Classifications of the 21st century topological phases of matter

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NIVERSITY OF

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MSTERDAM

MSc Physics

Theoretical physics

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Classifications of the

21

st

century

topological phases of matter

by Jorrit Kruthoff 10533958 July 2015 60 ECTS September 2014 - July 2015 Supervisor:

Prof. dr. Jan de Boer

Examiner: dr. Vladimir Gritsev

ABSTRACT

The classification of topological phases has become a very active field of research over the past 15 years. Several classification schemes dealing with or without interactions have been devel-oped from both a high-energy physics and condensed matter perspective. One of the purposes of this thesis is to merge those perspectives. The first classification scheme of topological phases was done for free fermions using K-theory by Kitaev (2009) in [1]. Despite its simpleness, these systems seem not to be realistic, because no crystal symmetry was taken into account. In 2012 Freed and Moore [2] resolved this and found that twisted equivariant K-theory does the classi-fying job instead. To illustrate their findings, we compute some of these K-theory groups and make a prediction for the number of topological phases. When interactions are present, a dif-ferent approach is needed. We use an approach considered by Kapustin in [3] to incorporate crystal symmetry in symmetry-protected topological (SPT) phases. We find that tom Dieck’s classifying spaces is the natural object to consider. Finally, we comment on the use of gauging of a global symmetry to classify bosonic topological phases which have a cancellable ’t Hooft anomaly. We consider various examples of both unitary and antiunitary symmetries and show that a wonderful bulk-boundary correspondence emerges from this viewpoint.

Populair wetenschappelijke samenvatting

Als we in de natuurkunde experimenten doen en natuurkundige fenomenen waarnemen, dan proberen we altijd te kijken naar andere situaties waarin dezelfde fenomenen optreden en hoe verschillende fenomenen in elkaar kunnen veranderen. Neem bijvoorbeeld het volgende ex-periment. Hang een spijker aan een magneet en warm deze op. Bij een bepaalde temperatuur zul je dan zien dat de spijker van de magneet loslaat. Dit kun je als volgt verklaren. Het ijzer in de spijker zorgt voor een ingebouwd magneetveld, waardoor de spijker aan de magneet blijft hangen. Bij een te hoge temperatuur verdwijnt dit ingebouwde magneetveld en laat de spijker los van de magneet. De temperatuur waarbij dit gebeurt heet de Curie-temperatuur en voor ijzer is deze ongeveer 770 ◦C. In de natuurkunde noemen we een materiaal met zo’n inge-bouwd magneetveld een ferromagneet en materialen zonder die eigenschap een paramagneet. IJzer is dus een ferromagneet voor temperaturen onder 770 graden celcius en een paramagneet voor temperaturen boven 770 graden celsius. In de natuurkunde noemen we dit fasen. IJzer kan dus in twee verschillende fasen zitten; een ferromagnetische fase of een paramagnetische fase. De overgang van de ene naar de andere fase, als gevolg van een verhoging van de temper-atuur bijvoorbeeld, noemen we een fasetransitie. Aan de hand van deze fasetransities kunnen we dus zeggen of een materiaal zich dan wel in een ferromagnetische fase of in een paramag-netische fase bevind. Nu hebben we hier vooral gekeken naar ferro- en paramagparamag-netische fase, maar op exact dezelfde wijze kun je de ook de fasen van water beschrijven.

Zoals je hierboven hebt kunnen lezen gaan fasen van materie samen met een natuurkundig fenomeen of fenomenen die die fasen karakteriseren. Hoewel we hierboven alleen naar fer-romagnetisme en paramagnetisme gekeken hebben, geldt deze uitspraak veel algemener. Om dus te weten te komen wat de natuur allemaal in petto heeft, is het belangrijk om erachter te komen welke verschillende fasen er in de natuur voor kunnen komen. Het antwoord op deze vraag zal niet betekenen dat de natuur is ’opgelost’; er zullen altijd fenomenen zijn die niet direct gerelateerd zijn aan een fase van een materiaal. Daarnaast is het natuurlijk onmogelijk om alle fasen op te noemen zonder te specificeren welke materialen je beschouwt of welke symmetrie¨en er aanwezig zijn. In deze scriptie zullen we ons daarom vooral bezig houden met isolatoren; materialen die geen electriciteit geleiden. Daarnaast zullen we vooral ge¨ınteresseerd zijn in robust fasen van isolatoren en niet in fasen die gemakkelijk verstoord kunnen worden door invloeden van buitenaf, zoals temperatuur. We zullen bijvoorbeeld niet kijken naar vaste, vloeibare of gasvormige isolatoren, maar eerder naar fasen in de trant van het voorbeeld hier-boven. Deze robuste fasen van isolatoren worden topologische fasen genoemd.

Aan de hand van deze aannames willen we weten welke fasen er allemaal bestaan in de natuur. Het opsommen van al die mogelijke fasen, gegeven de aannames, noemen we klassi-ficeren. Concluderend kunnen we dus stellen dat we topologische fasen van isolatoren willen klassificeren.

In de afgelopen 15 jaar zijn er onder een aantal extra aannames over bijvoorbeeld de sym-metrie¨en, topologische fasen van verschillende typen isolatoren gevonden en geklassificeerd. Door middel van geavanceerde wiskunde kun je eenvoudig bepalen welke topologische fasen onder die extra aannames mogelijk zijn. We zullen in deze populaire samenvatting niet dieper op het klassificeren zelf ingaan, maar een karakteristiek natuurkundige fenomeen behandelen welke in topologische fasen van isolatoren veelvuldig voorkoment. Dit fenomeen wordt het quantum Hall effect genoemd.

Het quantum Hall effect werd begin jaren 80 van de vorige eeuw door Klaus von Klitzing experimenteel ontdekt en daarmee was ook de eerste topologische fase gevonden! Om zijn ontdekking goed te kunnen begrijpen, moeten we eerste het gewone Hall effect toelichten. Dit effect is het volgende. Stel je neemt een dunne tweedimensionale plaat materiaal en je plaatst deze in een homogeen magneetveld zodanig dat de plaat loodrecht op de veldlijnen van het magneetveld staat. Als je nu over twee overstaande zijden een stroombron aansluit, dan zul je een spanningsverschil meten over de overige twee overstaande zijden. Iets precieser kunnen we zeggen dat een stroom in de x-richting zorgt voor een spanningsverschil in de y-richting. Dit wordt het Hall effect genoemd en kan, voor kleine magneetveldsterkte, met de Lorentzkracht verklaard worden en resulteert in de volgende vergelijking

Jx =σxyEy (1)

waarbij Jx de stroom in de x-richting, σxy de conductiviteit, dus hoe goed electriciteit geleidt,

en Eyhet electrisch veld dat onstaat door het spanningsverschil. Voor hoge

magneetveldsterk-ten, echter, gaat dit niet meer, omdat het systeem dan quantum mechanisch beschreven moet worden. Het anaologe effect treedt nog steeds op, maar wordt dan het quantum Hall effect genoemd. Het wonderlijke is dat vergelijking 1 ook nog steeds blijft gelden. We begrijpen nu wat het (quantum) Hall effect is en met deze kennis kunnen we bespreken wat Klaus von Klitzing zag in zijn experiment. Hij zag dat de σxy in vergelijking1gekwantiseerd was. Dit

betekent dat de conductiviteit alleen bepaalde waarden aan kan nemen en niet de waarden daartussen. Bovendien liggen die waarden allemaal even ver uit elkaar. Hij zag dat de conduc-tiviteit gegeven werd door

σxy =

e2

hn (2)

waarbij e de electronlading, h de constante van Planck en n een geheel getal. Met gekwan-tiseerd bedoelen we dus dat n niet 1.5 kan zijn, maar enkel 1, 2, 3, etc. Zulke gekwangekwan-tiseerde grootheden zijn heel karakteristiek voor topologische fasen en zijn sterk gerelateerd aan het feit dat het zulke robuste fasen zijn. Alleen sprongen van de ene naar de andere waarde zijn

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mogelijk; er zal geen geleidelijke overgang tussen die waarden plaatsvinden.

In het begin zeiden we dat we ons wilden focussen op isolatoren en dus materialen die geen stroom geleiden, maar toch praten we hier over het geleiden van stroom! Wat gebeurt er? In het quantum Hall systeem is de binnenkant, dus de binnenkant van de twee dimensionale plaat, nog steeds isolerend, maar het is de rand van het materiaal die geleidt. Hoewel het moeilijk is om uit te leggen wat er exact gebeurt, aangezien het quantum Hall toestand quantum mecha-nisch is, is het deze geleiding van de rand die ervoor zorgt dat we over conductiviteit kunnen praten.

Ga je nu terug naar de resultaten van de geadvanceerde wiskunde, dan vind je dat in twee dimensies en zonder extra symmetrie¨en, de quantum Hall fasen de enige fasen in de natuur zijn. Quantum Hall toestanden met verschillende n worden als verschillende fasen beschouwt. Het gehele getal n vertelt je dus met welke fase je te maken hebt en klassificeert topologische fase van isolatoren in twee dimensies. Hoewel dit het eerste voorbeeld van een topologis-che fase was, hadden natuurkundigen al snel door dat er een hele nieuwe wereld voor hen open ging. In 1980 is de deur naar deze wereld voor het eerst open gedaan, maar sinds 15 jaar beginnen we haar te betreden en de wild zoo van topologische fasen te ondekken. Er zijn verscheidene nieuwe fasen ontdekt en nieuwe klassificaties worden gevonden, waarbij steeds exotischere fasen van isolatoren worden voorspeld.

Met het quantum Hall effect hebben we proberen duidelijk te maken wat topologische fasen zijn en wat hun karakteriseert/klassificeert. Daarnaast hebben we ook besproken wat fasen in het algemeen zijn en dat deze met hele mooie natuurkunde gepaard kunnen gaan. Maar wat kun je met deze topologische fasen doen? Welke toepassing hebben deze fasen? Dankzij het feit dat topologische fasen heel robust zijn, zijn ze uitermate geschikt om te dienen in quantum computers. Dit nieuwe type computer heeft enorm veel meer rekenkracht dan de huidige computer en kan voor heel veel wetenschappen een uitkomst bieden. Het nadeel van dit type computers heeft erg kwetsbare bits (in een quantum computer heten deze qubits) en met de robustheid van topologische fasen kunnen ze een stuk minder kwetsbaar gemaakt worden.

Het hierboven besproken voorbeeld maakt ook direct duidelijk dat de natuur vol zit met fascinerende fenomenen die dusdanig intrigerend lijken dat je ze haast niet zou geloven. Daarom is het ook geweldig om deze fasen nader te onderzoeken en te bekijken welke wonderen er nog meer in de natuur te vinden zijn!

Acknowledgement IX

Preface XI

1 Topological phases 1

1.1 Why? . . . 1

1.2 To classify, or not to classify... . . 2

1.3 What is a phase?. . . 3

1.4 What are topological phases? . . . 6

1.5 Prototypical physics and host materials . . . 13

1.6 Explicit representatives: free theories . . . 20

1.7 Crash course in crystallographic groups . . . 27

1.8 Topological field theories . . . 30

1.9 Introduction to this thesis . . . 35

2 K-theory classification 39 2.1 K-theory `a la Kitaev: no crystal symmetry . . . 40

2.2 Classification with crystal symmetry . . . 46

2.3 Assumptions in K-theory classifications . . . 54

2.4 Explicit computation of Kν G(T2) . . . 56

2.5 Probing phases with crystal symmetry . . . 66

2.6 Summary . . . 71

3 Classification of interacting phases 73 3.1 Classifying interacting phases of gapped matter . . . 73

3.2 Bosonic topological phases. . . 85

3.3 Fermionic topological phases . . . 98

3.4 Overview of other classifications . . . 102

3.5 How to incorporate crystal symmetry? . . . 106

3.6 Summary . . . 112 VII

Contents VIII

4 Gauging of global symmetries 113

4.1 Gauging a symmetry . . . 113

4.2 Examples and relation with condensed matter . . . 125

4.3 General comments on gauging a global symmetry . . . 135

4.4 Summary . . . 135

5 Conclusion and discussion 137 5.1 Conclusion . . . 137

5.2 Discussion and outlook . . . 138

Bibliography 143 Appendices 153 A An introduction to twisted equivariant K-theory 153 A.1 Introduction in K-theory . . . 153

A.2 Twisted equivariant K-theory . . . 168

B An introduction to group cohomology 173 B.1 Introduction to group cohomology . . . 173

B.2 Relation to topology . . . 178

During my master project I had a lot of support from various people. First of all I want to thank Jan for having so much time during the discussions we had over the past 10 months. We started with a study of Wen’s papers, but after a while we gave up on him and turned to more high-energy physics related attempts to the classification of topological phases. For me, this was a real relief, because a much more cleaner classification was used there. Immediately from the start I also benefited from discussions with Vladimir. His view on condensed matter is a lot different from that of Jan and really helped making a bridge between classifications set up by high-energy physicists and condensed matter theorists.

Furthermore, I really enjoyed my time in the Master room with all other theoretical physics masters. During discussions with Vincent, Jonas and Giulio, I understood some things from a more refreshing point of view. Besides all the physics, I have also learned a great deal of Italian this year due to the efforts of Giulio, Dave and Bernardo.

Classification has always been a prime research area in mathematics. Numerous results about classifying manifolds, Lie algebras or sporadic groups have been obtained. However, in gen-eral, physics is about understanding nature and a priori not so much about classifications. Var-ious assumptions have to be made in order to start a classification in physics and it is therefore not as conclusive as it can be in mathematics. Nevertheless, classifications in physics have risen recently in various contexts. The most notable work in this area is the classification of unitary minimal models in 1+1 dimensions. These theories are conformal field theories with a finite number of primary fields, [4]. Another example is the work of Witten in which he classifies D-brane charges using Klassen-theory or K-theory for short [5].

In this essay, however, I want to focus on yet another and more recent attempt to classify physical phenomena, namely the classification of topological phases of matter. The basic ques-tions are of course what is being classified under what assumpques-tions and how can these phases be probed. Despite the fact that this branch of physics is more condensed matter oriented, we will see that a lot of connections with other elegant and fascinating areas of both physics and mathematics can readily be made.

K-theory

Ever since the development of Laundau-Ginzburg theory, physicists have been trying to under-stand phases of matter. Numerous methods to distinguish two phases have been developed over the past decades and have led to a better understanding of second-order phase transition. This changed, however, with the discovery of the quantum Hall effect. Such a phase of matter does not break any symmetry and is thus outside the realm of Landau-Ginzburg theory as this theory only deals with spontaneous symmetry-breaking.

The understanding of such quantum Hall phases or at least the nature thereof became clearer by the work of Thouless, et al [6]. They showed that a quantity that characterises the topology of the space of states, the first Chern number, is directly related to the Hall conductiv-ity. This revealed a strong interplay between the conductivity and the topology of a quantum Hall phase. Moreover it provided the first example of a topological phase. In contrast with the conventional second-order phase transition theory a topological phase will have a change in its topology but not its symmetry during a phase transition. Another novelty of these phases are the edge states which arise when the system is put on a manifold with boundary. This prop-erty is also shared by the quantum Hall phase. It has chiral gapless modes on the boundary.

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Moreover, a close connection between nonlinear sigma models with a target space associated to the global symmetry and Anderson localisation showed that these edge states evade Anderson localisation, [7,8]. This means that even in the presence of disorder the edge states are stable. However, if one breaks the global symmetry, for example in a time-reversal invariant system by adding a magnetic impurity, this rigidity is broken and the topological phase can be adia-batically connected to the topologically trivial phase. Thus, under every symmetry respecting perturbation, including disorder, the edge states are stable.

One of the basic questions one can have when talking about phases of matter or phases in general is that given a set of input data, what kind and how many phases are possible. The Landau-Ginzburg theory provides a reasonable good answer to this question. Beyond this description, so beyond spontaneous symmetry-breaking, no clear answer exists. However, in certain regimes beautiful results and classifications have been obtained. Most notably is the case of gapped free fermions. In [1], Kitaev showed under a certain notion of deformation, the classification of phases of gapped free fermions leads to the notion of K-theory of the Brillouin zone. This K-theory is a Abelian group and each element in this group signifies a different phase.

This classification scheme for free fermions was initiated by a paper Hoˇrava wrote in 2005 [9]. In this paper he states that the stability of gapless excitations is intimately related to K-theory and that only some of these excitations are stable. Alexei Kitaev took this idea a bit further and added elementary symmetries such as time-reversal, charge-conjugation and chi-rality to the description. By packaging these symmetries in a Clifford algebra he was able to find a classification of free fermions now referred to as the ten-fold way [1]. With this classifica-tion a lot phases could be identified. Most notably the integer quantum Hall effect and the Z2

topological insulator, which is in fact a time-reversal invariant system.

Topological field theory

To go beyond the free theory, we have to take a few steps back and look more closely at the assumptions. Especially, the gappedness of a system is of crucial importance, since only then we know that in general an effective action with local Lagrangian can be found. At long dis-tances the physics of any gapped system is believed to be governed by a topological quantum field theory (TQFT). The reason is quite simple. After applying renormalisation group flow, we constantly integrate massive degrees of freedom out and so for energies below the gap the theory has no dynamics. It is topological in the infrared. This means that the theory only de-scribes the ground states and so only a degeneracy or a manifold with boundary can introduce non-trivialities. The degeneracy introduces different topological sectors in the theory, because particles can be constructed from exciting different ground states. Due to the gap, these parti-cles cannot interact locally, but can interact topologically, [10]. Roughly, this means that when two particles wind around each other, the wavefunction acquires a Aharonov-Bohm phase. Usually, the ground state degeneracy is finite and thus the interactions between the different topological sectors is governed by a finite number of rules. These rules are called the braid-ing rules and the particles in the topological sectors are called anyons. A notable example of

these topological interactions occurs in three dimensions. This is a prototype for the theory of anyons and describes the low energy limit of the fractional quantum Hall phase; the interacting counterpart of the ordinary quantum Hall phase.

However, there are some important objects that we are forgetting here: non-local operators. Usually when thinking of a general quantum field theory, we think of local operators and their correlation functions. It was long thought that this was the only data needed to fully describe the theory, but this turns out to be wrong. Non-local operators such as Wilson lines or their duals ’t Hooft lines also give valuable information about the theory in question. In the late 1980s, Witten considered a Chern-Simons theory and he showed that a very rich and beautiful structure arises when one considers correlation functions of Wilson loops in such a theory. Even earlier Gerard ’t Hooft and Kenneth Wilson used non-local operators and the scaling behaviour of their expectation value to determine whether an SU(N)gauge theory is in a (de)confining or Higgs phase. These operators may therefore serve as an order parameter, although not in the sense of Landau-Ginzburg which uses local operators. In the classification of topological phases, these non-local operators are thus the prime objects to look for in order to probe the interesting features of the phase.

Gauging of global symmetries and interacting topological phases

As was shown recently by Seiberg et al. [11, 12], gauging a global symmetry in the UV can result in different theories in the infrared. When gauging a global symmetry, a gauge field for that particular symmetry is turned on. This procedure might succeed and result in a consistent theory, but when this is not the case, the theory is anomalous and has an ’t Hooft anomaly. In order to get a consistent theory this anomaly must be cancelled. There are several ways of doing so, but for now we will restrict our attention to the following. Suppose that in a d-dimensional boundary theory we want to gauge a global symmetry and that we end up with an anomalous theory. The projective realisation of such a global symmetry provides a possibility of such anomaly, [13,14]. In order to get rid of the anomaly, we use a bulk theory in d+1 dimensions, which has the same anomaly on its boundary as the original theory. The composite theory is thus a consistent and gauge invariant theory. In more mathematical terms, this procedure can be recasted in terms of cohomology and group extensions. The upshot is that the theories in one dimension higher that can cancel the anomaly are Dijkgraaf-Witten theories [15], which are classified by group cohomology; Hd+1(BG, U(1)). Each element in Hd+1(BG, U(1))thus gives a theory in d+1 dimensions with a gauged global symmetry G to make the whole system (boundary and bulk) anomaly free.

The beautiful and unexpected observation is that these cancellable ’t Hooft anomalies are in fact related to topological phases of interacting bosons. Wen et al.[16] have proposed a classification scheme for interacting bosons via group cohomology. In low dimensions (d =

2, 3, 4) and for unitary symmetries this is all there is to it. Moroever, in that case, it also gives the very natural explanation that symmetries at the boundary must be realised projectively in order to give rise to a non-trivial topological phase in the bulk. Only then the possibility of having an ’t Hooft anomaly opens [13,14].

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A unifying description: cobordism groups

As argued above, the classification of interacting bosons via group cohomology is not complete. There are phases of matter protected by time-reversal symmetry that are not captured by this classification. In order to capture those, we have to start with a few basic assumptions: the system has a gap, local Lagrangian and no degenerate ground states on a spatial slice.

Suppose you have a gapped system with a internal global symmetry and you want to probe the topological phase and determine the response of the system. The system can have different responses, but when the system undergoes a phase transition the response changes discontin-uously. So by determining the possible responses that cannot be continuously deformed into each other different phases can be identified and classified. In order to determine the response, a background gauge field for the internal global symmetry is turned on and the massive de-grees of freedom are integrated out so as to probe topological phases. By doing so an effective action is obtained and determines the possible responses of the system. The different phases are thus the different effective actions you can write down up to deformations. The data needed to write down such an action is the manifold your system is living on and the possible inequiv-alent background gauge fields for the global symmetry. This input data up to deformations turns out to describe the mathematical structure known as cobordism groups. Kapustin et al. have used these groups to find all possible topological phases under the assumptions described above [3,17].

This rather elegant classification scheme has several advantages. Firstly, bosons, fermions and anti-unitary symmetries (like time-reversal symmetry) can be described by one unifying description. Each case has its own cobordism group. For example, we can consider unoriented (or even unorientable) manifolds to describe time-reversal invariant systems and the topolog-ical phases are classified by the unoriented cobordism group. For fermions we simply restrict to manifolds with spin structure and the spin cobordism group gives us the required classifi-cation. Secondly, the assumptions and approach are very clean and offers clear sight of what is classified and how.

However, there are also disadvantages which make it difficult to develop intuition and to understand the theories conceptually. The main problem is that cobordism groups are very tedious to compute in general. One has to resort to certain mathematical constructions known as spectral sequences, which make the calculation possible but still hard. From a more physical point of view, the other disadvantage is that the actions one can construct are integrals over some cohomology class and offer very little physical intuition and make interpretation of the theory difficult. Ultimately one would like to relate these cohomology classes to physical data, just as Thouless et al. related the first Chern number to the Hall conductivity.

In the past decade, several attempts to classify topological phases have been developed and proven to be successful. For describing topological phases of free fermionic theories, K-theory turned out to be the ideal candidate and by now the structure of that classification is well understood. Despite this success, when interactions are turned on, it is not clear how to proceed. Many good and promising schemes have been developed, some more successful

than other. The group cohomology approach offers an elegant relation with gauging of global symmetries, but with the use of cobordism groups a more complete classification is realised. Moreover, it is not clear what representatives of the many exotic phases that arose from these approaches are actually physical and realisable. Further research needs to be done to clarify these questions and ultimately capture topological phases of both interacting fermions and bosons.

1

Topological phases

1.1 Why?

Topological phases of matter are robust phases of matter. They host some of Nature’s most fascinating physical phenomena, such as quantum Hall physics. Within these phases one can distinguish two classes. One class is such that no matter how the system is perturbed, the phase will not be destroyed. The other requires the perturbations to respect the symmetries in order to keep the phase intact. According to the symmetries these phases have, different phases can be constructed and due to their robustness they can also be classified. Most of this thesis will be concerned with this classification. We will consider both phases of interacting and free matter and find several ways to classify them. Before we can actually talk about those classifications and about topological phases, it is good to motivate why we are interested in this problem.

First of all, the physics of these phases is fascinating. For a theoretical physicist this is one of the prime motivations. The physics of topological phases spans a lot of branches of physics, ranging from plain vanilla construction of Hamiltonians to gauging of global symmetries and their anomalies. Although most of the topological phases are described by a low-energy ef-fective field theory, high energy techniques such as ’t Hooft anomalies and topological field theories naturally arise. However, this also poses another problem, because the language of high energy physicists differs a lot from that of a condensed matter theorist. A good bridge between those to languages is imperative for the understanding of topological phases. In this thesis we have therefore tried to make a first step towards such a bridge by comparing proto-typical models from classifications set up by high-energy theorists and models that exist in the condensed matter literature.

Second, classifications are always done by some fancy mathematical framework and are therefore amusing for the more mathematically inclined physicist. Various mathematical con-structions are used to preform classifications and in this way some old mathematical ideas find new meaning in topological phases. For a physicist it is a challenge to use the abstract mathe-matical results to say something about the real world and couple it to physical intuition. In this thesis we will try to do so and postpone any mathematical precise construction to the appen-dices. In this way, the reader will make him/herself comfortable with the language, without going through all mathematical theorems.

Third, topological phases form a good basis for quantum computers. Due to the existence 1

To classify, or not to classify... 2

of excitations with fractional statistics, called anyons, in phases with topological order (which we define below), information can be stored non-locally. Using the braiding and fusion prop-erties of these excitations, this information can again be extracted. This was first spelled out by Alexei Kitaev in [18]; a name we will encounter more in this thesis. Anyons can live in 2+1 dimensions or on the surface of a 3+1 dimensional materials and so when constructing a topological quantum computer one must know how to get these excitations. The classifica-tion of topological phases gives this informaclassifica-tion. It tells you what type of bulk or symmetry is needed for anyons to emerge. Nevertheless, we will not discuss this application in this thesis, but it is good to keep it in the back of your mind when reflecting on the abstract nature of these classifications.

1.2 To classify, or not to classify...

Classifications are a fairly old problem in science. We have always been intrigued by the objects that look the same and share common properties. To set up a classification it is essential to agree on what ”look the same” means and what properties are considered. The latter are the assumptions you make in a classification, it defines the category of objects that you consider. The former is needed to compare the objects within this category and enumerate classes of objects that are considered the same. Let us illustrate this with an example.

One could say, OK, I now want to classify cubes. No other information is specified. Ac-cording to your perception objects that are cubes have equal sides and a certain orientation. You try to classify all such objects and you come up with some list. Maybe you can even do some predictions of what cubes might also exist. On some day after you completed your list, a friend of yours comes along and shows you a cube. This is a cube according to your friend’s perception, but not according to yours. It has a different orientation, hence you do not include it in your list. This poses of course a problem. If your friend also wants to classify cubes, he or she will come up with a different list. Both you and your friend are thus not classifying cubes according to a general perception or agreement on what a cube is, but rather through what they think a cube is. To circumvent this problem, you and your friend should agree on what a cube is and what its key properties are. Extra information must thus always be specified and this extra informations are of course the assumptions in the classification.

In the classification of topological phases we should do exactly the same. We should agree upon a common definition of a phase and what we mean with topological. Further more, when extra properties of a system are assumed, such as a symmetry, we should also agree upon what we mean with these symmetries and how they act. When we agree upon such definitions we should never deviate from them as this will create unforeseeable confusion and no one will understand the proposed classification.

In the next few subsections of this chapter, we will define what we mean with a phase and what is meant with topological. It will be of utmost importance to make a distinction between all the different types of phases there exist in the literature, but that are not as clearly distin-guished there as one would hope. Furthermore, in this thesis, we will try to emphasise what

the assumptions within each classification scheme are an why they are used. Before we can agree upon all those definitions, we comment a bit on the general setting.

We will be interested in condensed matter systems with a gap. The fact that we are inter-ested in condensed matter systems is not so exciting, but the fact that we want a gap is! What we exactly mean with a gap will be made clear below. In low dimensions, these systems can, in principle, be constructed in a lab and via experiments the theoretical predictions of these material can be verified. The low dimensionality will not be a restriction of our classification, but rather a restriction on what is physically interesting. In reality, realising of these systems might be a very difficult job or even impossible. The reason is that, although, these systems are composed of electrons, materials with a specific type of emergent physics simply do not exist or might not be possible to engineer.

Furthermore, condensed matter systems are usually defined on a lattice. This lattice has a number of symmetries such as translational and rotational symmetry, which greatly affects the system under consideration. It is therefore important to take these symmetries into account as well. To do so, we will try to incorporate space group symmetry associated to the crystal. After we have agreed upon incorporating space group symmetries, we can take a long distance approximation and describe the system as being defined in the continuum with an the space group acting on the continuum.

1.3 What is a phase?

In this thesis we will consider two types of phases, which we will call ordinary and extraordinary phases. This first one is characterised by the fact that at a phase transition, the gap closes. For the second type this is not the case and the gap remains open, however, still no continuous (adiabatic) deformation (we will define this notion below) between the two phases is possible. Before discussion these to phases, we must first clarify what we mean with a gap.

What is a gap?

Simply put, a gap in a system is a region in the energy for which no states exist. This means that for some energy E inside the gap with size∆, there are no solutions to

H_|ψi =E|ψi. (1.1)

Of course, as is known from elementary quantum mechanics, all systems defined in some finite volume exhibit a gap. This gap is not only present between the ground state and the first excited states, but also between the first and second excited states and so on. To cure this issue, we will define a system with a gap as follows. A system with a gap is a system for which the gap persists to exist in the large volume or thermodynamic limit. In simple condensed matter systems this means that insulators and semi-conductors are systems with a gap and their valance and conductance bands form a continuum of states. These systems are also known as incompressible systems, since compression will force states to cross the gap which costs a lot

What is a phase? 4

of energy. Although this might seem a crude definition, since no system is defined in a infinite volume, for us this approximation will make perfect sense, because we are always interested in low energy behaviour anyway. This will be clearer when we discuss actual classification schemes.

Now that the notion of a gap is clear, let us move on to the main discussion of this section.

Ordinary phases

Everyone knows the classical example of a phase, namely in the Ising model. In one phase all spins are oriented randomly up and down and in the other phase spins are all parallel to each other. To distinguish these phases by some object in the theory, the order parameter is introduced. In this case it is simply the magnetisation, which is zero in one phase and non-zero in the other. As this phase is driven by an external magnetic field1, there must be some field strength at which the Ising model decides to align with the external magnetic field. This point is called a critical point. At this point the order parameter makes a jump and the susceptibility diverges. This is thus a perfect way to make sense of a phase and a phase transition. Another feature we will be needing later is the fact that at the critical point the correlation length di-verges and no obvious scale exists anymore. This will mean that the gap, which exists in both phases will close.

Despite the simplicity of this example, it is still a bit too specific, but gives a good flavour of the situation. Let us step back a bit and discuss what we really mean with a phase. A system is always in some phase, so let us assume that the system is in phase A and that our system is characterised by a number of parameters gi in a region A, a gapped Hamiltonian H(gi)and

a Hilbert space H . These parameters can be coupling constants or external parameters such as temperature. Now imagine that we tune the parameters gi in a continuous fashion, so we

find ourselves some one-parameter family of systems_{gi(t), H(gi(t), t),H(t)}. This is a path

in parameter space and by in a continuous fashion we simple mean adiabatically with respect to the size of the gap. Besides changing the the Hamiltonian H(g_i(t), t)adiabatically, we also track the adiabatic change of states, hence we have writtenH(t). If we now move along this path two things can happen. First of all, nothing can happen and we still remain in phase A. Second, by changing the parameter t slightly, the gap of the Hamiltonian will close and after further increase in t the system will be in another phase, which we call phase B. This phase is defined in a region B of parameter space. The system thus underwent a phase transition in region A_∩B= S and it is now obvious how to define a phase.

Extraordinary phases

In general we can view S just as some subspace of parameter space that defines a boundary between two phases which cannot be crossed by a continuous path. In the discussion above we saw how a phase is defined when the gap closes at S. For extraordinary phases this does 1_{Temperature is also an external parameter that can drive this system, but for now we restrict to an external}

not happen. There is another property of both phases between which one cannot continu-ously interpolate: the number of ground states. If the number of ground states changes the system underwent a phase transition. Examples are topological paramagnets or insulators of interacting bosons in d=3+1 dimensions, which have a gapped quantum spin liquid at their boundary. Why a phase transition occurs at the boundary of the system will be discussed when we consider boundary physics in the next section. In chapter3we will elaborate more on these phases. We will see that this naturally follows when we discuss the relation between topologi-cal phases and topologitopologi-cal field theories. For clarity, let us put this and the above discussion in a definition

Definition 1.1. (Phase) A phase is a path connected region in parameter space such that the Hamiltonian stays gapped. Different phases are connected through phase transitions, which are regions S in parameter space for which cannot be crossed with a continuous path. We distinguish two types of phase transitions

1. (Ordinary phase) The gap closes at S,

2. (Extraordinary phase) The gap does not close at S.

To get a better feeling for this definition, we have draw the following figure.

(

)

(

)

A

B

S

In this figure, we consider the two phases A and B, which are defined on regions A and B in parameter space. Within each phase a Hamiltonian and Hilbert space is present. The two phases are connected via S at which we have a phase transition. The blue paths represent continuous deformations of some initial pair(Hi,Hi)to some final(Hf,Hf). These paths are

therefore within one phase, they cannot enter another phase. This does happen for the red curve. This curve is not continuous, because it crosses the region S.

This definition will be enough for our purposes, but for the more mathematical inclined reader we refer to [2].

How to distinguish phases?

In the above discussion we have defined what a phase is an what separates two phases. How-ever, this does not help us distinguish two phases. In order to do so we need an operator

What are topological phases? 6

which has a different expectation value within each phase and thus jumps when going from one phase to the other. Such an operator is called an order parameter. In the Ising model, the or-der parameter is the average magnetization_hM_i. We will see below that for topological phases, this does not apply any more and we need to do something different. Basically it boils down to the fact that an order parameter is the expectation value of some local operator, but topology is characterised by global features, hence a local operator will not capture topological features. This gives already a hint of what a topological phases is, the subject of our next section.

1.4 What are topological phases?

Loosely speaking, a topological phase is a phase of matter which is characterised by the topol-ogy associated to the model describing the phase. We distinguish between free and interacting theories.

1.4.1 Free theories

We saw that in ordinary phases certain expectation values of operators change in the theory when going from one phase to the other. For topological phases this is not the case. Passing from one topological phase to the other will instead change the topology of the system. In order to define what a change in topology means, let us first discuss what a general system characterizes. A general condensed matter system in d spatial dimensions is defined on either a lattice or a continuum. This means that Brillouin zone (BZd) is either a torus or sphere, respectively, in d spatial dimensions. For each k in BZdthere exists a gapped Hamiltonian H(k)

and a Hilbert spaceH(k). Of course, we can view k as a parameter as well, but it is better to distinguish it from other parameters in the theory. 2 With this information we can construct a Hilbert bundle

H E

BZd

(1.2)

together with an action of the Hamiltonian, H :E →E which acts fibrewise and which causes the Hilbert bundle to split up in two piecesE_±, the conduction and valance band, respectively. The topology of this Hilbert bundle is determined by the way local patches are glued together to for the whole bundle. Locally it will always look trivial, but globally this leads to non-trivial topologies. Due to the fact that the conduction band is usually infinite dimensional3_and

thus cannot give rise to non-trivial topologies, we assume that valance band has finite rank. This part of the Hilbert bundle E₋ thus determines the topology of E . In the next chapter, we will embark on this in a bit more detail as well. To quantify these topologies, characteristic 2_{There could be possible values for k at which the gap closes, but for now we want to exclude this possibility.}

This will be important when discussing semi-metals.[19]

3_{There are always an infinite number of states available in a typical condensed matter system composed of}

classes, such as Chern, Stiefel-Whitney and Pontryagin classes are used. These called character-istic classes, because they are trivial for trivial bundles, but non-trivial for non-trivial bundles. By integrating these classes over the base manifold, we get a number which is a topological characterisation of the bundle. These numbers are called characteristic numbers and are usu-ally integers or integers modulo 2. For us, the most important property of these numbers is that they are topological invariants;homeomorphic bundlesgive rise to the same characteristic numbers. A continuous deformation of the Hilbert bundle will thus not change the topology of the Hilbert bundle and we will remain in the same phase. Only when a going to another phase, the topology of the E can change. For a more detailed discussion on characteristic numbers and their precise definition, we refer the reader to [20], [21] and [22].

Now we can define what we mean with a change in topology. A change in topology is simply a change in the topology of the Hilbert bundle, hence ofE₋. This means that the char-acteristic numbers change when going from one phase to the other. A classical example is the integer quantum Hall system. This system lives in two spatial dimensions and is characterised by the Hall conductivity σxy. In [6], Thouless et al. found that this Hall conductivity can be

calculated via an integral over the Brillouin zone:

σxy = ie 2 2πh Z BZ2 dkxdkyTr  ∂kxhψ a_| ∂ky|ψ b_{i −} ∂kyhψ a_| ∂kx|ψ b_i _(1.3)

The trace is over the states in the valance band, hence a sum over a =b. It is now also obvious why a gap is needed, since otherwise characteristic numbers would not be well-defined. The term in brackets is in fact the Berry curvature or the curvature of the valance bandE₋. Writing it as F_xyab, we end up with σxy = ie2 2πh Z BZ2Tr(F ab xy). (1.4)

However, this integral is related to the first Chern number of the valance bundle,

σxy =

e2

h Ch1(F). (1.5)

This characteristic number quantifies the topology of the valance bundle and hence we see a topological characterization of the quantum Hall phase at filling equal to the first Chern number. At different Hall plateau’s the filling is a different integer, hence a different first Chern number. Therefore, each Hall plateau defines a different phase of the quantum Hall state.

Now we are ready to state what we mean with a topological phase of gapped non-interacting matter. Combining the definition of a phase with that of the fact that topological invariants do not change under a continuous deformation, we obtain

Theorem 1.2. (Topological phases) Topological phases of gapped non-interacting matter are phases

of matter characterised by the topology of the valance band, E−. To each phase a different topological

invariant, such as a Chern number, is assigned and when crossing a phase transition, this invariant changes. The gap is between the valance and conduction band.

What are topological phases? 8

1.4.2 Interacting theories

Besides the ordinary characteristic numbers we have discussed above, there are also some quantum mechanical topological invariants that are important when discussion topological phases. In contrast to the other invariants, these are many-body invariants and are not useful for free theories. The utmost important one of such invariants, which we already mentioned above, is the ground state degeneracy, NGS. Due to this degeneracy, long-range interactions can

be established. Let us discuss why this is the case. When a non-trivial ground state degeneracy is present, excited states (particles) can be construct from different ground states. This gives rise to different sectors in the theory, labelled by the ground state from which the excitations originate. The gap suppresses any local interactions between excitations in different sectors ex-ponentially and only topological or long-range interactions are possible, [10]. Essentially, this means the following in 2+1 dimensions. Consider two excitations in a system with ground state degeneracy NGS. One of the particles is an excitation of ground state a and the other of

ground state b. We call these particles a-type and b-type particles, respectively. The wavefunc-tion that describes these two excitawavefunc-tions is denoted by ψ(a, b). Imagine now that we exchange the a- and b-type particle, which is a non-local process because no length scale is associated with it, then

ψ(b, a) =eπiθabψ(a, b). (1.6)

The wavefunction acquired a phase. This phase is a Aharonov-Bohm phase and this is what we mean with a long-range interaction. The angle θ_abis a statistical angle, determining the statistics between particles of type a and b. For example if θab = 1, they are fermions and bosons when

θab = 0. Actually, since there are a finite amount of ground states4, the topological interaction

can be described by a set of rules, called braiding rules. It is also in this sense that the long-range entanglement between the two particles of type a and b should be understood, [23]. Under any continuous deformation of the system this long-range quantum entanglement cannot change. Most examples with non-trivial NGS are found in 2+1 dimensions. There, the particles are

called anyons and their fractional statistics cause the long range entanglement, [24,25].

The difference between this invariant and the characteristic numbers comes from the fact that the latter ones really use the valance band, hence single particle physics, whereas the ground state degeneracy is more general. In fact, free theories do not have long-range quantum entanglement and so we always require such theories to have a unique ground state. A free field theory can have ground state degeneracy, which arises from a symmetry in the system, such as conservation of Sz, the z-component of spin. This decomposes the Hilbert space in a

direct sum and by putting the Hamiltonian in block-diagonal form in accord with the Hilbert space decomposition, we can focus on one block, which is again a free theory with a unique ground state. In an interacting theory, the direct sum decomposition is not possible, since there is mixing between the different sectors. Another thing that could cause ground state

degen-4_{We assume that N}

eracy is the existence of different topological sectors, just as described above. However, in a non-interacting theory, these sectors are independent and can be studied separately.

Moreover, we will see that for interacting theories, we look at energies well below the gap. So in these cases, we assume the gap to be between the ground state(s) and the first excited state. This is different from what we want to do for free theories, but since the notion of valance and conduction band does not exist in interacting theories, this is the most natural thing we can do. The topological phase is then described by a theory without any propagating degrees of freedom, hence by a topological field theory. In summary, with a topological phase of interacting matter we will mean

Theorem 1.3. (Interacting topological phases) Topological phases of interacting and gapped mat-ter are described by a topological field theory at low energies. To each phase a different topological field theory is assigned and when crossing a phase transition, the topological field theory changes. The gap is between the ground state and first excited state.

1.4.3 Incorporating global symmetries

Besides the topological structure we discussed above, symmetries can also enrich the topolog-ical phase. These symmetries can be either be global or local (gauge symmetries). The latter is rather a redundancy instead of a symmetry that constraints the system, which is typical for global symmetries. We will therefore restrict to global symmetries acting on the systems un-der consiun-deration. For us, the most important examples of global symmetries are time-reversal symmetry (TRS), particle-hole symmetry (PHS) and chiral or sublattice symmetry, which is a combination of the first two. Nevertheless, we will also look at other global symmetries, such as a Z2spin-flip symmetry. To implement these symmetries, we need to know how they act on the

Hamiltonian H. In the non-interacting case, we can simply use the single-particle Hamiltonian Hspto implement TRS, PHS and chiral symmetry. A system is invariant under time-reversal

symmetry if

TH(k)spT−1= H(−k)sp (1.7) with T a complex antilinear operator, which is usually decomposed as T = UTK . The

oper-atorK is the complex conjugation operator, acting on HspasK Hsp(k)K−1 = Hsp∗(k).

Time-reversal symmetry thus relates a Hamiltonian at k to a Hamiltonian to−k. This is what you want, since on the one hand time-reversal symmetry in second quantised picture is imple-mented in the usual way:[T, ˆH] =0. The action on the time-evolution operator is, on the other hand, simply eitH →e−itH. Combining the two we deduce that

T(₋i)T−1=i. (1.8) Hence, T is an complex antilinear operator. To justify the claim we made about the rela-tion between k and −k, which is are spatial momenta, we first assume that we start with an

What are topological phases? 10

Hamiltonian Hsp(x)5_{. Time-reversal symmetry invariant systems are then characterised by}

THsp(x)T−1 = Hsp(x). Upon taking a Fourier transformation of Hsp(x)we get

Z

dkeikxTHsp(−k)T−1−Hsp(k)



=0 (1.9)

which should hold for all x and where we have used1.8. Consequently, we get the relation we found in1.7. This relation is a reality condition on the Hamiltonian and constraints its form. For particle-hole or charge-conjugation symmetry we do a similar thing, but now (assuming that we have put the Fermi level at zero energy) the spectrum should be symmetric around zero. This symmetry between particles, which have momentum k and energy E, and holes, which have momentum₋k and₋E is thus implemented as

CHsp(k)C−1 =−Hsp(−k). (1.10) The combination of the two symmetries, chiral symmetry S = TC, therefore anti-commutes with Hsp(k):

SHsp(k)S−1 =−Hsp(k). (1.11)

For time-reversal symmetry only, Dyson determined in [26] three different classes of Hamil-tonians, which is now known as the three-fold way. Altland and Zirnbauer expanded this clas-sification by including the other two symmetries as well in [27]. Due to Wigner’s theorem (on which we elaborate more in chapter2) a symmetry (in the second quantized sense, so one which commutes with the Hamiltonian and preserves transition amplitudes) is lifted to a sym-metry on the Hilbert spaceH . Due to this lift, the lifted symmetry is complex linear or complex antilinear and in the latter case the lifted symmetry either squares to+1 or₋1, see [28] for a nice exposition of lifted symmetries. So for TRS and PHS we have T2 = ±1 and C2 = ±1. Chiral symmetry, is a unitary symmetry, hence can only square to +1. Now we can count the number of Hamiltonian classes with PHS, TRS or chiral symmetry. Denoting the absence of a symmetry as T2 = 0, C2 = 0 or S = 0 we immediately see that when S is absent, we have 9 classes. Within 8 of these classes, S is fixed (because it is determined by T and C), but when no symmetry is present S is not fixed, yielding one other possibility. Thus we have a total of 10 Altland-Zirnbauer classes. Another way of seeing it is by noting that if two symmetries are present, then the third one is also present. Hence only classes with zero, one or three symme-tries exist. We have listed the classes in table 1.1 in accord with the usual labelling found in the condensed matter literature, [8]. This is a simple implementation of three basic symmetries in a non-interacting theory. To incorporate global symmetries in an interacting theory, we take a completely different approach. Due to interactions, we cannot simply use the single particle Hamiltonian to implement the symmetries. Instead, one has to resort to an Lagrangian descrip-tion of the system in which we will implement global symmetries by gauging them. This will 5_{Time dependence of the Hamiltonian is excluded from our discussion, we are interested in time-independent}

A AI I I AI BDI D DI I I AI I CI I C CI T2 0 0 +1 +1 0 −1 −1 −1 0 +1 C2 0 0 0 +1 +1 +1 0 −1 −1 −1

S 0 1 0 +1 0 1 0 1 0 1

Table 1.1: The 10 Altland-Zirnbauer classes according to time-reversal (T), particle-hole (charge-conjugation) (C) and chiral symmetry (S).

provide a very general procedure for dealing with global symmetries. For antiunitary symme-tries, especially, time-reversal symmetry, the action on spacetime is considered, rather than the action on the Hamiltonian we discussed above. In chapters3and4we will discuss interacting phases in more detail and elaborate on what gauging of a global symmetry actually means.

Besides the global symmetries acting on a Hilbert spaceH , we will also consider crystal symmetries. These include not only translational symmetry, but also its extended versions: space group symmetry. These symmetries act on the lattice and can constrain the Hamiltonian even more. Due to their action on the lattice or spacetime, they need to be implemented dif-ferently. For translational symmetry, we already know what to do, since by Bloch’s theorem, our Hamiltonian and Hilbert space will then be defined above the Brillouin torus. The point group symmetry will then relate different point within the Brillouin torus to each other, allow-ing characteristic numbers to be constraint by symmetry, [29,2,30]. In the section1.7 we will consider crystal symmetry in some more detail, whereas in chapter2we will see how we need to deal with crystal symmetries in free theories. For interacting systems we propose some con-structions that can deal with these symmetries from a Lagrangian point of view. This will be done in section3.5.

Boundary physics

Perhaps one of the most prominent features of topological phases is the existence of bound-ary modes. These modes are topologically stable modes living on the boundbound-ary between two phases. Perturbations (either respecting the symmetry or not, depending on the type of topo-logical phase, see below) or disorder will not cause these modes to backscatter. They are pro-tected by symmetry and topology. Let us first consider free theories. If you happen to posses a piece of material in a topological phases and you put it in the open air, which is topologically a trivial phase, degrees of freedom will localise at the boundary of this piece of material. That is, the boundary of the material is in fact the region S, the region in parameter space where the phase transition occurs. If the gap closes at the phase transition, hence boundary of the slab of material, the boundary modes will be gapless. An interesting example is again the integer quantum Hall (IQH) state. In such a state, the boundary degrees of freedom are gapless and perfectly conducting. In fact, this is obvious. A IQH state is described, at low energies, by a Chern-Simons theory, which has chiral fermion or boson living at the edge in the form of a WZW model. In the next section, we will explicitly construct gapless modes at the boundary

What are topological phases? 12

of a class C and D system.

It is amusing to note that the classification of Ryu et al. in [8] is entirely based on the bound-ary physics. What they did is the following. Based on the fact that these boundbound-ary states must be topological, they cannot be sensitive to disorder, hence Anderson-localisation. To evade this type of localisation, topological terms need to be added to the boundary theory. Using the map between NLσMs and Anderson-localisation ([7,31]) and the fact that only certain topological terms are possible in a particular dimension, Ryu et al. figured out what possible topological phases where possible given the dimension and symmetries. They found exactly the same table as when only the bulk is considered, [1].

For interacting theories more exotic things can happen and various boundary theories are possible. For gapless modes at the boundary of interacting phases in 2+1 dimensions, Lu and Vishwanath considered various symmetric perturbations that could gap some modes on the boundary, but not all. According to these possible perturbations, they were able to classify non-chiral topological phases of interacting bosons and fermions, [32].

Gapped theories can also exist on the boundary of interacting topological phases as was shown in [33,34,35]. These boundary theories are rather exotic due their non-trivial ground state degeneracy and can occur at the boundary of extraordinary phases. We will see examples of this in later chapters, especially chapter4

From this discussion it seems strange that the region S in parameter space coincides with the boundary of a topological phase in real space. However, we can always think of a phase transition in this way, because if a material in a topological phase is assumed to be placed in vacuum, which is topologically trivial, there is a phase transition at the boundary of the material. Whether this is the most general type of phase transitions is not clear and needs to be studied more carefully.

1.4.4 Types of topological phases

After all the classifications found in the last 15 years, a lot of new types of phases of matter have emerged. Some of them are protected by their global symmetry, others are simply topological due to the existence of anyons. In this subsection we will list all the known classes of topological phases and how they are defined. Although, a lot of authors like to scramble these definitions around, we will pick one definition and stick to this definition. In the beginning of this section we already saw what is meant with a topological phase, but in this subsection we want to subdivide those as well. This list will therefore be a good reference for the coming chapters.

Trivial phase

A topologically trivial phase is a phase of matter for which the topology is trivial, meaning that the band structure is trivial, i.e. the Hilbert bundle is trivial, or in the interacting case, the topological field theory is the trivial theory.

Topological order

Topological phases with topological order are phases of matter which cannot be continuously deformed to the trivial phase by any local perturbation. These systems have a non-trivial ground state degeneracy, which causes long-range entanglement. This means that the degrees of freedom in such a theory can have fractional charge and statistics. Well known examples of these phases are the fractional quantum Hall states and gapped quantum spin liquids.

Symmetry enriched topological phases (SET)

These topological phases have topological order and are enriched by some global symmetry G, which can be continuous or discrete. Examples of these phases are gapped quantum spin liquids with additional symmetries such as time-reversal symmetry or an on-site Z2 global

symmetry. More details on quantum spin liquids can be found in [36,37].

Symmetry protected topological phases (SPT)

Suppose the global symmetry is group G, which might be continuous or discrete. Symmetry protected topological (SPT) phases are phases of matter such that when removing the symme-try G, the phase is topologically trivial. This means that when a perturbation does not respect the symmetry, the new system can be continuously deformed (i.e. via a continuous path in parameter space) to the a system in the trivial phase. The ground state of an SPT on a closed spatial slice is unique and respects the symmetry G. When this class of systems is put on a man-ifold with boundary, it has non-trivial edge modes, which can have topological order. Examples of these phases are topological insulators, which are the free theories we discussed above. In fact, most of this thesis will be about SPT phases and so we will encounter many examples of these types of phases. Most well known are the topological (band) insulators/superconductors, but we will also consider their interacting counterparts.

1.5 Prototypical physics and host materials

In this section we want to discuss some prototypical examples of topological phases to get a feeling of what physics we are talking about. Moreover, we will discuss some materials which can host these topological phases. We will be very brief and refer the reader to reviews such as [38,39,40,41] for more details.

Quantum Hall effect, 1980, d=2+1

The discovery of the quantum Hall effect [42] and the TKNN invariant [6] started new research area: topological phases of matter. This was in 1980 and it only concerned two dimensional physics. However, soon after this discovery an effective action was written down for this the-ory, which could explain the phases of a quantum Hall system particular successfully. This two dimensional physics turned out to contain a very rich structure and houses some of the most

Prototypical physics and host materials 14

fascinating phenomena in physics. The effective action that could capture all this was a simple Abelian Chern-Simons action:

S= k

4π

Z

A_∧dA, (1.12)

It accounts for the quantisation of the Hall conductivity, fractional statistics and the chiral edge states supported on the boundary of the two dimensional slap of material. The quantisation of the Hall conductivity sets the classification for the different phases to Z; the integers. This theory is actually very remarkable, since in such a theory, charge is equivalent to flux. Indeed, coupling to an conserved current gives the equations of motion

kεµνλ

∂µAν= Jλ (1.13)

and thus J0 _{= k(∂}

xAy−∂yAx) = kB, where B is the magnetic field. By introducing multiple

gauge fields at the effective action level to add flux to the electrons, fractional quantum Hall systems could also be described. The effective action is then a K-matrix Chern-Simons action,

S= K I J

4π

Z

AI∧dAJ. (1.14)

The N-by-N matrix KI J is an integer symmetric matrix describing the couplings between the

N gauge fields AI. The quasi-particles in this Hall fluid are vortices ([43]) and by introducing

a coupling between them and the gauge fields together with a coupling to an external electro-magnetic field α, we end up with an action

S= Z _KI J

4πAI∧dAJ+AI∧ ?j

I₋ 1

2πAI∧dα. (1.15) In this action, the second term is responsible for the coupling between the gauge field and the quasi-particles (vortices) and the third term describes a coupling to the external gauge field α. In order to find out the responses such as Hall conductivity, charge of the vortices and statistics of the vortices as a result of coupling to α, we integrate A out. Since the action1.15is quadratic in the fields, this is easily done and results in

S=π Z d3x j_µe f f_,IK−_{I J}1  εµνλ∂ν ∂2  je f f_λ,J. (1.16)

with effective current j_µe f f_,I given by

jµ_{e f f},I =jµ,I₋ 1

2πε

µνλ

∂ναλ. (1.17)

The first contribution is from the vortices and the second from the coupling to an external gauge field α. The action 1.16houses three interesting properties, which can be extracted by

expanding the effective current, S= Z d3x 1 4πε µνλ αµ∂ναλ

∑

∑

∑

The first term is the quantum Hall response and hence

σxy =

∑

K−_{I J}1. (1.19)

The second term simply states that the type I vortex has charge qI =∑JK−I J1. The last term is the

most interesting, due to its non-local nature. It describes the long-range interactions between vortices and is responsible for the fractional statistics. In fact, having a bound state of lI type I

vortices going around a bound state of lJ vortices of type J, the statistical angle is

θ

π =

∑

−1

I J lJ. (1.20)

Hence for single vortices, the diagonal elements of K−1 are responsible for self-statistics and the off-diagonal elements for mutual statistics. For bosons the diagonal elements must be even, whereas for fermions they must be odd. An example we will encounter later is the theory with KI J equal to K=   0 2 2 0  . (1.21)

Hence, σxy = 1 and q1 = q2 = 1/2. There are two particles l = (1, 0) and l = (0, 1), which

have self-statistics 0, hence they are bosons, but their mutual statistics is semionic θ = π/2.

A bound state of these two particles, l = (1, 1), is, however, fermionic. Thus the theory with the K-matrix in1.21is a theory of two bosons that are mutual semions and their bound state is fermionic.

More details on quantum Hall fluids can be found in [43]. As a final remark, we want to mention that this is exactly the type of physics that is expected for phases with topological order in 2+1 dimensions. When fractional statistics is present, the topological order phase is can be described by a Chern-Simons theory with an appropriate K-matrix. If there is no fractional statistics present, the topological phases is simply an SPT phase.

Usually, Chern-Simons theory is used as an effective theory for fermions, but recently, the interacting bosonic SPT phases [33] can also be described by such a theory as it is very ver-satile when it comes to statistics. The bosonic nature of such phases, then requires the Hall conductivity to be an even integer times the fundamental unit of flux e2_{/h, [44].}

As a final remark on the quantum Hall physics, we want to mention that no microscopic theory of quantum Hall physics exists. It is still unclear how the multiple gauge fields emerge in the low-energy regime from a UV theory. It is considered one of the Holy grails of condensed