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Momentum-space classification of topologically stable Fermi surfaces. Sheinbaum Frank, Daniel 2015

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Momentum-Space Classification ofTopologically Stable Fermi SurfacesbyDaniel Sheinbaum FrankB.Sc., Universidad Nacional Auto´noma de Me´xico, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Posdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2015c© Daniel Sheinbaum Frank 2015AbstractThe purpose of the present work is to derive a classification for topologicallystable Fermi surfaces for translationally invariant systems with no electron-electron interactions. To derive such a classification we introduce the nec-essary concepts in condensed matter and electronic band theory as well asthose in mathematics such as topological spaces, building up to topologicalK -theory and its connections with Fredholm operators. We further com-pute such classes when there is only translational invariance for dimensionsd = 1, 2, 3 and discuss the inclusion of other symmetries.iiPrefaceChapter 1 is an introduction to the subject of topological phases matterand topological Fermi surfaces. Chapter 2 is an introduction to Analysis,topological spaces, homotopy and K -theory. Chapter 3 describes the typeof physical systems we wish to describe and introduces the basics if elec-tronic band theory in condensed matter. Chapter 4 derives a classificationof topologically stable Fermi surfaces from the material introduced in pre-vious chapters. Finally, Chapter 5 is a summary of the present work.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . viii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Mathematical preamble . . . . . . . . . . . . . . . . . . . . . . 32.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.1 Hilbert space . . . . . . . . . . . . . . . . . . . . . . . 42.1.2 Bounded and compact operators . . . . . . . . . . . . 52.1.3 Fredholm operators . . . . . . . . . . . . . . . . . . . 52.2 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.1 Continuity, homeomorphisms . . . . . . . . . . . . . . 92.2.2 Compact and connected spaces . . . . . . . . . . . . . 102.2.3 Homotopy . . . . . . . . . . . . . . . . . . . . . . . . 112.2.4 Suspension, loop space, wedge sum and smash product 122.3 K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.1 Vector bundles and isomorphism classes . . . . . . . . 152.3.2 Direct sum and tensor product of vector bundles . . . 172.3.3 Pullbacks and universal bundle . . . . . . . . . . . . . 172.3.4 Semi-group and its Grothendieck completion . . . . . 192.3.5 K-groups and Bott periodicity . . . . . . . . . . . . . 22ivTable of Contents3 Physical setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.1 Free n electrons in a box . . . . . . . . . . . . . . . . . . . . 253.2 Bravais lattice . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2.1 Reciprocal Lattice . . . . . . . . . . . . . . . . . . . . 293.3 Bloch’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . 303.4 Fermi surface . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.1 K-theory and Fredholm operators . . . . . . . . . . . . . . . 354.2 K-theory and Fermi surfaces . . . . . . . . . . . . . . . . . . 404.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49vList of Tables2.1 Classifying spaces for real and complex K -theory. . . . . . . . 24viList of Figures1.1 (a) Experimental setup for the Quantum Hall effect. (b) Pla-teus observed at integer values of the Hall resistivity . . . . . 22.1 (a) How to construct an n-dimensional sphere from a an n-disk by quotienting the boundary of the disk to a point (b)Constructing an n-dimensional torus from an n-dimensionalcube by identifying opposite edges. Figures taken from [31]. . 83.1 (a) A 3-dimensional cubic Bravais lattice (b) A two-dimensionalHoneycomb or Hexagonal lattice. Figures taken from [1]. . . . 29viiAcknowledgementsI would like to thank both Professor A. Adem and Professor G.W. Semenofffor their guidance during the development and corrections to the work pre-sented here.I would also like to thank Mateus Arantes Fandin˜o, who pulled methrough most of our courses with shear intellect and Bernardo VillarrealHerrera for being very patient in correcting all my misconceptions in K -theory.I also thank Professor M.W Choptuik for his warm support during myfirst year at UBC.Finally, I would like to thank CONACyT, which provided a scholarshipthat payed both tuition and living expenses.viiiChapter 1IntroductionThe study and classification of topological phases of matter began in 1980with the unexpected experimental discovery by von Klitzing, Dorda and Pep-per [27] of exact quantization of the Hall conductance of a two-dimensionalelectron gas in a strong magnetic field, in integer units of e2/~, known asthe Integer Quantum Hall effect. Experimentally, it is the most precise mea-surement of quantization known to date, such that its used as the standardcalibration for resistivity world wide since 1990. Laughlin [29] related thiseffect to gauge invariance but it was not until a couple of years later thatThouless, Kohomoto, Nightingale and Den Nijs [36] explained the origin ofthe exact quantization of the Kubo formula for the Hall conductance (usingBerry’s phase [8]) as a Chern number, a topological invariant discovered bythe Chinese mathematician S.S Chern in 1946. This invariant is also knownto physicists as the TKNN invariant, named after the aforementioned au-thors.It was later realized that gapped systems such as the Integer Quantum Halleffect with different values for their topological invariants, represent differentphases of matter, so-called topological phases of matter, which have very ro-bust edge-states, that are insensitive to impurities and perturbations. Thismeans that one can not transit from one state to another without closing theenergy gap. Closing and reopening the energy gap is therefore tantamountto changing the value of a corresponding topological invariant, such as theTKNN invariant.This discovery is a paradigm shifting breakthrough in condensed matter,for it was thought that phase transitions only occurred through symmetrybreaking, according to the Landau paradigm[28]. Now it is known that thereare different topological phases protected by a given symmetry [11], mean-ing that one cannot transit from one phase to other without breaking andrestablishing the corresponding symmetry. Examples of such symmetriesare time-reversal, parity and chirality among others.Much later, in 2004 Kane and Mele [23] discovered a Z2- invariant for thethree dimensional Quantum Spin Hall effect, in which the system is natu-rally time-reversal invariant due to spin-orbit coupling. In 2006 Bernveg,1Chapter 1. Introduction(a) (b)Figure 1.1: (a) Experimental setup for the Quantum Hall effect. (b) Plateusobserved at integer values of the Hall resistivityHughes and Zhang [7] predicted such topological phenomena would be ob-servable in Mercury Telluride potential wells, which indeed was observedexperimentally.Soon after this developments, there has been what can only be described asan explosion of research, both theoretically and experimentally to study sys-tems in condensed matter which depend in one way or an other on topology,most notably gapped phases[18] but also topological defects[37], Majoranawires [13] and Fermi surfaces[21], to name a few. There are other kinds ofQuantum Hall effects known as the Fractional and Anomalous Quantum Hallefects. The former as the name suggests has fractional Hall conductance.It is understood that electron-electron interactions play a fundamental roleand distinguish it from the Integer case. The fractional case is much less un-derstood, but has been studied using very deep mathematical ideas knownas Chern-Simons topological quantum field theory. The anomalous Hall ef-fect has a long and controversial history, however, recent efforts such as thatof Haldane [17], have reinterpreted part of it in terms of Berry’s curvatureand the same invariants used for the integer case. This work in fact hassurprising connections with the kind of systems we shall classify.2Chapter 2Mathematical preambleIn this chapter we will introduce a vast array of mathematical definitions,objects and a few (but very powerful) theorems, most of which shall be em-ployed for the construction of our classification of topologically stable Fermisurfaces. There are several reasons for doing this. On the one hand, it makesthe present work sufficiently self-contained and easier to follow. This chap-ter is intended particularly for readers with a background in physics, most ofwhich might not have studied these concepts, which is why we provide themin such detail. These concepts also play a fundamental role in our remarkson inconsistencies lurking in the existing literature of classifications, that is,in order to understand why some constructions are inconsistent it is a goodidea to have the formal definitions at hand, for though some concepts aresimilar, the richness of the construction lies in their difference.First we introduce concepts of analysis since they are more familiar to physi-cists in general, then we go through the topic of what is known to math-ematicians as point-set topology and is not as common to encounter in aphysicist’s repertoire and finally to K -theory.We assume that our readers are familiar with the basics of set-theory as wellas finite-dimensional linear algebra, nevertheless we may reintroduce someof these notions as we see fit.2.1 AnalysisThe branch of mathematics known as analysis came into being with the workof Fourier on the now famous heat equation, with the byproduct inventionof his transform and series in the late 18th century but it was not untilthe 19th century that it was understood as an independent field throughthe work of Weierstrass, Cauchy, Fredholm and others. In the 20th centuryit was the work of Hilbert and Von Neumann that connected analysis andquantum physics.All of the definitions presented in this section can be found in [34], one ofthe standard texts in the literature of mathematical physics. Finally, weavoid unnecessary over-citation of [34].32.1. Analysis2.1.1 Hilbert spaceWe begin by introducing the concepts of a Cauchy sequence and a completemetric space,Definition 1 A sequence {xn} of points in a metric space M is a Cauchysequence if ∀  > 0 ∃N ≡ N() ∈ N such that ∀n,m > N, ‖xn − xm‖ < ,where ‖ ‖ denotes de norm of M .Definition 2 A complete metric space M is a metric space for whichevery Cauchy sequence converges.With these at hand we can define a Hilbert space,Definition 3 A Hilbert space H is a vector space over a field F which isa also a complete metric space.We will exclusively focus on the case were the field F = C, the set of complexnumbers, since (as we will later see) this choice is part of the foundation uponwhich quantum mechanics was built.Elements of H will be denoted as |Φ〉 using the Dirac bra-ket notation, |Φ〉being a ket. The bra 〈Φ| = |Φ〉† is the dual element to |Φ〉, where 〈Φ|Ψ〉 ∈ Fdenotes the inner product of H .We shall also restrict our attention to separable Hilbert spacesDefinition 4 A Hilbert space H is separable iff it has a countable or-thonormal basis {|Φn〉}∞n=1.Let us give an example of a complex separable Hilbert space.Example 1 Let L2([0, 2pi]) denote the set of square integrable functions f :[0, 2pi] −→ C ,∫ 2pi0|f(x)|2dx <∞. L2([0, 2pi]) is a complex separable Hilbertspace with orthonormal basis |Φn〉 = (2pi)−1/2einx.Note that a square integrable function f(x) is viewed as a vector and fol-lowing our convention, should be denoted as |f〉. Any square integrablefunction has a Fourier seriesf(x) =∞∑n=−∞(2pi)−1/2aneinx , (2.1)an = (2pi)−1/2∫ 2pi0f(x)e−inxdx , (2.2)which is why L2 (as is commonly abbreviated) has applications across a widevariety of fields of study.42.1. Analysis2.1.2 Bounded and compact operatorsLet us now study operators on our Hilbert space H . The first kind ofoperators are bounded operators:Definition 5 An operator B : H −→H is bounded if ∃ c such that∀ |Φ〉 ∈H , 〈Φ| B†B |Φ〉 ≤ c 〈Φ|Φ〉.c is known as the operator norm of B. The set of bounded operators of His denoted as B(H ).We shall also define the set of compact operators:Definition 6 K is a compact operator iff ∀ {xn} ⊂H bounded sequence,K({xn}) has a convergent subsequence in H . The set of compact operatorsis denoted as K(H ).Compactness is a topological property, but we seem to have gotten awaywith defining a ”compact” thing without defining topology. In reality it isthat we have implicitly used the notion of the norm of a Hilbert space inour definition and so have used indirectly the topology defined by the norm!For completion and for later use we should define the Calkin algebra [24],CAL(H ) ≡ B(H )/K(H ) . (2.3)where we have used the fact that K(H ) is a two-sided ideal (see [3]) inB(H ). CAL(H ) is known as a C∗-algebra, which are widely used, especiallyin what is known as non-commutative geometry [35] and there is a non-commutative approach to problems in condensed matter. We will discusssome of these in later chapters.2.1.3 Fredholm operatorsFredholm operators will play such an important role in our work that theyshould be discussed in greater generality. These operators are named afterSwedish mathematician Erik Ivar Fredholm and arose in his work on inte-gral equations which derived into what is known in analysis as the Fredholmalternative, which states the following:Let K be a compact operator in H and |Φ〉 ∈ H , then only one of thefollowing possibilities occurs• K |Φ〉 = λ |Φ〉,• (K − λI)−1 is bounded.52.2. TopologyA generalization to Banach spaces was developed by Alexander Grothendieckin his doctoral work [16].We are now in a position to define Fredholm operators, at least for a singleHilbert space, this will suffice for our purposes.Definition 7 An operator H : H −→H is Fredholm if• dimKerH is finite.• dimKerH† is finite.Let us denote the set of Fredholm operators of H as it is done in [3],[4]as F(H ). One of the classical results of Fredholm theory is that if K iscompact then I +K is Fredholm.We shall study many of the topological properties of the set of Fredholmoperators and various corresponding subspaces. As we will see they playa prominent role in K -theory through the work of Sir Michael Atiyah andIsadore Singer [3][4] and their work on the index theorem for elliptic op-erators, but more on this later. They have become fundamental in manyapplications outside analysis such as geometry through Donaldson’s workand in Conne’s non-commutative geometry as well as having applicationsin physics, including condensed matter but we do not discuss their role inthose subjects any further.2.2 TopologyThe subject of topology is one of the most incredible ones in mathematics,commonly described as the study of an object’s ”shape”, independent of itsgeometrical properties but it is much deeper than this claim leads us to be-lieve and fortunately in many important cases it is inextricably linked withgeometry, algebra and analysis themselves. Developed as an independentfield in mathematics until the early 20th century by Hausdorff, Poincare´and many others, it has increasingly played an important role in physics,in relation to statistical mechanics for classical and quantum systems, sym-metries in quantum field theory and the subjects discussed in this thesis, inwhich we shall particularly show its connections with analysis through thework of Atiyah, Bott and Singer.Readers can consult all the basic definitions of topology (topological space,continuous functions, compact spaces, etc) presented here in [31] and thoseof algebraic topology in [19] , which is the standard textbook reference on62.2. Topologythese subjects. Both of them are excellent references. Once again, we gen-erally avoid referencing them in the following sections.Let us start by defining a topological space in a formal wayDefinition 8 A topology on a set X is a collection T of subsets of Xhaving the following properties1. ∅ and X are in T .2. The union of elements of any subcollection in T is in T .3. The intersection of the elements of any finite subcollection of T is inT .Thus, providing X with such a T makes it a topological space.A few comments are in order, first a set X may be provided with differenttopologies, something that we will encounter in the classification chapterfor operator sets. However, there are some topologies which arise naturally(such as the topology induced by the metric of Rn) and in those cases wherethere is no confusion we will denote the topological space simply as X, eventhough formally, we should write a topological space as a pair (X,T ). Alsonote that by its definition T ⊂ P(X), the power set of X.Elements of T (subsets of X) are called open sets. Alternatively we candefine a closed sets, which are subsets of X (as a set) and give an equivalentdefinition of a topology in terms of closed sets, the relation between openand closed sets is that a set C is closed if Cc = X\C is open. We presentthree examples of topological spacesExample 2 Let X = B(H ), the set of bounded operators on a Hilbert spaceH , which we introduced earlier, with open sets U ∈ TB(H ), which are ar-bitrary unions and finite intersections of sets of the form Br(H) = {A ∈B(H )| ‖A − H‖ < r}. Br(H) is called an open ball of radius r centered atthe operator H.This topology is known as the metric topology and can be generalized to anymetric space, such as our well known acquaintance Rn.Example 3 Let X = Rn and let T = {U ⊂ Rn |Rn\U is finite}. Then(Rn,T ) is a topology in Rn, known as the cofinite topology Tcofinite.72.2. Topology(a) (b)Figure 2.1: (a) How to construct an n-dimensional sphere from a an n-diskby quotienting the boundary of the disk to a point (b) Constructing an n-dimensional torus from an n-dimensional cube by identifying opposite edges.Figures taken from [31].The only purpose of our last example is to show that a given set X can havedifferent topologies. The last example we present will be very useful lateron to define operations on topological spacesExample 4 Given a topological space (X,TX), we denote X∗ a set of equiv-alence classes of elements of X, that is a partition of X into disjoint subsetswhose union is X. Let P : X −→ X∗ be the function which takes an elementx ∈ X to the corresponding subset which contains it. We can endow X∗,with the quotient topology Tqby asking a set U to be open in (X∗,Tq) iffp−1(U) is open in (X,TX). X∗ is known as a quotient space.Constructions of such quotient spaces are shown in 2.1a,2.1b.Let us include the definition of Hausdorff topological space, which is thetype of spaces we are used toDefinition 9 A topological space (X,T ) is Hausdorff if ∀x1, x2 pair ofdistinct points in X, there exists open sets U1, U2 ∈ T , such that x1 ∈U1, x2 ∈ U2 and U1 ∩ U2 = ∅.This condition is necessary for sequence of points in a topological space toconverge at a unique point, hence this condition is necessary for the conceptof limit, fundamental to analysis. Indeed most spaces of interest, whicharise from other areas in mathematics (and physics) satisfy this condition.A slightly weaker condition known as T1 axiom, of the separability axioms,is the followingDefinition 10 A topological space (X,T ) is T1 if ∀x1, x2 ∈ X, ∃U ∈ Tsuch that x1 ∈ U but x2 ∈ U c.82.2. TopologyThis definition will only be necessary to us later, when we introduce theconcept of a topological group. Note that the Hausdorff condition is also aseparability axiom, known as T2.2.2.1 Continuity, homeomorphismsWe are now in a position to introduce the notion of a continuous functionDefinition 11 A function f : (X,TX) −→ (X′,T X′) is said to be contin-uous if ∀U ∈ TX′ , f−1(U) ∈ TX, where f−1 denotes the inverse image off .In other words, the inverse image of an open set under f is open in X. Thedistinction between inverse image and inverse function is very important inthis case. All functions have an inverse image (maybe empty) but not allfunctions have an inverse, let us give a simple example:Example 5 Let f : X −→ {x0}, then f is continuous for any topologicalspace X since f−1(x0) = X, which is open by definition. If X is differentfrom a point, then @ f−1 as a function, only as a relation.Also we can notice that the definition we have presented here is independentof whether our topological spaces are Hausdorff or not, since it plays no rolein the definition. We had discussed that to have a notion of limit it was nec-essary for our topological space to be Hausdorff, making this independentdefinition of continuity at odds with the conventional way one is taught in acalculus or analysis course, were we define continuity in terms of limits butfor more general spaces the notions of limit and continuity are independent!This definition allows us to present the definition of a topological groupDefinition 12 A topological group G is a group that is also a topologicalspace satisfying the following conditions1. G is a T1-space.2. The function (g, h) 7→ gh is continuous.3. The function g 7→ g−1 is continuous.Example 6 Let GLn(C) denote the group of nX n invertible matrices un-der multiplication with entries in C, We can view GLn(C) as subspace ofC2n with the subspace topology under the induced metric topology of C2n.GLn(C) with this topology is naturally a topological group under matrix mul-tiplication.92.2. TopologyThis example is basically the whole reason why we went through the troubleof introducing the notion of a topological group and it will be fundamentallater on when we study vector bundles and classifying spaces.We can define homeomorphisms between topological spacesDefinition 13 A bijective continuous function f : (X,TX) −→ (X′,T X′) isa homeomorphism if f−1 is continuous.The condition for a bijection f to be a homeomorphism is equivalent to fbeing continuous and open (sends open sets in X to open sets in X′).As far as it concerns topology, two spaces are indistinguishable if they arehomeomorphic. Thus, topological spaces can in principle be completelyclassified up to homeomorphism (analogous notions exists for groups, ge-ometries and other areas of mathematics). We state it as a principle sinceit is in general not possible to address whether two topological spaces arehomeomorphic, nevertheless, there are other less general but still elucidatingtopological properties we can study.2.2.2 Compact and connected spacesTo define compact spaces, we first need to introduce the notion of an opencover of a topological space, then we can immediately proceed to define acompact spaceDefinition 14 An open cover {Aα} is a collection of elements in T whoseunion⋃αAα = X.Definition 15 A topological space X is said to be compact if every opencover {Aα} of X, has a finite subcover, which covers X.A subcover is simply a subset of a given cover. It is usually not simple toprove a space is compact, however the following theorem, which we statewithout proof, is very useful for some casesTheorem 1 Every closed and bounded subset of Rn is compact.Thus, the n-dimensional sphere Sn = {(x1, .., xn+1) ∈ Rn+1|n+1∑m=1x2m = 1}is compact. In particular, the circle S 1 is compact and since the arbitrarycartesian product of compact spaces is compact (see [31], Tychonoff’s theo-rem), the n-dimensional torus Tn =n∏S 1 is also compact. This examples102.2. Topologyare not only extremely common and simple but they will also play a funda-mental role in what will follow since spheres are fundamental for K -theoryand in general the Brillouin zone of the condensed matter systems we shallstudy is an n-dimensional torus.Let us also introduce the notion of a connected spaceDefinition 16 A topological space X is connected iff the only sets whichare both open and closed are the total space X and the empty set ∅.This definition is very intuitive, for it simply means that a space is connectedif we cannot separate it into two pieces which are disjoint. Connectednesswill play a more subtle role but deep role in our further developments, par-ticularly in homotopy theory.2.2.3 HomotopyFrom here on we will write map instead of continuous function. The notionof Homotopy has basically driven the field of algebraic topology since the20th century, yet its definition is remarkably simpleDefinition 17 A homotpy is a family of maps ft : X −→ Y, t ∈ I suchthat the underlying map F : X × I −→ Y, F (x, t) = ft(x) ∀x ∈ X, t ∈ I iscontinuous, which we denote f0 ' f1.We say that two maps f0, f1 : X −→ Y are homotopic if there is a homotopy{ft} connecting them. Also there is a notion of two topological spaces beinghomotopicDefinition 18 A map f : X −→ Y is called a homotopy equivalence orthe same homotpy type if there exists g : Y −→ X such that fg ' IdX andgf ' IdY. We denote this as X ' Y.Spaces which are homotopic to a point x0 is called null-homotopic, examplesof null-homotopic spaces are Rn,Cn, Dn among many others. However, thereare also plenty of spaces which are not homotopic to a point, such as spheres,tori and higher genus surfaces.The set of all homotopy classes of maps f : X −→ Y will be denoted as[X,Y]. It is a pretty big set! If we choose basepoints x0 ∈ X, y0 ∈ Y makingX,Y pointed spaces, then we shall denote the set of all basepoint-preserving-homotopy classes as [X,Y]∗. If X and Y satisfy a few conditions (See [19])[X,Y]∗ is a group. If X = Sn, then we write [X,Y]∗ = pin(Y, y0), whichmathematicians call the n-th homotpy group. pi1(Y, y0) = [S 1,Y]∗ is knownas the fundamental group and is the only one which may be non-Abelian(but not necessarily so)112.2. TopologyExample 7 pi1(S 1) = Z, in general pin(Sn) = Z and pii(Sn) = 0 for i < n.Here we have abused of the fact that if the target space is connected then,then the choice of base-point is irrelevant up to isomorphism of the basedhomotopy classes, and since Sn is connected we can exclude the base-pointin our notation for its homotopy groups.Homotopy theory and its generalizations have driven algebraic topology thelast 100 years or so. Homotopy groups provide more information than coho-mology or homology groups (which we will only discuss a particular case of,namely K -theory) but they are in general much harder to compute, in facteven for simple enough spaces like spheres it is still open how to computeall their homotopy groups!Applications of homotopy groups to physics are many. In quantum fieldtheory they are used in the description of anomalies[32] and approaches todescribe real space topological defects when there is symmetry breaking areusually classified via homotopy groups[9].2.2.4 Suspension, loop space, wedge sum and smashproductThere are a number of operations one can perform on one or more topologicalspaces, in fact there are many but we will only make use of four distinct onesin this work. The first one is the suspension of a topological spaceDefinition 19 Let X be a topological space, the suspension of X, denotedas SX, is constructed by taking the product X × I and then quotientingX× {0} to a single point (south pole of SX) as well as quotienting X× {1}also to a single, different point (north pole of SX).The archetypal example of the use of suspension is with spheres whereSSm = Sm+1, where the equality is up to homeomorphism.One can also suspend maps by f × IdI : X × I −→ Y × I by making thesame identification, yielding Sf : SX −→ SY.There is also the concept of a reduced suspension of a spaceDefinition 20 The reduced suspension of a space X, denoted ΣX, isconstructed by first suspending, obtaining SX and then further quotientingthe line {x0} × I, with x0 ∈ X canonically identified with X× {12}.Reduced suspensions are particularly important for homotopy classes ofpointed spaces and so called loop spaces122.2. TopologyDefinition 21 Let (X, x0) be a pointed space, the set ΩX of loops S −→(X, x0), which is a subset of XI , the set of maps I −→ X which naturallyposses the compact-open topology.To see the connection between reduced suspensions and loop spaces, considera map f : ΣX −→ Y. We can identify the image of {x} × I/({x} × {0} ∼{x} × {1}) under f with the image of {x} under a map f˜ : X −→ ΩY,f({x} × I/ ({x} × {0} ∼ {x} × {1})) = f˜(x). Thus reaching the very im-portant result[ΣX,Y]∗ ≈ [X,ΩY]∗. (2.4)There is a much deeper connection between loop spaces and homotopyclasses with cohomology but we shall only make reference to it in the par-ticular case of K -theory, so the interested reader can explore this in [19].We shall now define two more operations on topological spaces, but theseones are between pairs of such. They are known as the wedge sum and thesmash productDefinition 22 Given two topological spaces X,Y consider their disjointunion X unionsq Y, and then quotient it by identifying a single point x0 ∈ Xwith a single point y0 ∈ Y(X and Y viewed canonically in X unionsq Y). Thisspace is called the wedge sum of X and Y, denoted X ∨Y.Definition 23 Given two topological spaces X,Y, the smash product ofXand Y is the quotientX×YX ∨Y, denoted as X ∧Y.The archetypal example for the use of the smash product is the fact thatSm ∧ Sn = Sm+n, where once again, the equality is up to homeomorphism.Notice that both the smash product and the wedge sum are symmetric un-der the ordering of the pair of topological spaces upon which they are beingemployed. These operations are widely use in many of the fundamental con-structions in algebraic topology, that is many spaces are at least homotopicto a mix of wedge sums and smash products of simpler ones. In our case,they are particularly necessary in connection with a reduced suspension forthe following homotopy relationLet X,Y be CW-complexes, thenΣ(X×Y) ' ΣX ∨ ΣY ∨ Σ(X ∧Y). (2.5)The definition of CW-complexes and the proof of this proposition can befound in [19]. Just to remove the possible sensation of being cheated by this132.3. K-theorystrange name (CW-complex), we mention that almost all nice spaces one canthink of such as tori, spheres, higher genus surfaces, etc are CW-complexes,but we avoid giving the proper definition since it will not be necessary forwhat will follow.2.3 K-theoryWe finally arrive at what is the area of algebraic topology we will employ themost to classify our physical systems, K -theory. Topological K -theory wasintroduced by Sir Michael Atiyah and Friedrich Hirzebruch in the 1960’s asan extraordinary cohomology theory, the word extraordinary having a pre-cise meaning in this case. It was based mainly on a fundamental construc-tion developed earlier by Alexander Grothendieck, known as Grothendieck’scompletion, obtaining an Abelian group out of a semi-group and it also de-pended crucially on Raoul Bott’s periodicity theorem on homotopy groupsof classical groups, which he had proved in 1957. Roughly speaking, a coho-mology theory is a functor (generalization of function) from the category (ageneralization of set) of topological spaces to the category of Abelian groups,assigning to a topological space X, a chain of groups, denoted H n(X), n ∈ Zwith connecting homomorphisms, usually denoted dn, H n(X)dn−→ H n+1(X)and use this functor to study topological properties of X in terms of the alge-braic properties of the groups H n(X) and sometimes the other way around!To portray a picture, we could say that Grothendieck’s completion con-structs the groups K n(X), where as Bott’s periodicity theorem constructsthe connecting homomorphisms, though in reality their roles are intertwinedin a deeper way. K -theory has many mathematical applications, some ofwhich can be studied in [24]. In our case we are interested in its applicationsto condensed matter systems, as first pioneered by Horˇava [21] and Kitaev[25], but that will come later.From now we start to develop the necessary tools to make such a con-struction, which can be studied more carefully in [20], which is incompletework but it conveys very beautifully the geometric constructions and weshall employ it repeatedly or [24], which is a more complete reference, atthe expense of developing the necessary ideas through more abstract notionsof Banach categories.142.3. K-theory2.3.1 Vector bundles and isomorphism classesWe will now study vector bundles by giving their precise definition and afew examples particularly relevant for physics. Let us beginDefinition 24 An n-dimensional vector bundle is map p : E −→ X satisfy-ing the following conditions1. p−1(x) ≈ Fn ∀x ∈ X, where Fn is an n-dimensional vector space overthe field F.2. There is a cover {Uα} of X such that for each Uα, there existshα : p−1(Uα) −→ Uα × Fn (2.6)p−1(x) 7→ {x} × Fn (2.7)hα homeomerphism and with hα(x) an isomorphism of vector spacesfor each x ∈ Uα.The last condition is known as the local trivialization condition, in plainwords it states that every vector bundle is locally isomorphic to the trivialbundle X× Fn. the space X is known as the base space of the bundle, E iscalled the total space and p−1(x) are called the fibers of the bundle. We shalloften only write the total space E to denote the whole vector bundle (withprojection map and base space included). Also notice that we are assumingthat Fn is a topological vector space. We will limit our selves to the casesF = R,Cor H.Example 8 Let F = R, n = 1 and X = S, a circle in R2. Let us constructour bundle E by taking I × R and identifying (0, t) ∼ (1,−t). This identifi-cation gives as base space S 1, where as the total space E is homeomorphic toa Mo¨bius strip without its boundary circle. This bundle is called a Mo¨biusbundle.The Mo¨bius strip and the corresponding Mo¨bius transformations are funda-mental to holomorphicity conditions in complex analysis, which is appliedto study the properties of propagators and other operators in quantum me-chanical systems. Let us present another example which is invaluable forphysics.Example 9 Let X = Sm ⊂ Rm+1, n = m and let E = {(x, v) ∈ Sm ×Rm+1 | 〈x, v〉 = 0, where 〈, 〉 denotes the inner product in Rm+1 and we areviewing points x of Sm as unit vectors in Rm+1. This bundle is known asthe Tangent bundle of Sm, denoted TSm.152.3. K-theoryThe construction of a tangent bundle can be generalized to manifolds, inclassical statistical mechanics, sections of the tangent bundle constitute thephase space of a system! If we take our space-time to be a more generalmanifold M as in general relativity, TM and the exponential map are fun-damental to study the properties of geodesics!Let us present an example which is not directly relevant for physics butrelevant for the whole subject of vector bundles, as we shall see it is quitetrivialExample 10 The n-dimensional trivial bundle over a base space X is sim-ply the product X× Fn, commonly denoted as ξnThe trivial bundle is indispensable for Grothendieck’s completion and thewhole area of topological K - theory.We now introduce the notion of vector bundle isomorphismDefinition 25 A homeomorphism F : E1 −→ E2 between vector bundlesover the same base space X is a vector bundle isomorphism if it takeseach fiber p−11 (x) of E1 to the corresponding fiber p−12 (x) of E2 by a linearisomorphism, for each x ∈ X.We will usually regard two vector bundles E1 and E2 as the same if they arein the same isomorphism class, which we denote as E1 ≈ E2. Isomorphismclasses of n-dimensional vector bundles over a base space X will be denotedas V ecnF(X). We shall also repeatedly make use of the following resultLet p : E −→ X× I be a vector bundle over X× I, where X is compactHausdorff, then the restrictions of p to X×{0} and X×{1} are isomorphic.Thus, vector bundles over compact Hausdorff spaces, which are connectedconnected by a homotopy are isomorphic.This definitions and results will be used throughout the rest of our con-structions, so one should keep them in mind as we go along.We shall skip many of the basic constructions and properties of vector bun-dles such as sections, clutching maps maps, inner products and cocycle con-ditions, for we will not use them directly in the construction of our classifi-cation, however readers are encouraged to consult [20] for examples and thedefinitions of these constructions, since these provide much of the geometricrichness of the subject.162.3. K-theory2.3.2 Direct sum and tensor product of vector bundlesGiven two vector bundles over the same base space X, p1 : E1 −→ X , p2 :E2 −→ X, we can define their direct sum in the following wayE1 ⊕ E2 = {(v1, v2) ∈ E1 × E2 | p1(v1) = p2(v2)} (2.8)Thus, the direct sum of two vector bundles is again a vector bundle, onewhere each fiber is the direct sum of the corresponding individual fibersE1,E2 at each point x ∈ X.Let us present the following factV ecnF(X ∨Y) = V ecnF(X)⊕ V ecnF(Y) . (2.9)where the direct sum of isomorphism classes is understood as isomorphismclasses of direct sums of vector bundles. The most important application weshall employ for the direct sum of vector bundles is the following:Let p : E −→ X be an n-dimensional vector bundle with X compactHausdorff, then ∃ p′: E′−→ X and m-dimensional vector bundle, for somem ∈ N, such that E⊕ E′is the trivial bundle ξn+m over X.This result will also be fundamental for Grothendieck’s completion.Given an n1-dimensonal vector bundle X, p1 : E1 −→ X and an n2-dimensional vector bundle p2 : E2 −→ X over the same base space , we candefine their tensor product in the following way:Let E1⊗E2 be as set, the disjoint union of vector spaces p−11 (x)⊗p−12 (x)for each x ∈ X. We now choose isomorphisms hi : p−1i (U) −→ U × Fn foreach open set U ⊂ X, such that E1 and E2 restricted to U are trivial. Thena topology Tp−11 (U)⊗p−12 (U)on the set p−11 (U)⊗ p−12 (U) is defined by makingthe fiber-wise tensor product h1⊗h2 : p−11 (U)⊗p−12 (U) −→ U × (Fn1⊗Fn2)a homeomorphism. the topology TE1⊗E2 is the union of all Tp−11 (U)⊗p−12 (U)for each U satisfying the conditions above.2.3.3 Pullbacks and universal bundleGiven two topological spaces X,Y, a vector bundle over one of them anda map between them we can construct a vector bundle over the other basespace in terms of the map and the bundle we already had172.3. K-theoryDefinition 26 Given a map f : X −→ Y and a vector bundle p : E1 −→ Y,then there exists a vector bundle p′: E′1 −→ X with a map f′: E′−→ Etaking the fiber of E′over each point x ∈ X by a linear isomorphism ontothe fiber of E over f(x). The bundle E′is called the pullback of E by f .The pullback is unique up to isomorphism of vector bundles, so we canconsider the map f∗ : V ecnF(Y) −→ V ecnF(X), taking the isomorphism classof E to the isomorphism class of E′. We will write the pullback as f∗(E)instead of E, making our notation clearer.Example 11 The tangent bundle TSm is the pullback of the tangent bundleTRPm under the quotient map Sm −→ RPm, identifying antipodal pointsin Sm.We enlist the properties which are satisfied by pullbacks1. (fg)∗(E) ≈ g∗ (f∗(E)).2. Id∗(E) ≈ E.3. f∗(E1 ⊕ E2) ≈ f∗(E1)⊕ f∗(E2).4. f∗(E1 ⊗ E2) ≈ f∗(E1)⊗ f∗(E2).Now that we have seen pullbacks and their properties, we can study theuniversal bundle, we first introduce some terminology and construct suchclassifying spacesDefinition 27 A Grassman manifold Gk(Fn) is the set of k-dimensionalF-planes which pass through the origin of Fn.We can define an action of the n-dimensional general F-linear group GLn(F)on Gk(Fn) asGLn(F)×Gk(Fn) −→ Gk(Fn) (2.10)(A, {e1, ..., ek}) 7→ span{Ae1, ...Aek}. (2.11)However, if A ∈ GLk(F) × GLn−k(F ⊂ GLn(F) then span{Ae1, ...Aek} =span{e1, ..., ek}, leaving any k-plane fixed. Thus GLk(F)×GLn−k(F) is theisotropy subgroup of the action. Gk(Fn) is an homogeneous space since foreach pair x, y ∈ Gk(Fn) there is an A ∈ GLn(F) such that Ay = x.Define the partition ofGLn(F) into left cosets GLn(F)/GLk(F)×GLn−k(F) ={g (GLk(F)×GLn−k(F)) | g ∈ GLn(F)}. We can give a bijection betweenGk(Fn) andGLn(F)/GLk(F)×GLn−k by identifyingGk(Fn) 3 y ⇔ A (GLk(F)×GLn−k(F)) ∈182.3. K-theoryGLn(F)/GLk(F)×GLn−k(F) such that (A, y) = x, where x ∈ Gk(Fn) arbi-trary but fixed.Thus, viewing GLn(F) as a topological group, we endow Gk(Fn) = GLn(F)/GLk(F)×GLn−k(F) with the quotient topology.Since we can view the action of GLn(F) as rotations of the k-planes,we lose nothing by contracting it to its maximal unitary subgroup U(n) forF = C, O(n) and Sp(n) for F = R,H respectively. Thus Gk(Fn) is thequotient subgroup of a compact Hausdorff space by a compact Hausdorffsubspace, hence it is compact Hausdorff.The inclusions F ⊂ F2 ⊂ F3 ⊂ .......Fn ⊂ .... induce the natural inclu-sions Gk(FK) ⊂ Gk(Fk+1) ⊂ Gk(Fk+2)........, hence when we take the limitGk(F∞) =⋃nGk(Fn) and give it the direct limit topologyDefinition 28 Given a union of topological spaces X∞ =⋃nXn, a set U isopen in X∞ iff U ∩Xn is open in Xn ∀n.We are finally in a position to construct a bundle p : Ek −→ Gk(F∞) whereEk = Gk(F∞)×F∞ and p(x, v) = x ∀ v ∈ x. Then we have the extraordinaryproposition, whose proof we must also omitFor every vector bundle L over a compact Hausdorff space X, there ex-ists a map f : X −→ Gk(F∞) such that L ≈ f∗(Ek). In other terms,[X,Gk(F∞)] ≈ V eckF(X).A similar but more general idea is that of a classifying space, which weshall use later, for our purposes, a classifying space F is a topological spacewhich we can use homotopy classes and loop spaces from a space X to F toconstruct cohomology groups of X for a given cohomology theory, but weshall see this later.2.3.4 Semi-group and its Grothendieck completionGiven [En] ∈ V ecnF(X), [Em] ∈ V ecmF (X) we can define a direct sum ofisomorphism classes as [En] ⊕ [Em] = [En ⊕ Em] ∈ V ecn+mF (X). Thus, wehave a natural inclusion of V ecnF(X) ⊂ V ecn+1F (X), [E] 7→ [E ⊕ ξ], makingthe union⋃nV ecnF(X) denoted V ecF(X) a semi-group, with identity ξ0, giventhat it satisfies the the axioms of a group except the inverse axiom.192.3. K-theoryWe can further define an equivalence relation between n-dimensional vectorbundles known as stable equivalenceE1 ∼s E2 ⇔ E1 ⊕ ξm ≈ E2 ⊕ ξm for some m ∈ Z+. (2.12)Notice that we loose some information under this equivalence relation, as inthe following exampleExample 12 Let our vector bundle be TSn, the tangent bundle of the n-dimensional sphere, then TSn ∼s ξn, for we can always add the normalbundle NSn, which is the one-dimensional vector bundle consisting of theposition vector x of each point in Sn. By construction NSn ≈ Sn × ξ andalso TSn ⊕NSn ≈ ξn+1. ThusTSn ⊕ ξ ≈ ξn+1,= ξn ⊕ ξ . (2.13)Thus stable equivalence is not without its consequences as we can see fromthis example, general relativity and classical statistical mechanics would bedestroyed by taking this equivalence.If X is compact Hausdorff, then ∼s satisfies the cancelation property, that isE1 ⊕ E3 ∼s E2 ⊕ E3 ⇒ E1 ∼s E2, (2.14)since on a compact Hausdorff base there is always a bundle LE3 such thatE3 ⊕ LE3 ≈ ξn+m.There is a universal way of constructing an Abelian group G out of a semi-group (S, ∗) and an equivalence relation ∼ in S. Let (a, b), (c, d) ∈ S × S,then we construct the equivalence relation(a, b) ∼Grothendieck (c, d)⇔ a ∗ d ∼ c ∗ b . (2.15)Example 13 Let (S, ∗) = (N ∪ {0},+) and define the equivalence relation∼ to be equality, then(a, b) ∼Grothendieck (c, d)⇐⇒ a+ d = b+ c . (2.16)We denote the pair (a, b) as a − d, and so the Grothendieck completion of(N ∪ {0},+) is (Z,+), the integers.202.3. K-theoryWe can hence construct an Abelian group from V ecF(X), ∼s and the can-celation property for a compact Hausdorff space as follows:Let ([E], [E′]) ∈ V ecF(X)× V ecF(X), we define an equivalence relation([E1], [E2]) ∼Grothendieck ([E3], [E4)]⇐⇒ [E1]⊕ [E4] ∼s [E2]⊕ [E3] . (2.17)We thus, denote elements of the Grothendieck completion of (V ecF(X),⊕)as [E1]− [E2] and, at long lastDefinition 29 the Grothendieck completion of the semi-group of stable iso-morphism classes of vector bundles over a compact Hausdorff space X isKF(X), the K -group of X.Given two elements [E1]− [E2], [E3]− [E4] of KF(X), we have besides theaddition, a multiplicative structure induced by teh tensor product of vectorbundles. The way to realize this operations is as follows([E1]− [E2]) + ([E3]− [E4]) = ([E1]⊕ [E3])− ([E2]⊕ [E4]) , (2.18)([E1]− [E2])([E3]− [E4]) = ([E1]⊗ E3])− ([E1]⊗ [E4])− ([E2]⊗ E3])+ ([E2]⊗ [E4]) . (2.19)As it can be seen from its definition, for a point {x0} we obtainKF({x0}) ≈ Z . (2.20)Thus K -theory can at best be an extraordinary cohomology theory, since itdoes not satisfy the dimension axiom [19], in which all cohomology groupsHn({x0}) = 0 ,∀nn 6= 0.Definition 30 We can define a homorphism induced by the projection of Xinto {x0} [24]K (X)→ K ({x0}) ≈ Z . (2.21)The kernel of this homomorphism is called the reduced K -theory of X,denoted as K˜F(X).Reduced K -theory plays a central role in our classification of topologicalFermi surfaces, though through a rather indirect route. It is also fundamen-tal for Bott periodicity, which we will discuss in the next section.212.3. K-theory2.3.5 K-groups and Bott periodicityAs we stated before, a cohomology theory on a space X gives a sequenceof Abelian groups, but so far we have only constructed one such group,namely K (X). Thus, without introducing properly the motivation for sucha definition we construct the higher K -groupsDefinition 31 K nF (X) ≡ K˜F(SnX+) ∀n ≤ 0, where once again Sn denotessuspending X+ n times and X+ denotes the union X∪ {+}, where {+} is adisjoint base-point.Roughly speaking, this definition is given because we can construct longexact sequences of these K -groups through a sequence of spaces built out ofunions of cones of both a space X and a closed subspace Y, and then iteratingthese as the new X and Y’s, so that the next space in the sequence is builtout of the previous two. We also use quotients of these, which yield SnX+and SnX/Y+ (The exact sequences induced by the inclusion and quotientmaps). We especially recommend chapter II of [20] for a much more detaileddiscussion.As one can see from our definition, we used the suspensions of X+ and notsimply X, this is obviously because otherwise we would not have the requiredlong exact sequence. It has important consequences when computing theseK -groups, for exampleSX+ ' SX ∨ S1 . (2.22)So that using our definition we haveKF−1(X) ≈ K˜F(SX)⊕ K˜F(S1) . (2.23)If F = C then, since (see below for this notation) K˜ (S 1) ≈ 0 this means thatK−1(X) coincides with the reduced K -group K˜−1(X), where as this is verydifferent for F = R, whereKO−1(X) ≈ K˜O(X)⊕ Z2 . (2.24)Hence , we have constructed half a cohomology theory, since we are stillmissing a definition for n > 0. For this we shall make use of a very deepresult in topology known as Bott periodicity, developed by Raoul Bott inthe end of the 1950’s.Starting from here it is necessary to differentiate the choice of field F. IfF = C, the K -group. simply as K (X) (In some old references it may be222.3. K-theorydenoted as KU (X) instead) and KO(X) for F = R. We shall ignore the caseF = H since it shall not be directly related to our classification.We begin by presenting the original statement of Bott periodicity, whichconcerns homotopy groups of classical spaces.Given the inclusions U(1) ⊂ U(2) ⊂ ..... ⊂ U , where again U =⋃nU(n) isthe direct limit, thenpin+2(U) ≈ pin(U) ∀n (2.25)Analogously, the inclusions O(1) ⊂ O(2) ⊂ ..... ⊂ O, O =⋃nO(n) generatespin+8(O) ≈ pin(O) ∀n (2.26)In terms of loop spaces we obtainΩ2U ' U , (2.27)Ω8O ' O . (2.28)Now its connection with K -theory comes about because the homotopy classesof a compact Hausdorff space X to a classifying space, either BO, BU forO and U respectively satisfyK (X) ≈ [X, BU × Z] , (2.29)KO(X) ≈ [X, BO × Z] . (2.30)Thus the periodicity in loop spaces induces a periodicity in K -groupsK−2(X) ≈ K (X) , (2.31)KO−8(X) ≈ KO(X) . (2.32)Allowing us to define groups for positive n as mod 2 for K (X) and mod 8 forKO(X).One can express complex and real K -groups as homotopy classes of classicalsymmetric spaces, whose full study was originally by H.Cartan. We presentthese spaces in ??Proving this results is quite a feat, but all details, specially for classifyingspaces can be seen in [24]. We should mention at this point that there is aconnection between Bott periodicity and Clifford algebras, but since we willnot use them directly, we defer them to be studied again in [24], however,we will review a little about them when we criticize the existing literatureon the classifications.232.3. K-theoryn Classifying Space (KO) Classifying Space (K )0 (O/(O ×O))× Z (U/(U × U))× Z−1 O U−2 O/U−3 U/Sp−4 (Sp/(Sp× Sp))× Z−5 Sp−6 Sp/U−7 U/OTable 2.1: Classifying spaces for real and complex K -theory. 8 classify-ing spaces for KO , where Sp =⋃nSp(n), is the direct limit of Sp(n)’s,where Sp(n) is the n-dimensional Symplectic group, the unitary group ofGLn(H). Complex K -theory only has 2 classiying spaces.This work was mostly done in the late 1950’s and early 1960’s. When wederive our classification of topologically stable Fermi surfaces, we will presentgeneralizations of this work due to Atiyah and Singer.As a reminder, the references we have employed through this chapter are[34], [31], [19], [20] and [24] and should be consulted for more details.24Chapter 3Physical setupThe most general area of physics one can fit to the kind of systems we shallstudy is condensed matter physics. It was formerly known as solid statephysics but over the past decades the field has widen its gaze. Condensedmatter can be portrayed as studying thermal, optical and magneto-electricproperties of materials through modeling the behavior of the electrons andatoms (or ions) which constitute the materials. It was in the 1930’s thatFelix Bloch, Eugene Wigner, Le´on Brillouin and others revolutionized thetheory of solids in terms of the periodic structure of crystals and quantummechanics. Nowadays it is perhaps the largest field in terms of its work-ing force and applications. All modern electronic systems which surroundus are in their majority understood by the theoretical developments of the30’s, 40’s and 50’s. We will use these building blocks to study propertiesof exotic states of matter, particularly Fermi surfaces of such states. Suchmaterials, as of yet, have no general application but perhaps will be of greatapplicability in the future, since as it can be deduced from our previousdiscussion, topological structures are very robust, stable against continuousdeformations, so we can conceive that topological properties of systems areinherently robust. Nonetheless we should not get ahead of ourselves, so farthese states arise (as far as experiment goes) in particular crystal structuresand at extremely low temperatures, often with the need of a strong mag-netic field such as is the case for the quantum hall effect, discussed in theintroduction.There are many references for condensed matter physics such as[1], [9],[26]we shall mostly follow [1].3.1 Free n electrons in a boxLet us study a system of non-interacting N electrons confined to a n-dimesional box of volume V . Since electrons are non interacting we canunderstand this problem by solving a single electron system and copy it Ntimes. Thus, we begin with a single electron Hilbert space Hs−e, and its253.1. Free n electrons in a boxstate vector |ψs−e(~r, t)〉, which satisfies the Scho¨dinger equationHs−e(t) |ψs−e(~r, t)〉 = i~∂∂t|ψs−e(~r, t)〉 , (3.1)where Hs−e(t) is the single electron Hamiltonian operator and ~ is the re-duced Planck’s constant. We will assume that our Hamiltonian is time-independent or equivalently that our system lies in a stationary state, thus|ψs−e(~r, t)〉 = |ψs−e(~r)〉 ei ~ t , (3.2)Hs−e |ψs−e(~r)〉 =  |ψs−e(~r)〉 . (3.3)For a one particle Hamiltonian, Hs−e = P22m = −~22m∇2, so that the solutionto the Schro¨dinger equation, ignoring boundary conditions is|ψs−e,~k(~r)〉 = V− 12 |ei~k·~r〉 , (3.4) ≡ (~k) = ~2~k · ~k /2m, (3.5)~~k = ~p . (3.6)The solution ei~k·~r is known as the plane wave solution and ~k is called thewave vector. The name plane wave comes from the fact that the solution isconstant for all planes orthogonal to ~k and periodic along lines parallel toit.Some remarks are in order. First, when writing equations in quantum me-chanics momentum, position and the wave vector can be referring to op-erators and/or vectors, so its important to distinguish from context whatmathematical object we are employing. Secondly, the V −12 factor is a nor-malizing factor, so that the probability of finding an electron with wavevector ~k always adds up to 1.∫Vd~r 〈~r, ψs−e,~k(~r)〉 〈ψs−e,~k(~r), ~r〉 = 1 . (3.7)Here we should stop to highlight an obscure mathematical detail which isnot mentioned in most textbooks on quantum mechanics. The single parti-cle Hilbert space is isomorphic to L2, with basis plane waves |ψ(~k)〉, but wesaw in the previous chapter that L2 is separable, so it has a countable basis.This is at odds with what is usually taught in introductory textbooks onquantum mechanics where we are told that our Hilbert space has as basis{|~r〉 , ~r ∈ R3}, which, by the nature of R, is uncountable. The correct inter-pretation is that the physical Hilbert space is not a Hilbert space perse´ as263.1. Free n electrons in a boxmathematicians define it but what is called by mathematicians themselvesas a rigged Hilbert space. That is a Hilbert space with additional structure.One of the motivations for physicists to employ a rigged Hilbert space is sothat experimental setups are better described in terms of ~r for systems of asmall number of particles. For condensed matter systems, the position of asingle particle is in general not relevant observable, thus, we shall continueto employ a separable Hilbert space and use all the standard mathematicalproperties mathematicians have derived for them.We now have to make a choice of boundary conditions, for our purposesit suffices to choose periodic boundary conditions, so that|ψs−e,~k(~r)〉 = |ψs−e,~k(~r + ni~li)〉 (3.8)where ~li is L = V13 in the i-th component and zero elsewhere, ni ∈ Z and wehave used Einstein’s summation convention. This conditions are equivalentto gluing the opposite sides of our box, making it T3 a 3-dimensional torus.This gluing is not realizable in R3 but as we will see later, this will not bea bad representation of our systems because of their periodicity.This choice of boundary conditions has a powerful implication sinceei~k·ni~li = 1 , (3.9)~k · ni~li = = 2pim ,m ∈ Z , (3.10)(3.11)and so~k =2piL~m , ~m ∈ Z3 . (3.12)Momentum and energy are quantized! Or, said differently, we can only havediscrete, whole values for momentum in terms if ~ and L. Given that weare dealing with electrons, which obey Fermi-Dirac statistics and at its corePauli’s exclusion principleNo two fermions may occupy the same one particle state |ψs−e〉. Thusfor every wave vector ~k there are 2 electrons, one with spin ~2 and one with−~2 .Thus, we can construct an N electron system by filling states corre-sponding to the lowest energy level (~k = 0), 2 electrons for each ~k until wehave used up our N electrons, filling a region in ~k-space. In our case, since273.2. Bravais lattice(~k) ∼ ~k2 and typically N ∼ 1023, our region is indistinguishable from theclosed disk BkF (0), of radius kF centered at 0 ∈ ~k-space. The boundary ofthis disk ∂BkF (0) = SF is called the Fermi sphere.Notice that we have used two facts, one is that ~k is quantized so that kF isnot arbitrarily small, in fact it may be quite large since for our casekF = (3pi2NV)13 . (3.13)The other fact is that though it is discrete we are approximating it by acontinuum of points. This will be used extensively, particularly to studytopological properties of such continuum approximation. It would perhapsbe interesting to study in a more rigorous manner this continuum limit,using a tool called persistent homology [38] but we will not discuss this anyfurther.For more general systems there is an analogous construction of filling aregion in ~k-space, its boundary known as the Fermi surface. In general itdoes not yield a sphere but a surface with a much richer topological nature.It may not even be a connected surface nor its components posses the samedimensions. Materials which posses a Fermi surface behave like metals, wereas those where the filled region has no boundary, e.g no Fermi surface areknown as insulators. We shall present later on a more general and usefuldefinition of a Fermi surface.3.2 Bravais latticeTo describe the periodic structure of a crystal, it is necessary to define aBravais latticeDefinition 32 A Bravais lattice consists or all points with position vec-tors of the form~R = ni~ai , {~ai} L.I, ni ∈ Z . (3.14)L.I stands for linearly independent.One can see examples of Bravais lattices in 3.1a,3.1b. Real crystals are, ofcourse, not infinite and there is a very important area of study of surfaceeffects and edge states, which are in fact particularly important for topolog-ical phases of matter, such as the IQHE. Nevertheless since we have aroundN ∼ 1026 − 1027 ions in a crystal it is a very good first approximation, par-ticularly to describe what is known as the bulk of the crystal. If we consider283.2. Bravais lattice(a) (b)Figure 3.1: (a) A 3-dimensional cubic Bravais lattice (b) A two-dimensionalHoneycomb or Hexagonal lattice. Figures taken from [1].periodic boundary conditions , then there is no necessity for an infinite ar-ray, restricting the ni’s to be bounded.We will now define a primitive cell which is the most common way of parti-tioning a Bravais latticeDefinition 33 The Wigner-Seitz primitve cell is the region in Rd whichis closest to a specific point in the Bravais lattice. Since the array is periodicall points in the lattice have the same Wigner-Seitz primitive cell.Hence, Rd can be completely foliated by the Wigner-Seitz cell. We shall notmake much use of the Wigner-Seitz cell of the direct Bravais lattice of ourcrystal, but as we will see, a similar construction will be the basis of ourclassification.3.2.1 Reciprocal latticeThe reciprocal lattice is also a fundamental construction underlying thestudy of crystals. Even though it is constructed from the direct Bravaislattice, it is this structure which is employed in the analysis of materialsand ironically not the direct lattice (so in that sense the name is misleadingbut there is perhaps no better alternative).Definition 34 The reciprocal lattice is set of wave vectors { ~K}, such293.3. Bloch’s theoremthate~K·~R = 1 , (3.15)Where ~R = ni~ai is an element of the Bravais lattice, as defined above.Notice that we have defined the reciprocal lattice { ~K} in terms of a choiceof direct lattice {~R}. Since the only allowed coefficients of the generatorsof the direct lattice are integers, each element ~K = kj~bj of the reciprocallattice must satisfy~K · ~R = nisj(~ai ·~bj) = 2pim, m ∈ Z ,∀ ~R. (3.16)The simplest way to construct such a lattice is to choose ~bj so that~ai ·~bj = 2piδij , (3.17)sj ∈ Z . (3.18)Hence our reciprocal lattice is also a Bravais lattice!The Wigner-Seitz cell of the reciprocal lattice is called the first Brillouinzone and we shall denote it by X. We will see in our classification that theBrillouin zone will play the role of our topological base space! We will seein Bloch’s theorem that for periodic boundary conditions the Brillouin zoneis always a torus.3.3 Bloch’s theoremConsider an electron in a crystal array described by a Bravais lattice {~R}.Because of the periodicity of the lattice, the potential in the Schro¨dingerequation satisfiesU(~r + ~R) = U(~r) , ∀ {~R}. (3.19)For such systems we have Bloch’s theoremBloch 1 Given the one-electron Schro¨dinger equation with a periodic po-tential and periodic boundary conditionsH(~r) |ϕ(~r)〉 = ε |ϕ(~r)〉 , (3.20)For each eigenvector of H(~r) there exists ~k such that|ϕn(~r + ~R)〉 = ei~k·~R |ϕn(~r)〉 . (3.21)where ~k is known as the crystal momentum.303.3. Bloch’s theoremHere we can deduce the fact that our Brillouin zone is topologically a torus,constructing the reciprocal lattice { ~K} corresponding to the direct lattice{~R} and multipliying by 1 = e ~K·~R, we have|ϕn(~r + ~R)〉 = ei~k·~R |ϕn(~r)〉 ,= ei~K·~Rei~k·~R |ϕn(~r)〉 ,= ei(~k+ ~K)·~R |ϕn(~r)〉 . (3.22)Thus, for each ~k ∈ X , ~k + a ~K labels the same quantum state for a ∈ Z,Since the Brillouin zone translated by the reciprocal lattice foliates ~k-space,we can and should restrict ourselves to points in X.We shall hence forth, restate the problem at hand in manner that exploitsthis periodicity to maximum. Define the Bloch Hamiltonian (also known asthe crystal momentum Hamiltonian) asH(~k) =∑~re−i~k·~rH(~r)ei~k·~r , (3.23)with corresponding eigenvectors|Φn(~k)〉 =∑~re−i~k·~r |ϕn,~k(~r)〉 , (3.24)where∑~rshould properly be something similar to1V∫d~r but we avoidthis to simplify notation.Together, they satisfyH(~k) |Φn(~k)〉 = εn(~k) |Φn(~k)〉 . (3.25)Where {εn(~k)} are known as energy bands and are assumed to be continuousfunctions of ~k, this assumption will be fundamental for our construction.Energy bands acquired their name from the tight-binding approximation,where the potential that the electrons are subject of is assumed to be thatof a single ion and is related to the atomic levels s, p, d and so on. We willnot have much to say about the tight-binding approximation other than thatit is a very successful model and has many applications.A few but very important remarks are in order. Since X is a torus, whichis compact and εn(~k) is continuous, εn(~k) must be bounded! Nevertheless,εn(~k) is a countable set for each ~k and bounded operators on a Hilbertspace are locally bounded functions because of the norm of the Hilbert313.4. Fermi surfacespace. The crystal momentum ~k plays the analogous role of the momentumoperator ~p but they are not equivalent, since in general the eigenstates ofthe Hamiltonian are not eigenstates of momentum.3.4 Fermi surfaceWhen the system under study is in a metallic phase, it means that some ofthe energy bands are partially filled. For these partially filled bands, therewill be a surface embedded in X, separating the filled ~k from those whichare not.The derivations we have employed so far assume zero temperature. Exper-imentally it is possible to describe systems at very low temperatures to agood approximation using this schemes. In those cases one can experimen-tally control the chemical potential of the system, which is the equivalent ofthe Fermi energy εF , and so it is equivalent to setting the Fermi energy ofthe system.The Fermi surface are the set of points ~k in X such that at least ∃m ∈ N, sothat εm(~k) = εF . There may be more than one such m for a given ~k, thesedifferent m’s are called the branches of the Fermi surface, the Fermi surfacebeing the union of all such branches.In mathematical terms we can define the momentum-space propagator (orparametrix) of a constant energy ε-manifold asG(~k, ε) |Φn(~k)〉 =1εn(~k)− ε|Φn(~k)〉 , (3.26)(H(~k)− εI)G(~k, ε) = I , (3.27)where the last equation holds except at the poles of G(~k, ε), which corre-sponds to the kernel of (H(~k)− εI).Thus, the Fermi surface of system corresponds to the set of points of itsBrilluoin zone such that(H(~k)− εF I) = 0 , (3.28)or equivalently the collection of all ~k ∈ X for which G(~k, εF ) has a polesingularity. Hence our operator of interest is (H(~k) − εF I) and we wish tostudy its kernel, which has a physical interpretation. An important remarkwe can make at this point is that this is consistent with the freedom ofchoosing the zero of the energy scale in quantum mechanics, for εF hasitself a geometric interpretation, since it tells us if a ~k ∈ X are occupied or323.4. Fermi surfacenot. Any rescaling of the energy can change the numerical value of εF butcannot change whether a given ~k is occupied or not! Nor the degeneracydepends on such value. We have the freedom to choose our energy scalesuch that εF = 0 and we shall only write H(~k) instead. This definitionsare useful for they generalize in some cases where there is electron-electroninteractions and a ~k- space but not a Brillouin zone. Non the less, we willrestrict our selves to the framework we have elucidated here, where there areno electron-electron interactions and hence, we have a well defined Brillouinzone with a topology in the continuum limit approximation.33Chapter 4ClassificationWe are now ready to develop a classification of the different topologicallystable Fermi surfaces which can arise in metallic phases of condensed mattersystems. Let us refrain from discussing results obtained in our constructionuntil 4.4. First, we need to introduce some more mathematical conceptswhich are necessary but the basic tools for their constructions were intro-duced in 2. We will also employ the conventions of [34], [3], [4] and [33].Let us remind ourselves of the topology we will be applying to our op-eratorsDefinition 35 Let B(H ) be the Banach algebra of bounded operators on aseparable Hilbert space H . The Norm topology of B(H ) is the one inducedby the norm on H , where for T ∈ B(H ), it is given by‖T ‖B(H ) = supv∈H‖T v‖H‖v‖H, (4.1)and the norm topology is just as defined in 2.One of the key properties we shall use of the norm topology for B(H ) isthat the composition of operatorsB(H )× B(H ) −→ B(H ) , (A,B) 7→ AB (4.2)is continuous.There are other types of topologies for B(H ) such as the so-called weakand strong operator topologies. We should mention that the norm topologyis the strongest topology for B(H ), it is also more natural than the othertwo from a classical picture point of view. We should also mention that in[15] they employ a different topology which is for more general operatorsthan bounded operators, called the compact-open topology and it is fromtheir point of view a more natural topology for operators on the Hilbertspace under study. That being said, in condensed matter systems with noelectron-electron interaction, were we have Bloch’s theorem, the eigenvalues344.1. K-theory and Fredholm operatorsof our Hamiltonian are assumed to be continuous over the Brillouin zone X,which is compact. Therefore, the eigenvalues of our Hamiltonian (energybands εn(~k)) must be bounded, a fact whose satisfaction is not evidentlyclear by the choice of topology in [15].4.1 K-theory and Fredholm operatorsIn what follows we shall relate K -theory and Fredholm operators, essentiallyfollowing the work of Atiyah [3]. Let us endow the set of Fredholm operatorsF(H ), defined as a set in 2 by first defining a topology on the Calkinalgebra CAL(H ) by giving B(H ) the norm topology and quotienting byset of compact operators K(H ). Then, we restrict to the set of invertibleoperators in CAL(H ), denoted as CAL∗(H ) and consider the projectionpi : B(H ) −→ CAL(H ) . (4.3)Then the set of Fredholm operators is pi−1 (CAL∗(H )). Since F(H ) isopen in B(H ), the composition of Fredholm operators is Fredholm (inducedby the norm topology on B(H )), that adjoints of Fredholm operators areFredholm, since every open set in B(H ) is open in B(H ) and the adjointrelation is defined in terms of the norm on H and, finally that the additionof a compact operator and a Fredholm operator is again Fredholm.Associated to a given Fredholm operator H is its indexIndexF = dim kerF − dim kerF† , (4.4)The index satisfies the following very useful propertiesIndexF ◦ F ′ = IndexF + IndexF ′ , (4.5)IndexF +K = = IndexF , (4.6)for F , ′F Fredholm and K compact.If we now study a family of Fredholm operators F(x), where x ∈ X andX is compact, it happens that the dimension of the kernel of F(x) is onlysemi-continuous with x, jumping in N, however its index is a locally constantfunction (may change for different path components of X).We can generalize the index function in order to describe elements in K (X).First, given an ONB {e0, e1, ....} of H , we define the following subspaces,354.1. K-theory and Fredholm operatorstogether with their corresponding projection operator asHn = {span{ei, ei+1, .....}, i ≥ n} , (4.7)H ⊥n = H /Hn , (4.8)Pn : H −→Hn , (4.9)IndexPn = 0 . (4.10)The last equation being true since Pn is self-adjoint. Thus, for any contin-uous Fredholm family F(x) we can defineFn(x) = Pn ◦ F , (4.11)IndexFn(x) = IndexF(x) . (4.12)Since Fn(x)(H ) = Hn ∀x and dim kerF†n(x) = dimH ⊥n = n for all x ∈ Xfor sufficiently large n by construction (fact proven at the beginning of [3]),we obtainIndexF(x) = d− n , (4.13)where d = dim kerFn(x) and now it is only d which is an unknown.Consider a family of vector spacesV (x) = kerFn(x) . (4.14)and consider the topological spaceE =⋃x∈XV (x) ⊂ X×H . (4.15)E is a locally trivial vector bundle with base space X!Imagine that we now wish to study homotopy classes of families of Fredholmoperators, parametrized by a compact space, then we have the followingincredible theorem due to AtiyahAtiyah-Fredholm 1 Let X be a compact space and let [X,F(H )] denotethe set of homotopy classes of continuous maps F : X −→ F(H ). Then theoriginal index map F(x) 7→ IndexF(x) induces an isomorphismIndex : [X,F(H )] −→ K (X) , (4.16)IndexF = [kerFn]− [X×H⊥n ] . (4.17)Thus, F(H ) ' BU × Z.364.1. K-theory and Fredholm operatorsWe should clarify that, though the construction seems to depend on a choiceof orthonormal basis for H and a choice of n for the subspaces Hn, howeverit can be showed thatkerFn+1 ≈ Fn ⊕ ξ , (4.18)kerF†n+1 ≈ F†n ⊕ ξ , (4.19)were the trivial line bundle ξ is generated by the extra element in the kernel,en. Also the former property it was only used as having any ONB, hence itis independent of the choice of ONB.Let us now move on to study a particular kind of Fredholm operators,known as skew-adjoint Fredholm operators, denoted as Fˆ(H ). The topologyof these operators was first studied by Atiyah and Singer [4]. Skew-adjointFredholm operators are a subset of Fredholm operators F with IndexF = 0= BU by the preceding theorem.First we split Fˆ(H ) into 3 disjoint components• Fˆ+(H ) = {iεn ≤ 0 , only for finite number of n’s .}• Fˆ−(H ) = {iεn ≥ 0 , only for finite number of n’s .}• Fˆ∗(H ) = Fˆ(H )/(Fˆ+(H ) ∪ Fˆ−(H ))Elements of Fˆ+(H ) are called essentially positive and elements of Fˆ−(H )are called essentially negative. Equivalently we can say that Fˆ+(H ) andFˆ−(H ) have invariant subspaces of codimension n, where either the eigen-values are always positive or always negative.We shall now prove that Fˆ+(H ) and Fˆ−(H ) are null-homotopic . ConsiderH ∈ Fˆ+(H ) and the following homotopyHt = (1− t)H− itI , (4.20)H†t = −Ht . (4.21)(4.22)Since Ht is skew-adjoint for all t, it has an ONB in H , written in terms of{|ϕn〉}, ONB associated to H, we can see thatHt |ϕn〉 = [(1− t)εn − it] |ϕn〉 . (4.23)374.1. K-theory and Fredholm operatorsThus,iεn(t) = (1− t)iεn + t ≥ 0 , (4.24)and so, for every t, Ht ∈ Fˆ+(H ), thus there is path from every operatorH ∈Fˆ+(H ) to −iI ∈ Fˆ+(H ), so we can finally conclude that Fˆ+(H ) is null-homotopic. Interchanging itI for −itI we obtain an equivalent retraction ofFˆ−(H ) to iI.The remaing non-trivial component of Fˆ(H ), Fˆ∗(H ) is of great importanceto us because of the following theoremAtiyah-Singer-Skew 1 The mapα : Fˆ∗(H ) −→ Ω (F(H ), I) (4.25)A 7→ (I cospit+A sinpit , t ∈ [0, 1]) , (4.26)Is a homotopy equivalence.At first sight, the map given above is not even a loop since it starts at Iand ends at −I, however since GL(H ), the set of invertible operators of ourseparable Hilbert space is null-homotopic, we can (through a very intricateway) deform this path into a loop in F(H ). The original proof of this is dueto Atiyah and Singer [4]. There is no simple way to summarize this proofwithout taking too great a detour of our interest, nevertheless we suggestour readers to attend to the the lecture notes of Dan. S. Freed [14]. Wewill only give a few remarks and examples concerning this proof, which areemployed in these notes.First, because of the previously sketched theorem of Atiyah, we knowthat F(H ) ' U/ (U × U) × Z, then, the above theorem would imply thatFˆ∗(H ) ' U , since Ω (F(H ), I) ' Ω (U/ (U × U)× Z) ' U . Notice thatpi0 (F(H )) = Z with each component determined by the index, thus Ω (F(H ), I)are simply loops in the component containing I, that being Fredholm op-erators of index 0. Hence we only need to proove Fˆ∗(H ) ' U . Now weretract Fˆ∗(H ) to Fˆ∗(H ), where H ∈ Fˆ∗(H ) satisfies• H is Fredholm.• ‖H‖ = 1.• IndexH = 0.• H† = −H.384.1. K-theory and Fredholm operators• H is neither essentially positive nor essentially negative.The deformation retraction is given byHt = H[(1− t) +t‖H‖], (4.27)which is well defined since ‖H‖ 6= 0 because H is Fredholm. Now let usconsider a new subset of B(H ), namelyGLK(H ) := {P ∈ GL(H ) | P − I compact}We then retract GLK(H ) to UK(H ) using the same homotopy we used forFˆ∗(H ) and Fˆ∗(H ). One can prove (using methods for infinite dimensionalmanifolds developed by Pais) that UK(H ) is homotopic to U , the classifyingspace for K−1( ). This is our connection to Ω(F(H ), I)' U and so, weonly need to prove Fˆ∗(H ) ' UK(H ) viaβ : Fˆ∗(H ) −→ −UK(H )J 7→ epiJ . (4.28)and we can conclude. We will do no such thing except give an example.Let T = P + I (P ∈ UK(H )) compact, furthermore assume T has finiterank, (a special subset), meaning dimH /T (H ) = ∞, then naming L =kerT we haveH = L ⊕L ⊥ . (4.29)Suppose Q ∈ β−1(P), so that P = epiQ. Note that the exponential for infi-nite dimensional operators is not injective. At L ⊥, which is by constructionfinite dimensional, the restriction Q|L⊥ is in one to one correspondence withP|L⊥ , since ‖Q‖ = 1 and Q = −Q†, then the eigenvalues of Q must lie in[−i, i], which maps almost homeomorphically to S 1 ⊂ C, except at the endpoints, both of which go to −i. With this in mind one can prove that Qsplits L = L+⊕L−, where Q(L+) = iL+ and Q(L−) = −iL−. Thus, β−1(P)is homeomorphic to U(L)/ (U(L+)× U(L−)), however, due to Kuiper’s the-orem all three spaces at hand are infinite dimensional Hilbert spaces, so allof them are contractible. and thus we can retract β−1(P) to P, viewingUK(H ) as ”base space” for Fˆ∗(H ) and β as a projection. This is the gen-eral strategy employed, where it is shown that an inverse image of β retractsto an element of UK(H ), however, we must be careful for the inverse imageβ−1 is not always homeomorphic to U(L)/ (U(L+)× U(L−)) (This was onlytrue for P finite rank).394.2. K-theory and Fermi surfacesWith this we have portrayed the path to proving both the theorem ofAtiyah and the theorem of Atiyah-Singer for skew-adjoint Fredholm op-erators. We will now proceed to show the connection between Fredholmoperators and Fermi surfaces.4.2 K-theory and Fermi surfacesUp to now, we have introduced, independently, many mathematical con-cepts leading up to K -theory and the basic condensed matter concepts ofelectronic transport theory in systems with discrete translational invariance,that is the Brillouin zone X and Bloch’s theorem. Furthermore, with a fewexceptions, we have not made any connections between K -theory and Fermisurfaces, though we have claimed that we would classify topologically stableFermi surfaces. We will immediately amend this. We will reintroduce manyof the necessary physical and mathematical concepts in a summary leadingup to the classification. This will be useful for readers who already knewthe concepts presented in the previous sections and chapters and are onlyinterested in the connections and results.Let X be the Brillouin zone of a non-interacting fermionic system. Foreach crystal momentum ~k ∈ X there is a Bloch Hamiltonian operator H(~k)acting on a complex separable Hilbert space H . H(~k) is hermitian [5],furthermore, there exists a complete orthonormal basis {|Φn(~k)〉}∞n=1 of H ,such thatH(~k) |Φn(~k)〉 = εn(~k) |Φn(~k)〉 , (4.30)where the ~k-dependent eigenvalues εn(~k) are known as energy bands.Define the momentum-space propagator (or parametrix) of a constant energyε-manifold asG(~k, ε) |Φn(~k)〉 =1εn(~k)− ε|Φn(~k)〉 , (4.31)(H(~k)− εI)G(~k, ε) = I , (4.32)where the last equation holds except at the poles of G(~k, ε), which corre-sponds to the kernel of (H(~k)− εI) (see chapter 3).The Fermi surface [1] of a non-interacting fermionic system corresponds tothe set of points of its Brilluoin zone for which one or more energy bandsare equal to the Fermi energy εF(H(~k)− εF I) = 0 , (4.33)404.2. K-theory and Fermi surfacesor equivalently the collection of all ~k ∈ X for which G(~k, εF ) has a polesingularity. Hence our operator of interest is (H(~k) − εF I) and we wishto study its kernel, which has a physical interpretation 1. We have thefreedom to choose our energy scale such that εF = 0 and we shall onlywrite H(~k) instead. If we restrict ourselves to the n-th energy band εn(~k),the corresponding surface embedded in the Brillouin zone is called the n-thbranch of the Fermi surface, the Fermi surface being the union of all suchbranches.We shall only consider Fermi surfaces which consist of a finite number ofbranches, implayingdim Ker H(~k) <∞ . (4.34)As in [25], we shall also make the physical assumption that a Hamiltoniandescribing this type of system is bounded. This is a generalization of modelswith a finite number of energy bands, which are inherently bounded opera-tors. Thus, we allow an infinite number of bands and as we shall see, thisinfinity will be of utmost importance for the physical interpretation of ourresults.Enlisting the assumed physical properties of our Bloch Hamiltonianin mathematical terms,• H(~k) is bounded,• dim Ker H(~k) <∞ ,• H(~k) is self-adjoint.Operators satisfying these conditions are known in the mathematical lit-erature as self-adjoint Fredholm operators [4],[33], where the set of Fredholmoperators [3] are bounded operators in H with finite kernel and cokernel2. We shall denote the space of all complex Fredholm operators F(H ) andself-adjoint Fredholm operators as Fsa(H ). Let us endow F(H ) with thenorm topology 2, as we did in 4.1.This choice of topology comes at a price, for we proved in 4.1 that op-erators which are essentially negative (finite number of conduction bands,εn(~k) > 0) or essentially positive (finite number of valence bands, εn(~k) < 0)are topologically trivial in the norm topology, so we must assume• H(~k) is neither essentially positive nor essentially negative.The former assumption adds an infinite number of trivial core bands and iscommon to all references we have cited here, that attempt at a classification.1No rescaling of the energy will change this kernel2In [15] they employ instead the compact-open topology for gapped systems414.3. ResultsAssuming an infinite number of valence bands is new, however, the behaviorof bands below (or above) the Fermi energy is physically relevant for thebehavior of our Fermi surface only when they cross, adding a branch to it,making them irrelevant otherwise. Let us denote the subset of operatorsthat satisfy all our assumptions Fsa∗ (H ).H(~k) is a continuous function of ~k and we wish to put in the same equivalencerelation all systems which can be adiabatically evolved 3 into one another[6] or equivalently, ∀ H0,H1 : X→ Fsa∗ (H ) we haveH0 ∼ H1 ⇐⇒ ∃ g : X× I → Fsa∗ (H ) continuous,g(~k, 0) = H0(~k), g(~k, 1) = H1(~k) ∀ ~k ∈ X . (4.35)Thus, all types of Fermi surfaces separated by this equivalence relation aregiven by the set of homotopies [X,Fsa∗ (H )].The set of skew-adjoint Fredholm operators Fˆ(H ) ≡ iFsa(H ) is triviallyhomeomorphic to Fsa(H ) in the norm topology, and so are Fsa∗ (H ) andFˆ∗(H ), the corresponding non-trivial component of Fˆ(H ). As we por-trayed in ??, Atiyah and Singer proved in [4] that Fˆ∗(H ) ' ΩF(H ), theloop space of the set of Fredholm operators F(H ). Combining this withAtiyah’s proof [3] that F(H ) is a classifying space for complex K -theoryand the suspension isomorphism [19] we get[X,Fsa∗ (H )] ≈ [X, Fˆ∗(H )] ,≈ [X,ΩF(H )] ,≈ [SX ,F(H )]∗ ,≈ K−1(X) , (4.36)where (2)K−1(X) ≈ K˜ (SX) denotes the Grothendieck completion [20], ofthe semi-group VectsC(SX) of stable isomorphism classes of complex vectorbundles over SX, the suspension of our Brillouin zone. It is an abelian groupand its elements are not vector bundles but virtual vector bundles!Notice that Grothendieck’s completion popped out naturally from our con-struction!4.3 ResultsWe will now compute K−1(X) for X = Td, d = 1, 2 and 3. To do this, weshall employ the formula seen in 2Σ(X×Y) ' ΣX ∨ ΣY ∨ Σ(X ∧Y). (4.37)424.3. ResultsThus for d = 1K−1(S 1) ≈ K˜ (SS 1) ,≈ K˜ (S 2) ,≈ Z. (4.38)For d = 2K−1(T2) ≈ K˜ (ST2) ,≈ K˜(S (S 1 × S 1)),≈ K˜(SS 1 ∨ SS 1 ∨ S (S 1 ∧ S 1)),≈ K˜(SS 1)⊕ K˜(SS 1)⊕ K˜(S (S 1 ∧ S 1)),≈ K˜(S 2)⊕ K˜(S 2)⊕ K˜(S (S 2)),≈ K˜ (S 2)⊕ K˜ (S 2)⊕ K˜ (S 3) ,≈ Z⊕ Z . (4.39)For d = 3, we have a more complicated computation sinceK−1(T3) ≈ K˜ (ST3) ,≈ K˜(S (S 1 × S 1 × S 1)),≈ K˜(SS 1 ∨ ST2 ∨ S (S 1 ∧ T2)),≈ K˜(S 2)⊕ K˜(ST2)⊕ K˜(S (S 1 ∧ T2)). (4.40)Now,S 1 ∧ T2 = T3/(S 1 ∧ T2). (4.41)In [15], theorem 11.8, it is shown thatT3 ' S 1 ∨ S 1 ∨ S 1 ∨ S 2 ∨ S 2 ∨ S 2 ∨ S 3 . (4.42)Thus,K˜(S (S 1 ∧ T2))≈ Z , (4.43)which yieldsK−1(T3) ≈ K˜(S 0)⊕ K˜(S 2)⊕ K˜(ST2),≈ Z⊕ Z⊕ Z⊕ Z. (4.44)Equations 4.38, 4.39 and 4.44 materialize the fruits of our labor. We willproceed to discuss part of the physical interpretation of these results in ourfinal section.434.4. Discussion4.4 DiscussionWhat is the physical interpretation of 4.38, 4.39 and 4.44 ? For d = 1,a class in K−1(S 1) ≈ pi1(Fsa(H ) is determined by the spectral flow [33].This invariant has been used to study the so called chiral anomaly in quan-tum field theory [32] but to our knowledge, had not been used previouslyto study Fermi surfaces. Quantum anomalies are, in short, the breaking ofsymmetries of the classical system by its mere quantization, never the less,they are independent of the choice of quantization procedure and play a fun-damental role in the standard model of particle physics. For an accessibleintroduction to anomalies, we refer to [22]. We will defer to discuss furtherinterpretation of the spectral flow for higher dimensions but will commentthat it is very likely to be related to a chiral current, which is an effect thatin principle, could have already been detected in experiments.Our results differ from those obtained in [30] and those presented in [12].We believe that this is because they restrict themselves to lower dimensionalsphere surrounding a single path-component of the Fermi surface. The rea-sons why they employ such a restriction is due to so called nesting insta-bilities. This instabilities are related to the Fermion doubling theorem [10].However, this types of argument do not seem to apply precisely when thereis chiral symmetry-breaking, which is exactly the result we obtained withthe spectral flow invariant! Also, such arguments apply to lattice modelswhere the field equations are discretized and their is a problem when takingthe continuum limit. This lattice argument applies to tight-binding modelswhere the number of energy bands is finite but in our case we have no dis-cretization and the number of energy bands is infinite as expected of the fullsingle particle Hilbert space.Our work is thus in the same rationale as [17],where we allow for all possible topologies for the Fermi surface and wherechirality plays an important role. This is also what [21] seems to have hadin mind as a generalization of his work. At the level of the derivation of theclassification, in which we obtained K−1(X) as the group, we could general-ize this approach in the spirit of [25], [15] and [35] by including symmetries,such as time reversal or reflection symmetry, which are both operators act-ing on H . As it is very well explained in [15], care must be taken for thereis an ambiguity in the choice of representation for these symmetries, and totake it into account it is necessary to modify the isomorphism classes of ourvector bundles, yielding a different kind of K -theory for each symmetry andeach choice of its representation! Thus, for the case of time reversal symme-try where the representation θ is anti-unitary, we would expect to modifyour isomorphism classes of vector bundles to isomorphism classes of Real444.4. Discussionvector bundles, in the sense of [2], were as for reflection symmetry, where Ris unitary, we would expect what is known as Z2-vector bundles (also see [2].The corresponding K -theory is known as KR- theory or Real K -theory andfor Z2-vector bundles is called KZ2- theory or Z2-equivariant K -theory. Itis of great importance to developed a framework to encompass all possiblesymmetries and their representations as the one employed in [15] for gappedsystems, however, our classification does not readily admit such a general-ization, for the topology employed in [15] is the Compact-Open topology andFredholm operators are trivial in this topology! Non the less, there is noimmediate contradiction between our classification and that of [15], as theirframework was developed to study gapped systems, where there is no Fermisurface. This generalization is of great interest but will be deferred to futurework.45Chapter 5ConclusionsIn this thesis, we made a great effort to introduce a vast array of physicaland mathematical concepts from the ground up, in order to later employ inan integral manner for the development of a classification of topologicallystable Fermi surfaces.We first introduced our readers to the subject with a brief recount of thehistory of topological phases of matter, starting with the discovery of Von Kl-itzing and his collaborators of the exact quantization in the integer quantumHall effect in the early 80’s, up to its interpretation as the Chern numberof a line bundle over the Brillouin zone due to the TKNN group, the Z2topological invariant of Kane and Mele in the spin quantum Hall effect andthe explosion of research that developed there after. We later restricted tothe literature of topological properties of Fermi surfaces, where the earliestwork is due to Horˇava and generalizations developed there after, togetherwith some of the criticisms on the general use of K -theory in condensedmatter systems due to Thiang.We plunged in deeply into the mathematical aspects, starting at the root ofthe matter with the introduction of analysis and the different kinds of oper-ators such as bounded, compact and Fredholm operators, then proceedingto topological spaces, moving through what is denoted as point-set topol-ogy (continuity, compactness, connected, quotient and Hausdorff spaces,together with topological groups), then to algebraic topology (homotopy,loop-spaces, suspensions, homotopy groups, cohomology) and all the wayup to vector bundles,isomorphism classes of vector bundles Grothendieck’scompletion, K -theory and Bott periodicity, as well as the powerful connec-tion it has with Fredholm operators and its self-adjoint subset, discoveredand developed by Atiyah and Singer.Thereafter, we introduced the basic concepts of condensed matter systemsfor N electrons in a box, arriving at the Fermi sphere, and then studying sys-tems with discrete translational invariance due to a periodic array of ions (orpossibly something else) called lattices, then the reciprocal lattice in crystalmomentum space and the first Brillouin zone. We later introduced Bloch’stheorem and used it to describe the notions of band theory of electrons, and46Chapter 5. ConclusionsFermi surfaces, which are the generalization of the Fermi sphere for discretetranslational invariant systems with no electron-electron interactions. Wealso discussed adiabatic evolution and its interpretation for gapless systems.We then proceeded to make the connection between the physical subjectswe introduced and the mathematics, by showing that the Hamiltonian of adiscrete translational invariant system with no electron-electron interactions(also called Bloch Hamiltonian) and taking out of it the Fermi energy yieldsa self-adjoint Fredholm operator, whose kernel has a precise physical inter-pretation concerning the Fermi surface of our system. Using the fact thatthe Bloch Hamiltonian is parametrized by the Brillouin zone of the systemand the connection shown by Atiyah and Singer between self-adjoint Fred-holm operators, Loop spaces and K -theory, we arrive at our classificationwhere it is the elements of K−1(X), which constitute the distinct kinds oftopologically stable Fermi surfaces, such distinct classes cannot be connectedby adiabatic evolution or small perturbations, in the appropriate sense.Once we had our classification for the case with no symmetries, we computedsome K -groups of systems with dimension d = 1, 2 and 3, and we presentedthe results in ??. After surveying the literature, we discussed the d = 1 case,were the topological invariant which determines the class of a Fermi surfaceis known as the spectral flow. The spectral flow had been used previously bySemenoff and Niemi and also by Haldane to describe what is known as chiralanomaly. Anomalies are of fundamental importance in quantum field theoryand our work shows connections between anomalies and Fermi surfaces.We also discussed briefly why our results differ from those presented byChing-Kai and other sources, were the arguments presented there seem tobe more restricted in the topological nature allowed for their surfaces andtheir arguments do not seem to apply as it was precisely the connectionwith chiral symmetry breaking which renders them inadequate on physicalgrounds.We further discussed how to incorporate symmetries into our classification,we believe they should yield different kinds of vector bundle isomorphisms,such as those of Real vector bundles or Z2- vector bundles, yielding for ev-ery choice of representation of a symmetry a different kind of K -theory, inthe spirit of the framework developed by Freed and Moore for topologicalphases of gapped systems. It is however, not possible to directly extend ourframework to the one employed there, and we have pointed out were thetroubles lie.Thus, we have developed a classification of these strange phases of matter,employing the sophisticated mathematical framework of K -theory, arrivingat unexpected connections with the profound concepts of quantum anoma-47Chapter 5. Conclusionslies. We have also pointed out what future work would have to surmountin order to have a classification which is fully consistent with the differentsymmetries and their representations. We should keep in mind however thatwe have specified our limitations and assumptions in our derivation, suchas ignoring electron-electron interactions which may be relevant for manysystems, as well as working in the zero temperature approximation. Nonethe less, for many systems these approximations are shown by experimentsto be good.We hope to develop elsewhere a classification consistent with the one devel-oped by Freed and Moore, and to have a better connection with experimentalefforts, a fact which every physicist must always keep in mind, our classifi-cation yields results of the physical kind that have already been observed inexperiments and so, such a connections seem a reasonable expectation forthe near future.48Bibliography[1] N.W. Ashcroft and N.D. Mermin. Solid State Physics. HRW interna-tional editions. Holt, Rinehart and Winston, 1976.[2] MF Atiyah. K-theory and reality. Quart. J. Math, pages 367–386, 1966.[3] MF Atiyah. Algebraic topology and operators in hilbert space. In Lec-tures in Modern Analysis and Applications I, pages 101–121. Springer,1969.[4] Michael Francis Atiyah and Isadore Manuel Singer. Index theoryfor skew-adjoint fredholm operators. Publications Mathe´matiques del’IHE´S, 37(1):5–26, 1969.[5] J. E. Avron, R. Seiler, and B. Simon. Homotopy and quantization incondensed matter physics. Phys. Rev. Lett., 51:51–53, Jul 1983.[6] Joseph E. Avron and Alexander Elgart. Adiabatic theorem without agap condition. Communications in Mathematical Physics, 203(2):445–463, 1999.[7] B Andrei Bernevig, Taylor L Hughes, and Shou-Cheng Zhang. Quan-tum spin hall effect and topological phase transition in hgte quantumwells. Science, 314(5806):1757–1761, 2006.[8] Michael V Berry. Quantal phase factors accompanying adiabaticchanges. In Proceedings of the Royal Society of London A: Mathemati-cal, Physical and Engineering Sciences, volume 392, pages 45–57. TheRoyal Society, 1984.[9] P.M. Chaikin and T.C. Lubensky. Principles of Condensed MatterPhysics. Cambridge University Press, 2000.[10] Shailesh Chandrasekharan and U-J Wiese. An introduction to chiralsymmetry on the lattice. Progress in Particle and Nuclear Physics,53(2):373–418, 2004.49Bibliography[11] Xie Chen, Zheng-Cheng Gu, and Xiao-Gang Wen. Local unitary trans-formation, long-range quantum entanglement, wave function renormal-ization, and topological order. Physical Review B, 82(15):155138, 2010.[12] Ching-Kai Chiu, Jeffrey CY Teo, Andreas P Schnyder, and Shinsei Ryu.Classification of topological quantum matter with symmetries. arXivpreprint arXiv:1505.03535, 2015.[13] Marcel Franz. Race for majorana fermions. Physics, 3:24, 2010.[14] Dan S. Freed. Math263y: Topics in geometry and physics: K-theory,September 2014.[15] Daniel S. Freed and Gregory W. Moore. Twisted equivariant matter.Annales Henri Poincare´, 14(8):1927–2023, 2013.[16] Alexander Grothendieck. La the´orie de fredholm. Bulletin de la Socie´te´Mathe´matique de France, 84:319–384, 1956.[17] FDM Haldane. Berry curvature on the fermi surface: Anomalous halleffect as a topological fermi-liquid property. Physical review letters,93(20):206602, 2004.[18] M Zahid Hasan and Charles L Kane. Colloquium: topological insula-tors. 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