Topology in band theory

Report of bachelor project Physics and Astronomy (15EC)

Conducted between 29/03/2016 ans 13/07/2016

Topology in band theory

July 13, 2016

Rien Vanneste (10656561)

University of Amsterdam

rien_94@hotmail.com

Supervisor:

Robert-Jan Slager

Examinator:

Jasper van Wezel

Second reviewer:

Eric van Heumen

Institute:

ITFA

Abstract

Topology is a branch of mathematics which also has applications in the physics of crystalline solids. In this thesis I will explicitly show the topological features of the Haldane model and Kane-Mele model, which both are based on a tight-binding model of graphene. I will show that the Haldane model is a topological Chern insulator with a Chern number of +1 and with edge states similar to the edge states of a quantum Hall state. The Kane-Mele model is a Z2-topological insulator with a

Z2-invariant of ν = +1. This model describes quantum spin Hall state with spin filtered topological

edge states. The edge states of both models are topologically protected and the conductivity of these edge states is invariant under disorder. Furthermore I will briefly generalize the notions of the topology of the Haldane and Kane-Mele to general Chern insulators and 2D and 3D Z2-topological

insulators. I will conclude discussing a Weyl semimetal and numerically show the existence of Fermi-arcs on its surface. I also show numerically that the topology is linked to zero-eigenvalues of the Greens functions defined in the bandgap at a 2D impurity.

Populaire Nederlandse Samenvatting

Het vakgebied ‘gecondenseerde materie’ houdt zich bezig met de eigenschappen van vaste stoffen, waaronder de elektrische eigenschappen. Van oudsher kunnen we de vaste stoffen uit ons dagelijks leven onderscheiden op basis van hun geleiding in twee categoriën: geleiders die stoom geleiden zoals koper en isolatoren die geen stoom geleiden, zoals plastic. Dit blijkt echter een te eenvoudige weergave van de werkelijkheid. Er zijn materialen die niet in een van deze categorieën vallen, zoals supergeleiders en halfgeleiders.

In deze thesis ga ik in op een nieuw soort materiaal: een topologische isolator. Dit materiaal lijkt op een isolator omdat in het materiaal (de bulk) geen stoom kan lopen. Toch gedraagt dit materiaal zich anders dan een isolator omdat de oppervlakte geleidend is, en wel op een bijzondere manier. Om deze bijzondere eigenschap te onderzoeken, heb ik gebruik gemaakt van twee modellen: het Haldane model en het Kane-Mele model. Beide modellen berekenen de toegestane toestanden van een elektron in een materiaal. Deze toestanden bepalen de energieën van een elektron als functie van zijn impuls.

Het resultaat van deze berekeningen heb ik vergeleken met de toegestane toestanden van een gewone isolator. Het verschil tussen een gewone isolator en een topologische isolator kan worden begrepen door middel van topologie. Dit is een wiskundig vakgebied dat zich bezig houdt met eigen-schappen (topologische invarianten) van een oppervlak, die niet veranderen als het oppervlak wordt vervormd. Gewone isolatoren kunnen in elkaar worden getransformeerd zonder dat de topologische invariant verandert. Maar een topologische isolator heeft een andere topologische invariant dan een gewone isolator. Daarom kan een topologische isolator niet zomaar transformeren in een gewone isolator. Er is een faseovergang nodig waarin het materiaal tijdelijk geleidend is. In die geleidende fase kan de topologische invariant veranderen.

Wanneer een topologische isolator tegen een gewone isolator aanligt, dan moet de topologisch invariant veranderen op het grensvlak. Het grensvlak moet daarom geleidend zijn, zodat de topol-ogische invariant kan veranderen. Lucht en vacuüm kunnen ook worden beschouwd als een gewone isolator. Het oppervlak van een topologische isolator kan dus worden beschouwd als een grensvlak en moet geleidend zijn. Deze geleiding moet er zijn om de topologische invariant te laten veranderen, ook als er vervuiling of verstoringen op het oppervlak zijn. Daarom is deze geleiding topologisch beschermd.

Contents

1 Introduction 3

2 Second Quantization 4

2.1 Fockspace . . . 4

2.2 Creation- and annihilation operators . . . 4

2.3 One- and Two-body Operators . . . 5

2.4 Tight-binding model . . . 6 3 Graphene 6 3.1 Dirac points . . . 8 4 Chern insulator 10 4.1 Haldane model . . . 10 4.2 Chern invariant . . . 11 4.3 Bulk-boundary correspondence . . . 12 5 Z2-Topological Insulator 13 5.1 Kane-Mele model . . . 13 5.2 Z2invariant . . . 14 5.3 Bulk-boundary correspondence . . . 15 5.4 3D topological insulators . . . 16 6 Weyl Semimetals 17 6.1 Numerical model of the Fermi-arcs . . . 18

6.2 Green’s functions . . . 19

7 Conclusion and outlook 21 A Formal definition Berryphase 22 A.1 Berry phase in band theory . . . 23

1

Introduction

Topology is a branch of mathematics which studies properties of objects that are invariant under smooth deformations. A sphere, for instance, can be smoothly deformed into many different shapes, for example a disk or a bowl. But it cannot be smoothly deformed into a donut, because that would involve making a hole. Therefore a sphere and a disk are topological equivalent and are part of the same topological class, while a sphere and a torus are not. In this example, these topological classes can be distinguished by a integer topological invariant called the genus, which essentially counts the number of holes.

The Gauss-Bonnet theorem

Z

M

KdA = 2πχ(M ) (1.1)

provides an interesting addition to the notion of topology. It shows that the locally defined Gaussian curvature K integrated over the total manifold M, in this case the surface of the object, gives us the Euler characteristic χ(M ) = 2 − 2g, which is directly connected to the genus of the manifold.

Both the notion of topology and band theory exists already for a long time. But only one decade ago, it was realized that topology could be used in band theory of insulators. Ever since it has been a hot topic, both in theoretical physics and experimental physics [8, 18]. More recently, it became apparent that the topological notions of insulators can also be explored in the context of semi-metals [17, 26, 2, 9, 29, 21].

This thesis in the result of a literature study of topology in band theory, in which I will show how topology including the Gauss-Bonnet theorem can be defined in gapped band structures of crystalline solids. To do this I will first explain second quantization to be able to deal with the many electrons living in a solid. Then I will work out a tight-binding model of graphene. From there I will continue with the Haldane model and the Kane-Mele model which I will show to be topologically non-trivial. Then I will briefly discuss how these ideas can be generalized to three dimensions in 3D topological insulators and Weyl-semimetals. I will conclude with the link between zero-eigenvalues in Green’s functions and topology, complemented by a numerical calculation of the Green’s functions in Weyl-semimetals.

2

Second Quantization

Band theory is the study of the allowed quantum states (bands) inside a crystalline material. There are many (∼ 1023) electrons in a material, which are best described using quantum mechanics. But simply solving the Schrödinger equation by finding the eigenvalues and wave functions of all the particles at once is practically impossible. We need to develop a simple method which allows us to predict and describe all the physical quantities without the need to write down the explicit wave function for all the electrons. The method we will use, is called Second Quantization, with which we will be able to make tight-binding models in a easy way.

2.1

Fockspace

Let us assume we know the Hilbert space H1of a single particle described by a complete basis {|αi}.

Then the Hilbert space HN of N identical particles is described by the product of all the single

particle Hilbert spaces.

HN = N

O

i=1

Hi1 (2.1)

which is spanned by the complete basis {|α1, α2, . . . , αN)}, with αithe state of particle i. But the true

wave function of a N indistinguishable particles system will not simple be of the form |α1, α2, . . . , αN),

since the total wave function has to be symmetric for bosons or antisymmetric for fermions under particle exchange. For example the properly symmetrized and normalized wave function |α1, α2i for

two particles is:

|α1, α2i = 1 √ 2  |α1, α2) ± |α2, α1)  (2.2)

where the + sign is for bosons and the − for fermions. For two particles this will barely cause extra work. But, since the number of terms scales with N !, the wave function will become unbearable when N increases.

So the indistinguishably of the particles causes a problem, but also provides a solution. Due to this indistinguishably, the wave function does not care about which particle occupies which state, but rather the number of particles per state. So rather than assigning a state to each particle we can count the number of particles in each individual state. Let us assume our one-particle basis consists of Ω different states: {|αi} = {|α1i , |α2i , . . . , |αΩi}. Then we can define ni to be the number of

particles in state |αii. A N particle state |ψi can be written as:

|ψi = |n1, n2, . . . , nΩi (2.3)

with P

ini = N . For bosons ni can be every positive integer value. For fermions ni = 0, 1 due

to the Pauli exclusion principle. Writing a state in this way is equivalent to writing a state in the ‘standard’ way as equation (2.2). {|n1, n2, . . . , nΩi} forms the basis for the so-called Fock space F ,

which is essentially the sum over the Hilbert space for every number of particles:

F =

∞

M

i=0

Hi=C ⊕ H1⊕ H2⊕ . . . (2.4)

So the Fock space contains all the possible states for every number of particles.

2.2

Creation- and annihilation operators

In this Fock space, we can now define the so-called creation and annihilation operators ˆc† and ˆc . I will define them for fermions but the same operator can be defined for bosons, in a very similar way [6] . As the name suggests, the creation operator ˆc†_i creates and the annihilation operator ˆc_i destroys a particle in state |αii. So with the proper prefactor they are defined as [6]:

ˆ c†_i|n1, . . . , ni, . . . , nΩi = (1 − ni)(−1)i|n1, . . . , ni+ 1, . . . , nΩi ˆ c_i|n1, . . . , ni, . . . , nΩi = ni(−1)i|n1, . . . , ni− 1, . . . , nΩi (2.5) where i =P i−1

j=inj. The prefactor 1 − ni makes sure that no particle can be added if that state is

negative. From this definition the factor anti-commutation relations can be derived:

{ˆc†_i, ˆc†_j} = ˆc†_icˆ†_j+ ˆc†_jcˆ†_i = 0 {ˆc_i, ˆc_j} = ˆc_icˆ_j+ ˆc_jcˆ_i = 0 {ˆc†_i, ˆc_j} = ˆc†_icˆ_j+ ˆc_jcˆ†_i = δij

(2.6)

Note that the first two anti-commutators are 0 because the factor (−1)i _{causes a sign differences}

between the first and second term if i 6= j. These anti-commutation relations can be considered as the formal definition of the fermion creation and annihilation operators and contain all the information about these operators. The definition in equation (2.5) can be derived from these anti-commutation relations.

With these new operators, we can rewrite our N particle states as creation operators working on the empty state |∅i = |n1= 0, . . . , nΩ= 0i:

|n1, . . . , nΩi = (ˆc†1)

n1_{. . . (ˆc}† Ω)

nΩ_{|∅i (2.7)}

Note that the anti-commutation relations of these operators automatically take care of the indistin-guishably and the Pauli-exclusion principle. Furthermore they allow us to do computations by only using the anti-commutation relations. For example:

hαi|αji = h∅|ˆcicˆ †

j|∅i = h∅|δij− ˆc†jˆci|∅i (2.8)

= δijh∅|∅i = δij (2.9)

2.3

One- and Two-body Operators

Now we have defined the creation and annihilation operators which can construct the whole Fock space. What remains is the use of them to calculate physical observables. First we will sort the physical observables by the number of particles involved. There are so-called one-body observables which measure only one particle at a time, like the momentum operator or the potential energy while two-body operators usually describe interactions between particles. One could also think of operators which involve more than two particles at a time, but they are rarely relevant in solid state physics.

Let us start with describing a one-body operator ˆO. Because a one-body operator only acts on one particle at a time, ˆO can be written as a sum over single particle operators ˆO(1)_:

ˆ O = N X i=1 ˆ O(1)_i (2.10)

Note that ˆO_i(1) implies11⊗ . . .1i−1⊗ ˆO (1)

i ⊗1i+1. . .1N. For simplicity let us assume we have only

one particle described by a complete basis {|αi} = {|α1i . . . |αΩi}. Now we will add the identity

1 = Pα|αi hα| and simplify:

ˆ O = ˆO(1)=X α,β |αi hα| ˆO(1)|βi hβ| (2.11) =X α,β hα| ˆO(1)|βi |αi hβ| =X α,β hα| ˆO(1)|βi ˆc†_{α|∅i h∅| ˆcβ} (2.12) =X α,β hα| ˆO(1)|βi ˆc†αˆcβ (2.13)

The physical interpretation of this formula is quite simple. The operator ˆc_β destroys a particle in state β while ˆc†_α creates one in state α. This can be viewed as the jumping of a particle from state α to β. The matrix element hα| ˆO(1)|βi gives the hopping amplitude. When ˆO is working on a one-particle state, then the state between ˆc†

αand ˆcβ has N − 1 = 0 particles. |∅i h∅| can be viewed

as the identity operator for a N − 1 = 0 particle system and can therefore be omitted. The same result is true for an arbitrary number of particles [4].

The formulation of a two-body operator can be treated the same way and will result in a very similar formula:

ˆ

O = X

α,β,γ,δ

The hopping amplitude hα|O(1)_{|βi or the interaction strength hαβ|O}(2)_{|γδi are usually hard to}

calcu-late, because the calculation involves integrating over the wave functions |αi. Therefore we will not explicitly perform these integrations, but simply assume these integrals are finite complex numbers, often denoted by tαβ. In this thesis the specific values of these amplitudes are not imported, since we

are only interested in describing general phenomena. If, however, the numerical values are needed, then they can be determined experimentally or with numerical simulations.

2.4

Tight-binding model

In this thesis I will discuss binding models of crystalline materials like Graphene. A tight-binding model is based on the assumption that the potential of the lattice closely resembles separate atomic potentials on the lattice sites and that the atomic wave functions can be chosen as the basis of our Hilbert space. So {|αii} can be chosen as the basis of the single particle Hilbert space,

where αi denotes the atomic wave function at atom i. In this basis the creation operator can be

denoted as ˆc†_i which creates a particle at atom i or ˆc†_R which creates a particle at the atom at coordinates R. Note that with these second quantization operators automatically incorporate the translational symmetries of the lattice into the wave functions. In our tight-binding models we will also presume periodic boundary conditions. Therefore we can also work in Fourier space with the basis {|ki =P

Re ik·R_|α

Ri} and the creation operator ˆc†k =

P

Re−ik·Rˆc † R.

3

Graphene

With this notion of second quantization, we can now make a tight-binding model of graphene. Graphene is a interesting material, which got allot of attention the last decade. Besides it’s interesting properties it is also a nice stepping stone for models which show topological behavior.

Figure 3.1: Hexagonal lattice of graphene

Graphene is a material made of a single atomic layer of carbon atomes arranged in a hexagonal (honeycomb) lattice (see figure 3.1). This hexagonal lattice can be regarded as a triangular lattice with 2 atoms per unitcell, atom A and B. The lattice is then defined by the latticevectors ~a1 and ~a2

with length a: ~a1= aˆx, ~a2= a 2x +ˆ √ 3a 2 yˆ (3.1)

The reciprocal lattice is also hexagonal and defined by the reciprocal lattice vectors ~b1and ~b2, which

can be found by requiring ai· bj= 2πδij:

~b1= 2π a x −ˆ 2π √ 3ay,ˆ ~b2= 4π √ 3ayˆ (3.2)

We will also need the vector from atom A to atom B: ~ δ1= − a 2x −ˆ a 2√3y,ˆ ~ δ2= a 2x −ˆ a 2√3y,ˆ ~ δ3= a √ 3yˆ (3.3)

The carbon atoms in graphene contain 6 electrons per atom, with a configuration of 1s2_{, 2s}2_{, 2p}2_.

Due to the covalent bonds in graphene, 5 of the 6 electrons are tightly bound and occupy completely filled bands. The only relevant orbital, with states around the Fermi level is the pz orbital pointing

out of the plane. In this simplified model I will only include nearest-neighbor terms of these pz

electrons in our Hamiltonian. In the nearest-neighbor approximation there is only a hopping from atom A to B and B to A. Since the electrons are described by pz orbitals, the hopping amplitude is

rotationally symmetric in the x-y plane. Therefore the Hamiltonian can be described with only one coupling strength t: H = −t X < ~RA, ~RB> ˆ c†_~ RA,A ˆ c_~ RB,B + ˆc†_~ RB,B ˆ c_~ RA,A (3.4)

where subscript A and B denote on which sublattice the electron is created or destroyed and ~RAand

~

RB denote their coordinates. To diagonalize this Hamiltonian, it is useful to go to Fourier space.

Going to Fourier space is usually a good idea when solving a Hamiltonian with periodic boundary conditions, since a periodic Hamiltonian is usually diagonal in ~k but not in ~R. For this part I will denote the second term of the Hamiltonian as ‘h.c.’ since this term is simply the hermitian conjugate of the first term.

H = −t X < ~RA, ~RB> ˆ c†_~ RA,A ˆ c_~ RB,B + h.c. = −tX ~ RA,j ˆ c†_~ RA,A ˆ c_~ RA+δj,B + h.c. = − t N X ~ RA,j  X ~ k ˆ c†_~ k,Ae −i~k· ~RA  X ~ k0 ˆ c_~ k0_,Be i~k0·( ~RA+~δj)  + h.c. = − t N X ~ k,~k0_,j ˆ c†_~ k,Aˆc~k0_,Be i~k0·~δjX ~ RA ei(~k0−~k)· ~RA | {z } =N δ~k0 ,~k +h.c. = −tX ~ k,j ˆ c_~† k,Aˆc~k,Be i~k·~δj_{+ ˆc}† ~_k,Bˆc~_k,Ae −i~k·~δj (3.5)

Now we can recognize the Hamiltonian as a 2x2 matrix:

H =X ~_k  ˆ c_~† k,A, ˆc † ~_k,B  0 −tP je i~k·~δj −tP je−i~ k·~δj ₀ ! | {z } h(~k) ˆc_~ k,A ˆ c_~ k,B ! (3.6)

where the matrix h(~k) is called the ‘Bloch Hamiltonian’. We can compactify this Bloch Hamiltonian even further by introducing a vector ~d, which allows us to write the Bloch Hamiltonian as a sum of the Pauli matrices. This can always be done for a hermitian 2x2 matrix, since the Pauli matrices together with the identity form a basis of the hermitian 2x2 matrices.

dx= Re(−t X j ei~k·~δj_{) = −t}  2 cos ( a 2√3ky) cos ( a 2kx) + cos ( a √ 3ky)  (3.7) dy = −Im(−t X j ei~k·~δj_{) = t}_{2 sin (} a 2√3ky) cos ( a 2kx) + sin ( a √ 3ky)  (3.8) dz= 0 (3.9) Then: h(k) = dx 0 1 1 0  + dy 0 −i i 0  + dz 1 0 0 −1  (3.10) = ~d(~k) · ~σ (3.11)

The energy eigenvalues can now easily be found by calculating h2_{(~k), due to the properties of the}

index.

h2(~k) = ( ~d(~k) · ~σ)2= didjσiσj= didj(δij1 + iijkσk) (3.12)

= ~d(~k) · ~d(~k) (3.13)

So h2_{(~k) has eigenvalues | ~d(~k)|}2_{. The eigenvalues of h(~k) are:}

E±(~k) = ± q ~ d(~k) · ~d(~k) = ±t s 3 + 2 cos(kxa) + 4 cos( √ 3 2 kya) cos( kxa 2 ) (3.14)

3.1

Dirac points

Figure 3.2: The two energy bands graphene, which show six Dirac cones at the corners of the brillouin zone.

The eigenvalues calculated in equation (3.14) give the two energy bands shown in figure 3.2. These bands are half filled, since there is only one pzelectron per atom. So the Fermi level is in the middle

at E = 0. In figure 3.2 can be seen that the energy bands cross the Fermi level only at 6 points at the corners of the first Brillouin zone, of which only two points are truly inequivalent. They are denoted with K at 4π

3ax and K’ at −ˆ 4π

3ax. The other 4 points are actually the identical to K or K’ˆ

since they are connected to K or K’ by the reciprocal lattice vectors ~b1and ~b2, see figure 3.3.

Figure 3.3: First brillouin zone of Graphene, with a Dirac cone at every corner. There are only two inequivalent Dirac points K and K0, since the other Dirac points are connected to K or K0 by the lattice vectors ~b1 and ~b2.

These 2 points K and K0, known as ‘Dirac points’ or ‘Dirac cones’, cause many interesting electronic properties of graphene. Only at these two Dirac points the Fermi level is crossed. Therefore this band structure is quite different from band structures of ordinary metals or insulators. Graphene does not have a bandgap and therefore it does not behave like a insulator or a semiconductor. But

it also does not have many states around the Fermi level like metals have. Therefore it is called a semimetal or a zero-gap semiconductor. To see what happens around these Dirac points and understand why they are called ’Dirac points’, we can expand the Hamiltonian up to linear order around a Dirac point by the transformation ~k → ~K + ~q. ~d(~k) will become:

dx( ~K + ~q) = √ 3at 2 qx (3.15) dy( ~K + ~q) = √ 3at 2 qy (3.16) By recognizing vf= √ 3at

2 as the Fermi velocity, the Hamiltonian can be written as:

hK( ~K + ~q) = vf(+qxσx+ qyσy) (3.17)

Expanding around K’ gives the same answer up to a minus sign:

hK0( ~K0+ ~q) = v_f(−q_xσ_x+ q_yσ_y) (3.18)

These Hamiltonian’s can be written as one by introducing a second Pauli matrix τz in the {K,K’}

basis:

h(~q) = vf(qxτzσx+ qyσy) (3.19)

Again the eigenvalues can be found by calculating h2(~q):

E(~q) = ±vf|q| (3.20)

So around the Dirac point the dispersion in linear in k and therefore the effective mass of the electrons around the Dirac points is 0(see figure 3.4). We can Fourier transform ~q back to real space by ~q → −i∇:

h(~q) = −ivf(τzσx∂x+ σy∂y) = ivfγi∂i (3.21)

With γx= −τz⊗ σxand γy= −τ0⊗ σy. Since γxand γyobey the Clifford Algebra, this Hamiltonian

can be recognized as a two dimensional massless Dirac equation, where c is replaced by vf and ‘spin’

is replaced by the sublattices of atom A and B. Hence, the points are called ‘Dirac points’.

Now we have written out the most basic tight-binding model of graphene and expanded it around the Dirac points. The degeneracy at the massless Dirac points is protected by inversion (I) and time-reversal (T ) symmetry. I sends ~k to −~k and also interchanges atom A and B. So I sends σz

to −σz. T sends ~k to −~k. So IT will send dz(~k)σz to −dz(~k)σz. Therefore dz(~k) has to be zero.

By breaking one of these symmetries the degeneracy can be lifted and the massless Dirac cone will become a massive one.

The most normal and easiest way to add a mass-term is to break inversion symmetry. Inversion symmetry is broken if for some reason the electrons at atom A have a slightly different energy than the electrons at atom B. Because we defined the energy to be 0 at the Fermi level, we can give atom A energy +mI and atom B energy −mI.

HmI = −t X ~ RA. ~RB mIcˆ†_~ RA,A ˆ c_~ RA,A − mIˆc†_~ RB,B ˆ c_~ RB,B (3.22)

After Fourier transforming this will indeed become term proportional to σz:

hmI(~k) = mI

1 0 0 −1



= mIσz (3.23)

So essentially equation (3.11) remains correct but with dz = mI instead of dz= 0. After expanding

around the Dirac points the Hamiltonian have the form of a massive Dirac Hamiltonian:

h(~q) = vf(τzqxσx+ qyσy) + mIσz (3.24) With eigenvalues: E(~q) = ±qv2 fq2+ m 2 I (3.25)

So breaking inversion symmetry gives rise to a mass-term which splits the degeneracy at the Dirac point and opens the bandgap (see figure 3.4). Therefore graphene will become a ordinary insulator.

Figure 3.4: Left: A normal massless Dirac point. The bandgap is closed. Right: A Dirac point with a small mass-term. The degeneracy at the Dirac point is split and the bandgap is opened.

4

Chern insulator

4.1

Haldane model

Haldane considered lifting the degeneracy at the Dirac points by breaking time reversal symmetry T [7]. He considered a model of graphene with the normal nearest hopping as discussed earlier at which he added a next-nearest neighbor term (i.e. the nearest neighbor on the same sublattice: from atom A to the nearest other atom A and the same for B atoms). Furthermore he considered a magnetic field in the ˆz direction perpendicular to the plane, with a specific configuration which obeys the full symmetry of the lattice and averages to zero for the complete unit cell. So this magnetic field respects the inversion symmetry and leaves the nearest neighbor hopping intact. The next nearest neighbors terms, however, obtain an extra phase eiφ _{or e}−iφ _{depending on the direction of the hopping. This}

extra phase breaks T , because time reversal flips the direction of the hopping. The Haldane term can be written in the following form:

HHaldane=

X

i,j

t2eiνijφˆc†iˆcj+ h.c. (4.1)

where νij = −νjidepending on the direction of the hopping and i denotes the the atom at ~Ri. Note

that i goes over all the atoms A and B. Only the imaginary part of the phasefactor will be interesting so we choose φ = π. HHaldane= X i,j t2eiνijπˆc†iˆcj+ h.c. = −it2 X i,j νijˆc†iˆcj+ h.c. (4.2)

We can choose vij positive for a counter clockwise hopping around the center of the hexagonal

and negative for a clockwise hopping. From every atom there are 6 next-nearest neighbors, three counter clockwise and three clockwise (see figure 3.1). Then we can Fourier transform the operators and due to νij we will get sinus like bands. The term for the B sublattice will get an extra minus

sigh, since the clockwise and counter clockwise hoppings interchange for atoms A and B. This extra minus sigh is also expected, because the Haldane term does not break inversion symmetry. When I sends ~k to −~k, the sinus bands will obtain a minus sign, a extra minus sign from the exchange A ↔ B is needed to cancel this minus sign. The Haldane term will become:

HHaldane= −t2

X

k

2[sin(~k · ~a1) − sin(~k · ~a2) − sin(~k · (~a1− ~a2))](ˆc†_k,Aˆc_k,A− ˆc†_k,Bcˆ_k,B) (4.3)

Then the Bloch Hamiltonian hHaldaneis proportional with σz: hHaldane= dzσzwith dz= −2t2(sin(~k ·

~a1) − sin(~k · ~a2) − sin(~k · (~a1− ~a2))). Expanding around the Dirac points up to linear order gives a

mass-term independent of ~q:

where mh= 3

√

3t2. From equation (4.4) can clearly be seen that the mass-term has a opposite sign

at the two Dirac points, due to τz. Haldane showed that this gapped system is no ordinary insulator

but a topological insulator with a quantized Hall conductivity σxy= e 2

h and Chern number +1 [7].

Let us now look to graphene were both the normal and the Haldane mass-term are present. Around the Dirac point the Hamiltonian will look like:

h(~q) = vf(τzqxσx+ qyσy) + ˜mσz (4.5)

with ˜m the total mass:

˜ m = (mI + mhτz) = ( mI+ mh at K mI− mh at K0 (4.6)

If mh  mI, then the mass-term is positive on both Dirac points K and K0 and bandgap at the

Dirac points is 2mI. If we imagine slowly increasing mh, then the total mass ˜m at K will get bigger

and remain positive, but the ˜m at K0will become smaller. When mh= mI, ˜m = 0 and the bandgap

will be closed at K0. If we increase mh further, ˜m will become negative and the bandgap will open

again, see figure 4.1. The system is an insulator for both ˜m > 0 and ˜m < 0, but the system cannot be deformed from ˜m > 0 to ˜m < 0 without closing the bandgap. This can be interpreted as topological distinction between the two situations.

Figure 4.1: Three Dirac cones with a different mass-term. Both the left and the right systems are insulators with | ˜m| > 0. Yet the positive mass system is separated from the negative mass system, by a gab-closing phase transition at ˜m = 0. Therefore graphene with a negative mass-term is a topologically non-trivial insulator, where the Chern number n is one and graphene with a positive mass-term is a trivial insulator with n = 0.

We can call to insulators topologically equivalent if their Hamiltonian’s are smoothly deformable into one another without closing the bandgap. It follows that connecting that two topological in-equivalent insulators involves a phase transition where the bandgap vanishes. These topological different states can be distinguished by a topological invariant, in this case by the so-called the Chern invariant n. The Haldane state with with ˜m < 0 has n = +1 and is called a Chern insulator. The normal insulator with ˜m > 0 has n = 0.

4.2

Chern invariant

I have stated that the topological invariant characterizing the Haldane state is the Chern number n, or sometimes called the TKNN-invariant, and that the Haldane state has n = 1, while a ordinary insulator has n = 0. Just like genus of a surface can be calculated using the Gauss-Bonnet theorem, the value of this Chern number can be calculated by integrating over a locally defined function called the Berry curvature Ω(k).

n = 1 2π

I

BZ

Ω(k)d2k (4.7)

In appendix A I will give the formal definition of this Berry curvature. For a two-level Bloch Hamiltonian h(k) = d(k) · σ the Berry curvature reduces to [8]:

with ˆd = _|d|d , which can be viewed as a mapping from the brillouin zone to the unit sphere. So the Chern number is:

n = 1 2π I BZ ˆ d · (∂kxd × ∂ˆ kyd)dˆ 2_k (4.9)

which simply counts the number of times d(k) wraps around the unit sphere. At the Dirac points dy and dz are zero. So at the Dirac points ˆd = _|ddz

z|= ±1, depending on the sign of dz= ˜m. Thus ˆd

visits the north-pole or south-pole of the sphere depending on the sign of dz. For mI > mh, dz= ˜m

is always positive. So ˆd wraps around the north-hemisphere once for each Dirac point. Because dx has a different sign at the two Dirac points, both times ˆd wraps around the north pole, have a

opposite direction. So n = +1₂ − 1

2 = 0. If mI < mh, then ˆd wraps once around the north-pole

at the K points and once around the south-pole at the K0 point. So n = +1₂ − (−1

2) = 1. Note

that the interesting part of this integration happens around the Dirac points. So to understand the sign differences of the different parts of the integral, we can use d from the Dirac equation (5.2) to calculate Ω. Then we see that Ω ∼ τzif ˜m = mI. Therefore the integral will get two identical terms

with opposite sign. If ˜m = mhτz, then Ω ∼ τz2=1. So the integral will get two terms with the same

sign.

The Chern number n can be any integer depending on the number of Dirac point. For every Dirac point it obtains ±1₂. Therefore it is essential that the Dirac points always come in pairs, which is guaranteed by the fermion doubling theorem, which states that for a time reversal invariant system the Dirac points always come in pairs. (Note: The basic model of graphene in T invariant.)

4.3

Bulk-boundary correspondence

The Chern number is defined in the bulk of the material. But both an topological insulator and a trivial insulator are insulating in the bulk, in other words the bulk does not show if the material is a topological non-trivial insulator or not. However, a fundamental consequence of the topological classification of gapped band structures is the existence of gapless conducting state at interfaces where the topological invariant changes. These gapless states have to be there because the bandgap has to close when the topological invariant changes. This link between the topological invariant defined in the bulk and the edge states in called the bulk-boundary correspondence. This can easily be seen in this Haldane model.

Let us imagine that there is a boundary at y = 0 between a topological Haldane state with n = 1 and ˜m < 0 and a trivial insulator with n = 0 and ˜m > 0. Note that also vacuum can be seen as a trivial insulator with ˜m > 0 because of the mass of the electrons. Then we can imagine ˜m → ˜m(y) as a function which smoothly changes sign at the boundary. So somewhere around y = 0 the mass is zero and the bandgap is closed. The Dirac Hamiltonian with the mass-term ˜m(y)

− ivf(σx∂x+ σy∂y) + ˜m(y)σz (4.10) has a solution ψ ∼ eiqx_e− 1 vf Ry 0m(y˜ 0_)dy0_1 1 

with eigenvalues E(~q) = vfqxand a positive group

veloc-ity _dqdE

x = vf. This eigenstate only exists at the edge, because e − 1 vf Ry 0 m(y˜ 0_)dy0 decreases exponentially far from the edge. The real edge state for this Haldane model is schematically depicted in figure 4.2.

Figure 4.2: Schematic picture of the band structure at the edge of graphene with an Haldane term. The two massive Dirac cones (purple and pink) are the same Dirac cones present in the bulk. The red line shows the edge states, which cross the bandgap [11].

This edge state has to exist because the Chern number changes, therefore this edge state is topologically protected. The shape can be deformed by perturbations of the edge, but the edge state cannot be gapped and always has to cross the Fermi level. In other words, the difference NR− NL

between the number of modes with right moving group velocity and modes with left moving group velocity cannot we changed and is determined by the difference in Chern number ∆n at the interface. Therefore the bulk-boundary correspondence for the Chern insulator can be summarized with the equation:

NR− NL= ∆n (4.11)

Due to this topologically protected edge state, the edge is always a conductor, even though the bulk is insulation. The edge has a conductivity of σxy = ne

2 h =

e2

h, because the Chern number n

which describes the topological classification of this Haldane model is the same Chern number which describes the integer quantum Hall state. When a material is placed in a strong magnetic field, the atomic lattice can be neglected and the system can be described as a free electron gas in a stong magnetic field. The Hamiltonian can then be solved by casting the Hamiltonian in the form of the quantum harmonic oscillator. It’s band structure is then described by so-called Landau levels. At the edge of a quantum Hall state, edge states will appear. It is possible to explicitly calculate the Hall conductivity of these edge states [1]. The conductivity is quantized by a integer times e_h2, where the integer is given by a integral over a certain function [14]. It was discovered that this function is exactly the Berry curvature. Thus the integer can be interpreted as the Chern number.

σxy = e2 h Z BZ Ω(k) · d2k = e 2 hn (4.12)

Furthermore it is also possible to rewrite the squared of the Dirac Hamiltonian (5.2) with the trans-formation q → −i∇+e_cA into the the Hamiltonian of a quantum harmonic oscillator plus some extra terms [1]. Where A = xB ˆy is the vector potential in the Landau gauge. Then the eigenvalues will be Landau levels. When ˜m < 0 one extra Landau level is filled, which results in the same conductivity of a Hall state with one filled Landau level. Thus the Haldane model gives a topological state which is very similar to the quantum hall state. But the quantum hall state exists on a free electron gas with a strong magnetic field, while the Haldane state exists on a a normal lattice without a strong homogeneous magnetic field.

The fact that the conductivity in equation (4.12) is always constant, is quite remarkable, since the conductivity of normal metallic states is usually dependent on the basic properties of the metal, but also quality of the material. This can be understood by realising that elastic backscattering is forbidden. During a elastic scatting, a particle does not lose or gain energy. So a particle can only change it’s momentum. But for a particle occupying a egde state, there is no other state with the same energy and an other momentum available, except on the other edge of the material. Since the wavefunction of the edge state decays exponentially when going into the bulk, the particle has a very small change to jump to the other edge in a macroscopic material. Therefore elastic backscattering is forbidden and the conductivity does not feel disorder.

5

_Z

-Topological Insulator

5.1

Kane-Mele model

In section 3.1 we argued that the Dirac points in graphene are protected by time reversal and inversion symmetry, and that one of these symmetries has to be broken to split the degeneracy and open the bandgap. However in that model of graphene spin was not included. Kane and Mele considered our basis model of graphene and added two Haldane terms with opposite sign. The one term for spin up electrons and one for down electrons [12]:

HSO= −it2 X i,j X α,β νijSzαβcˆ † iαˆcjβ+ h.c. (5.1)

where Sz _{a Pauli matrix in the spin space and α and β denote the spin. This term respects all the}

symmetries of graphene. Even though a Haldane term breaks time reversal, this term does not. Time reversal flips the direction of the hopping, which gives a minus sign, but it also flips the spin which gives an extra minus sign. Since this term respects all symmetries can exist in normal graphene and it corresponds to the spin-orbit coupling (SO). Since the Hamiltonian remains diagonal in spin (i.e. there is no hopping which flips spin), the bands from the individual Haldane models remain unchanged. The bands for the spin up electron will be identical to the bands discussed in section 4.1, with a positive mass at K and negative mass at the K0 and a Chern number n↑= +1. The bands for

the spin down electrons have an extra minus sign in front of the mass-term, so the mass is negative at K and positive at K0 and the Chern number n↓= −1.

h(~q) = vf(τzqxσx+ qyσy) + mhSzτzσz (5.2)

There are also two topological protected edge states from the Haldane model: one with spin up and a positive group velocity and one with spin down and a negative group velocity (see figure 5.1). These edge state are so-called spin filtered edge states.

Figure 5.1: Schematic picture of the the bandstructure at the edge of the Kane-Mele model of graphene. The two massive Dirac cones (purple and pink) are the same Dirac cones present in the bulk. The red and the green line represent the edge states for spin up and spin down respectively [11].

Even though the two independent conducting edge state have a non-zero Chern number, the total Chern number n = n↑+ n↓ is zero. Thus the hall conductivity is σxy = ne

2

h = 0, which is

also required by time reversal symmetry. Nevertheless the spin Hall conductivity σxys = 2e 2 h 6= 0.

Therefore the Kane-Mele model describes a quantum spin Hall insulator.

In the Haldane model the edge state were protected against backscattering because there were no state with the energy but opposite momentum at the edge. For the Kane-Mele model, the edge states are also protected from backscattering on disorder, even though the electrons are allowed to travel in both directions on the edge. If a particle wants to occupy a state with opposite momentum, is has to flip his spin. This 180◦rotation of the spin of can be done in two ways, one with a clockwise rotation and one with a counter clockwise rotation. Both rotations will acquire a phase of eiφ_{2 = e}±iπ

2,

depending on the sign of the rotation φ. So the phase of these two rotations differ by a factor of eiπ

and therefore the backscattering is suppressed by destructive interference [18]. This argument holds as long as the disorder conserves time reversal symmetry, which nonmagnetic disorder usually does.

5.2

_Z

invariant

We know the Kane-Mele model describes a topological state, since it has topological protected edge states. However, due to the time reversal invariance, the Chern number is 0 for the Kane-Mele model. Therefore there is a other topological invariant ν, which describes the topology of the Kane-Mele model and other time reversal invariant topological systems [13]. This invariant can only be 0 or 1 and therefore it is called the Z2-invariant. Note that this implies there are only two different topological

classes for systems with time reversal symmetry. For the Kane-Mele model the Z2-invariant is simply

the difference in the Chern numbers: ν = n↑−n↓

2 = 1. Kane and Mele, however, gave a much more

general definition of this Z2-invariant. They constructed a unitary matrix

wmn(k) = hum(k)|Θ|un(k)i (5.3)

from the occupied Bloch functions |um(k)i, with Θ is a the anti-unitary time reversal operator. In

the brillouin zone there are special points Λa where k coincides with −k. At these special points

w(Λa) is a antisymmetric, for which the determinant is the square of its Pfaffian. This allows us to

define:

δa =

Pf[w(Λa)]

pDet[w(Λa)]

= ±1 (5.4)

Then ν is connected to the sign of the product of these δa’s for all the special points Λa.

(−1)ν =Y

a

δa (5.5)

This general formulation of the Z2 can be used in two dimensions but also in three dimensions.

perpendicular spin Sz is conserved, the up and the down spins have independent Chern numbers n↑,

n↓. T requires n↑+ n↓ = 0, but the difference nσ = (n↑− n↓)/2 can be non zero and defines the

quantized spin Hall conductivity. Then the Z2-invariant is:

ν = nσ mod 2 (5.6)

In the presents of inversion symmetry the calculation of δa can be reduced to the product of the

parity eigenvalues ξmof the bands at Λa, which have to be ±1 due to inversion symmetry.

δa=

Y

m

ξm(Λa) (5.7)

5.3

Bulk-boundary correspondence

Just as for the Chern insulator, a topological non-trivial insulator cannot be easily distinguished by looking at the electronic properties of the bulk of the material, since both the trivial as the non-trivial material are insulator. But as the topology of a system changes at the boundary, this causes topologically protected edge stats. We have already seen this in the Kane-Mele model, where there are spin filtered edge state on the boundary. But we can also describe these edge states more generally for Z2-topological insulator.

A first important thing to realise is that a Z2-topological insulator obeys time reversal symmetry,

which gives us constrains on the edge states. A T -invariant system of spin-1₂ particles has to obey the so-called Kramers’ theorem, which states that every eigenstate of time reversal symmetry operator Θ has to be double degenerate. This can be proven by contradiction: The anti-unitary time reversal operator Θ can be represented as Θ = eiπSy_{K, with S}

zthe spin operator and K complex conjugation.

For spin-1₂ particles Θ has the property Θ2= −1. Let us assume |χi is a non-degenerate eigenstate of Θ. Then Θ |χi = c |χi. Thus Θ2_{|χi = |c|}2_{|χi, which is not allowed because |c|}2_{6= −1. Therefore}

each eigenstate has to be at least twofold degenerate.

For a system without spin-orbit coupling, Kramers theorem is always obeyed because of the degeneracy between spin up and spin down. Spin-orbit coupling breaks this the spin degeneracy. However, the Kramers theorem requires that the states with a time reversal invariant wavevector k = −k retain their spin degeneracy. These two degenerate states are called a Kramers’ pair.

Figure 5.2: (a) shows the non-topological edge states which connect pairwise between the double degenerate kramer points at Γa and Γb. These edge states are not topologically protected, because they can be pushed out of

the Fermi level by deforming the Hamiltonian at the edge. (b) shows the edge states of aZ2-topological insulator,

which don’t connect pairwise and cannot be pushed out of the Fermi level. These edge states are topologically protected and have a characteristic linear dispersion at the Γ-points [8].

The 1D edge of a 2D insulator, has Γa at k = 0 and Γb at k = π_a = −π_a as special time reversal

invariant points. Depending on the details of the Hamiltonian, edge states may or may not be present. If they are present, they are degenerate at these special points. But away from these points, their degeneracy is broken by spin-orbit coupling. Their are two ways the states at k = 0 and k = π_a can connect. They can connect pairwise as in figure 5.2(a). Then between k = 0 and k = π_a the Fermi level always intersects the edge states a even number of times. In this case the edge states can be pushed away from the Fermi level by small deformations of the Hamiltonian. So the edge are not protected and can be conducing or insulating depending on the specific details of the Hamiltonian.

In the other case the edge states are not connected pairwise, see figure 5.2(b). In this case the the Fermi level crosses the edge state a odd number of times. The edge states cannot be gapped and are therefore topologically protected. So the bulk-boundary correspondence relates NK, the number of

times the edge states crosses the Fermi level, to the change in the Z2-invariant across the interface:

NK = ∆ν mod 2 (5.8)

Note that the edge states of a topological insulator cross eachother at the Kramer points with a characteristic linear dispersion similar to the dispersion of a one dimensional massless Dirac particle. The number of these one dimensional Dirac points is always odd just like NK. This number is allowed

to be odd, because the fermion doubling theorem does not hold at the edge.

5.4

3D topological insulators

Unlike the Chern insulator, the Z2topological insulator has a natural generalization to three

dimen-sions. In 3D, there are different kinds of systems which show topological properties. A 3D topological insulator is characterized by four Z2 invariants (ν0; ν1ν2ν3) [8]. ν0 is then defined the same as ν in

equation (5.5):

(−1)ν =Y

a

δa (5.9)

where a goes through all the 8 special points in a 3D lattice. So ν0 = 1 if an odd number of

time reversal invariant points Γa have δa= −1. A system with ν0= 1 is called a strong topological

insulator. A strong topological insulator has topologically protected surface states with a odd number of massless Dirac cones below the Fermi level. A example can be seen at the right figures of figure 5.3. ν0= 1 because there is only one time reversal invariant point with δa= −1. At the surface is a single

Dirac point (black dot) enclosed by the Fermi surface (grey) at one of the time reversal invariant points on the surface.

Figure 5.3: The upper left figure shows a cubic brillouin zone with two time reversal invariant momenta with δa= −1. So the strong Z2-invariant is zero, but the topological invariant (ν1ν2ν3) = (011). The lower left figure

shows the corresponding Fermi surface (grey), which encloses two Dirac cones. The upper right figure shows a cubic brillouin zone with one time reversal invariant momenta with δa = −1. So the strong Z2-invariant ν0 is

one. The lower right figure shows that the corresponding Fermi surface encloses one Dirac cone [18]

In three dimensions there also exist so-called weak topological insulators, where a even number of the time reversal invariant momenta have δa = −1, but where some of the planes can be viewed

as a two dimensional topological insulator, with an odd number of time reversal invariant momenta with δa = −1 in the plane. So a weak topological insulator can be viewed as a stacking of 2D

topological insulators, described by the topological invariant (ν1ν2ν3), which can be seen as miller

indices denoting the Z2-invariant of the plane. Because ν0 = 0, the surface has an even number

of Dirac cones below the Fermi level (see the left side of figure 5.3). The surface states of a weak topological insulator are therefore not protected by time reversal symmetry, so they are sensitive to disorder on the surface.

However, the weak topological insulator can only be defined consistently by taking time reversal symmetry into account, but neglecting all the other information about the lattice. More generally

each the band structure necessitates a lattice, having a well-defined space group symmetry (i.e. lattice symmetries). Taking these space group symmetries into account, topological insulators can be classified more generally [23]. This classification generalizes the ideas of the weak invariants and reduces to their notion in special cases where the space group allows for the definition of a stacking axis.

I will not discuss these more general classification of topological insulators. However I will note that if the surface states are not protected by time reversal symmetry in a weak topological insulator, the insulator can still show topological features, caused by specific kinds of disorder at the surface and/or the presents of certain symmetries like particle-hole symmetry in a superconductor. The topological features of three dimensional systems is still a active field of research. In this thesis I will not discuss all the developments that have been made, but a lot of information about new topological systems can be found in literature [21, 1, 18, 8].

6

Weyl Semimetals

In the previous sections we have discussed Chern insulators and Z2-topological insulators, which

both are insulating in the bulk with topological protected edge states. It was recently discovered that so-called Weyl semimetals can also show topological features [29]. A Weyl semimetal is a three dimensional semimetal described by two bands crossing the Fermi level at so-called Weyl nodes. These Weyl nodes are points in the Brillouin zone with a linear dispersion in the three lattice momenta. They are called Weyl nodes, because the Hamiltonian around the Weyl nodes has the form of an effective Weyl equation (see equation (6.2)) [27], which describes three dimensional massless spin-1₂ particles. So in some sense the Weyl nodes are similar to the massless Dirac points discussed in graphene and just like graphene, a Weyl semimetal is a semimetal because there is no bandgap, but also a zero density of states at the Fermi level. These Weyl nodes, however, have remarkable topological features, in contrast with the Dirac points which don’t show topological behavior by themselves.

First of all, one could think that a small permutation could open the bandgap and remove the Weyl node, just like we could open the bandgap in graphene. But this is not the case. The two bands of the Weyl semimetal can be described by a hermitian 2x2 matrix:

h(k) = d · σ (6.1)

Let’s assume we have two Weyl nodes on the x-axis at ±k0. Now this Hamiltonian can be expanded

up to linear order around the Weyl nodes at k0. Then every Pauli matrix will acquire a orthogonal

direction of the momentum q as a prefactor such that the dispersion is linear in every direction:

H(k0+ q) = vxqxσx+ vyqyσy+ vzqzσz (6.2)

where viare constants. In graphene, only two of the three Pauli matrices are linked to the momentum,

therefore it is possible to open the bandgap by adding a term proportional with the third Pauli matrix. Here, however, all the three Pauli matrices are already in use. Therefore there is no matrix left to open the bandgap. All the terms which can be added to the Hamiltonian will be proportional to linear combination of the Pauli matrices and will merely shift the Weyl node k0 in k-space or alter

the constants vi. So a Weyl node is robust for small perturbations of the Hamiltonian. The only

way to make a Weyl node disappear is if the node meets another node with a opposite chirality. The two Weyl nodes together form a fourfold degenerate node, of which the degeneracy is no longer protected.

Moreover, a Weyl node is remarkable because the Weyl nodes can be seen as a monopole of the Berry connection A(k). This can be seen by looking at the Berry curvature Ω(k), which has a singularity at the Weyl nodes similar to the singularity of magnetic field at a magnetic monopole [20]:

∇ · Ω(k) = ±δ3_{(k ∓ k}

0) (6.3)

So the Weyl nodes can be thought of as sources or sinks of the Berry curvature. Therefore the Berry curvature can be interpreted as a flux going from the source to the sink, while penetrating through all the two-dimensional planes in momentum space between the Weyl nodes. Therefore all the ky-kz

planes between the Weyl nodes at kx = ±k0 have a Chern number of n = 1, whereas the planes

not between the Weyl nodes have a zero Chern number, see figure 6.3 where the Chern number is denoted by C.

Figure 6.1: Weyl semimetal with two Weyl nodes on the kx-axis. The layers between the Weyl nodes have a

Chern number C = 1 and have corresponing edge states. These edge states are connected to each other in kx

and form a Fermi-arc between the Weyl nodes [9].

So for each kxbetween the Weyl nodes, the ky-kz plane is a Chern insulator. Thus each individual

plane has topologically protected edge states as discussed in section 4.3. These edge state are then connected to each other in the kx-direction and form a so-called Fermi-arc at the surface [20]. As

can be see in figure 6.3 these Fermi-arcs only appear on the x-y and the x-z surface, because those surfaces are parallel to the x-axis. The y-z plane does not show surface states. The Fermi arcs at the surface only appear between −k0 < kx < k0 with a linear dispersion against the other k direction

defined on the surface, similar to the dispersion of the edge state of a Chern insulator.

6.1

Numerical model of the Fermi-arcs

Let us now look at a specific model of a Weyl semimetal, which I will use to show the Fermi-arcs on a two dimensional impurity inside the Weyl semimetal. I will use the Hamiltonian

h(k) = (cos kx− cos k0+ M (2 − cos ky− cos kz))σx+ sin kyσy+ sin kzσz (6.4)

with Weyl nodes on the kx-axis at kx = ±k0. The energy eigenvalues of equation (6.5) are again

easily calculated by calculating h2_{and are shown in figure 6.2. Note that I plotted the energy against}

kx and ky and chose kz = 0, because it is not possible to plot a 4D picture. Plotting the energy

against kxand kz will give exactly the same figure. For this and the following plots I chose M = −3₂

and k0=π₂.

E = ± q

(cos kx− cos k0+ M (2 − cos ky− cos kz))2+ sin ky2+ sin kz2 (6.5)

Figure 6.2: Weyl semimetal

Now I want add a two dimensional impurity to the model. I chose to add a potential V to a x-z plane inside the material. We cannot simple add a potential term to the Hamiltonian of equation (6.5), because if we add a potential, the periodicity of the lattice is broken and ky is no

Hamiltonian back to real space coordinates. This Hamiltonian can now be viewed as a matrix containing all the hopping terms from a atom and a spin to another atom and spin. To make the calculation faster and easier, I fourier transformed the x and z components back to their momentum space and I mapped the 3D system to a 1D chain of N atoms in the y-direction, where a hopping in the z or x direction can be viewed as hopping to the same atom on the chain with a hopping amplitude depending on kx and kz. The two dimensional impurity V of the x-z plane can now be

added by adding a hopping from one specific atom on the chain to itself with a hopping amplitude of V . Note that the 1D chain still has periodic boundary conditions, so edge states will only appear at the impurity.

The eigenvalues of the matrix corresponding to the 1D chain can be determined numerically for specific values of kxand kz. In figure 6.3 I have calculated the eigenvalues as function of kxfor both

with and without the potential V for which I chose V = 30, N = 30 and kz = 0. Note that the

multiple lines represent the 2N = 60 different eigenstate of the matrix. If V = 0, the eigenvalues are also eigenvalues of the momentum and each line corresponds to a value of ky in the first brillouin

zone. In the upper figure the eigenvalues are calculated for V = 0. This figure corresponds to the front view of figure 6.2. In the lower figure the eigenvalues are calculated for V = 30. The shape of most bands are more or less identical to the bands with V = 0. These bands can be recognised as the bands from the bulk. There are however two extra bands on top of eachother in the bandgap between the Weyl nodes at kx= ±π₂. This are the Fermi-arcs. There are two of them because the

impurity is inside the material, so we will get an edge state for each side of the impurity.

3 2 1 0 1 2 3 4 2 0 2 4 Energy

Band structure of a Weyl semimetal

kz=0.0, V= 0

V=30

Figure 6.3: The energy bands for a Weyl semimetal without an impurity (upper figure) and with a two dimensional impurity (lower figure). The lower figure has two extra states between the Weyl nodes. These states correspond to the Fermi arcs of both sides of the impurity.

6.2

Green’s functions

At last I will shortly discuss the link between Greens functions and impurity bound states. Slager et al. [25] showed mathematically that Greens functions defined in the bandgap at a codimension-1 or -2 impurity show the topological nature of the material. If the Greens function has zero eigenvalues, the material has a non-trivial topology. If the Greens function does not have zero eigenvalues, the material has a trivial topology. A codimension-1 or -2 impurity in a d dimensional system is an impurity with a dimension of d − 1 or d − 2. The Greens function in the bandbap of a material is defined as:

G0(ω, k) =

1

ω − h(k) (6.6)

where ω denotes the energy in the bandgap. Furthermore Slager et al. supplemented their math-ematical prediction with numerical calculations of the Greens functions in topological insulators. They however, did not do numerical calculations in Weyl semimetals. Therefore I will calculate the Greens functions in a Weyl semimetal.

With the model of the Weyl semimetal used above, I have calculated the Greens function as function of ω and k according to equation (6.6). The eigenvalues of the Greens function at the point (kx = 0, kz = 0) are shown in figure 6.4 as function of ω. The bandgap at (kx = 0, kz = 0) is

from −1 to +1, so I have plotted the eigenvalues for −1 < ω < +1. The upper figure shows the eigenvalues of Greens functions without an impurity (V = 0). As expected, the Greens function does not have zero eigenvalues. The lower figure shows shows the eigenvalues of Greens functions with the two dimensional impurity. This figure shows an extra line at zero, which is the zero eigenvalues caused by the bound states (i.e. edge states) at the impurity. So we have shown that the relation between the Greens function and the topological nature, described by Slager et al. holds for of a Weyl semimetal. Note that the weird jump of all the eigenvalues in the middle is not physical but an artifact from the numerical calculation, since the ordering of the eigenstates changes at that point, which causes the program to connect the wrong points to eachother.

0.5 0.0 0.5 4 2 0 2 4 Eigenvalue

Eigenvalues greens function

kx=0.0, kz=0.0, V=0

kx=0.0, kz=0.0, V=0

Figure 6.4: The eigenvalues of the Greens function at (kx = 0, kz = 0), as function of the energy ω in the

bandgap. The upper figure shows the eigenvalues without a impurity and the lower figure shows the eigenvalues with an impurity. It shows that the bound state at the impurity causes a zero eigenvalue of the Greens function. Note that the weird jump of all the eigenvalues in the middle is not physical but an artifact from the numerical calculation, since the ordering of the eigenstates changes at that point.

7

Conclusion and outlook

I this thesis I have have discussed the topological nature of the Haldane model and the Kane-Mele model. The Haldane model is a model of graphene with an extra term breaking time reversal symmetry. This term can be regarded as a mass-term splitting the degeneracy of the Dirac points in graphene. I have shown that this Haldane model is topological non-trivial and has a non zero Chern number, which can be seen and the topological invariant characterizing the topology of the Haldane model. An insulator with a non-zero Chern number is called a Chern insulator. On the interface between a trivial insulator and a Chern insulator, conducting edge state will appear similar to the edge states of the integer quantum Hall effect. This connection between the topological invariant and the edges states is called the bulk-boundary correspondence, which is in general an important charateristic of topological non-trivial materials.

The Kane-Mele model is a model of graphene with two Haldane mass-terms, one for each spin. This term corresponds to the spin orbit coupling. The Hamiltonian does no longer break time reversal symmetry and therefore the Chern number is zero. Yet an other topological invariant, the Z2-invariant ν can be defined, which is +1 for the Kane-Mele model. Thus, the Kane-Mele model is a

two dimensional Z2-topological insulator with spin filtered edge state. The notion of the Z2-invariant

and the Z2-topological insulator can be generalized to three dimensions.

Also non insulating materials can have topological feature. A Weyl semimetals is a three dimen-sional material of which the bands cross the Fermi level in an even number of single points. These Weyl nodes have a non zero Chern number and can be regarded as a source or sink of the Berry curvature. Therefore the layer between the Weyl nodes in momentum space can be regarded as two dimensional Chern isolators. The edges states of these Chern insulators connect to each other in the momentum space between the Weyl nodes and form a Fermi-arc at the surface. I showed numerically that these Fermi-arcs are indeed present.

I concluded my research with the link between the topology of a material and the eigenvalues of the Greens functions at an impurity. If there are zero-eigenvalues, the material is topological. If not the material is a trivial material. I showed numerically that this link, as predicted, is also valid for the topology of a Weyl semimetal.

Both in experimental and theoretical physics, a lot of research is done to better understand and generalize the topological systems that we have already seen and to search for new kinds of topological features in new materials. And currently, it’s still an active field of research. In this thesis I only discussed a small portion of what is already known and a lot more is still to be discovered. Maybe one day we will find topological materials which can be used in technology, for instance to make quantum computers or to make use of the spin Hall conductivity in spintronic devices.

Appendix

A

Formal definition Berryphase

For a system described by a Hamiltonian that depends on a set of time dependent variables R = (R1, R2, ...), the set of eigenstates {|n(R)i} are defined by:

H(R) |n(R)i = E(R) |n(R)i (A.1)

These states {|n(R)i}, however, are only defined up to a phase factor eiφn(R)_{, with φ}

n(R) a smooth

real function. This is called a U(1) gauge freedom. Any physical observable is not influenced by the choice of a specific gauge. Therefore the gauge can always be chosen freely.

Now lets assume we will adiabatically evolve the system from R(0) to R(T ) along a path C, in such a way that |n(R(t)i remains a eigenstate of the Hamiltonian. If we start with |ψ(0)i = |n(R(t)i, then the wavefunction at time t will be of the form |ψ(t)i = e−iθn(t)_{|n(R(t)i. Now we can solve the}

time-dependent Schrödinger equation to find θ(t):

H(R(t)) |ψ(t)i = i~∂t|ψ(t)i (A.2)

⇒ θn(t) =

Z t

0

En(R(t0))dt0− γn (A.3)

The first part is called the dynamical phase factor, which is gauge invariant and is therefore observable in for instance interference experiments. γC can be written as:

γn= i Z t 0 hn(R(t0)| d dt0|n(R(t 0_{)i dt}0 _(A.4) = i Z R(t) R(0) hn(R)| ∇R|n(R)i · dR (A.5) = Z R(t) R(0) A(R) · dR (A.6)

Where A(R) = i hn(R)| ∇R|n(R)i is called the Berry connection or the Berry vector potential.

A(R) is obviously not gauge invariant and for a long time for a long time physicists thought that γCwas also not gauge invariant. Therefore this term is not relevant and was often neglected. To see

this let us make a gauge transformation:

|n(R(t)i → eiφn(R)_{|n(R(t)i (A.7)}

Then A(R) and γn

A(R) → A(R) − d

dRφn(R) (A.8)

γn → γn+ φn(R(0)) − φn(R(t)) (A.9)

Because φn(R) can be chosen freely, it can be chosen in such a way it exactly cancels γn. In other

words, γn can be absorbed into the gauge and is therefore a irrelevant quantity. However, this is not

always true. Berry considered a cyclic evolution along a closed path C with R(0) = R(T ) [3]. The gauge we choose for our state |n(R)i has to be single-valued for every R, so:

eiφn(R(0))_{= e}iφn(R(T )) _(A.10)

Which implies:

φn(R(0)) − φn(R(T )) = 2πn (A.11)

With n an integer. Now the berry phase γn can only be changed by an integer multiple of 2π under

a gauge transformation. Therefore e−iγn _{becomes a gauge invariant quantity. Now equation (A.6)}

for the Berry phase can be rewritten:

γn=

I

C

Now we define the Berry curvature Ωn

µν (or sometimes called the Berry flux)

Ωn_µν(R) = ∂ ∂RµA n ν− ∂ ∂RνA n µ (A.13)

similar to the gauge field tensor Fµν in electrodynamics. Now we can use Stokes’ theorem to rewrite equation (A.12) into a surface integral:

γn= I S dRµ∧ dRν1 2Ω n µν(R) (A.14)

When the parameter space is three-dimensional, equations (A.12) and (A.14) can be written in vertor form: Ωn(R) = ∇R× An (A.15) γn= Z S dS · Ωn(R) (A.16)

This vector form shows that the berry curvature Ωn can be viewed as an analog of a magnatic field

in parameter space.

A.1

Berry phase in band theory

This general definition of the berry phase curvature and connection can also be used in band theory. The Hamiltonian of an electron in a solid is diagonal in k-space due to the periodic boundary condition. Therefore the Hamiltionian can be Fouriertransformed to the Bloch Hamiltonian h(k) with eigenfunctions |un(k)i defined on the first brillouin zone, where n denotes the ntheigenfunction.

We can now identify the brillouin zone as the parameter space of the Bloch Hamiltonian with the eigenfunctions |un(k)i. When k is changed along a closed path C in the brillouin zone, the Berry

vector potential, the Berry curvature and the Berry phase can be written as:

An(k) = hun(k)|i∇k|un(k)i (A.17)

Ωn(k) = ∇k× An(k) (A.18) γn = I C d~k · An(~k) = Z S d2k · Ωn(k) (A.19)

For a specific Hamiltonian the eigenfunctions |um(k)i can be calculated, with which the Berry phase

can be calulated. When we let the path C walk along the edge of the brillouin zone, the Chern number n is defined as:

n =X m γm 2π = X m 1 2π Z BZ d2k · Ωm(k) (A.20)

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