eq. (3.1.1) with respect to t and taking the inner product with h m; t |, for m 6= n, we find hm; t | ˙H(t) | n; ti + hm; t | H(t)∂ ∂t| n; ti = ∂En(t) ∂t hm; t | n; ti + En(t) hm; t | ∂ ∂t| n; ti , (3.1.8) which simplifies to hm; t | ˙H(t) | n; ti = En(t) − Em(t) hm; t | ∂ ∂t| n; ti . (3.1.9) Note that we have used that H(t) is hermitian such that hm; t i H(t) = hm; t i Em(t). With Eq. (3.1.9) we can rewrite Eq. (3.1.7) as

˙cm(t) = −cm(t) hm; t | ∂ ∂t| m; ti − X n6=m ei[θn(t)−θm(t)]_c n(t) hm; t | ˙H(t) | n; ti En(t) − Em(t) , (3.1.10) which is a formal solution to the general time-dependent problem. Equation (3.1.10) demonstrates that states n 6= m will mix with | m; ti, as time goes on, due to the time-dependence of H(t). We now want to apply the adiabatic approximation in order to neglect the second term in Eq. (3.1.10), and this is possible if the inverse natural frequency of the state-phase factor is much smaller than the time scale τ for changes in the Hamiltonian. That is, for the approximation to be valid we need hm; t | ˙H(t) | n; ti En(t) − Em(t) ≡ 1 τ hm; t | ∂ ∂t| m; ti ∼ Em ~ (3.1.11)

to hold for all n 6= m. In that case, Eq. (3.1.10) reduces into a simple differential equation and we obtain the solution

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3.1. BERRY CONNECTION, BERRY CURVATURE 27 where the phase γn(t)is defined by

γn(t) :=i Z t 0 hn; t0| ∂ ∂t0 | n; t 0_i_dt0 _(3.1.13)

Since the time derivative is an anti-hermitian operator, i.e. ∂ ∂t

†

= −_∂t∂, we note that the quantity γn(t)is real: γ∗_n(t) = −i Z t 0 hn; t0| ∂ ∂t0 † | n; t0idt0=i Z t 0 hn; t0| ∂ ∂t0 | n; t 0_i_dt0_{= γ} n(t) .

Therefore, in the adiabatic approximation, the eigenvectors | n; ti obtain a phase-factor γn(t) such that the state vector Eq. (3.1.3) becomes

| α; ti =X n

cn(0)eiγn(t)eiθn(t)| n; ti . (3.1.14) It turns out that this additional phase can be observed experimentally. It manifests itself in many physical phenomena that involve quantum systems which are cyclic in time. Therefore, the accumulated phase Eq. (3.1.13) is of great interest when studying such cyclic systems [13, page 50-55].

Suppose that the Hamiltonian depends on a set of parameters

λ(t) = λ1(t), λ2(t), . . . , λn(t) , (3.1.15) which change in time adiabatically. In addition, we assume the corresponding parameter space to be a smooth manifold. Then we have En(t) = En(λ(t))and | n; ti = | n; λ(t)i, and also

hn; t | ∂

∂t| n; ti = hn; t | ∇λ| n; ti · dλ

dt , (3.1.16)

where ∇λis the gradient operator with respect to λ. Inserting Eq. (3.1.16) into (3.1.13) yields γn(T ) =i Z T 0 hn; t | ∇λ| n; ti · dλ dt dt =i Z λ(T ) λ(0) hn; t | ∇λ| n; ti ·dλ . (3.1.17) In the case when T is the period of one cycle, such that λ(T ) = λ(0), where the vector λ traces a curve C , we have γn(C ) ≡ I C An(λ) ·dλ , (3.1.18) where An(λ) ≡i hn; t | ∇λ| n; ti . (3.1.19) The quantity An(λ) is called the Berry connection and the phase γn(C ) is known as the

geometrical phase or Berry phase [14, page 224]. Using Stokes’s theorem on manifolds [15,

page 111] we can express the Berry phase as γn(C ) = I C An(λ) ·dλ = Z Z S Fn(λ) ·da , (3.1.20)

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28 CHAPTER 3. TOPOLOGICAL NATURE OF THE HALL CONDUCTANCE where S is a surface that is bounded by the closed loop C and the measure da is the area element on the surface. The quantity Fn(λ)introduced above, defined as

Fn(λ) ≡ ∇λ× An(λ) , (3.1.21)

is known as the Berry curvature. Note that, similarly to how we showed that the Berry phase is real, both the Berry connection and Berry curvature are real-valued quantities. Equations (3.1.19) and (3.1.21) have remarkable properties which we will exploit in the next section.

3.1.2 Symmetries of the Berry connection and Berry curvature

The following analysis of the Berry connection and Berry curvature is valid for any arbitrary parameters λ. In the upcoming sections, however, we will treat condensed matter systems in reciprocal space and the Berry connection and Berry curvature will naturally appear in our expressions. We therefore limit our attention to a system in which the parameters can be taken to be λ = k. That is, we let the wave-vector k of a particle to be viewed as slowly changing [14, page 223]. We also let the state

| n; ti = | un,k(r)i , (3.1.22)

where un,k(r)is a Bloch function with band index n. Inserting this into Eq. (3.1.19) yields An,k=i hun,k| ∇k| un,ki . (3.1.23) First, we note that under the local phase transformation

|un,ki 7→ eiφn(k)|un,ki , (3.1.24) with φn(k)a smooth function of k, An,k transforms as

An,k7→ ˜An,k=i hun,k| e−iφn(k)∇keiφn(k)| un,ki

=i hun,k| ∇k| un,ki − ∇kφn(k) hun,k| un,ki

= An,k− ∇kφn(k) , (3.1.25)

meaning that the Berry connection transforms like a gauge field. Furthermore, this transformation leaves the Berry phase Eq. (3.1.20) invariant. In other words, the Berry phase does not explicitly depend on the path C , but rather only on the geometry of the curve C traced out by k [10, page 350]. This explains why the Berry phase is also called the geometric phase.

3.1.3 Berry connection and curvature under time reversal

transform-ation and symmetries

Let us now study what happens to the Berry connection and Berry curvature under the time-reversal transformation T . The Bloch functions transform under time-time-reversal as (Eq. (5.171) in [16])

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3.1. BERRY CONNECTION, BERRY CURVATURE 29 Here, T transforms k 7→ −k and changes the wavefunction into its complex conjugate (compare with Eq. (2.3.17)). For the case when the system is invariant under time-reversal we therefore have

T un,k= eiφn(k)un,k. (3.1.27)

The phase factor appears since a system can be described by a wavefunction up to an arbitrary phase. Comparing Eq. (3.1.26) and Eq. (3.1.27) we obtain the relation

u∗_n,−k= eiφn(k)_u

n,k. (3.1.28)

The Berry connection transforms under time-reversal transformation as T An,k= −i hun,k| T ∇k

† T†_{| u}

n,ki =i hun,−k| ∇−k| un,−ki = An,−k. (3.1.29) The minus sign in the first equality comes from T being anti-linear. In the second equality we used that the gradient is an anti-hermitian operator, ∇k

†

= −∇k, and that the time-reversal transformation changes ∇k7→ ∇−k. If the system is invariant under time reversal, we may insert Eq. (3.1.28) into Eq. (3.1.29) and obtain

=i hun,k| ∇k| un,ki + ∇kφn(k) . (3.1.30) We see that for systems with time reversal symmetry An,−k and An,k only differ by a gauge transformation.

Lastly, let us consider the Berry curvature under time reversal transformation. This analysis simplifies by expressing the Berry curvature in components, where we represent the cross-product with the Levi-Civita symbol and employ the summation convention:

=iεkijh∂kiun,k| ∂kjun,ki . (3.1.31)

In the last equality we used that partial derivatives commute so that εkij_∂

ki∂kj= 0. The Berry curvature under time reversal transformation becomes (Eq. (5.179) in [16])

T Fn,k=iεkijhT un,k| T ∂ki † ∂kj † T†_{| T u} n,ki =iεkijhun,−k| ∂−kj † ∂−ki| un,−ki =iεkijh∂−kjun,−k| ∂−kiun,−ki

= −iεkjih∂−kjun,−k| ∂−kiun,−ki = −Fn,−k. (3.1.32) For the case when the system is invariant under time reversal we can in addition use Eq. (3.1.30), which yields

T Fn,k = −Fn,−k= ∇k× An,−k = ∇k× An,k− ∇k× ∇kφn(k)

= ∇k× An,k= Fn,k. (3.1.33)

The φn(k)-term vanishes because, again, εkij∂ki∂kj = 0, resulting in T Fn,k = Fn,k. Equation (3.1.33) shows that the Berry curvature must be an odd function of k, i.e.

Fn,−k= −Fn,k, (3.1.34)

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30 CHAPTER 3. TOPOLOGICAL NATURE OF THE HALL CONDUCTANCE

3.2 Semiclassical equations from wave packets

We now want to study how the Berry connection and Berry curvature are incorporated into the semiclassical equations of motion for electrons in the presence of an electromagnetic field. The resulting formulas will allow us to define the quantized Hall conductance in later sections.

3.2.1 Wave packets

In situations described by wave equations in which it is possible to speak of ”particles” one is referring to a wave packet. A general way to derive the semiclassical equation of motion for electrons is to consider how wave packets evolve with time. It is necessary to carefully define a wave packet Wrckc centred in space at rc and dominated by wave vector kc.

Consider the situation of a crystal with volume V and Brillouin zone ”BZ”. Letting A be the vector potential representing the presence of a magnetic field, the proper definition of a wave packet is (Eq. (16.65) in [14]) Wrckc = 1 √ N X k∈BZ wkkce −ieA(rc)·r/~c_e−ik·rc_ψ k(r) . (3.2.1)

Here, N is the number of unit cells in the system, k is summed over a single Brillouin zone and ψk(r) are solutions to the Schrödinger equation for a periodic potential. The first phase factor is due to the presence of a magnetic field, whereas the second phase naturally occurs for Bloch functions. The incorporation of the phase factors is needed in order to ensure that Wrckc varies

slowly as rc and kc vary (compare to the result of the adiabatic approximation), eliminating potential oscillations. In order for the wave packet to be normalized we need

1 = hWrckc| Wrckci = 1 N X k,k0∈BZ Z dr ei(k0 −k)·rc_w kkcw ∗ k0_k cψ ∗ k0(r)ψk(r) = X k,k0_∈BZ wkkcw ∗ k0kcδkk 0 = X k∈BZ wkkc 2 , (3.2.2)

where we have used that ψkis normalized over the volume V of the crystal. The shape of the wave packet depends on the normalized weight function w. The amplitude w_k−k

c, which depends on

k − kc, determines the range of wave numbers involved in the packet and its spatial extent. The phase-factor of w contains information about the spatial location of the wave packet, and must be chosen carefully in order to be centred at rc. In particular (Eq. (16.68) in [14]):

wkkc =

w_k−k

ce

i(k−kc)·An,kc_, (3.2.3)

where An,kc is the Berry connection defined in Eq. (3.1.19) with λ = kc and the state | n; ti is

represented by a Bloch function uk (Eq. (7.45) in [14]): An,kc=i Z Ω dr u∗ kc(r) ∂ ∂kc ukc(r) , (3.2.4) with uk(r) = e−ik·rψk(r) . (3.2.5)

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3.2. SEMICLASSICAL EQUATIONS FROM WAVE PACKETS 31 Let us now demonstrate that the wave packet is centred at rc. We will use Eq. (3.2.5) for the periodic function uk to replace ψk in Eq. (3.2.1) and that sums of functions Fk can be converted into integrals over a continuous function according to (Eq. (6.11) in [14])

X k Fk= V (2π)3 Z dk Fk. (3.2.6)

We also need the identity (Eq. (A.29) in [14]) X

R

eiq·R= NX K

δq,K, (3.2.7)

where the two sums are over all Bravais lattice vectors R and reciprocal lattice vectors K, respectively, which are related by eiR·K _{= 1}_{[14, page 49]. Then we get}

hWrckc| r | Wrckci − rc= hWrckc| r − rc| Wrckci = Z dr N X k,k0 w_kk∗ _cwk0_k cu ∗

k(r)uk0(r)ei(k

0_{−k)·(r−r} c) _{r − r} c = Z dr N X k,k0 w_kk∗ _cw_k0_k cu ∗ k(r)uk0(r) ∂ ∂ik0e i(k0_{−k)·(r−r} c) = Z dr N X k w_kk∗ cu ∗ k(r) Z _V (2π)3dk 0_w k0kcuk0(r) ∂ ∂ik0ei(k 0_{−k)·(r−r} c)_. (3.2.8) Next, we perform an integration by parts, where the boundary term is zero. Then the k0_integral is turned back into a sum via Eq. (3.2.6):

hWrckc| r − rc| Wrckci = − Z dr N X k w∗_kk_cu∗_k(r) Z _V (2π)3dk 0_ei(k0_{−k)·(r−r} c) ∂ ∂ik0 wk0_k cuk0(r) = − Z dr N X k,k0 w∗_kk_cu∗_k(r)ei(k0−k)·(r−rc) ∂ ∂ik0 wk0_k cuk0(r) . (3.2.9) We apply Eq. (3.2.7) where no reciprocal lattice vectors K are needed since both k and k0 lie within the same Brillouin zone. The integral can therefore be performed over a single unit cell Ω. hWrckc| r − rc| Wrckci = − Z dr X k,k0 δ_kk0w∗_kk cu ∗ k(r) ∂ ∂ik0 w_k0_k cuk0(r) = − Z dr X k w 2 k−kcu ∗ k(r) 1 wkkc ∂ ∂ik wkkcuk(r) =i Z Ω dr u∗ kc(r) ∂ ∂kc ukc(r) − ∂ ∂ikln wkkc _k=k c. (3.2.10)

In the calculations we used that |u|2 integrates to 1 and |w|2 sums to 1. At this point, we have not used the explicit form of wkkc. Furthermore, this calculation does not suggest that the

expectation value is zero. However, if we now insert Eq. (3.2.3) and (3.2.4) and use ∂ w k−kc ∂k = 0, because of the maximum at kc, we obtain

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32 CHAPTER 3. TOPOLOGICAL NATURE OF THE HALL CONDUCTANCE or equivalently

hWrckc| r | Wrckci = rc. (3.2.12)

The expectation value of r is now localized at rc. This result shows why it is important to define the weight function wkkc via the Berry connection Eq. (3.2.4).

3.2.2 Dynamics from a Lagrangian

The dynamics of the expectation values of position rc and crystal momentum kc for the wave packet can in principle be derived from the Schrödinger equation. However, it will be more convenient for us to use a Lagrangian description. This can be achieved by time-dependent

variational principle. The main idea is to consider an action S = R dt L with the Lagrangian of

the form L = hΨ | i~d

dt− H | Ψi, with H the Hamiltonian of the slowly perturbed crystal. From the requirement that δS = 0 for any arbitrary variation of Ψ one obtains the time-dependent Schrödinger equation [17, page 1-2][18, page 2-3]†_.

Since the wave packet is now given, the equations of motion for rc and kc can be obtained by evaluating the Euler-Lagrange equation with the Lagrangian (Eq. (16.79)-(16.81) in [14])

L = hWrckc|i~ ∂ ∂t| Wrckci − hWrckc| ˆH − eV (r) | Wrckci , (3.2.13) with ˆ H = 1 2m h ˆ_{P +} eA(r) c i2 + U (r) . (3.2.14)

For the case when A(r) = 0 the Schrödinger equation is hPˆ

2

2m+ U (r) i

ψk = Ekψk, (3.2.15)

where we have suppressed the band index n in ψk and Ek. The magnetic field B and the static electric field E are included through the vector potential A(r) and scalar potential V (r), respectively, via

B = ∇ × A , (3.2.16)

E = −∇V . (3.2.17)

I will omit the details for evaluating the expectation values needed to obtain L and take the following result as given (Eq. (16.82) in [14]):

L = erc c ·

dA(rc)

dt + ~kc· ˙rc+ ~ ˙kc· Akc− Ekc+ B · mkc+ eV (rc) , (3.2.18)

with the orbital magnetic moment mkc of the wave packet given by (Eq. (16.82c) in [14]),

mkc = − e~ 4mc Z Ω drh∂u∗_k c ∂ikc − Akcu ∗ kc i ×h ∂ ∂irc − kc i ukc+c.c. , (3.2.19)

†_{For more information about time-dependent variational principle I recommend reading Ref. [18] and [17].}

These articles, however, are somewhat old and I could not find any modern texts that presents the subject in a more illustrative way.

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3.2. SEMICLASSICAL EQUATIONS FROM WAVE PACKETS 33 where ”c.c.” stands for ”complex conjugate”. The following identity will be useful when evaluating the Euler-Lagrange equations:

D × ∇ × Cr = ∇ D · Cr − D · ∇ Cr, (3.2.20) where Cr is a vector function of r and D is a vector independent of r. Furthermore, we may recall that the dynamics of a system is unchanged when any total time derivative term is added to the Lagrangian [19]. Matters simplify if one studies L 7→ L +dF

dt, with F = −ecrc· A(rc), such that Eq. (3.2.18) is replaced by

L = −e ˙rc c · A(rc) + ~kc· ˙rc+ ~ ˙kc· Akc− Ekc+ B · mkc+ eV (rc) . (3.2.21) We have that ∂L ∂rc = − ∂ ∂rc e c˙rc· A(rc) + e ∂ ∂rc V (rc) = − ∂ ∂rc e c˙rc· A(rc) − eE(rc) , (3.2.22) d dt ∂L ∂ ˙rc = d dt h −e cA(rc) + ~kc i = −e c˙rc· ∂ ∂rc A(rc) + ~ ˙kc = −e c h ∂ ∂rc ˙rc· A(rc) − ˙rc× ∂ ∂rc × A(rc) i + ~ ˙kc = − ∂ ∂rc e c˙rc· A(rc) +e c˙rc× B + ~ ˙kc, (3.2.23) and ∂L ∂kc = ~ ˙r c+ ∂ ∂kc ˙kc· Akc −∂Ekc ∂kc + ∂ ∂kc B · mkc , (3.2.24) d dt ∂L ∂ ˙kc = d dt h ~Akc i = ~ ˙kc· ∂ ∂kc Akc = ∂ ∂kc ~k˙c· Akc − ˙kc× ∂ ∂kc × Akc. (3.2.25)

The resulting equations of motion are

~k˙c= −eE(rc) − e c˙rc× B , (3.2.26) and ˙rc= 1 ~ ∂ ˜Ekc ∂kc − ˙kc× Fkc, (3.2.27) where ˜ Ekc = Ekc− B · mkc, (3.2.28) Fkc = ∂ ∂kc × Akc. (3.2.29)

We now see how the semiclassical equations have an explicit dependence on the Berry curvature Fkc, which, in this context, is also called the anomalous velocity.

The anomalous velocity Fkc and the magnetic moment mkc can sometimes be omitted from the

semiclassical equation, often due to symmetry. Under time-reversal we have

T Fk= −F−k, (3.2.30)

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34 CHAPTER 3. TOPOLOGICAL NATURE OF THE HALL CONDUCTANCE whereas under spatial inversion P, r 7→ −r, we have

PFk= F−k, (3.2.32)

Pmk= m−k, (3.2.33)

For crystals with both time-inversion symmetry and spatial inversion symmetry, both Fk and mk must vanish for all k in order for the two symmetries to be compatible. It follows that for many monoatomic solids these terms can be omitted. They cannot be omitted, however, for crystals such as GaAs, which has no spatial inversion symmetry, or iron, which has spontaneous magnetic moments [14, page 469]. Due to the inclusion of the anomalous velocity term, let us consider how this would affect transport properties such as conductivity.

3.3 The Hall Conductivity

3.3.1 Derivation of the Hall conductivity

In this section we explain how the Hall conductivity is obtained. From now on, we drop the subscripts c in the equations of motions. Inserting Eq. (3.2.26) into Eq. (3.2.27), eliminating ˙k, yields ˙r = 1 ~ ∂ ˜Ek ∂k + e ~E × Fk+ e ~c ˙r × B × Fk. (3.3.1)

In the original experiment, in which the Hall effect was found, an external magnetic field is applied in the ˆz-direction and an electric field driving a current along the ˆx-direction. For boundaries in the ˆy-direction, where current can flow across, the electric field along ˆx induces a transverse current jy [14, page 500]. For materials with spontaneous magnetic moments, however, no ex-ternal field is needed for producing a Hall current. The electric field in the ˆx-direction leads to an electric current in the ˆy-direction, a phenomenon called the anomalous Hall effect [14, page 503]. Considering the case when B = 0, Eq. (3.3.1) reduces to

˙r = 1 ~ ∂ ˜Ek ∂k + e ~ E × Fk. (3.3.2)

For a system with N electrons the average velocity v of all particles can be expressed as (Eq. (4.123) in [20]) v = 1 N X n,k∈occupied vn,k, (3.3.3)

where vn,k is the velocity of the Bloch electron with band index n and wavevector k. Only occupied states contribute to the current, as indicated in the sum. The current per area is given by (Eq. (4.122) in [20]) j = −e AN v = − e A X n,k∈occupied vn,k = − e A X n,k∈occupied ˙rn,k = −e A X n,k∈occupied 1 ~ ∂ ˜En,k ∂k + e ~E × Fn,k. (3.3.4)

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3.3. THE HALL CONDUCTIVITY 35 The first term of Eq. (3.3.4) leads to a current parallel to E [14, page 503]. The second term however produces a current perpendicular to E, the Hall current, given by

j_H = −E × e 2 ~A X n,k∈occupied Fn,k . (3.3.5)

Letting E = E ˆx and Fn,k = Fn,kzˆwe identify E jH ≡ σxy= e2 ~A X n,k∈occupied Fn,k. (3.3.6)

Next, using Eq. (3.2.6) the sum over k can be turned into an integral: X k∈occupied Fn,k= A (2π)2 Z k∈BZ d2_{k F} n,k. (3.3.7)

The bands that contribute to a current, which the electrons can occupy, are either completely filled bands or partially filled bands. The latter situation is the one that occurs in metals. In the case of an insulator, which has no partially filled bands [21, page 157-158], the sum over n runs only over band indices which correspond to fully occupied bands. The Hall conductance then becomes σxy= e2 h X n∈fully occupied 1 2π Z k∈BZd 2_{k F} n,k. (3.3.8)

The above relation shows that the conductance is proportional to the Berry curvature integrated over the 2D Brillouin zone†_.

Considering the case where the Brillouin zone of a 2 dimensional system is without boundaries, the Hall conductance can be written as

σxy= e2 h X n∈fully occupied Cn, (3.3.9) where Cn= 1 2π Z BZd 2_{k F} n,k. (3.3.10)

The resulting Hall conductance is now in terms of the quantum e2

h times a sum of coefficients Cn. Equation (3.3.10) is of great importance within the theory of topological insulator. In fact, the topological nature of σxy has manifested itself via Eq. (3.3.10). To understand this issue more conceptually it is beneficial to mention concepts from topology, which we can apply to the Hall conductance.

3.3.2 Gauss-Bonnet theorem and Topological Index

If a manifold M1 can be smoothly deformed into manifold M2, they are said to have the same topology [23, page 47]. For example, topologically equivalent surfaces would be a sphere and an

†_{The Hall conductance Eq. (3.3.8) can be derived by a different approach. For example, taking the average}

over the so called Kubo formula for σxy [22, page 274-280]. In this derivation the magnetic field B 6= 0, as we

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36 CHAPTER 3. TOPOLOGICAL NATURE OF THE HALL CONDUCTANCE ellipsoid, whereas the sphere and the torus are topologically different surfaces. For 2 dimensional manifolds M2, i.e. surfaces, the topological classification can be achieved by studying the

topo-logical index of M2 called the Euler characteristic χ(M2₎ [24, page 3-6]. This index is an

integer and topologically equivalent surfaces has the same value of χ(M2₎, which is given by [15, page 428]

χ(M2) = 2 − 2g , (3.3.11)

where g is the genus of the surface M2. The genus is the number of handles that arises when the surface is embedded in 3 dimensional space. In Fig. 3.1 we illustrate some examples of surfaces with different genus g. Topology (global structure) originates from geometry (local structure)

g = 0

g = 1

g = 2

Figure 3.1: A sphere (g = 0), a torus (g = 1) and a genus-2 surface (g = 2).

which describes manifolds in R3 (later generalized to higher dimensions and other abstract sur-faces) [23, page 47]. There is a link between geometry and topology which we will examine briefly. Let x = x(s) define the curve C, parametrized by arc length, on a surface M2 in R3. The unit tangent at x(0) is then T = dx

ds. The curvature vector for C, as a space curve, at x(0) is κ = κn = dT

ds , (3.3.12)

where n is the principal normal to C. The component of the curvature vector κ in the direction of the unit surface normal N, which we denote B(T , T ), is given by (Eq. (8.9) in [15])

B(T , T ) = h κn, N i , (3.3.13)

where h , i denotes the inner product. There are infinitely many curves passing through x(0) with tangent T which, as space curves, may have different curvatures. However, Eq. (3.3.13) tells us that the component of the curvature vectors normal to the surface depends only on the tangent T. In particular, let T be a unit tangent vector to M2 at a point p. Let P be the plane spanned by T and N at p. The plane P cuts out the curve C on M, whose unit tangent is T , and is a normal section of M2. The curvature vector κ points from p towards the centre of curvature at a distance κ−1 _{(see Fig. 3.2). From Eq. (3.3.13) we have, for this normal section,}

B(T , T ) = ±κ , (3.3.14)

where the + sign is used if the curve C is ”curving” towards the chosen surface normal; the case of B(T , T ) = −κ is depicted in Fig. 3.2. If p ∈ M2 is kept fixed but T rotated in the tangent

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3.3. THE HALL CONDUCTIVITY 37

κ

T

N

C

p

Plane P

M

centre of curvature

Figure 3.2: The plane P , spanned by T and N at point p, cuts out the normal section C of M. The curvature vector κ points from p towards the centre of curvature (at a distance of κ−1_). plane M2

p the curvatures B(T , T ) will change in general. For unit T ∈ M 2

p, we define (Eq. (8.10) in [15])

κ1(p) =max B(T , T ) , (3.3.15)

κ2(p) =min B(T , T ) , (3.3.16)

to be the principal (normal) curvatures of M2 at p. With these two quantities the Gaussian

curvature K, a measure of curvature of a surface M2 at p, is defined as [15, page 207]

K = κ1κ2. (3.3.17)

Using the Gaussian curvature and letting M2be a closed Riemannian surface, then (Eq. (17.21) in [15]) 1 2π Z Z M KdS = χ(M2) . (3.3.18)

The Euler characteristic χ(M2₎ is independent of the Riemannian metric used on M2, which implies that the left-hand side of Eq. (3.3.18) must be independent of the metric. This is known

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38 CHAPTER 3. TOPOLOGICAL NATURE OF THE HALL CONDUCTANCE as the Gauss-Bonnet theorem and provides a remarkable observation. Namely, a smooth deformation of the surface M2 can change K pointwise and likewise the area form dS, yet the total integrated curvature RRMKdS remains unchanged, implying that the integer χ(M

2₎ remains constant. Equivalently, the Euler characteristic is invariant under any continuous change in the metric, i.e.

δχ δgαβ

= 0 , (3.3.19)

and depends only on the topology of M2 [11, page 16]. That is, it is a topological invariant.

3.3.3 Chern number and symmetries

Consider the Hall conductance σxy given by Eq. (3.3.9). Considering a 2 dimensional mani-fold without boundaries, the total Berry curvature RR_BZFn,kd2k, analogously as the Gaussian curvature, is proportional to an integer. In fact,

σxy = e2 h X n∈fully occupied Cn = σxy= e2 hC , (3.3.20)

and Cn, given by Eq. (3.3.10), is a topological index called the first Chern number for band n[23, page 48]. The factor C defined in Eq. (3.3.20) plays the role of a topological index for an insulator. Then, a trivial insulator has C = 0, whereas a topological insulator has C 6= 0. Consider the behaviour of the Chern number Cn under time reversal transformation. Using Eq. (3.1.32) we have T Cn= 1 2π Z Z BZT Fn,kd 2_{k = −} 1 2π Z Z BZFn,−kd 2_k = − 1 2π Z Z BZFn,kd 2_{k = −C} n. (3.3.21)

In the third equality we make the change of variables k 7→ −k and the integral is unchanged. For the case when the system is invariant under time reversal; i.e. T Cn = Cn, Eq. (3.3.21) implies that Cn = 0for all band indices n. Hence, insulators with time-reversal symmetry have a total Chern number C = 0 and are trivial insulators. Therefore, in order to have a nontrivial Chern number, the system must break time-reversal symmetry (for example, using external magnetic fields).