A new look at edge modes, in the context of the bulk-boundary correspondence

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MSc Physics & Astronomy, Track theoretical

physics

Master Thesis

A new look at edge modes, in the

context of the bulk-boundary

correspondence

Author: Supervisor:

T.M. Vonk

dr. J. van Wezel

Examination date:

Juli 1, 2018

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Abstract

The bulk-boundary correspondence is a major, but underrated, part of the topological classification of solids. In this thesis, bulk topology and an example of the correspondence are reviewed. A recent method for finding boundary modes is discussed and implemented to investigate the SSH chain and the charge-density wave. This method reveals level repulsion between bulk- and boundary modes.

Title: A new look at edge modes, in the context of the bulk-boundary correspondence Author: T.M. Vonk, tomvonk1@gmail.com, 10534806

Supervisor: dr. J. van Wezel

Second Examiner: prof. dr. J.-S. Caux Examination date: Juli 1, 2018

Instituut voor Theoretische Fysica Amsterdam University of Amsterdam

Science Park 904, 1098 XG Amsterdam uva.nl

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1. Introduction; the shape of

physics to come

1.1. Topological insulators - the parable of the boring

duckling that became a topologically non-trivial swan

I don’t blame you if you find insulators boring. What’s interesting about a piece of matter that is defined by its inability to react to electromagnetic fields? Their only hope is the clich´e “still waters run deep” and believe David Foster Wallace when he says that clich´es express great truth in their banality. Luckily, these still waters do run deep. The discovery of topological insulators propelled insulators to center stage of condensed matter. Under the surface of stoic unreactiveness lie interesting phenomena governed by modern mathematics.

The classic example of topology is another clich´e that comes surprisingly close to what’s actually going on; the donut and the coffee cup. These are regarded as the same because they can be deformed into each other without tearing, gluing or removing ma-terial. Two topological insulators are the same if their band structures can be deformed into each other without closing the gap or breaking a symmetry. Topologists do not use clay for their proof that the donut is the same as a coffee cup. They use numbers that classify topologies, such as the number of holes for the donut example and the Chern number for band structures.

This thesis is not about the classification of objects like coffee cups and donuts. That is because the topological classification of household objects does not really matter. This is in stark contrast with the topological classification of topological insulators. The topology of the band structure has important and measurable consequences. This thesis is about some of those consequences.

1.2. The Bulk-Boundary Correspondence

This thesis is mainly about one of the aforementioned “interesting physical phenomena”: the bulk-boundary correspondence. A quick list of definitions:

• bulk The bulk is a vague concept. The naive idea of the bulk part of a system1 is the ‘middle bit’: far away from the edges. In practice the bulk system is defined as the same interactions but with periodic boundary conditions. This definition is useful: the typical Hamiltonian is translationally invariant if there are periodic

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boundary conditions. As a consequence, the bulk Hamiltonian is block diagonal in the k-space basis and can therefore be viewed as an operator defined at every k with discrete eigenvalues. These eigenvalues at every k form the band structure. The bulk is not a part of a system with boundaries. It is a separate but similar system.

• boundary, or edge On the other hand, the boundary of a system is usually pretty clearly defined. In field theory terms: if the system is described by a field theory on a d-manifold M, the boundary is the manifold ∂M . Especially interesting is an effective theory on the boundary; a model that lives purely in d − 1 dimensions and describes the behavior of those modes that are localized on the boundary. • correspondence quantities in one system are in relation with properties of another

system.

The central claim of the bulk-boundary correspondence is that the topology of the bulk system—the Chern number for example—corresponds to properties of the edge states. Among those properties: their existence and whether they conduct electricity or not. That’s right, even though we call them topological insulators they are capable of transporting charge on their edges.

Let me be clear, the bulk-boundary correspondence has not been deduced from first principles. The bulk-boundary correspondence doesn’t even have a precise form across all kinds of topological insulators. Of course, for the classic examples like the Chern-insulator and the quantum Hall effect there exist a precise formulation and proof of the correspondence, but for other systems—for example, torsion invariants [1]—it is not at all clear what the edge states are supposed to be. It is not even clear if there should be edge states in those systems.

As a naive 41₂th year student I thought the correspondence was great! There is this system which seemed to be topological, but I didn’t know in what way. I hoped, or at that point, expected that I could turn the correspondence around and let the edge states tell me what kind of topology I needed to look for in the bulk. As an overly cynical 45₈th year student the correspondence was all meaningless: there is no precise form of the bulk-boundary correspondence outside of the Chern-number case. It is not clear that, in general, there is a correspondence between the bulk and the boundary.

This anecdote serves as motivation for the main goal of this thesis: finding more precise expressions of the bulk-boundary correspondence.

1.2.1. A word of warning

This is a hard problem for a very fundamental reason: the Heisenberg uncertainty prin-ciple. By definition, certainly by my definition, the bulk theory is formulated in k-space. The bulk side of the correspondence consists of information written in the momentum basis: the band structure and eigenstates. The topological invariants are not written in any basis, but are usually calculated in the k-basis. The Chern number is usually calculated as an integral over the Brillouin zone. On the other side of the bulk-boundary

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correspondence one points at a region in x-space and asks: what happens there? The Heisenberg uncertainty principle says that those questions are difficult because you can’t know the momentum and position of a mode at the same time. That is why in practice you tend to get stuck at expressions involving sums over all k when calculating stuff about edge modes.

1.3. A new look at edge states

I hope that this introduction motivates the need for something more precise when it comes to the bulk-boundary correspondence. So you can imagine I was excited to come across a paper titled “Unified bulk-boundary correspondence for band insulators” by Rhim, Bardarson and Slager [2]. Like articles do, they claimed a lot. (Some of these claims will be evaluated in Chapter 4)

The idea is pretty simple; go slowly from the bulk to a system with a boundary and count the modes that enter the gap in the band structure. Those in-gap modes are the consequence of the introduction of a boundary; they are edge modes. Take a one dimensional periodic lattice (a chain) and add a potential (Vb) that breaks up one of the

links, so that you end up with a chain with two edges (the ends). What can happen to the energy eigenvalues? A mode that was previously below the gap can deform and go above the valence band into the energy gap. Conversely, a eigenvalue above the gap can cross the conductance band into the gap. Finally, a mode already in the gap can move out. To find the number of in-gap edge modes, you need to count the number of modes moving in and out of the gap. This can be done by analyzing the poles in the Green’s function [2].

Consider the Green’s functions of the two Hamiltonians: there is G0 corresponding to

the bulk hamiltonian and Gβ, where the boundary is turned on a bit (β ranges from 0

to 1—from a closed to an open chain, resp.). Poles in the Green’s function are bound states and Gβ has a pole whenever

Aβ = det(1 − βG0Vb) = 0. (1.1)

To see when this ‘poledeterminant’ is zero and what that means for the poles (are they moving into the gap or out of the gap?) we’ll need some more technical details. Fans of technical details can find them in Chapter 4.

1.4. Is the bulk-boundary correspondence like AdS/CFT?

No.

The bulk-boundary correspondence is different from AdS/CFT. Nobody claims that there is a duality between bulk and boundary, and it would be weird if there was one. The reason why a gravitational theory can be dual to a lower dimensional field theory is because gravity has surprisingly little entropy. The entropy of gravity scales with the area, and therefore with the volume of the boundary. This is not the case in a

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topological insulator; there is just regular quantum mechanics on both the bulk and the boundary. The bulk-boundary correspondence is also much smaller; in it’s most precise form it claims that one number in the bulk is equal to one number on the boundary. That pales in comparison with AdS/CFT, where in principle all correlation functions can be determined from the boundary.

1.5. About this thesis

This is maybe a less traditional master thesis. That is on purpose, because I wanted to write something that is interesting to read, interesting to write and something that introduces some interesting concepts. That is why I make a joke now and then and why I experiment with bullet points and loose(ish) language.

In order to make this document a bit more useful, I have included a guide to the literature and my Mathematica code.

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2. In the bulk my beautiful

dark twisted band structure

This chapter is about the bulk side of the bulk-boundary correspondence. As stated in the introduction, the bulk system is defined as having periodic boundary conditions. According to the general form of the correspondence, topological properties of the bulk correspond to states on the boundary. This chapter will introduce bulk topology.

Topology is a term a lot of people use. It doesn’t mean the same thing to everybody, especially within physics. Here are the three main uses:

• general shape of something. People sometimes say that a thing has the topology of a triangle, for instance.

• topological in the sense that the main properties are invariant under continuous, smooth deformations. This is related to the often used donut - coffee cup analogy

• topological in the sense of topological field theory. A field theory is called topo-logical if it doesn’t depend on the metric of the base manifold, but only on the topological properties.

I tend to avoid the first use. The second and third notion are confusing because some-thing can be topologically trivial in the third sense but interesting in the second sense. Xiao-Gang Wen expands on the difference between these two notions in [3].

The idea that the properties of some physical system are invariant under smooth deformations is very interesting and powerful, and this is what I will focus on in this chapter. I will introduce the Chern number in great detail as that is the prototypical band invariant. It is important to realize that the Chern number is not the only invariant. I will stress this again in section 2.3.

2.1. Berry phase and Chern number

I have two eyes. They allow me to analyze something in depth, so to analyze the Berry phase I will give two perspectives. I’ll start by casting everything in a more mathematical light so that I can use the machinery of fiber bundles. In the second subsection I’ll go back to earth and use words like Hamiltonian and wave function, and I’ll use explicit parameters.

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ε

ε_F k

Figure 2.1.: An illustration of a gapped band structure. The fermi energy is denoted with F

and the dashed line. Adapted from http://www.ioffe.ru/SVA/NSM/Semicond/ AlN/bandstr.html

2.1.1. High and mighty

In this subsection I consider a 2+1 dimensional system on a torus with radiuses Lx

and Ly and discrete translational symmetry1. Because of translational invariance, the

momentum of the electrons is also periodic, with period 2π. The momentum space is a torus, the Brillouin zone. At each point on the Brillouin zone electrons can be in different energy eigenstates and together these states form the band structure. It is assumed that there is a gap in our system and that the Fermi level lies in that gap, see Figure 2.1. This means that the bands below the gap (valence bands) are fully filled while the conductance bands above the gap are empty.

The notion of a Brillouin zone can actually be used in a more general sense. Instead of viewing the Hamiltonian as an operator that depends on the momentum, consider a Hamiltonian that depends on some parameters ~α. These parameters can in principle be anything, from terms you add to the Hamiltonian to the mass of electrons. As long as the space of α’s is periodic the Chern number can be defined.

There is something special about k-space because it is an internal parameter; it is summed over in the full Hamiltonian: H = P H(k)c†_kck. If someone says that they are

varying the Hamiltonian with respect to k they are varying this H(k), not H. From now on I will use α as the parameter to be varied.

Eigenstates are not uniquely defined: multiply it with a complex phase and the energy will be the same2. While an overall phase does not matter, a difference in phase at different alpha does matter. If there is a Berry phase, the phase of an eigenstate will

1

I put the ‘lattice spacing’ to 1. Note that it is required that the system is periodic for translational invariance.

2_{The eigenstates could also be degenerate. Degenerate eigenstates lead to a phenomenon analogous}

to the Berry phase, called the Wilczek-Zee [4] holonomy. In this discussion, I will assume that the eigenstates are non degenerate. This is a simplification, in general (and in Fig. 2.1) this is not true.

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vary gradually as you change α. Consider a path in α space; the Berry phase is defined as the phase acquired by an eigenstate after transport of the state along that path.

At this point it might not seem that impressive. Phases are often irrelevant and this one doesn’t seem to say anything about the properties of the system or about topology. But there is a lot hidden in this phase, and to see that it is useful to put this into a more mathematical setting3.

I collect all the phases of the eigenstates into a fiber,

Fn,α =

n

eiφ|n(α)i | φ ∈ [0, 2π)o, (2.1)

that is equivalent to U (1). |n(α)i is the n-th eigenstate of H(α). If you consider all fibers you get a fiber bundle.

Pn=

[

α∈T2

Fn,α (2.2)

Now I ask the question: what happens when I transport the eigenstates around a loop in α-space. I start in one eigenstate with one particular phase and look what happens if it completes a loop in the base space. For the movement in the fiber bundle you need a rule for parallel transport4, or a connection. But which connection should be used? Luckily there is a natural connection that comes from the notion of adiabatic evolution. The idea is that the path in α-space is traversed slowly so that if a system is prepared in an eigenstate, say |n(α)i, the system will stay in that eigenstate. ‘Slowly’ means, in this context, that the time it takes to complete a loop (T ) is much larger than the inverse of the size of the energy gap, 1/∆. This condition is enough to determine the connection, which is:

An= ihn(α)|d|n(α)i. (2.3)

This connection, in which d is the exterior derivative, is derived in the box 2.1.1. It is a gauge field and can have a curvature F = dA. Now that we have set up all of the fiber bundle machinery results come pretty quickly. The result of parallel transport of a state |ni, along a loop C is

eiHCAn|ni . (2.4)

Using Stokes’ theorem:

eiH An _{= e}i R

SdAn _{= e}i

R

SFn _(2.5)

If we choose the surface S to be the whole 2-torus, the integral is determined by a generalization of the Gauss-Bonnet5 theorem:

Z

T2

F = 2πc1 (2.6)

3

This does then require some mathematical background, which can be found in Nakahara [5].

4

This is analogous to the transport of four-vectors in GR. There, the base space is spacetime. The rules for parallel transport come from the connections, which are called Christoffel symbols.

5

A more familiar version of this theorem states that the integral over the curvature of a two-dimensional manifold is equal to 2π times the euler characteristic.

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c1 is an integer known as the Chern number. Some comments on this result:

• The integral over F is manifestly gauge invariant.(A gauge transformation is: A → A + df .)

• The fact that the Chern number is an integer makes the topological underpinnings clear: if you change parameters of the Hamiltonian a little bit (for instance, you could change mass of the electron), the Chern number can’t change a little bit because it has to be an integer.

• I haven’t said much about the choice of parameter space, but in practice it is often momentum space. This is realized in the following way. First you decompose the Hamiltonian in it’s Fourier components: H = P

kH(k)c †

kck. Then you view the

H(k)’s as a normal Hamiltonian with parameters {~k}. Because of the periodic boundary conditions these parameters form a torus like the αs did, {~k} = T2_.

Behind the Fourier decomposition is the assumption that there are no interactions. This is a central assumption behind the idea of the band structure.

• A mathematical description of the Berry phase is that it is the holonomy of a line bundle. That line bundle is the n-th band in the band structure.

• The Chern number is additive in some sense. For a band structure like in Figure 2.1 it makes more sense to calculate the Chern number of all the bands below the gap and that can be done by tracing over all the states below the gap.

c1 = 1 2π Z S tr F (2.7)

• When all topological invariants are zero, the sample is said to be topologically trivial. Be aware though, that there are ways of being topologically nontrivial other than the Chern number. (See section 2.3)

2.1.1: Adiabatic Evolution

The idea of adiabaticity is that you evolve the system slowly. To make that more precise, let T be the time scale associated to the gap in the band structure and scale the time coordinate with T: s = t/T . The Schr¨odinger equation is expressed in this new coordinate:

i∂sψT(s) = T H(s)ψT(s). (2.8)

Adiabatic evolution corresponds to taking the limit T → ∞. In this limit an eigenstate that starts below the gap can not evolve to a state above the gap[6]. In other words, projecting the Hilbert space onto the ‘sub-gap’ subspace and evolving within that subspace results in the same as first evolvinga, then projecting onto

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the sub-gap subspace;

P (s)U (s) = U (s)P (0). (2.9)

Now I follow some insight by Kato ([7]), who noticed that this evolution is also described by the following Hamiltonian:b

HKato= i

1

T[∂sP (s), P (s)] (2.10)

Using this Hamiltonian, the time evolution of a sub gap state follows:

∂sψ = [∂sP, P ]ψ. (2.11)

I do some algebraic manipulations to get to a simple expression for the adiabatic evolution of ψ: P ∂sψ = 0. I use that P⊥ψ = (1 − P )ψ = 0.

∂sψ = ((∂sP )P − P (∂sP ))ψ (2.12)

= ((∂sP )P − P (∂sP ) + P (∂sP )P − P (∂sP )P )ψ (2.13)

= (P⊥(∂sP )P − P (∂sP )P⊥)ψ (2.14)

= P⊥(∂sP )ψ (2.15)

P ∂sψ = P P⊥(∂sP )ψ = 0 (2.16)

If there is one state below the gap it is possible to write this as

|ψi hψ| ∂s|ψi ds = 0, (2.17)

or, more simply:

hψ(s)| d

ds|ψ(s)i = 0. (2.18)

This has a corresponding connection 1 form:

An= i hφ| d |φi . (2.19)

This can be confusing. Any |ψ(s)i that satisfies Eq. 2.18 is parallel transported. Take a state φ, it doesn’t have to satisfy Eq. 2.18. If you multiply it with the phase ei

Rs

s0A_{, this new state will satisfy Eq. 2.18.} a

the evolution operator is given by U = T e−iR0sds 0_H(s0₎

b

I have omitted the proof for this statement; it can be found in [6]

2.1.2. Low and slow

The fiber bundle language of the previous section is very natural and useful, but it can seem a bit too hoity-toity. The following approach will be more down-to-earth and

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provide a more physical reason for the quantization of the Chern number. In this section I’ll start out in a more boring way; I just look at the time evolution of some state.

That state is |ψi = eiθ(t)|n(α(t))i. In this notation |ni is an eigenstate of the Hamil-tonian H(α(t)), which depends on time through a set of varying parameters α(t). It is multiplied with a time-dependent phase θ. The time evolution is dictated by the Schr¨odinger equation in the adiabatic limit. The other thing you should remember is that α is varied in time in such a way that α(T ) = α(0) for some T , or in other words, α(t) describes a loop C.

id

dt|ψi = H(α(t)) |ψi (2.20)

In the adiabatic limit, both |n(α(t))i and |ψi are eigenstates of H(α) at each time. I use this to expand Eq. 2.20 into an equation for the evolution of θ.

En(α) |n(α)i = ˙θ |n(α)i + i

d

dt|n(α)i (2.21)

The next step is taking the inner product.

d dtθ = En− hn| d dt|ni (2.22) θ(t) = Z t 0 dt En− i Z t 0 dt hn| d dt|ni (2.23)

The first part of his equation is not all that interesting; it is just the phase you acquire because |ni has energy. This is called the dynamical phase and is usually ignored. The other bit is what is known as the berry phase and denoted as γn.

So, we get the same formula as in the previous section:

γn= Z C An (2.24) An= i hn(α)| d dαi |n(α)i dα = i hn(α)| d |n(α)i (2.25)

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imaginary part the states would decay6. Luckily there are two good arguments against this. One, the above is the result of unitary time evolution, so decay is not possible. Two, it’s easy to show that the real part of hn| d |ni is zero with the following trick.

0 = d(1) = d hn|ni = (hn| d |ni)∗+ hn| d |ni (2.26)

Now that we’ve dealt with the realness of the Berry phase, how about taking a look at its integerness. Of course I already did this in section 2.1.1, but a different perspective can’t hurt. It might even clear up some confusing concepts.

Such as: a gauge. Consider this, under the addition of phase, |n(α)i → eif (θ)_|n(α)i,

the gauge field transforms in the way you would expect.

A → i hn| d |ni + idf (2.27)

Choosing a phase (for each α) is the same as fixing the gauge for A. That is why people tend to use the two terms interchangeably. Let’s think again about the Berry phase. It is usually explained as a phase you acquire if you go around a loop in parameter space. But could you not just absorb this phase into the definition of |ni? No. At least not in a smooth way, and this is one way to view the Chern number—as an obstruction to defining a global gauge. It is not always possible to pick the right f because there is no reason why i hn| d |ni should always be purely gauge.7 Consider the ideal f : it ensures that |ni gets the right phase after a loop around the Brillouin zone. If A is not pure gauge, f (α) must have a singularity at some point (α0) in the zone. That singularity

forces you to pick a different gauge, say g(α), in the neighborhood around α0. These

pure gauge fields are integrated over the Brillouin zone. This is illustrated in Figure 2.2.

2πc1 = Z T2 dA = Z T2−Dα0() d(df ) + Z D_α0() d(dg) (2.28)

Usually, you’d say that d(dh) is zero, but here there is a problem at the boundary between the g region and the f region. Stokes theorem results in

2πc1= Z ∂D_α0 dg − Z ∂D_α0 df. (2.29)

section by promising to show that c1 is an integer. For that, the single-valuedness of the

wave functions is important. Without loss of generality, the neighborhood can be chosen

6_{Note that many definitions involve an i. This makes it hard to keep track of what should be real and}

what not. In the definition of |ψi there is an i in front of θ so θ should be real. Consequently the berry phase and A should both be real. In the definition of A there is another i so hn| d |ni should be purely imaginary

7

‘pure gauge’ means that it can be written as the d of something: A = dg. The more mathematical term would be ‘exact’.

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𝜖

f g

𝜕D_𝛼 𝛼0

Figure 2.2.: A schematic overview of the Brillouin zone. The gauge is different in the white region (f ) than in the purple disk (g) around α0.

to be a disk. In the formulas I already assumed this: Dα0() is a disk of size around α0.

The edge of the disk can be parameterized by φ ∈ [0, 2π) and in this parameterization single-valuedness determines that

f (2π) = f (0) + 2πnf,

g(2π) = g(0) + 2πng.

(If these demands are not met the wave function at 2π differs from the wave function at 0, even tough 0 and 2π are the same point.) This shows that c1 is an integer; the

integral in Eq. 2.29 is 2πc1= [g(φ) − f (φ) 2π− [g(φ) − f (φ) 0= 2π(ng− nf). (2.30)

Let’s step back out of the weeds and look at this result. It is of course the same as we had before, but from a different perspective.

• Here, we interpret the Chern number as the number of obstructions to defining a global gauge. In the previous section it was the result of the Gauss-Bonnet theorem.

• I think that a real fiber bundle expert can figure out that this actually the same, but I can’t.

2.2. The quantum Hall effect

The quantum hall effect has many claims to fame. It is used to measure the electron charge to record breaking accuracy [8] and as an example of topology in condensed matter physics. I want to introduce it because it is a setting in which the bulk-boundary correspondence is actually well understood, unlike basically everywhere else.

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fundamental physics8: the fractional quantum hall effect.

In this section I will introduce a Hamiltonian that features the quantum Hall effect, solve it, and show that the quantized conduction is an effect of topological origin.

2.2.1: The point of this section

Strictly speaking, it is not really necessary to understand the quantum Hall effect from the microscopic description to follow the arguments in this thesis. I still wanted to include it here because I think that it is good to have an example in mind to be referred to whenever you think about abstract stuff. In this section I will build up the QHE from a microscopic model and show that it is equivalent to a Chern-insulator. This also sets up the next chapter on Chern-Simons theories because a U (1) Chern-Simons action is an effective action for the QHE system.

2.2.1. QHE from microscopics

The microscopic model for the quantum hall effect is that of electrons in a magnetic field, in two dimensions. The electrons live in the xy-plane, the magnetic field is in the z direction.

H = (p +

e cA)2

2m (2.31)

As is often done in physics texts, I have written this in a suggestive way that makes the following step seem much more logical than if I had written the Hamiltonian in the usual form of H = K + V . Because I did not, it makes sense to interpret

Π = p + e

cA (2.32)

as the momentum operator. The two components of this momentum operator have the following commutation relation.

[Πx, Πy] = −imωc (2.33)

The cyclotron frequency ωc is eB/mc This reminds me of the commutation relation

between x and p. The Hamiltonian also looks like the harmonic oscillator if I write it in another suggestive way.

H = Π2_x+ Π2_y (2.34)

Analogous to the harmonic oscillators, creation and annihilation operators can be de-fined.

a = √ 1 2mωc

(Πx− iΠy) (2.35)

8_{In my opinion: quantum gravity, FQHE, High-T}

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These have the usual commutation relations and Hamiltonian.

[a, a†] = 1 (2.36)

H = ωc(aa†+

1

2) (2.37)

The eigenstates can be written in the x-basis, in the gauge Ay = 0 they are[6]:

ψnk(x, y) = 1 pLy eikye−(x+eBk ) 2_/2eB Hn x + k eB (2.38)

This is a plane wave in the y direction that is localized in x by a Gaussian. Hn is the

n-th Hermite polynomial. It is not important to know that the eigenstates are Hermite polynomials, so feel free to forget that. These wave functions assume periodicity in y: the electrons are in a square of size Lx×Ly with periodic boundary conditions. The different

n denote the different bands; Landau levels. The energy levels are highly degenerate, each k has the same energy eigenvalue. The degeneracy is equal to

deg = BLxLy Φ0

(2.39)

2.2.2. Hall conductance

In this subsection I’ll look at the conductance. It is hard to overstate the importance of periodic boundary conditions; the first part of this section is devoted to constructing those in the quantum hall setting. For that a different definition of translation is needed. Instead of looking at one Lx× Ly square, consider the plane tiled with those squares.

Then the Hamiltonian is invariant under the magnetic translations9,

Taψ(x) = e−i e

2B·(a×x)ψ(x + a), (2.40)

if a is a lattice vector. That is to say a = (nxLx, nyLy, 0) with the n’s integer10. These

magnetic translations have interesting commutation relations. They commute with the Hamiltonian but not among themselves.

TaTb= e 2πiΦ

Φ0_T_b_T_a _(2.41)

I introduced the elementary quantum of flux, Φ0 = 2π_e ; and Φ, the total flux through the

parallelogram spanned by a and b. I showed this by using the Baker-Campbell-Hausdorff formula for operators such that [A, [A, B]] = 0.

eAeB= e[A,B]eBeA (2.42)

9_{The idea of these translations is that instead of translating with the normal generator (}∂ ∂x) you

translate with the canonical momentum, Eq. 2.32

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To use it you need to have the whole operator in exponential form. The translation of the argument of the wave function is generated by the momentum operator. So the translation can be written like this:

Ta= exp h −ie 2 ijk_B iajxk+ ialpl i . (2.43)

The commutation relation of the exponent is

[A, B] =h−ie 2 ijk_B iajxk+ ialpl, −i e 2 ijk_B ibjxk+ iblpl i (2.44) =he 2 ijk_B iajxk, blpl i −he 2 ijk_B ibjxk, alpl i (2.45)

= ieijkBiajbk = eB · (a × b) = 2πi

Φ Φ0

. (2.46)

This completes the proof of Eq. 2.41.

The usual translation operators, pi = i_∂x∂i,commute with each other. If a system

is periodic under discrete translations Bloch’s theorem applies11. So it is reasonable to expect that Bloch’s theorem applies when magnetic translations commute. This happens when the flux through a unit cell is a multiple of the flux quantum Φ0. Let’s assume

that this is the case. That means that the QHE system can be treated with the usual periodic-boundary-conditions tools. I start by introducing Bloch wave functions. The eigenfunctions are written as:

ψ(k,n)= eik·xu(k,n)(x) (2.47)

with k on the Brillouin-zone torus of radiusses 2π/Lx and 2π/Ly. A similar

decompo-sition can be performed on the Hamiltonian, and the part of the Hamiltonian that acts on the u bit is:

H(k) =eik·x−1H eik·x (2.48)

This Bloch decomposition is crucial; because now there is a periodic Brillouin zone and therefore a (possible) Chern number. This Chern number is important because of the following claim. The (average) Hall conductance is equal to the Chern number. Sadly, there is not really a clever way to show this, one just uses the Kubo formula to find the Hall conductance and rewrite it until it looks like this [6].

hσxyi = e2 h i 2π Z BZ ∂un,k ∂kx ∂un,k ∂ky − ∂un,k ∂ky ∂un,k ∂kx dkx∧ dky. (2.49)

The expression in the brackets is the berry curvature

dA = d hn| d |ni = ∂ hn| ∂α1 ∂ |ni ∂α2 −∂ hn| ∂α2 ∂ |ni ∂α1 dα1∧ dα2.

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The integral of this curvature over the Brillouin zone is quantized and so is the Hall conductance.

hσxyi =

e2

hc1 (2.50)

Sometimes it is not the destination but the journey. Not in this case; a pretty boring derivation gives a very important result. Not only shows it that the integer quantum Hall effect is really a topological effect but also that the reverse: the topology of the bands can have real, measurable(!) effects. We also found a system with a tunable Chern number, because the Chern number here is equal to highest filled Landau level (see section 2.2.1). It also proposes a question that will be answered in the next chapter: what is transporting electric charge in this gapped system?

2.2.2: Bloch decomposition–some thoughts

I want to write down some thoughts on Bloch wave functions, as they are used a lot in this thesis and can be confusing. If the Hamiltonian is invariant under discrete translations it is possible to write the eigenfunctions as

ψn,k(x) = eik·xun,k(x) with (2.51a)

un,k(x + a) = un,k(x) (2.51b)

and k in the Brillouin zone defined by the periodic translation symmetry. I tend to think about this as writing in a particular basis:

|ψni = |ki |un,ki . (2.52)

What I mean is that you separate the Hilbert space into different subspaces labeled by k. Those subspaces are much smaller—the dimension is the number of different n—and spanned by the |un,ki states.

From here it is a small step to seeing this as fiber bundles. The Brillouin zone is the base space and there is a subspace for every k, the fiber. If there is only one n the fiber bundle is a U (1) fiber bundle and the fibers are the space of phases. To construct an actual fiber bundle I would need to specify a lot of other stuff, like the projection map and the transition functions, but I hope this picture is clear. Formally, the bundle of states over the Brillouin zone is a vector bundle. The topology of vector bundles has been classified with K-theory, a (complicated) field of mathematics [1].

Another good thing to keep in mind is that the Bloch’s theorem states that such a decomposition is possible, but not all states that are written like eik·xvn,k(x)

will have all the Bloch function properties. A case I encountered in my research was: after some perturbation of the Hamiltonian the original Bloch functions are no longer periodic, even though they can still be used to write the eigenfunctions of the Hamiltonian in a way similar to Eq. 2.51a.

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2.3. There are more invariants

It is good to note that there is more to topology in condensed matter than the Chern number.

• Wilcek-Zee Wilcek and Zee generalized the known U (1) berry phase to U(N) by considering degenerate bands [4].

• Zak phase There is a one-dimensional equivalent of the Berry phase and its called the Zak phase [9]. It is controversial, because it can be ambiguously defined. See [10] or section 4.6.

• Spiral Holonomy For very particular band structures another type of holonomy can be defined: ‘spiral’ holonomy. [11]

• Representation- and torsion Invariants Kruthoff et al. showed that there are other topological invariants of the band structure if you consider the lattice symmetries [12, 1].

• For systems without any crystal symmetries all topological phases (that are robust against impurities) have been classified [13, 14]; for review see [15].

Not much is known about the edge states of these torsion invariants in particular. Be-cause these invariants are (also) classified by K-theory12 _{there is some interesting,but}

hard to understand, research from that point of view, see for instance [16, 17, 18, 19].

2.4. Conclusions

In this chapter I have presented the textbook topological invariant: the Chern number. Although there are many other topological invariant I think that the Chern number and the quantum Hall effect are solid examples of topology in condensed matter that can help shape your intuition.

Of course, none of this would matter if there were no consequences attached to the topological invariants. In this chapter these consequences have been ignored; I have stayed in the safe space of periodic boundary conditions, where translational invariance holds up and Fourier transforms make sense.

In the next chapters I will look at the consequences of topological invariants in the real world, without period boundary conditions!, in the hope of finding out more about the connection between the two worlds. This connection is the bulk-boundary corre-spondence.

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3. From bulk to boundary;

Chern-Simons and the QHE

2+1 dimensions1 are special. It’s the only place where anyons can be found [20]. The Chern number and QHE are 2 dimensional effects. This chapter is about Chern-Simons effective theories and those are 2+1 dimensional, although 5- and 7-dimensional gener-alizations exist.

The reason I want to write about CS is that it is an effective field theory of Chern-type insulators and the integer quantum Hall effect. Versions of Chern-Simons can even describe the behavior of the fractional quantum Hall effect. Even more interesting, in my opinion, is that CS gives a concrete realization of the bulk-boundary correspondence. I’ll explain that in section 3.3. Chern-Simons theories are interesting in their own right and have many application beyond condensed matter. They are examples of topological field theories. I have mentioned those before; topological means here that the partition function (and all the correlation functions) are independent of the metric of the spatial 3-manifold. The only thing CS theories ‘notice’ about the space they live on is the topological information; holes, genus, defects, things like that.

This chapter is built up in the following way. First I’ll give a quick argument of why you’d expect CS to be an effective theory of some topological insulators and introduce the action. The main focus is on the integer quantum hall effect, but sometimes it’s possible to generalize statements to the fractional QHE. Then I’ll present a proof of the fact that CS is an effective field theory for a Chern insulator. Finally I’ll get to Chern-Simons on a manifold with a boundary and extract a concrete formulation of the bulk-boundary correspondence.

3.1. Chern-Simons in action

In our regular 3+1 dimensions the action for electromagnetism is

Z

V×R

d4x 1 2(E

2_{+ B}2_). _(3.1)

The usual method of guesstimating actions works in this case. Under the requirement of gauge invariance, the term in Eq. 3.1 has the smallest possible mass dimension. However,

1

There is a very annoying convention difference between high energy and condensed matter physicists. The high energy community generally includes the temporal dimension while the condensed matter folks don’t. In this thesis I try to use the condensed matter convention, but in this chapter the total number of dimensions makes more sense to use.

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in 2+1 dimensions, there is another possible term, that has an even lower dimension. kCS = k 2π Z Σ×R A ∧ dA (3.2)

A is the electromagnetic gauge potential and k is called the level. What is a gauge transformation? I’d say that

A → A + dφ (3.3)

is a gauge transformation. However, there is some choice in φ, it can either be single- or multi-valued: the wave function does not see the difference between φ and φ + 2π.

eiφψ = ei(φ+2π)ψ (3.4)

The Dirac string thought experiment shows that multi-valued gauge transformations are the natural choice [21]. The multi-valuedness of φ means that we cannot show gauge invariance of the Chern-Simons term by expanding the transformation and recognizing a total derivative.

kCS → kCS + k(dφ ∧ dA) = kCS + d(φ ∧ dA) (3.5)

Because φ is not single-valued the total derivative does not vanish after integration, even though the manifold has no boundary. So what do we accept as being ‘gauge invariant’ ? Just like the wave function, the Lagrangian only appears in exponents. It is reasonable to accept changes to the Lagrangian of the form 2πn. I take Σ = S2 and compactify the time part of the manifold to a circle of circumference β as an example, and look at the consequences of a non single-valued gauge transformation φ(β) = φ(0) + 2πn. Then the integral over the total derivative (Eq. 3.5) is 2πikn, which we can only ignore if kn is an integer. Therefore we demand that k is an integer.

This was a very specific example, but it is true in general[22]. In fact, we can go a lot more general. If we step away from the connection to electrodynamics and only care about consistent, gauge-invariant theories there are two extensions to the action in Eq. 3.2: more fields and non-U(1) fields. For instance I could add a term like

Z

Σ×R

ai∧ dA, (3.6)

with ai another U (1) gauge field. Also possible is

Z

Σ×R

a ∧ da +3

2a ∧ a ∧ a (3.7)

with a not a U (1) but some other group, for instance U(N). For all of these there exists demands similar to the demand that k be integer. The 3₂ comes from those demands, for example. As I said before, these theories are very interesting (one particular type of Chern-Simons is closely related to three dimensional gravity). In section 3.4 I’ll introduce

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a Chern-Simons theory that describes the fractional quantum Hall effect.

3.2. k = c

1

In the last section I argued that the Chern-Simons action is low dimensional and con-sistent (if k is an integer). But does it describe the behavior of a topological insulator? There is an easy test: does it have the same conductivity as the quantum Hall system?

Jx= δS δAx = k 2π xjk_∂ jAk = − k 2πEy (3.8)

The Hall conductivity is then

σxy =

k

2π. (3.9)

This is the right answer; k is even integer just like the Chern number. I would like to present a derivation of ‘level = Chern number’ that is a bit more formal. My main task is then to show that k = c1. It is a bit of a weird task; the Chern number is defined by

the behavior of fermions under adiabatic changes of the Hamiltonian, while the whole idea of an effective theory is to integrate out the fermions. Instead of calculating the Chern number of the quantized Chern-Simons theory, I treat the action semi-classically and analyze the effect that the kCS action has on itself under topological variations. I find a Berry-type phase.

This is the structure of the proof of k = c1. First I’ll argue that varying A1 and A2

results in the same Chern number as varying the momentum, then I’ll investigate the phase from the CS action when A1 and A2 are varied.

I have a choice in the way I vary A and I exploit this by making the choice that simplifies the analysis. I choose to turn on an electromagnetic background potential such that the magnetic field vanishes. This field influences the system by imparting a phase on fermions going around the x1 and x2 directions. (Recall that in this bulk analysis

periodic boundary conditions are imposed, so the real space is a torus with circumferences L1 and L2.) The following is a simple field configuration that accomplishes both.

A1= α1 L1 (3.10) A2= α2 L2 (3.11)

The αi are the phases picked up by fermions completing a cycle in the i direction and

are defined to be between 0 and 2π. In principle you could allow a greater range of α, but a gauge transformation relates α to α + 2π. The α form a torus, which I’ll call Tα.

The effect on the fermions is a shift in momenta:

ki = 2πn Li + α Li (3.12)

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number of states in a momentum band. The Chern number was earlier on defined as: c1 = 1 2π Z BZ tr F (3.13)

Because of the band structure the trace is really just a sum over the Berry curvatures of the different bands.

c1= 1 2π Z BZ X bands Fi (3.14)

The integral over the Brillouin zone can be split up.

c1 = 1 2π X ki=2πni_Li Z Tα X bands Fi (3.15)

Now I change perspective. I look at the α space as if it is the Brillouin zone. Then, the sum over discrete k is not that different from the sum over bands I already had. So I can change the order of sum and integral,

c1= 1 2π Z Tα X ki=2πni_Li X bands F_i, (3.16)

and use the trace to sum over the different k.

c1 = 1 2π Z Tα tr F (3.17)

This shows that the Chern number obtained from varying A is the same as the Chern number from varying k. Now I will turn to the effective theory and investigate what variations in A mean for phases of the field itself.

First, an example of how this is supposed to work. If you consider the action of a test particle coupled to some connection A,

I = 1 2 Z dtgij dxi dt dxj dt + Z A_idx i dt − Z dtV (x) + ..., (3.18)

in the adiabatic regime the action reduces2 to

Z dtAi

dx

dt. (3.19)

The action determines the phase a particle acquires over a path. Equation 3.19 is

2

In the adiabatic regime the time derivatives are very small and the energy of the test particle can be neglected.

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independent of the path taken and can be written like this.

Z p0

p

A_idxi (3.20)

This illustrates that the berry connection can be read from the term with one time derivative. In the CS action this term is

k 4π Z A1∂0A2− k 4π Z A2∂0A1= k 2π Z A1∂0A2dx2dt (3.21)

I fill in the choices for A1 and A2(Eqs. 3.10 & 3.11) and integrate over space to get:

k 2π

Z

α1∂0α2dt (3.22)

From which the Berry connection

D Dα1 , D Dα2 = ∂ ∂α1 , ∂ ∂α2 + ikα1 2π (3.23)

can be read off. This connection has curvature3 _2πk, so the Chern number is

c1 = 1 2π Z Tα k 2π = k (3.24)

This concludes the proof that a U (1) Chern-Simons theory of level c1 is an effective

theory for a topological insulator with Chern number c1.

3.3. Boundary modes

This is where we’ll see the usefulness of the effective theory: it allows us to say something about the boundary modes. In last two chapters I assumed periodic boundary conditions, but no longer. Instead I’ll assume that for a topological insulator with boundaries the effective theory will be: a Chern-Simons theory with integer level k, modified in such a way that it is still gauge invariant. I want to stress that this will not be the same theory as we had before, but that is a good thing; a system with non-periodic boundary conditions is fundamentally different from one with periodic boundary conditions. This work was initiated by X-G Wen [23].

Let’s see what a gauge transformation looks like if the manifold Σ has a boundary.

k 4π Z Σ×R AdA → k 4π Z Σ×R AdA + k 4π Z Σ×R d(f dA) (3.25)

This should be familiar to readers of Eq. 3.5. However in this case I can use Stokes

3 The curvature ish_DαD 1, D Dα2 i

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theorem. δS = k 4π Z ∂Σ×R f dA (3.26)

Oh boy. This contribution never vanishes! The only way out is to change what I mean by gauge invariant. I weaken the demand of gauge invariance to the following. The action should be invariant under

A → A + df (3.27)

with

f |_∂Σ×R= 0. (3.28)

Problem solved! The theory is invariant under these gauge transformations. On the boundary there are field configurations that are now not constrained by gauge invariance, so some degrees of freedom become dynamical (see box 3.3). To study these degrees of freedom I choose A0 = 0 (on the boundary) and use the equation of motion for A0 as

a constraint4. The equations of motions for A0 are dA = 0. Then I express A (on the

boundary) as the exterior derivative of something else.

A = dφ (3.29)

I substitute the A in the Chern-Simons action for dφ.

− k 4π Z Σ×R dφ ∧ ddφ = − k 4π Z Σ×R d(dφ ∧ dφ) (3.30) = − k 4π Z ∂Σ×R dφ ∧ dφ (3.31)

For concreteness I take Σ to be a manifold with as boundary the x-axis. Derivatives with respect to y are not possible on the boundary so the action is:

− k 4π

Z

∂Σ×R

∂tφ∂xφ dxdt (3.32)

This is a boundary mode with velocity zero. That is weird and not really physical5 and I should have picked a different gauge condition (not A0 = 0). If I pick

A0+ vAx = 0 (3.33)

and go through the same procedure, I’ll get

S = − k 4π

Z

dtdx (∂t+ v∂x)φ∂xφ (3.34)

This is the action for a 1 + 1 dimensional chiral boson! I’ll review those in the next

4

You use A0 as Lagrange multiplier.

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section. First I want to comment on the weirdness from the section above.

What you just read was not a derivation. I did not investigate the physical conse-quences of an action. Instead I asked what would be a consistent, physical theory that looks like Chern-Simons with periodic boundary conditions. There is no guarantee that the theory I ended on is actually physically realized. That is why this v parameter enters through choice; in reality it would be determined from the physical properties of the topological insulator. Here there is no reason to pick one v or another. The different boundary modes are not related to each other through a gauge transformation.

Finally, these boundary modes cannot be integrated out; the result would be an effec-tive action for A and that can’t be gauge invariant. It is not possible to reformulate the theory so that it is gauge invariant under transformations that act on the boundary.

These boundary modes are purely a consequence of topology. That implies that a change in the bulk Hamiltonian has no effect on the boundary modes unless the topology changes. This defines the adjective ‘protected’ in ‘protected boundary modes’: the states are robust against topological deformations of the bulk Hamiltonian. Most importantly, the edge modes have to exist throughout the gap; if there is a gap in the spectrum of the edge modes, they can be integrated out. That is not allowed.

3.3.1: Gauge theories from a Hilbert space point of view

One way to view gauge invariance is from a Hilbert space perspective. You imagine a system with states, or field configurations. To say that there is a gauge symmetry is to demand that certain states are actually the same; that there can not be a way of distinguishing these states. The (complete) labeling scheme you had before you demanded gauge invariance is now overcomplete. This is a more modern view of gauge theories championed by people like Witten [24] en X-G Wen[25].

From this Hilbert space point of view I look at what happens at the boundary in Chern-Simons theory. The new requirement on gauge transformations (that they are zero on the boundary) means that less states are identified than with the old gauge transformations. It makes sense that there are now more degrees of freedom on the boundary; the states that were related by a gauge transformation that is no longer allowed.

3.3.1. Chiral bosons

To quantize the action (Eq. 3.34) I need to decide what the position and momentum coordinates are. A choice by [23] is the following. First the action is transformed to momentum space (note that the −k appears because action needs to be real; φ2 means φφ∗). S = m 2π Z dtX k>0 ik ˙φkφ−k− vk2φkφ−k (3.35)

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The level is now m instead of k to reduce confusion with the momentum k. I choose φk

(k > 0) as the position coordinate, so the momentum coordinate is

δS δ ˙φk

= m

2πikφ−k (3.36)

Now the system is quantized by the commutation relation

[φk, πk0] = iδ_kk0. (3.37)

Fortunately the action can be expressed in less awkward variables. Namely

ρ = 1 2π∂xφ (3.38) with [ρk, ρk0] = [ 1 2πikφk, 1 2πik 0_φ k0] (3.39) = i 1 2mπk[φk, m 2πik 0_φ k0] (3.40) = k 2mπδk+k0. (3.41)

The Hamiltonian is then:

H = 2πmvX

k>0

ρ†_kρk (3.42)

This is a fully determined quantum theory of the boundary modes of the quantum Hall system! One thing that might be interesting to see is the effects of tunneling between two edges. I tried to calculate that but I ran into an interesting issue. The theory in this section is an effective theory; meaning that you don’t know what ρ is in terms of electron fields. So it is hard to say what happens if an electron tunnels from one boundary to the other. I did calculate what happens if you allow quanta of ρ to tunnel with a constant coupling, but that can all be absorbed in v.

If you were to guess an interaction based on the expected result then you might construct the following Hamiltonian.

H =X k>0 vρ†_kρk− vR†kRk+ α kρ † kRk+ α∗ k R † kρk (3.43)

This mixes the eigenstates.

vk α

α∗ −vk

(3.44)

With new eigenenergies (to reproduce this, keep in mind the unconventional commuta-tion relacommuta-tions):

±= ±

p

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This results in the expected opening of a gap between the new, mixed edge modes. This is illustrated in Figure 3.1.This kind of behavior seems to occur in systems analyzed in section 4.7.3.

Figure 3.1.: Level repulsion between boundary modes. When two chiral bosons are coupled (with a specific coupling, see Eq. 3.43) a gap opens.

3.4. The fractional quantum Hall effect

I hope I have made clear that Chern-Simons theories are subtle and that many things are complicated to do right. In this section I will be less careful in order to not get lost in details about gauge invariance but still give a taste of the fractional quantum Hall effect.

A theory for the fractional quantum Hall effect seems hard to construct. There has to be a gap, there is no ‘regular’ conductivity, and it has to be gauge invariant. Above, I argued that this would only leave a CS theory for Aµ. Here I will weaken that statement;

that there is a gap does not mean that there can’t be anything, just that whatever there is has to have no degrees of freedom.

Luckily we know a theory that has no local degrees of freedom while still being inter-esting: Chern-Simons. Consider the following action.

S = 1 2π

Z

Ada − m

2ada (3.46)

Here, A is the regular electromagnetic potential, now without CS term, and a is some other U (1) field. In a real material it is presumed that this field is an emergent gauge field. For the same reasons as before, m is an integer.

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to A—as I did in section 3.1. δS δAx = 1 2π xjk_∂ jak (3.47)

I now use the equations of motions to substitute a. The equations of motion say that dA = mda, thanks to a convenient factor of two. A solution for this equation is a = _m1A.

δS δAx = 1 2mπ xjk_∂ jAk= 1 2mπEy (3.48) And correspondingly, σxy = 1 2πm (3.49)

The conductivity is now a fraction. To get more complicated fractions, the action can be extended with multiple ai’s.

S = KIJ 4π Z aIdaJ+ qI 2π Z AdaI (3.50)

This is an effective theory for the fractional quantum Hall effect. It is not a microscopic theory because it is not at all clear how the gauge fields emerge. However it is possible to study the edge states. The arguments I used in section 3.3 carry over, it is just more complicated. Details can be found in [23, 26].

3.5. Conclusions

As I have noted before, this thesis aims to clarify and improve my (and your) under-standing of the bulk-boundary correspondence. How did the construction of edge states from Chern-Simons theory help achieve this goal?

The lore of the bulk-boundary correspondence, that topological invariants of the bulk imply the existence of protected edge states, is mostly based on the quantum Hall effect and its Chern-Simons effective action. Sadly it seems hard to generalize the construction from this chapter. For instance there is no known effective theory for the SSH chain6, a one-dimensional topological system which has boundary modes.

However I think this derivation has been useful to set our expectations and gain some intuition. It gave a definition for ‘protected’: edge states have to ‘connect’ the bands no matter the (topological) deformation of the Hamiltonian.

From this point onwards, this thesis will switch gears and take a more concrete look at edge states.

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4. On the boundary;

the Dresden method

... ...

Figure 4.1.: The bulk system and that same system after a boundary has been established On the left, the bulk system: a periodic chain with interactions between sites. A periodic chain is, in the Dresden method, assigned β = 0. On the right: the interaction between the N th site and the first site has been removed by increasing β to 1. This system has two boundaries. Adapted from [2].

4.1. The big picture

The idea of the bulk-boundary correspondence is that there is a fundamental link between topological invariants in the bulk and states on the boundary. But this link is vague or unknown in basically any system other then two-dimensional Chern insulators or (F)QHE systems. To explore the bulk-boundary in other systems we need new tools to investigate edge states. That is what this chapter is about: a new tool to examine edge states (primarily) in one-dimensional systems.

This method, which I will call the Dresden1 method, works likes this. It is assumed one basically knows everything about the bulk2 system. In particular, the bulk Green’s function is assumed to be known. The poles of a Green’s function correspond to energy eigenstates. This new method can find the existence of poles of the full (bulk and boundary) Green’s function without having to calculate the explicit eigenvalues and eigenstates. A detriment of avoiding explicit eigenvalues is that it is not clear where the edge state are in the k-plane, just that they exist. It is this existence of edge modes that is connected to bulk invariants, according to the bulk-boundary correspondence.

1

It was proposed by a group of physicists from Dresden [2]

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To find edge states I consider ‘portals’ in the band structure and count the poles that move through those portals as I turn on the boundary. These portals are located just inside the gap, as is illustrated in Fig. 4.2. This method was proposed by Rhim, Bardarson and Slager [2].

The goal of this work was to expand on the result of [2] and gain some new insight into the bulk-boundary correspondence. This chapter documents the results of that work.

4.2. The Dresden Method, a detailed look

ε εC εV k β=0 β=1 ε εC εV k β_ii

Figure 4.2.: A sketch of a band structure. There are two bands, separated by a an energy gap. The dashed lines indicate the energy portals v and c, placed just inside the gap. The figure on the right shows a mode that enters the gap at β = βi.

The system consists of:

• a one dimensional chain of finite size with Hamiltonian H1

• boundary conditions, these define the edge

• a bulk Hamiltonian (H0): this is H1 with periodic boundary conditions.

• The bulk Hamiltonian is known and has an energy gap. • There are two ‘portals’, as indicated in Figure 4.2

• The difference between the full Hamiltonian (H1) and the bulk Hamiltonian is the

edge potential Vb. So, H1 = H0+ Vb.

One can imagine this setup as a circular chain of N atoms with some interactions between the sites (Fig. 4.1). The chain is slowly broken by turning on the edge potential. The modes that move (presumably into the gap) in response to the appearance of a boundary are the edge modes. ‘Slowly’ is formalized in the following way.

Hβ = H0+ βVb. (4.1)

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4.2.1. Green’s functions

Each of these Hamiltonians have corresponding Green’s functions which are defined as follows.

Gβ() = ( − Hβ+ i0+)−1. (4.2)

Here, I used the iη prescription to regulate the Green’s function. This Green’s function is an operator on the Hilbert space. A few lines of algebra reveals a handy way to relate the different Green’s functions.

Gβ() = ( − Hβ+ i0+)−1 (4.3)

= ( − H0+ i0+− βVb)−1 (4.4)

= (G−1₀ − βV_b)−1 (4.5)

= G0(1 − βG0Vb)−1 (4.6)

Because the bulk Hamiltonian is gapped, G0 does not have poles in the gap. Therefore,

I ignore G0 and find poles in the gap by figuring out when (1 − βG0Vb)−1 is singular.

This is the case if the ‘poledeterminant’ is zero.

Aβ() = det [1 − βG0()Vb] = 0. (4.7)

Facts!

• A is a polynomial in β with order rank(Vb).3

• This is because 1 − βG0Vb can be represented in a NV dimensional basis. I’ll show

this in section 4.4.1

• A is real-valued even though 1 − βG0()Vb is not. That is because you can

decom-pose the determinant: det(1 − βG0Vb) = det(Vb) det V_b−1− βG0

and use that Vb, Vb−1 and G0 are all Hermitian.

• A is not singular in because G0 is non-singular in the gap.

4.2.2. Tracking poles

To determine the number of edge states it is necessary to gain some more information than the zeros of A. After all, if Aβ(c) = 0, that only tells you that there is a pole at

that portal. The ‘extra information’ is in the derivatives of A.

If a pole moves into the gap from the valence band, going through the portal v, the product of ∂βAβ() and ∂Aβ() is negative. Let me explain. The derivatives denote in

which way the function grows: if ∂βA > 0 then Aβ+δ > Aβ. A zero of A that moves up

3_{The rank of a linear operator is the number of dimensions of the image of the operator. Intuitively, it}

is ’dimension- degeneracy’. Consider a N × N matrix with two values on the diagonal, a11 and a22.

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in energy as β increases can only happen if A+δ < A. Otherwise the increase in A will

not be canceled and A cannot be zero.

A mode coming from the conduction band will have a product of derivatives that is positive. A negative product indicates a mode moving out of the gap. I combine the above into the following counting formula.

M =X

i

piai sign[∂βA∂A]βi,i (4.8)

This formula sums over all zeros, both at the valence portal and the conduction portal. For a zero at the valence window ai = −1 and at the conduction portal ai = 1. The

multiplicity is taken into account by pi. Later, I will put portals in places in the band

structure other than C and V. In general, the rule is that if the product of derivatives is negative the pole moves ‘up’ in the k-plane; if the product is negative the pole moves down.

4.2.3. Expansion to two dimensions

The Dresden method can be used to find edge states of higher dimensional models. The extra momenta that come with higher dimensions can be interpreted as extra parameters of the Hamiltonian. Then, the Dresden method is applied in the same way as before. My study of the charge-density wave is an example of this, I treat φ as some parameter and study the edge state at various values of φ. [2] expands on this. They can determine the chirality of the edge states, although it is not clear to me how that is done, exactly. This is a topic for further study.

4.3. Examples, some go wrong in an enlightening way.

4.3.1. Trivial example

The following example is a bit trivial, but it is good to see the above in action. The bulk Hamiltonian is

N

X

i=1

t |ii hi + 1| + h.c. + δt (−1)i|ii hi| (4.9)

and the boundary potential is

Vb = lim V0→∞

V0|1i h1| . (4.10)

This is a simple hopping model with a staggered potential. Instead of removing one of the links of the chain, here I’ve just made one site inaccessible. A general formula for the Green’s function is:

G0() = X k |ki hk| − k (4.11)

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In the basis4 {|1i} the poledeterminant is easy to write and the determinant is even easier to calculate. A = h1| 1 − βG₀Vb|1i = 1 − βt X k | h1|ki |2 − k = A (4.12)

Now I am supposed to find the zeros of this function. β and t are both positive, so the only way A can be zero is if the sum over the energies is positive. This sum is dominated by its divergent term: the k such that k is very close to either cor v.5 At

the conductance band c< k so the sum is negative and there are no zeros in A. At the

valence band there will be a zero: the sum is positively divergent. Now we now that a mode traverses the portal, but we don’t know which way. The derivatives of A will give us that information. ∂βA = −t X k | h1|ki |2 − k < 0 (4.13) ∂A = βt X k | h1|ki |2 ( − k)2 > 0 (4.14)

The product is negative, so the mode moves into the gap and this system will have one edge mode. This mode is not ’topological’ in any sense of the word; it is always there.

4.3.2. The need for numerics

Looking back at section 1.2.1, one might wonder if this method evades those concerns. To reiterate, the difficulty in finding edge states comes from the Heisenberg uncertainty principle. The k-space description of the bulk is not suited for questions about local phe-nomena, i.e. the edge states. The bulk has, per definition, periodic boundary conditions and this allows for translational invariance of the Hamiltonian. Let’s assume that this is the case, and let’s also assume that the system is noninteracting. Then, the Hamiltonian is diagonal in the k-space basis.

H = X

k∈BZ,n

k,n|k, ni hk, n| (4.15)

The Green’s function is also diagonal in the k-space basis.

G() = X k∈BZ,n |k, ni hk, n| − k,n (4.16) 4

Section 4.4.1 explains why this small basis is sufficient.

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In this example I will leave the regularization implicit. The edge potential is, again per definition, localized in space. Generally, it has the following form.

Vb =

X

i,j∈‘edge’,m

Vij|i, mi hj, m| (4.17)

The ‘edge’ is tautologically defined as all the states that appear in the edge potential. These states form a basis in which I can represent 1 − βGVb. This is explained in

This is not great. The sum over k has no closed-form expression. This is in line with the idea that the Heisenberg principle is a major obstruction to finding edge modes, as I put forth in section 1.2.1.

When the system size N is not too big, numerics offer a way out. Equation 4.19 is still not ideal because it requires adding lots of phases with different amplitudes, which can be difficult in silica. A different approach is to calculate the bulk Green’s function in real space.

4.4. Practical implementation

In order to help the understanding of this new method I will go over some practical details.

4.4.1. Lower-dimensional representation

This subsection is dedicated to the claim that 1 − βG0Vb can be represented in a basis

of dimension rank(Vb). First, G0 is per definition invertible—it is the inverse of ( −

H0+ i0+). The rank of a product of two matrices, A and B is equal to the rank of B if

A is invertible. The rank of a matrix is the column space of that matrix. That means that you can reshuffle the columns to get a matrix with zeros everywhere except the NB

leftmost columns. The determinant is then the determinant of the NB× NB upper-left

part of the matrix. So we get that the rank of βG0Vb is the rank of Vb (NV).

It is probably instructive to walk through an example. Consider the SSH chain6 with 14 sites, it has the following real-space Green’s function at energies near the