Universiteit van Amsterdam
Verslag van Bachelorproject Natuur- en Sterrenkunde
Generalization of Haldane’s
Chern Insulator model
Auteur:
B.A. van Voorden
Studentnummer:
10193685
Begeleider:
prof. dr. C.J.M. Schoutens
Tweede beoordelaar:
Dr. P.R. Corboz
Omvang 15 EC
Uitgevoerd in april, mei en juni 2014
Ingeleverd op 3 juli 2014
Faculteit Natuurwetenschappen, Wiskunde & Informatica Instituut voor Theoretische Fysica Amsterdam
Scientific Abstract
In this bachelor thesis the quantum Hall effect (QHE) is introduced. In materials with a strong external magnetic field it leads to an insulating bulk with conducting edges. These edge states are visible in the band structure as lines crossing the band gap and thereby connecting the valence and conducting bands with each other. There is a close link between this QHE, the Berry phase and the topology of the system. The Haldane model is shown which has band gap crossing edge states between two energy bands, without the use of an external magnetic field. A new model, the ABC-model, is made that has three energy bands. It is shown that this model has band gap crossing edge states in both band gaps for specific values of the parameters of the system, when using a Haldane-like tight binding Hamiltonian. Suggestions are made for other models and possible continuations of this project.
Populair wetenschappelijke samenvatting Generalisatie van Haldane’s Chern isolator model
Als elektronen in een sterk magnetisch veld bewegen, zullen ze door de Lorentzkracht cirkelbewegingen gaan maken. Normaal gesproken kan zo’n cirkel elke grootte aan-nemen, maar in de quantummechanica kan dit niet. De banen van elektronen blijken namelijk gekwantiseerd te zijn en kunnen daardoor slechts bepaalde groottes aannemen. De energie die de elektronen hebben hangt ook af van de de beweging die ze maken en is daarom dus ook gekwantiseerd. Dit betekent dat een elektron behoorlijk wat extra energie moet krijgen om in een grotere baan te kunnen bewegen. Dit energieverschil tussen twee energieniveaus, ook wel energiebanden genoemd, heet de bandkloof, zie figuur 1a.
In de fysica van de gecondenseerde materie worden elektrische isolatoren beschreven door een energiediagram met een bandkloof tussen het hoogste bezette energieniveau (de valentieband) en het laagste onbezette energieniveau (de geleidingsband). Bij een elektrische geleider zit er geen bandkloof tussen en kan een elektron met een heel klein energieverschil in de geleidingsband komen. Je zou dus verwachten dat een materiaal met elektronen in een sterk magnetisch veld een elektrische isolator is, omdat er een bandkloof zit tussen de gekwantiseerde energiebanden. Toch is dit niet waar en de oorzaak hiervan ligt bij de randen van het materiaal.
Zodra een elektron in een cirkelbaan bij de rand komt, zal hij namelijk terugkaatsen om vervolgens weer een cirkelbaan te maken. Zo kan een elektron zich langs de rand verplaatsen, zie figuur 1b. Maar dit kan hij slechts in ´e´en richting doen, aan de andere rand moet hij de andere kant op bewegen. Dit betekent dat er wel elektronen-transport is op de randen van het materiaal en het daar dus een elektrische geleider is. We hebben dus een materiaal gemaakt dat een geleider is aan de buitenkant, maar een isolator aan de binnenkant. In een energiediagram kan je deze randtoe-standen zien als een lijntje die de valentieband en geleidingsband met elkaar verbindt.
(a)Energiebanden met bandkloof. (b)Een geleider aan de buitenkant en een isolator aan de binnenkant.
Dit effect treedt niet alleen op bij sterke magneetvelden, maar komt ook van nature voor in bepaalde materialen. In dit verslag worden een aantal modellen van materi-alen beschreven, die uiteindelijk leiden tot het Chern isolator model van Haldane. In dit model bestaan er randtoestanden die de bandkloof tussen twee energiebanden oversteken in grafeen, een hexagonaal rooster van koolstofatomen, zie figuur 2a. Deze randtoestanden bestaan echter alleen voor bepaalde waarden van de massa’s van de atomen. Dit is een gevolg van de zogenaamde topologie (grof gezegd de vorm) van de bandenstructuur.
Het doel van dit onderzoek was om te kijken of er ook zulke bandkloof-overstekende randtoestanden gemaakt kunnen worden in een model met drie energiebanden. Daarvoor is het model van Haldane aangepast. Dit model, het ABC-model genoemd, bestaat uit een driehoekig rooster van drie verschillende atomen. Met dit ABC-model is het inderdaad gelukt om ook de bandkloof-overstekende randtoestanden te cre¨eren in zowel de onderste als de bovenste bandkloof, zie figuur 2b.
(a)Energiebanden met randtoestanden in het Haldane model. (b)Energiebanden met randtoestanden in het ABC model.
Contents
Scientific Abstract 2
Populair wetenschappelijke samenvatting 3
1. Introduction 6
2. Time Reversal and Inversion Symmetry 7
3. Berry Phase 8
4. Two Level Systems 10
5. Three Level Systems 12
6. Second Quantization 14
7. Quantum Hall Effect and Topology 15
8. Graphene 18
9. Haldane Model 21
10. Edge States 24
11. The ABC model 28
12. Conclusion and Discussion 34
References 35
1. Introduction
The quantum Hall effect, an effect that occurs when an electron system is placed in a strong magnetic field, leads to materials with insulating bulks and conducting edges. This strange state is of course interesting for physicists, so there were a lot of studies devoted to this subject. But it didn’t seem to be more than an interesting theoretical novelty without actual practical uses, since you’d need a strong magnetic field to create this state. This changed in 1988, when Haldane published a paper in which he introduced a lattice model, based on graphene, that has a nonzero Hall conductance, without the use of a strong magnetic field. This model would lead to a lot of new research in these models with a special role for the topology of the systems and the breaking of several symmetries, one of which is time reversal. It would eventually lead to the Z2 topological insulator of Kane
and Mele in 2005. In this model the spin-orbit coupling allowed for a nonzero Hall conductance without the time reversal symmetry breaking. These new phases were quickly observed in real materials and are still a very active subject of research today. In this project, that basically served as an introduction to topological insulators, I studied the necessary theories to understand the Haldane model and get an insight in the models that would lead to the discovery of topological insulators. I also made a new lattice model, that I call the ABC-model, that is a variation on Haldane’s model and exhibits the same effects, but then in a band structure with three bands instead of two.
I wrote this thesis with the intent that a four month younger version of myself could understand it. The first couple of sections will therefore explain theories that I didn’t know about at the start of the project. These theories, including the Berry phase, second quantization and the quantum Hall effect, are necessary to be able to understand the remainder of this paper. These are all pretty well known theories and therefore my goal in these sections is not to discuss them in excruciating detail, but to really only give and explain the formulas that are important. There were also a few theories that I had to rehearse during this project (such as Fourier transformations and some concepts from condensed matter physics, like the Brillouin zone and band structures), but I chose to omit them from this report, since I had studied them earlier. After the theoretical sections I’ll discuss graphene and the Haldane model from the literature and also add a few figures I made in Mathematica. The last part of this thesis consists of the entirely new ABC-model.
Finally, just a small note on notation: in this thesis I chose to use a bold font for vectors, instead of an arrow. I also chose to write all fractions in the text as a/b instead of ab, because some fractions (certainly those with parameters that have a subscript or superscript) became almost unreadable. The same logic applies to the few rare cases where I used a vector in the text, so with (a, b, c) I just mean the normal vector and not the transposed one.
2. Time Reversal and Inversion Symmetry
In this thesis I’ll use a few arguments based on time reversal and inversion symmetry. These symmetries are not particularly important, but they can be useful, as we’ll see later on. Therefore I’ll quickly describe these symmetries in the current section. The information in this section comes mainly from chapter 5 of Bernevig[1]. The time reversal (TR) operator T changes the sign in front of the time t:
T : t → −t (1)
Physical variables can either be even (unchanged) or odd (multiplied with −1) under time reversal. Even variables include the position, acceleration, force and the energy of a particle. Odd variables include the velocity and angular momentum (both orbital and spin) of a particle and the magnetic field. The product of two odd variables becomes, of course, an even variable (for example, the even classical kinetic energy depends on the odd velocity squared) and a combination of an even and an odd variable is an odd variable. Quantum mechanical operators are even when they commute with T and odd when they anticommute with T :
Even: T ˆOT−1= ˆO Odd: T ˆOT−1= − ˆO (2) The TR operator also works as a conjugation operator (changes i to −i and vice versa), this can be seen by looking at the commutator of the operators of x and p. The position operator x is even and the momentum operator p is odd, so the two combinations xp and px also have to be odd. We therefore expect [x, p] = (xp − px) to be odd under TR: T [x, p]T−1= −[x, p]. We know that [x, p] = i~ and therefore we get that T (i~)T−1= −i~, which can only be true if T iT−1= −i, because ~ is just a constant. So T not only changes the sign of t, but also of i.
The inversion operator P (for parity) works in a similar way as the TR-operator, but then for position:
P : r → −r (3)
It can also be called ‘point reflection’, since the whole system is inversed in the point (0, 0, 0). You can once again divide physical quantities in even and odd under the P -operator. Energy, angular momentum and time are all examples of even variables. Examples of odd operators are position (of course), velocity and force. The inversion operator does not, like the TR operator, work like a conjugation operator. This can be seen in a similar way as before with the commutator of x and p. Both x and p are odd under the P operator, so the commutator (and thus i) has to be even. A physical system that remains the same under time reversal is said to have TR symmetry and one that remains the same under inversion symmetry has P symmetry.
3. Berry Phase
One of the physical quantities that is essential for understanding topological insula-tors is the Berry Phase γ. In this section I will describe a compact version of the derivation of the Berry phase, as is given in Griffiths [3] and Bernevig [1].
If we have a system with a Hamiltonian that depends on some parameters R (for example position, momentum or the magnetic field) the eigenvalues and eigen-functions will in general also depend on this R. These eigenvalues and eigeneigen-functions will therefore evolve in time when R evolves with time:
H(t) |ψn(t)i = En(t) |ψn(t)i (4)
The adiabatic theorem states that a system that starts in the eigenstate |ψn(t = 0)i
of H(t = 0) will stay in the eigenstate |ψn(t)i of H(t) when the parameters R are
varied very slowly (adiabatically) with time. The final state Ψn(t) can however pick
up some phase factors:
Ψn(t) = ei(θn(t) + γn(t))ψn(t) (5)
The first factor θn(t) is called the dynamical phase and isn’t really interesting.
It just generalizes the phase e−iEntthat a time independent state has to the same
phase for a time dependent state: θn(t) = − 1 ~ Z t 0 En(t0)dt0 (6)
The second phase γn is called the geometrical phase and this is the phase that is
important for topological insulators. It can be calculated with γn(t) = i Z t 0 ψn(t0) ∂ ∂t0ψn(t 0) dt0 (7)
There is a slight problem with this formula. It depends on time, but it’s only valid when we vary the system very slowly. It would be easier to have this function depend on R instead. To do this we change ∂ψn/∂t0 to (∇Rψn) · dR/dt0, after which γn
has become independent of time: γn = i
I
hψn|∇Rψni dR (8)
The γn in equation 8 is called the Berry phase, named after Micheal Berry, who
introduced this notation in a paper published in 1984[5]. This can also be written as: γn = I An· dR with An = i ψn ∂ ∂R ψn (9) An is called the Berry Connection or the Berry vector potential, since it acts
An important property of the Berry phase is that it is gauge independent. When the initial state |ψni is multiplied with some phase eiχ(R), a term ∇R χ will be
added to A, just as an electromagnetic vector potential will change. When this new A is integrated over a closed path, the extra term will vanish and γn is therefore
gauge independent.
The Berry curvature Fn is defined as the curl of A:
Fn = ∇ × An (10)
And application of Stokes’ rule to formula 9 shows that γn can also be calculated
via
γn=
Z Z
Fn· dR (11)
which is, of course, still gauge independent. All these formulas so far have made use of derivatives of eigenstates. Even in a somewhat simple Hamiltonian, these can become very complicated to compute. Another way of computing the Berry phase, is by taking the derivative of the Hamiltonian as follows (see for the derivation reference [1], page 9):
γn = −
Z Z
Vn· dS (12)
Where Vni= ijkFjk and after some algebra you get that:
Vn= X m6=n hn |∇H| mi × hm |∇H| ni (En− Em)2 (13)
In the next section we’ll calculate the Berry phase for a generic two dimensional system with both formula 11 and formula 12.
4. Two Level Systems
In this section we’ll follow the calculation of the Berry phase for a generic two dimensional Hamiltonian as done by Bernevig[1]. A generic Hamiltonian of a two level system can be written as:
H = (R) I2+ d (R) · σ (14)
Where I2is the two dimensional identity matrix, d some three dimensional vector
depending on the parameters of the system and σ the vector consisting of the three Pauli matrices: σx= 0 1 1 0 σy= 0 −i i 0 σz= 1 0 0 −1 (15) The eigenvalues of this Hamiltonian can easily be calculated and are E± =
(R) ±√d · d. The -term is the same in both eigenvalues and can therefore be ignored in the further calculations, since it doesn’t affect the eigenstates nor the difference between the two eigenvalues. So we can take E± = ±|d|. To find
the Berry phase we can calculate it using the Berry curvature formula 11 or using the Hamiltonian derivative approach of formula 12. First we’re going to do the calculation with the Berry Curvature formula.
The eigenstates of the Hamiltonian are needed in order to use formula 11. We write d in spherical coordinates as d = |d| (sin(θ) cos(φ), sin(θ) sin(φ), cos(φ)). The eigenstates are then:
|−i =sin(θ/2)e −iφ − cos(θ/2) |+i =cos(θ/2)e −iφ sin(θ/2) (16) With eigenvalues of respectively − |d| and + |d|. The Berry connections Aθ and Aφ
can then easily be calculated via formula 9 and are:
Aθ= 0 Aφ= ± sin2(θ/2) (17)
The + or − in Aφdepends on if we calculate it with the |−i or |+i state. The Berry
curvature is then:
Fθφ= ∂θAφ− ∂φAθ= ±
sin(θ)
2 (18)
With the use of Vni= ijkFjk we get that V±= ±d/2d3.
Another way of obtaining the same V± is with the Hamiltonian derivative
for-mula 12. We can take for our Hamiltonian H = R · σ since we can translate and rotate the system to get to this form. Formula 12 then becomes pretty simple, because ∇H = σ and the Pauli matrices have the simple eigenfunctions:
|+i =1 0 |−i =0 1 (19) with of course σx|±i = |∓i, σy|±i = ±i |∓i and σz|±i = ± |±i.
After some algebra you can find that V+= (0, 0, 1/2R2) and V− = (0, 0, −1/2R2) =
−V+. When we rotate back to the general case, where R doesn’t lie on the z-axis,
we get that V+ = R/2R3and V−= −R/2R3.
This field has the form of a monopole, compare it for example with the elec-tric field of an elecelec-tric charge E = er/4π0r3 or the gravitational field of a point
mass G = Gmr/r3. To calculate the Berry phases we have to take the surface
integral of the fields V− and V+. This integral becomes pretty simple, because it
rep-resents the flux through an area that has a monopole with strength ±1/2 within. So this integral simply becomes ±Ω/2, where Ω is the solid angle subtended at the origin. For a parameter space confined to two dimensions (for example: lattice momentum kx and ky) we get two different situations when we make a closed path in the
parameter space: we either enclose the origin, or we don’t. In the first case we get:
γ±= ±π → eiγ± = −1 (20)
In the second case:
γ± = 0 → eiγ± = 1 (21)
So the final state is multiplied by either −1 or 1, based on whether the path in the parameter space encloses the origin or not.
Note that the nonzero value of the Berry phase is due to the origin. At the origin d = 0 and the eigenvalues of the Hamiltonian (±|d|) are the same, so there is a degeneracy. Notice also that the eigenfunctions in equation 16 are not well defined at all points. When θ = π both eigenfunctions become (e−iφ, 0), which is ill defined, because φ is ill defined at this point. This can be fixed by taking a new gauge in which the eigenfunctions are multiplied by eiφ, but then the eigenfunctions are ill defined at θ = 0 for the same reason as before. As it turns out the Berry curvature integrated over a surface is always zero when the eigenfunctions are well defined at all points. A nonzero value is obtained if there is some degeneracy in the parameter space.
5. Three Level Systems
In this section the steps of the last section are repeated with the Hamiltonian of a three level system. The standard eigenfunctions of a three level system can be represented by: |+i = 1 0 0 |0i = 0 1 0 |−i = 0 0 1 (22)
The equivalent of the Pauli matrices in a three level system are the τ -matrices. They are given by:
τz= 1 0 0 0 0 0 0 0 −1 τ+= √ 2 0 1 0 0 0 1 0 0 0 τ−= √ 2 0 0 0 1 0 0 0 1 0 (23) τx= 1 2(τ++ τ−) = 1 √ 2 0 1 0 1 0 1 0 1 0 (24) τy= 1 2i(τ+− τ−) = 1 √ 2 0 −i 0 i 0 −i 0 i 0 (25)
We make the same sort of Hamiltonian as in the last section, but this time with τ = (τx, τy, τz) instead of σ and I3 instead of I2: H = (R) I3+ d (R) · τ . The
part with I3 can be ignored for the same reason as last time. To calculate the Berry
curvature we switch to spherical coordinates and get the following Hamiltonian:
H = √1 2
√
2 cos(θ) sin(θ)e−iφ 0 sin(θ)eiφ 0 sin(θ)e−iφ
0 sin(θ)eiφ −√2 cos(θ)
(26)
The eigenvalues of this Hamiltonian are ± |d| and 0. First we’re going to look at the Berry curvature. The three eigenfunctions can be calculated with a lot of algebra, the |+i eigenvector with eigenvalue + |d| is given by:
|+i = e−iφcos2(θ/2) √ 2 sin(θ/2) cos(θ/2) eiφsin2(θ/2) (27)
With this eigenfunction I find that:
Aθ= i h+| ∂θ|+i = 0 and Aφ= i h+| ∂φ|+i = − cos(θ) (28)
It follows from those two Berry connections that Fθφ = ∂θAφ− ∂φAθ = sin(θ),
which is twice as big as the Berry curvature of equation 18. This means that V+ is also twice as big and is thus equal to d/d3. In the same way we get that
Finally we can once again get the same result when using formule 12. Just as in the case for the two dimensional system we translate and rotate the system to get H = R · τ , which is in matrixform: H = ~τ · ~R = √1 2 z√2 x − iy 0 x + iy 0 x − iy 0 x + iy −z√2 (29)
The fields can then be calculated with the eigenfunctions from equation 22 and the τ -matrices from equations 23, 24 and 25 and they give the following result:
V+= X m6=+ h+ |~τ | mi × hm |~τ | +i (E+− Em) 2 = i R2zˆ (30) V0= 0 V−= − i R2zˆ (31)
We see that the fields are once again twice as large as the fields from the two level system. The effect is that the Berry phase is therefore also twice as large. This has the effect that eγ± = 1 or 0 now and can never become −1 in contrast to the two
6. Second Quantization
In this section the concept of second quantization and its use in lattice models will be explained. The information for this section comes mainly from the book “Condensed Matter Field Theory” by Altland and Simons [2].
Working with the traditional wave functions in a system of multiple indistinguishable particles is hard and cumbersome, because the total wave function ψ(r1, r2, r3, ...)
consists of all possible combinations of the individual wavefunctions and the particles. For example ψ(r1, r2) = ψ1(r1)ψ2(r2) ± ψ2(r1)ψ1(r2). For a m-particle system you’ll
get m! different terms, so the wave functions get complicated really quickly. Second quantization introduces a better way of handling a system with a lot of indistinguishable particles. It uses a basis in which we count the number of particles in a given state, which is called a Fock State. The wavefunctions then look like |ψi = |n1, n2, n3, ...i where ni is the number of particles in state i. We work on
these states with two important operators in the lattice models. Those operators are the creation and annihilation operators, written as c†i and ci respectively. These two
operators work similarly to the ladder operators in the quantum harmonic oscillator:
c†i|.., ni, ...i =
√
ni+ 1 |.., ni+1, ...i ci|.., ni, ...i =
√
ni|.., ni−1, ...i (32)
So c†i adds one particle in state i to the system, while ci removes one. Note that
letting ci work on a state with ni = 0 results in a factor 0 before the final state.
This ensures that we can’t get a state with a negative number of particles. The Pauli exclusion principle dictates that ni can only be 0 or 1 for electrons and other
fermions. In this case we need the extra constraint that we can’t create a state with some ni = 2, so we need to have that c†i|.., 1, ...i = 0.
A combination of a creation and an annihilation operator can be used to write down in a short way the hopping of a particle from some state to another state. For example a simple Hamiltonian for a lattice where the particles can only hop to the nearest neighbor can be written as follows:
H = X <i,j> tijc†icj+ X i Vic†ici (33)
The first term describes the hopping from site j to site i, because the annihilation operator removes a particle on site j and the creation operator creates a particle on site i. The hi, ji indicates that we sum over all the nearest neighbors and the tij is
the hopping strength. The second sum describes the energy of particles that reside at site i. Vi is the energy of one particle on that site and c†ici gives the number of
7. Quantum Hall Effect and Topology
In this section the quantum Hall effect will be explained. It leads to some interesting physics: materials that are an insulator in the bulk, but a conductor at its edges. The close connection between the quantum Hall effect, the Berry phase and the topology of the system will also be explained. The information that is contained in this section comes from a few different sources, mainly Bernevig’s book [1], Kane and Hasan’s colloquium article [6] and chapter 3 of some college notes / syllabus I found online, made by Kai Sun [11].
The classical Hall effect is the result of applying a magnetic field perpendicu-lar to the current in a two dimensional material. When an electric field Exin the
x-direction is applied to a two dimensional system with electrons, the electrons will start to move along this field and this results in a current jx. If we then
apply a magnetic field B perpendicular to this plane, the electrons will take more complicated paths, because they’re pushed to the side by the Lorentz force. This leads to a pile up of electrons on one edge of the plane and a shortage of electrons on the other edge. The charge difference between the two edges results in an electric field Ey in the y-direction. When this Ey equals vB, the net force on electrons
in the y-direction becomes 0 and the electrons will once again only move in the x-direction.
The conductivity σ is equal to 1/ρ = j/E, where ρ is the resistivity. In this case we can define a conductivity σxx= jx/Ex and a conductivity σxy = jx/Ey,
which is called the Hall conductance. Due to rotational symmetry of this system σyy = σxx and σyx = σxy. In classical mechanics the Hall conductance can be
calculated and is σxy= en/B, where n is the electron density. A measurement of the
Hall conductance is one way to accurately measure the electron density in a material. In the quantum Hall effect (QHE) the Hall conductance is quantized. This quantization can be explained as follows. When an electron is placed in a two dimensional plane with a magnetic field perpendicular to the plane, it will move in circles: the cyclotron motion. The circular orbits are quantized, which leads to a quantized energy m = ~ωc(m + 1/2), where ωc is the cyclotron frequency.
These energy levels are called the Landau levels. They can be viewed as a band structure, where each Landau level represents a band. These Landau levels give the impression that this system is an insulator, because there is an energy gap between the highest occupied level and the lowest unoccupied level, see figure 3b. But this impression is false, because the argument of the rotating electrons fails at the edge. Electrons at the edge will try to make a circular motion, but they can’t continue making this movement when they hit the edge. The electrons will bounce back and start a new semicircle until they once again hit the edge and repeat the previous steps, see figure 3a. This results in an electron transport along the edge, so the material is certainly not an insulator at the edges. These edge states can be seen in the band structure as lines that cross the band gap and connect the valence and conduction band with each other.
(a)Model of the situation. [13] (b)Sketch of the Landau levels and edge states.
Figure 3. The system is an insulator in the bulk, but has conducting edge states.
The electrons at an edge can only move in one direction. This means that the velocity of electrons at the top edge is opposite to the velocity of the electrons at the lower edge. When there is a potential difference between the two edges, the electron density at the two edges will be different and this leads to a Hall conductance. This Hall conductance is given by equation 34, when ν Landau levels are filled.
σxy= ν
e2
h (34)
So far this section can be summarized as “moving electrons in a two dimensional system with a strong magnetic field lead to an insulating bulk and conducting edges”. These special edge states can, however, also occur naturally in a material, without the use of a strong magnetic field. The lattice model of Haldane, which we’ll study later, was one of the first models that exhibited this nonzero Hall conductance without the use of a magnetic field.
This nonzero Hall conductance can be explained by the topology of the system, as was first done in the paper of Thouless at al [7]. Basically the topology of the band structure with band gaps can be classified by the so called Chern numbers. Two states have the same topology, and thus the same Chern number, if the band structure of the first material can be continuously transformed to get the band structure of the second material, meaning that the band gaps are never closed during the transformation process. A difference in Chern number between two materials leads to the band gap crossing edge states between the two materials. The reason is that there has to be a conducting state between the two insulating states, since the difference in topology tells you that you have to close the band gap to get from the one insulating state to the other insulating state. So there is a link between the Chern number and the existence of edge states (and thus a nonzero Hall conductance). The so called bulk-boundary correspondence shows that the number of edge states at an edge is equal to the difference of the Chern numbers of the two materials. The vacuum has a Chern number of 0, so the num-ber of edge states in a material adjacent to the vacuum is equal to the Chern numnum-ber.
The Chern number also turns out to be closely related to the Berry phase and can be calculated by 1/2π times the integral of the Berry curvature over the Brillouin zone, which will always give a multiple of 2π. The Hall conductance can then be calculated by: σxy = e2 2πh Z Z dkxdkyFxy(k) (35)
Where, from the section on the Berry phase: Fxy(k) = ∂Ay(k) ∂kx −∂Ax(k) ∂ky An = −i X a ∈ filled bands a ∂ ∂kn a (36)
In most lattice models this integral will be equal to zero. In fact, it is only nonzero when the eigenfunctions have a singularity at some point in the parameter space, as we had seen earlier in the sections in which the Berry phase of the generic two and three dimensional Hamiltonians was calculated.
Notice that the sum in formula 36 goes over all the occupied bands. The Hall conductance has to be zero when the band structure is completely filled with elec-trons, because then there are no states for a conducting electron to occupy. This means that the sum of the Chern numbers of all bands has to be zero, this fact will be used later in this report.
A last important property of σxy is that it’s odd under both the time reversal
operator and the inversion operator. A model of a system that has TR symmetry and/or inversion symmetry can therefore only have a zero Hall conductance. This is not completely true, a nonzero Hall conductance can be obtained in a system with TR symmetry by taking into account the spin of electrons. Kane and Mele showed this in a paper in 2005[8].
8. Graphene
As noted earlier Haldane made a lattice model that exhibits the band gap crossing edge states without the use of a strong magnetic field. For this model he added a few terms to a simple nearest neighbor hopping Hamiltonian on the hexagonal lattice of graphene. I will therefore first introduce graphene and the nearest neighbor hopping model in this section, before we get to Haldane’s model in the next one.
The graphene lattice is a hexagonal lattice made up of carbon atoms, as can be seen in figure 4. The lattice consists of two sublattices of the different sites A and B. There is one A-site and one B-site in a unit cell.
Figure 4. Left: the hexagonal lattice with the basis lattice vectors. Right: the reciprocal lattice with the Brillouin zone. [1]
Each site is connected to its nearest neighbors with the three δ-vectors and to its next to nearest neighbors with combinations of the translation vectors a, as can be seen in the above figure. The reciprocal vectors that belong to the a-vectors are the b-vectors. All these vectors are given in terms of a, the closest distance between two atoms, as: δ1= a 2 1√ 3 δ2= a 2 1 −√3 δ3= −a 0 (37) a1,2= a 2 3 ±√3 b1,2= 2π 3a 1 ±√1 3 (38) The Brillouin zone can be taken by the parallelogram formed by b1 and b2 (see
figure 4), but it is often taken to be the hexagon with vertices at K and K0, whose coordinates in the reciprocal space are given by:
K, K0= 2π 3a 1 ±√1 3 (39)
The simplest Hamiltonian involves only the hopping of electrons from one site to one of the nearest neighbor sites, therefore called the nearest neighbor (NN) Hamiltonian:
H = −tX
hi,ji
a†ibj+ b†jai (40)
where the a-operators are the creation and annihilation operators for A-sites and b for the B-sites. The hopping strength t is the same for all hoppings in this model. When we Fourier transform this model to k-space we get the following Hamiltonian:
X k a†k 3 X j=1 teik·δjb k+ b†k 3 X j=1 te−ik·δja k (41)
Which we could also write in the basis of a†k and b†k in a matrix representation: H =X k a†k b†k 0 P3 j=1te ik·δj P3 j=1te −ik·δj 0 ! ak bk (42) = a†k b†k h(k)ak bk (43) All the important properties of tight binding Hamiltonians reside in this h(k), which, from now on, I will just call “the Hamiltonian” of the system. The eigenvalues of h(k) as a function of k can easily be calculated in Mathematica and they give the following band structure in the Brillouin zone (see for the code the appendix):
Figure 5. The band structure in the Brillouin zone of graphene in the nn-hopping model.
It can be clearly seen in figure 5 that there are two energy bands with a band gap between them, although there are multiple points at which the two bands touch each other. In graphene the complete lower band is filled, so the Fermi level lies within this band gap at the E = 0 level. Graphene is called a “semimetal” because of this band gap that closes at only a few points. Its properties are different than an insulator, because an electron that is close to these points can go to the conduction band with a minimal amount of extra energy, that is always present in a real material. It appears that there are six of those points in the Brillouin zone. But since all six points lie on the corners of the Brillouin zone, multiple points are actually the same because they’re connected by the reciprocal lattice vectors b1 and b2 from equation
Zooming in around the point K gives us figure 6:
Figure 6. The band structure close to the point K.
It can be seen in this figure that the energy goes linearly with k, like a massless particle in a Dirac Hamiltonian. Expanding the Hamiltonian around point K using a two dimensional Taylor expansion shows that the Hamiltonian can indeed be written as a Dirac Hamiltonian of the form κxσx+ κyσy, where κ = K − k. The
same expansion around K0 gives −κxσx+ κyσy. K and K0 are therefore called the
Dirac Points of graphene.
This simple model of graphene has both TR and inversion symmetry, so it will be no surprise that the Hall conductance is zero. Furthermore the band gap is not completely open, so there are no band gap crossing edge states required to get conducting electrons. In order to get a nonzero Hall conductance we thus need to have some extra terms that break the TR and inversion symmetries and also opens the band gap, forcing the material to become an insulator. Haldane provided these terms, as we’ll see in the next section.
9. Haldane Model
In this section the model proposed by Haldane in 1988[4] will be discussed. This model showed that a nonzero quantized Hall conductance can occur in a system without the presence of an external magnetic field. Haldane added to the NN hopping Hamiltonian of graphene (see equation 42) two terms that break the TR symmetry and the inversion symmetry and opens the band gap.
The first term is a second real hopping term between the next nearest neigh-bor (NNN) sites (which are the nearest neighneigh-bors on the sublattice of that particular site). With this hopping term he also added a local periodic magnetic field, with a total zero flux through the unit cell, perpendicular to the graphene plane. This field multiplies the hopping strength between next nearest neighbors with a phase φ, whose sign depends on the direction of the hopping (clockwise or anticlockwise in the graphene cell). This magnetic field with the extra phase breaks the TR symmetry of the Hamiltonian.
The second term he added is an inversion symmetry breaking mass term. He added a mass M to the A-sites and a mass −M to the B-sites. The potential ener-gies on the different sites are then different, thus breaking the inversion symmetry of the Hamiltonian. The total Hamiltonian can be written as:
H = tX hi,ji c†icj+ t2 X hhi,jii e−ivijφc† icj+ X i iM c†ici (44)
Here vij = ±1 depending on the direction of the hopping and i= ±1 depending
on the site being an A or a B site. To get the band structure we once again have to Fourier transform this equation to the k-space. The nearest neighbor terms are of course still the same as in formula 42. There are six different hopping directions for each lattice site to its next nearest neighbors, namely the vectors a1,
a2, a2− a1, −a1, −a2 and a1− a2. So for the NNN-terms we get sums over all
these directions. The mass-terms just add an extra M and −M on the diagonal entries of the matrix. After the Fourier transformation and some algebra (with some help from Mathematica) the Hamiltonian can be written as:
h(k) = h0I2+ hxσx+ hyσy+ hzσz (45)
The different hi’s in this formula are given by:
h0= 2t2cos(φ) (cos(k · a1) + cos(k · a2) + cos(k · (a1− a2)) (46)
hx= cos(k · a1) + cos(k · a2) + 1 (47)
hy = sin(k · a1) + sin(k · a2) (48)
hz= M + 2t2sin(φ) (sin(k · a1) − sin(k · a2) − sin(k · (a1− a2)) (49)
These can be written in cosines and sines instead of e-powers, because the sum over all the six directions eliminates either the imaginary or real parts of each term.
The eigenvalues of this Haldane Hamiltonian give the following band structure in the Brillouin zone:
Figure 7. The band structure of the Haldane model with t2=18, M = 1, φ = 1.
As can be seen in figure (7) there are still two energy bands. However, unlike the band structure from the previous section, the two bands never touch each other, at least for the chosen values of t2, M and φ. The two bands do still touch each
other at the Dirac points K and K0 for specific combinations of t2, M and φ, see
for example figure 8.
Figure 8. The band structure with closed band gap; t2= 13, M =
√
3, φ = π/2.
The hxand hy from equation 49 become, just as in the NN hopping model, 0 at
the Dirac points. The two eigenvalues then become 0 at these points when hz is
also equal to 0. Filling in the values of K and K0 in hz results in hz = 0 when
M = 3√3t2sin φ at K and M = −3
√
3t2sin φ at K0. So it is possible to open or
close the band gap by changing the value of M . In the band structure of figure 8 the band gap is closed at the points K.
To obtain the value of the Hall conductance I’ll use a very short version of the method Bernevig used. For a more thorough method see chapter 8 of his book [1]. We start at M → ∞ and lower the value of the mass until we get to another topological class. We know that in the situation where M = ∞ the eigenfunctions have to be constant, since the wave function is localized on either site A or B due to the infinite potential energy. Because the eigenfunctions are constant, the Hall conductance has to be 0. The band gap is open in this case.
First let’s look at the case where t > 0 and φ > 0. We lower the value of M until we get to the value of M = 3√3t2sin φ, when the band gap closes and then opens
again at K. We’re then in a different topological class, since we’ve closed a band gap. The Chern number then becomes +1. When we keep lowering the value of M we’ll eventually close and reopen the gap at point K0 at which σ becomes 0 again. We can do the same thing for a negative t2or a negative φ, both multiply the Chern
number by −1.
The different areas with the different values of the Hall conductance are displayed in figure 9, which Haldane himself used in his paper. The masses at which the band gap closes are plotted in this diagram as a function of φ. It becomes clear in this diagram that there are topologically different areas, separated by the lines at which the band gap closes.
10. Edge States
So far we’ve looked at models of graphene that have periodic boundary conditions in both the x- and y-direction. There’s one problem with the approach we have taken thus far. We’ve calculated that we have a non zero Hall conductance in the Haldane model for certain values of M , but we can’t see any band gap crossing states in figure 7, where the band structure of the Haldane model is shown. It’s easy to explain why: the band gap crossing states are the edge states and so far we’ve only looked at a model that doesn’t have any edges, but is an infinite plane. Such an infinite sheet of graphene is, of course, not realistic and the energy bands haven’t explicitly shown the Hall conductance. For these two reasons we’ll now look at a more realistic model: a graphene ribbon that has a finite number of sites in the y-direction but still an infinite number of sites in the x-direction, see figure 10a. This model will show us the band gap crossing edge states.
First we’re going to take a look at such a ribbon in the NN hopping model. The basic Hamiltonian for this new model is the same:
H = −tX
hi,ji
c†icj (50)
The ribbon doesn’t have periodic boundary conditions in the y-direction anymore, since the ribbon has edges. The translational symmetry is thus broken and we can therefore only make a Fourier-transformation in the x-direction:
c†x,y =X
k
eikxc†k,y (51)
Here k is just the momentum in the x-direction.
Because of the periodic boundary conditions in the x-direction all sites with the same y-value must be the same. And therefore we can index the sites by counting them from the top edge to the bottom edge in a zigzag line as in figure 10b. We note that there are two different hoppings, one has only an y component, the other has also an x component. The e-power in formula 51 is equal to 1 in the vertical hop and e±ika1/2 (with a
1 still the translational vector in the x-direction from equation
38) for the diagonal hoppings. We can also classify two different sites in the lattice: A-sites with a nearest neighbor directly above them and B-sites with a nearest neighbor directly beneath them, see figure 10a. An electron can hop to an A-site at index j from one site at index j − 1 with strength t and from two different sites at index j + 1 with total strength t(eika1/2+ e−ika1/2). The Hamiltonian for the
model in figure 10b, can therefore be written in this index basis as:
H =c†1,k c†2,k c†3,k · · · 0 t0 0 · · · t0 0 t · · · 0 t 0 . . . .. . ... ... . .. c1,k c2,k c3,k .. . (52)
(a) The ribbon continues infinitely in the x-direction. You can clearly see the difference be-tween the A and B states.
(b) All the different sites on a ribbon with 16 sites in the y-direction.
Figure 10. A graphene ribbon[1].
The Hamiltonian of equation 52 can easily be altered to get the Hamiltonian of a slightly different model. The t and t0 factors are always next to the diagonal in the nearest neighbor model. For example the Hamiltonian of a ribbon with extra rows of A and B-sites is just a larger matrix with still the t and t0 elements next to the diagonal. When the ribbon starts with a B site instead of an A site, we have to exchange the t and t0 in the Hamiltonian. Finally we can also have a model with an odd number of sites, where both the top and bottom edge consists of ei-ther A or B sites. In that case we can just cut the last row and column of this matrix. I made the matrices of all these possible different Hamiltonians in Mathemat-ica and plotted the resulting energy bands (the eigenvalues of these Hamiltonians). This turned out be quite difficult, since the code has to calculate the eigenvalues of very big matrices. I finally found a small guide on the internet, where someone had also made the energy bands of a graphene ribbon [12], so I used his or her way of calculating the resulting figures. The code and all the graphs can be seen in the appendix, but I show a couple different situations in the graphs below. The band structure for an even number of chains, starting with an A-site and ending in a B-site is shown in figure 11a. In figure 11b the band structure of an odd number of sites, starting and ending with an A-site, is shown.
We already saw in figure 5 that the two bands must touch each other, which luckily is still the case in the two subfigures in figure 11. The edge states are the lines with zero energy.
(a)Model with even number of sites. (b)Model with odd number of sites.
Figure 11. Band strucure of a graphene ribbon with nn-hopping.
Then I added a mass-term −m to the A-sites and +m to the B-sites. These masses break the inversion symmetry and therefore open the band gap in the band structure. These masses are called the Semenoff mass. In figure 12a the band structure of the model with an even number of sites is shown with the added masses and in figure 12b the model with an odd number of sites. It can be clearly seen in these figures that the band gap is now opened. The edge states still don’t cross the band gap, so we have created a normal insulator. This can be explained by the model still having TR symmetry.
(a)Model with even number of sites. (b)Model with odd number of sites.
Figure 12. Band structure of graphene ribbon with nn-hopping and Semenoff masses.
Finally I’ve also made the Hamiltonians of graphene ribbons in the Haldane model, so with the next nearest neighbor hoppings added. NNN-hoppings to a site with index j can come from sites with index j − 2, j or j + 2. So we get extra matrix elements on the diagonal and two entries to the right and left of the diagonal. Each of these hoppings have hopping strength t2, get a phase due to the Fourier
transform of either e±ika1/2 or e±ika1, depending on their position and a phase of
e±iφ, depending on whether the hopping is clockwise or counterclockwise in the graphene cell. The complete calculation can again be seen in the Mathematica code in the appendix. As an example, the Hamiltonian of the small ribbon with only four sites, that starts with an A-site and where I’ve set a1/2 = 1 for clearity is:
−m + 2t2cos(2k − φ) −t(1 + e−ik) 2t2cos(k + φ) 0
−t(1 + eik) m + 2t
2cos(2k + φ) t 2t2cos(k − φ)
2t2cos(k + φ) t −m + 2t2cos(2k − φ) −t(1 + e−ik)
0 2t2cos(k − φ) −t(1 + eik) m + 2t2cos(2k + φ) (53)
I have plotted the band structure of four different models in Mathematica: those with an even number of sites starting with either A or B and those with an odd number of sites with either A- or B-sites at the edges. The band structure in figure 13a is of the model with an even number of sites, starting with an A-site and the band structure in figure 13b is with an odd number of sites, also starting with an A-site.
(a)Model with even number of sites. (b)Model with odd number of sites.
Figure 13. Band structure of a graphene ribbon in the Haldane model.
The band gap crossing edge states are clearly visible in the two band structures of figure 13. There are two of these edge states, one on each edge. One of them moves to the right (since v ∝ ∂E/∂k is positive) and the other moves to the right. They connect the two bands with each other at the Dirac points. These figures show clearly that there is a nonzero Hall conductance in the Haldane model between two energy bands.
11. The ABC model
All the things I’ve done so far have been adapted from a couple of sources and while I did do quite a lot of calculations in Mathematica myself, nothing so far has been entirely new. That changes with this section. The idea for the rest of my project was to construct and study a system that would have three bands and to try to get band gap crossing edge states in the two band gaps. I originally started working with a model comparable to CuO2. In that model there are two different
sites (the Cu and the O), but three sites in a unit cell, one Cu and two O-sites. I started working on this model, but I quickly encountered some problems. The main problem was that it didn’t look enough like the graphene from the last sections, so I was unsure how to continue on certain points. I also tried making the edge states figures for this model in a similar way as I had done in the previous section, but it didn’t result in the figures I wanted to have. Although later on, when working on the edge states for the ABC model, I realized that I had made the edge state model in a wrong way, almost certainly explaining why I got those strange results. I therefore started trying to make a model that looks more like graphene, in the sense that it has three different sites per unit cell and the same translational and rotational symmetries. This model became the triangular lattice, as shown in figure 14. This model has three different sites, A, B and C, in a unit cell and I will therefore call it the ABC-model. It can be seen as graphene with an extra site within each graphene ring.
Figure 14. The lattice of the ABC model [10].
I will immediately jump to a Haldane-like model for this lattice. The Hamiltonian of this model looks as follows:
H = tX hi,ji c†icj+ t2 X hhi,jii e−ivijφc† icj+ X i iM c†ici (54)
This is exactly the same as formula 44 from the Haldane model. The difference is that we now have to sum over three different sites, instead of the two in the Haldane model and we now have for respectively the A, B and C-sites = +1, 0, −1 and vij = +1, 0, −1 instead of = ±1 and vij= ±1. The B-sites are the sites that
are added to the Haldane model and they have zero mass (the average of the two others) and no extra phase for the NNN hopping terms. This is because the phase φ is added clockwise to the rings consisting of A- and C-sites, so a NNN hopping between two B-sites goes perpendicular to this phase and thus isn’t affected by it.
The Hamiltonian of equation 54 can be written, after Fourier transforming and simplifying, as follows in the basis of creation and annihilation operators on the three different sites:
H = a†k b†k c†k p + q + M r∗ r r p (φ = 0) r∗ r∗ r p − q − M ak bk ck (55) where
p = 2t2cos(φ) (cos(k · (a1− a2)) + cos(k · a1) + cos(k · a2)) (56)
q = 2t2sin(φ) (sin(k · (a1− a2)) − sin(k · a1) + sin(k · a2)) (57)
r = t eik·δ1+ eik·δ2+ eik·δ3 (58)
The different ai’s and δi’s are of course still the same vectors as in equations 37 and
38 from the section on graphene. I calculated the eigenvectors of this Hamiltonian in Mathematica and they give the following band structures for two different values of the mass and the phase:
(a)Model with M = 3 and φ = π/3. (b)Model with M = 0 and φ = 2π/3.
Figure 15. Band structure of the ABC model in the Brillouin zone.
It can be seen in both subfigures that we do indeed get three different energy bands. It is clear in subfigure 15a that there are two band gaps in between the three bands. In subfigure 15b the lower two bands are more folded in to each other, making the lower band gap harder to see. This happens for a lot of combinations of the mass and phase.
By varying the values of the mass and the phase of these band structures I got the strong idea that the bands, once again, only close at the Dirac points K and K0. The Hamiltonian of equation 55 is at point K equal to:
H = M + t2 −3 cos(φ) + 3√3 sin(φ) 0 0 0 −3t2 0 0 0 −M + t2 −3 cos(φ) − 3√3 sin(φ) (59)
At point K0 the Hamiltonian is the same, but with an extra minus in front of the two sin(φ)-terms.
The eigenvalues of these two matrices give the energies at those points as a function of M , φ and t2. I calculated these eigenvalues and the eigenvalues become the same
(so the bands touch each other) for the following masses: M1= −3t2 1 − cos(φ) +√3 sin(φ) M2= −3 √ 3t2sin(φ) M3= −3t2 −1 + cos(φ) +√3 sin(φ) M4= 3t2 −1 + cos(φ) +√3 sin(φ) M5= 3 √ 3t2sin(φ) M6= 3t2 1 − cos(φ) +√3 sin(φ) (60)
The masses on the left are the values at which the bands touch each other at K and the masses on the right for when they touch at point K0. At M2and M5 the upper
band gap closes and at the other four masses the lower band gap is closed. I plotted M/t2 as a function of φ between 0 and 2π, just like Haldane did:
Figure 16. M/t2as a function of φ for all the six M ’s.
(a) For the higher band gap: M2and M5. (b)For the lower band gap: M1, M3, M4 and M6.
Figure 17. M/t2 as a function of φ.
These diagrams have, in comparison to the phase diagram of Haldane (figure 9), a lot more areas, each bounded by lines at which the band gap is closed. The obvious goal is then to calculate the Chern numbers (and thus the Hall conductance) in these different areas. Unfortunately, I didn’t succeed in doing so. I tried a few different things.
First of all I tried making a calculation like Bernevig did in his book[1] and which I quickly described in the section on Haldane’s model. He first approximates the Hamiltonian for k close to the Dirac points and then bases his calculation on Hamiltonians that can be written like d · σ, where σ are the Pauli matrices. I did approximate the Hamiltonian of equation 55 for k close to K and K0 (which took a lot of work, by the way), but that’s as far as I could get. Since the ABC-Hamiltonian is a 3 × 3 matrix, we can’t, of course, use the Pauli matrices. I tried writing it as H = d · W, where W is some vector consisting of three 3 × 3 matrices, but that attempt failed. You can write the approximated Hamiltonian as d · λ, where λ are the Gell-Mann matrices, after which the Hamiltonian looks like:
H = (−t2− 2t2cos(φ)) I3 + −3 √ 3 4 kx− 3 4ky (λ1+ λ4+ λ6) + 3 4kx− 3√3 4 ky (λ2− λ5+ λ7) + 3√3t2sin(φ) − 3t2cos(φ) + 3t2+ M λ3 2 + (3√3t2sin(φ) + M ) √ 3 2 − (3t2− 3 cos(φ)) 1 2√3 λ8 (61)
The approximation around point K0is the same, but with ky→ −ky. This
Hamilton-ian does look similar to the HamiltonHamilton-ians that are studied in the paper “Three-Band Model for Quantum Hall and Spin Hall Effects” by Go et al [9]. They split the three band Hamiltonian in a spin-1 part and a nematic part. Then they use as an example the so called Kagome lattice, which can be written as a term with (λ2, −λ5, λ7) and
a term with (λ1, λ4, λ6), both of which are also in equation 61. The factors before
these terms are however much easier than those in my model, so I couldn’t reproduce their calculations with this Hamiltonian. And unfortunately, they don’t show a way to calculate the Chern numbers for a general three band model. However, the paper does, in my opinion, show that it must be possible in some way to calculate the Chern numbers of the ABC-model by approximating the Hamiltonian and writing it as d·λ. I also tried calculating the Chern numbers in Mathematica using formula 35. But this also didn’t lead to a result. The eigenvectors of this Hamiltonian become quite long and complicated and I couldn’t get Mathematica to give a credible result for these eigenvectors. But I am quite inexperienced with Mathematica, so I’m quite sure that someone with more knowledge of the program could get a credible result from it. There is one last theory we can use to at least see if we have a zero or nonzero Hall conductance in each area of figure 16. Because we know that the Chern number is equal to the number of edge states. So I can make a model of a lattice with edges, in the same way as I’ve done in the previous section, and enter the values of M/t2 and φ for each area and see if there are any band gap crossing edge states.
The big problem with this method was that the different energy bands sometimes overlapped, what I could have expected from the band structure diagrams like figure 15b. After a morning of making matrices and at least an afternoon of making band structure diagrams I settled on the following result (see next page):
Figure 18. The phase diagram with the three different areas.
There are, as far as I can see, three different areas. The first area, which I made light blue in figure 18, is the area that is outside of M2and M5, the two masses at
which the upper band gap closes. There are no band gap crossing edge states in this area. All the band structures I made in this area look like 19a, although the bands do overlap in some regions. Since there are no band gap crossing edge states, the Hall conductance will be zero. This is somewhat surprising, since it is inside the lines at which the lower band gap closes. Apparently these band gap closings do not change the topology of the situation.
(a)First area, M = 6 and φ = π/3. (b)Second area, M = 0.1 and φ = 2.9.
The second area, which I made light pink in figure 18, can only be reached after closing the upper band gap, because it is inside the area enclosed by M2 and M5.
The band structures in this area all look like the one in figure 19b. In this subfigure you can clearly see that there are two band gap crossing edge states in the upper band gap. This means that the Hall conductance is either e2/h or −e2/h, since
there are two edge states and the model has two edges. This is only true if the electron filling in the lattice is 2/3, because only then the Fermi level is within this band gap.
The third and last area, light yellow in figure 18, is the most interesting, since it is also enclosed by lines that close the lower band gap. The band structure in this area looks like the one in figure 20. It can be clearly seen that there are two band gap crossing states in both the upper and the lower band gap. The Hall conductance will thus be either e2/h or −e2/h if the electron filling is either 1/3 or 2/3.
Figure 20. The band structure within the third area, M = 0 and φ = π/3.
There is one last piece of information that can be gained from these figures. We know from the section on the QHE that the sum of the Chern numbers of all the three bands should be equal to 0. In both the lower band gap and in the upper bandgap from figure 20 are two edge states, so a Hall conductance of ±e2/h as
noted earlier. A possible combination of the Chern numbers of the three bands is then 1 for the lowest band, −2 for the middle band and 1 again for the highest band. This leads to a total Chern number of 0 and also to two edge states in both band gaps. An alternative is, of course, all these values times −1.
12. Conclusion and Discussion
The last section, basically the results of my project, can be summarized in a few sentences as follows: with the ABC-model I succeeded in making a lattice model based on the Haldane model that has band gap crossing edge states between three energy bands. Using a qualitative approach, by counting the number of edge states in different areas, I made a phase diagram, showing the Chern number and Hall conductance for different values of the mass and phase. Unfortunately I didn’t succeed in actually calculating the Chern numbers by, for example, integrating the Berry curvature.
I can see a couple of possible directions on how this project could theoretically continue. First of all I am quite sure that there has to be some way to calculate the Chern numbers for the ABC-model, as I also wrote in the last section. Someone who knows more about Mathematica than me might actually succeed doing the calculations I was trying to do. There is also one other way of calculating the Chern number, via the derivative of the Hamiltonian (see formula 12), that might be successful. Unfortunately I didn’t find the time to try this calculation.
Secondly, in the paper by Go et al [9], on the QHE in a three band model, the Chern numbers of the so called Kagome lattice, that actually looks pretty similar to the triangular lattice of the ABC-model, were calculated using an approach very similar to what I tried with the Gell-Mann matrices. I think that the Kagome lattice might have been a better lattice to study the Hall conductance in a three band model, since the terms in its Hamiltonian are easier. And of course, that paper could have helped me during the process.
A third possibility is to further study the lattice originally proposed for the three band model, the CuO2-lattice. I made some mistakes in calculating the edge states
of this lattice, which led to me believe that I wouldn’t get band gap crossing edge states. And the Hamiltonian also didn’t look familiar enough for me to be able to do certain calculations. I should be able to at least make the band structure of the ribbon model to see if there are any band gap crossing edge states in this model. At the start of the project I was also planning to study the model of Kane and Mele [8], in which they succeed in getting a nonzero Hall conductance, while the TR symmetry is preserved, by taking the spin-orbit coupling into account. It would have been interesting to study a far more realistic model that has lead to the experimental realization of topological insulators.
References Books
[1] Bernevig, A.: Topological Insulators and Topological Superconductors, Princeton University Press, (2005)
[2] Altland, A., Simons, B.: Condensed Matter Field Theory, Pearson Education, (2013)
[3] Griffiths, D.: Introduction to Quantum Mechanics, 2nd ed., Cambridge University Press, (2013)
Papers
[4] Haldane, F. D. M.: Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the “Parity Anomaly”., Physical Review Letters
volume 61, number 18, (1988)
[5] Berry, M.: Quantal Phase Factors Accompanying Adiabatic Changes., Proceedings of the Royal Society A 392, (1984)
[6] Hasan, M. Z., Kane, C. L.: Topological insulators., Reviews of Modern Physics 82, (2010)
[7] Thouless, D. et al: Quantized Hall Conductance in a Two-Dimensional Periodic Potential., Physical Review Letters 49, (1982)
[8] Kane, C.L., Mele, E.J.: Z2 Topological Order and the Quantum Spin Hall Effect.,
Physical Review Letters 95, (2005)
[9] Go, G., Park, J., Han, J.: Three-Band Model for Quantum Hall and Spin Hall Effects, Physical Review B: Condensed Matter and Materials Physics 87, (2013) [10] Sen, A. et al: Variational wavefunction study of the triangular lattice supersolid.,
Physical Review Letters 100, (2008)
Other sources / Internet
[11] Sun, K.: Lecture notes Advanced Condensed Matter Physics, University of Michigan, viewed on http://www-personal.umich.edu/~sunkai/teaching/Fall_2013/ [12] M.W. Daniel: Tight-binding model for graphene ribbons., viewed on https://www.
andrew.cmu.edu/user/mwdaniel
[13] Edited figure, original from http://www.u-tokyo.ac.jp/en/utokyo-research/ research-news/a-new-material-for-dissipationless-electronics
Appendix
I placed all the above papers and books in a Dropbox folder, viewable at https:// www.dropbox.com/sh/29j73hofnybz4jd/AABDE4NyYmpQMbtY5JIZUtEsa (clickable link in the pdf-version). In this folder are also a couple of papers and a book that I at least skimmed through at one point during the project, but that I didn’t deem useful in the context of this report.
I also placed all the Mathematica files I made in a Dropbox folder, that can be found on https://www.dropbox.com/sh/0zmmrr5o8sd9l0q/AAB7HDjUAIQm_i7ENMg7ObR_ a (again, clickable hyperlink in the pdf-version). They include the files I used to make all the figures in this report of the band structures of graphene, the Haldane model and the ABC-model as well as all the ribbon versions of these models. There are also a lot of figures of different band structures I didn’t use in this paper. Some Mathematica files were only used for my understanding of the subjects and there are some files with a few unfinished and/or incorrect calculations. I was originally planning to make the code more readable and add some comments, but I didn’t had enough time for that.
