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Exotic phenomena in topological states of matter Vazifeh, Mohammad Mahmoudzadeh 2014

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Exotic Phenomena in Topological States of MatterbyMohammad Mahmoudzadeh VazifehM.Sc., Sharif University of Technology, 2008A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Physics)The University Of British Columbia(Vancouver)October 2014c©Mohammad Mahmoudzadeh Vazifeh, 2014AbstractElectronic states in band insulators and semimetals can form nontrivial topologicalstructures which can be classified by introducing a set of well defined topologi-cal invariants. There are interesting experimentally observable phenomena tied tothese topological invariants which are robust as long as the invariants remain well-defined. One important class manifesting these topological phenomena in the bulkand at the edges is the time reversal invariant topological band insulators first dis-covered in HgTe in 2007. Since then, there have been enormous efforts from boththe experimental and the theoretical sides to discover new topological materialsand explore their robust physical signatures.In this thesis, we study one important aspect, i.e., the electromagnetic responsein the bulk and at the spatial boundaries. First we show how the topological ac-tion, which arises in a time reversal invariant three dimensional band insulator withnontrivial topology, is quantized for open and periodic boundary conditions. Thisconfirms the Z2 nature of the strong topological invariant required to classify time-reversal invariant insulators. Next, we introduce an experimentally observable sig-nature in the response of electronic spins on the surface of these materials to theperpendicular magnetic field. We proceed by considering electromagnetic responsein the bulk of topological Weyl semimetals in a systematic way by considering alattice model and we address important questions on the existence or absence of theChiral anomaly. In the end, we show how a topological phase in a one dimensionalsystem can be an energetically favourable state of matter and introduce the notionof self-organized topological state by proposing an experimentally feasible setup.iiPrefaceI have written this thesis based on the results my supervisor and I have obtainedby working on various problems in the field of topological insulators throughoutmy program. Some sections of this thesis are based on the notes written by myresearch supervisor Professor Marcel Franz, and also publications authored by meand him.Chapter 1 gives a brief introduction to the subjects that we have studied duringthe course of my program in the field of topological insulators. This is entirelybased on my understanding of the topics which is of course limited but I havetried to convey the most important concepts of the field to people who are notvery familiar with these novel topics. I have also tried to show why we think theproblems that we have considered in this thesis are among important ones in thefield.Chapter 2 starts with an introduction to the topological Axion response. I alsoprovide a simple proof of the quantization of the associated topological action termwhich are based on the short article entitled “Quantization and 2pi Periodicity ofthe Axion Action in Topological Insulators” which we have published in Physi-cal Review B, 82, 233103 (2010). The quantization proof is based on the noteswritten by my supervisor and I. Providing a proof for the quantization based onthe electromagnetic field decomposition was my supervisor’s idea. He proved thatusing a certain field decomposition one can write a second Chern number as aproduct of two first Chern numbers which are quantized. I provided an argumentfor the quantization of the first Chern numbers on tori based on the fact that single-valued electronic wave-functions must be well-defined. I wrote the initial draft ofthe manuscript based on my supervisor’s and my own notes. He then edited andiiimodified the short report before we submitted it to Physical Review B.Chapter 3 is based on my calculations while working on magnetic response ofthe mobile electrons at the surface of a topological insulator and some sections arebased on the article we have published in Physical Review B 86, 045451 (2012).The problem statement was provided by my supervisor, I carried out the calcu-lations. He provided extensive guidance during the time I was carrying out thecalculations and resolved all the problems I encountered. I wrote the initial draft ofthe manuscript. He then edited the entire manuscript and we submitted the paperentitled “Spin Response of Electrons on the Surface of a Topological Insulator” toPhysical Review B.Chapter 4 is based on my notes while working on this project. The ideawas mine and I carried out the calculations and wrote the paper entitled “WeylSemimetal from the Honeycomb Array of Topological Insulator Nanowires”, pub-lished in Euro Physics Letters 102, 67011 (2013). My supervisor encouraged mea lot during the course of this project and he provided some calculations and hintsto put me in the right direction. In the end, he suggested that I publish this arti-cle on my own since he thought it was my idea even though his contribution wassignificant in this project and he helped me by reading the paper and providingconstructive comments before the submission.Chapter 5 is based on my supervisor’s and my own notes while working on thisproject. We addressed a controversial question which was an open question at thetime. We published a paper entitled “Electromagnetic Response of Weyl Semimet-als” in Physical Review Letters 111, 027201 (2013) based on our numerical andanalytical results. I carried out most of the numerical computations and we workedon the theory together. In the end, he wrote the manuscript and then I read it andgave my comments before the submission. Most of the sections in this chapter arebased on this published paper.Chapter 6 is again based on the noted written by my supervisors and I. Thisproject started with my supervisor’s idea to find the phase diagram of a 1D super-conductor model which could potentially be in a topological phase that supportsMajorana fermions at its ends motivated by some previous works on similar se-tups [1, 2]. I carried out part of the calculations and also the numerical computa-tions. My supervisor wrote the paper (based on our notes and results), which wasivthen published as a letter entitled “Self-Organized Topological State with MajoranaFermions” in Phys. Rev. Lett. 111, 206802 (2013). Again most of the sections inthis chapter are based on the text of this published paper.The calculations given in the appendices provide supplementary material to ex-plain some of the concepts, approximations and calculations discussed in chapters1 to 6.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii1 Introduction to Topological Phases of Matter . . . . . . . . . . . . . 11.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Topological classification via topological invariants . . . . . . . . 21.3 Topological invariants in condensed matter systems . . . . . . . . 31.4 Quantum Spin Hall phase . . . . . . . . . . . . . . . . . . . . . . 71.5 Topological insulators in three dimensions . . . . . . . . . . . . . 121.6 A lattice model for topological insulators . . . . . . . . . . . . . 141.7 Weyl and Dirac semimetals . . . . . . . . . . . . . . . . . . . . . 171.8 One dimensional topological phase with Majorana fermions . . . 212 Axion Angle Periodicity in Topological Insulators . . . . . . . . . . 242.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24vi2.2 Electromagnetic field decomposition . . . . . . . . . . . . . . . . 252.3 Quantization of the Chern numbers . . . . . . . . . . . . . . . . . 282.4 Open boundaries case . . . . . . . . . . . . . . . . . . . . . . . . 302.5 Abelian versus non-abelian gauge fields . . . . . . . . . . . . . . 312.6 Effect of magnetic monopoles . . . . . . . . . . . . . . . . . . . 322.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 Response of Surface Dirac States . . . . . . . . . . . . . . . . . . . . 343.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.2 Dirac modes in out-of-plane applied magnetic field . . . . . . . . 363.3 Spin susceptibility and magnetization . . . . . . . . . . . . . . . 383.4 Knight shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484 Models for Weyl Semimetals . . . . . . . . . . . . . . . . . . . . . . 494.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2 Multiple Weyl pairs . . . . . . . . . . . . . . . . . . . . . . . . . 504.3 TI nano-wires . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.4 Honeycomb lattice of parallel wires . . . . . . . . . . . . . . . . 534.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595 Electromagnetic Response of Weyl Semimetals . . . . . . . . . . . . 615.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.3 Anomalous Hall current . . . . . . . . . . . . . . . . . . . . . . . 635.4 Can CME exist in an equilibrium state? . . . . . . . . . . . . . . 655.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676 Self Organized Topological phase with Majorana Fermions . . . . . 696.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716.3 Stability analysis of the spiral ground state . . . . . . . . . . . . . 766.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81vii7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857.2 Future works . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89A Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101A.1 Berry phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101A.2 TKNN invariant for massive Dirac states . . . . . . . . . . . . . . 103A.3 Electrons in a periodic potential . . . . . . . . . . . . . . . . . . 104A.4 Generalized Bloch states . . . . . . . . . . . . . . . . . . . . . . 105A.5 From discrete to continuum limit . . . . . . . . . . . . . . . . . . 106A.6 Fourier transform of the step function . . . . . . . . . . . . . . . 108A.7 Fourier transform of the product . . . . . . . . . . . . . . . . . . 108A.8 Green’s function for Schrodinger equation . . . . . . . . . . . . . 109A.9 Density of states function from the Green’s function . . . . . . . . 110A.10 Green’s function for a perturbed system . . . . . . . . . . . . . . 111A.11 Quantum linear response theory . . . . . . . . . . . . . . . . . . 111A.12 Coupled systems and the effective theory . . . . . . . . . . . . . 112A.13 Topologically protected semi-metal . . . . . . . . . . . . . . . . . 114A.14 Mean-field treatment of Hint . . . . . . . . . . . . . . . . . . . . 117viiiList of TablesTable 2.1 Chern Character for Abelian and non-Abelian gauge fields . . . 31ixList of FiguresFigure 1.1 Topological classification of differentiable manifolds . . . . . 3Figure 1.2 Chiral edge modes in a QH state . . . . . . . . . . . . . . . . 5Figure 1.3 Energy spectrum in a QH system . . . . . . . . . . . . . . . . 6Figure 1.4 Schematic of the edge modes in a QSH state . . . . . . . . . . 7Figure 1.5 Energy spectrum of Graphene in QSH phase . . . . . . . . . 10Figure 1.6 Longitudinal four terminal resistance in HgTe quantum wells . 11Figure 1.7 Low energy spectra in Dirac and Weyl semimetals . . . . . . 19Figure 1.8 1D topological superconductor with Majorana fermions . . . 22Figure 2.1 First Chern tori . . . . . . . . . . . . . . . . . . . . . . . . . 26Figure 3.1 Energy spectrum of the surface states in a TI . . . . . . . . . 37Figure 3.2 Surface spin magnetization versus the chemical potential . . . 40Figure 3.3 Surface spin magnetization versus the applied magnetic field . 41Figure 3.4 Spin susceptibility of the surface states . . . . . . . . . . . . 42Figure 3.5 Intrinsic magnetization gap versus the applied magnetic field . 44Figure 4.1 Energy spectrum of a TI nano-wire . . . . . . . . . . . . . . 51Figure 4.2 Schematic of two adjacent parallel TI nano-wires . . . . . . . 53Figure 4.3 BZ before and after separating the Weyl nodes . . . . . . . . 56Figure 4.4 Haldane’s configuration of the modulating magnetic flux . . . 58Figure 5.1 Energy spectrum of a Weyl semimetal . . . . . . . . . . . . . 63Figure 5.2 Charge density in the presence of a magnetic flux . . . . . . . 64Figure 5.3 Accumulated charge as a function of bz . . . . . . . . . . . . 65xFigure 5.4 Chiral current as a function of energy offset b0 . . . . . . . . 66Figure 6.1 Chain of magnetic atoms on a superconducting substrate. . . . 70Figure 6.2 Pitch of the spiral and the topological phase diagram . . . . . 82Figure 6.3 Spectral function of the adatom chain . . . . . . . . . . . . . 83Figure 6.4 Spin Susceptibility of the 1D wire . . . . . . . . . . . . . . . 84xiGlossaryOI Ordinary InsulatorTI Topological InsulatorTR Time ReversalARPES Angle Resolved Photo Emission SpectroscopySTM Scanning Tunnelling MicroscopyQH Quantum HallLL Landau LevelQSH Quantum Spin HallTKNN Thouless-Kohmoto-Nightingale-NijsBZ Brillouin ZoneQCD Quantum ChromodynamicsMF Majorana FermionsCME Chiral Magnetic EffectSO Spin-OrbitNN Nearest-NeighbourxiiAcknowledgmentsIt has been a wonderful journey from beginning to end of my PhD program inphysics department at UBC. I am deeply grateful to awesome family members,friends and mentors who helped me during this time. Before anyone else, I wouldlike to express my sincere gratitude to my research supervisor Prof. Marcel Franzfor his tremendous guidance and encouragement. He played a key role in all ofthe academic accomplishments I have had since I started my PhD. I also wishto thank the members of my advisory committee, Prof. Ian Affleck, Prof. RobKiefl, and Prof. Moshe Rozali. My grateful thanks are also extended to Prof.Allan H. MacDonald, Prof. Abdollah Langari, and Dr. Gilad Rosenberg for theirsupport and assistant in my Four-Year Fellowship (4YF) application. Being on a4YF enabled me to carry out research with the least amount of financial pressure.During my years in Vancouver I made wonderful friends whom I shared someof the best moments of my life. I would like to specially thank, Dr. Kamran Albafor being an amazing friend for me during this time.Finally, I would like to dedicate this thesis to my mother, MS. Farah Emami, avery special person whose support and encouragement is beyond what words canexplain. I have been extremely lucky to have great relatives and friends. Spe-cial thanks goes to Mr. Khalil Vazifeh for being a great caring father, Mr. EhsanVazifeh, Ms. Sahar Vazifeh, Dr. Alireza Emami, Mr. Mohammad Reza Emami,and Ms. Mahjabin Emami for their support and love. Special thanks also goes toAtomic Racing cycling club members particularly Mr. Andrew Wilson for admit-ting me to their great community of cyclists in Vancouver.Let me also extend my gratitude to the rest of family and friends for theirsupport and guidance. My special thanks goes to Dr. Azizeh Mahdavi, Mr. RaminxiiiHatam, Dr. Reza Behafarin, Dr. Reza Asgari, Dr. Haik Koltoukjian, Dr. MasoudJazayeri, Dr. Rouhollah Aghdaee, Dr. Hessameddin Arfaei.xivChapter 1Introduction to TopologicalPhases of Matter1.1 OverviewTopology is one of the important fields of mathematics which has far-reaching ap-plications in major fields of science such as biology, computer science and physics[3, 4]. One can use the basic concepts developed in this field to classify and char-acterize seemingly different systems. The main idea is that usually an importantglobal property of a system under study remains invariant under a class of “smooth”deformations and this makes it possible to successfully distinguish between distinctstructures by means of the theory of topological classification. A simple exampleis classifying closed differentiable surfaces in three spatial dimensions. It is easy tosee that one cannot smoothly deform a donut-shape manifold to a sphere withouttearing or puncturing it; therefore, the number of holes in a closed surface can beinterpreted as a global property of the system which is robust under smooth defor-mations and infinite possible variations from the ideal sphere can still be consideredas one distinct class of manifolds. The same simple idea applies to other examplesof topological classification in science including condensed matter physics as wedescribe here. In physics, these classifications become important especially whenthe global property used to categorize possible phases is tied to some importantphenomena that can be observed directly or indirectly in experiments. This con-1cept will become more clear once we introduce the first example in the next section.As we will see, establishing a good topological classification requires defining rel-evant global features in a proper quantitative fashion in order to understand whythey are robust under local changes in a system. To see how this is possible in asystematic way and in the language of mathematics, in the next section we describethe concept of topological invariants by discussing Gauss-Bonnet theorem used toclassify closed differentiable surfaces.1.2 Topological classification via topological invariantsTopological invariants are the building blocks of any classification based on basicconcepts of topology. In the case of closed two dimensional differentiable mani-folds this invariant is the genus which is defined as the number of existing holesin the closed manifold embedded in a three dimensional space. It is important tonote that this property is a global feature of the object under study and does notmanifest itself directly in any local property of the system. However, as we willsee one can establish a connection between the topological invariant in this caseand the integral of a local property over the entire system. The Euler-Poincarecharacteristic is a good example to better understand this concept [5]. Euler firstdefined this invariant for polyhedra. A polyhedron is a solid geometrical objectin three dimensions which has flat faces, straight edges and sharp corners. Eulershowed that for all polyhedra, subtracting the number of edges (E) from the sumof the number of vertices (V) and faces (F) yields the number 2. Surprisingly, heobserved that although this quantity which is now called Euler character is relatedto local features, it remains invariant under adding edges, faces and vertices.A generalization of this invariant in the case of differentiable two dimensionalmanifolds is a special case of the more general Gauss-Bonnet theorem which canbe expressed as∫MKdS = 2pi(2−2g), (1.1)this relates the genus to the local curvature, K which is defined for all thepoints forming the closed manifold. However, even though this connection exists,the genus remains invariant if one locally changes the curvature (see Fig. 1.1). The2g = 0g = 1Figure 1.1: Both manifolds on the left side have a genus equal to one, i.e.,g = 1 and those on the right have g = 0 which is equal to the number oftheir holes which is a robust global property under smooth deformations.only requirement is that the deformation must be smooth in such a way that themanifold remains differentiable at all times during the deformation.This is indeed the essence and the beauty of topological classifications fromwhich one or several topologically ”robust” quantities emerge. In the next section,we describe how this concept entered on to the scene of condensed matter systemsin the context of Quantum Hall (QH) systems more than two decades ago andrevisited the scene again during the last years.1.3 Topological invariants in condensed matter systemsFirst discovery of the quantized Hall conductance in two dimensional electronicquantum-well heterostructure in a strong magnetic field by von Klitzing, Dorda,and Pepper in 1980 [6] stimulated a series of theoretical works aimed to describethe robust precisely quantized values of the transverse conductivity. The open ques-tion at the time was why these conductance values are exactly multiples of e2/hwith extremely high precision although the experimental setups were prone to im-perfections which exist in all real crystalline structures. [7–10] The robustness tolocal impurities and perturbations suggested that there might be some fundamen-tal connections to a global structure in the underlying many-body electron systemwhich remains unchanged in the course of local changes in the same fashion as a3topological invariant does. Of course this is not as obvious as it is in the case ofdifferentiable manifolds. Thouless and collaborators established this connectiontwo years after the original discovery of the QH effect [11]. They studied a twodimensional electron gas in a periodic potential and calculated the transverse con-ductivity by performing a linear response (Kubo) [12] calculation and they foundthat the transverse conductivity has a simple expression as an integral over theentire Brillouin Zone (BZ) given asσxy =ie24pih ∑occn∮d2k∫d2r(∂u?n(~k,~r)∂k1∂un(~k,~r)∂k2− ∂u?n(~k,~r)∂k2∂un(~k,~r)∂k1), (1.2)in which un(~k,~r)s are the generalized single electron Bloch functions (see A.4)and the sum is over the occupied electron bands. They showed that the abovequantity is always an integer multiple of e2/h whenever the chemical potential isin the gap, consistent with the values reported in the experiments. This equationwhich relates the transverse conductivity to a global structure of the underlyingBloch states is a milestone of the topological classification in condensed matterphysics. To understand how this quantization arises, we can use the Dirac notationand show that the above quantity is an integral of the Berry curvature (see appendix.(A.1)) over the two dimensional torus of the BZ.One can define the following gauge field~A(~k) = i ∑occ α< u~kα |∇~k|u~kα > . (1.3)From the properties of the Bloch states one can argue that the total flux corre-sponding to the pseudo-magnetic field, ~B(~k) = ∇~k×~A(~k), is quantized∫~B(~k) ·d2~k = 2pin, (1.4)in which the integer n can be thought of as the number of the pseudo-magneticmonopoles that can be imagined to exist inside the BZ torus responsible for thequantized net flux of the pseudo-magnetic field. In other words, the phase of theBloch wave-functions can form U(1) vortices in the BZ and the transverse conduc-4BFigure 1.2: Two dimensional electron gas under applied perpendicular mag-netic field in a QH phase with topological invariant n = 1. According tothe bulk-edge correspondence, there are doubly degenerate chiral edgemodes propagating counter-clock wise. The conductance of this edgemode is quantized and local impurities cannot change the conductivityas long as the bulk remains gapped.tance is linearly proportional to the total vorticity [13].It is easy to realize that Eq.(1.2) and Eq.(1.4) exactly describe the same quan-tity. This observation explains the quantization of the transverse Hall conductance.The robustness of the transverse conductivity arises since the above quantity is in-deed a topological invariant and does not change as long as the bands are filled andthe chemical potential is inside the gap keeping the gauge field well-defined [14].In the language of mathematics this classification distinguishes different topolog-ical structures on a tangent bundle formed on the BZ torus as the base manifoldand the tangent space is the Hilbert space of the electronic Bloch states. The aboveinvariant classifies all the possible nontrivial topological structures on this tangentbundle which can arise due to the vortices in the U(1) Berry phase of the Blochstates [11, 15].This profound observation is important because conventionally in condensedmatter physics the categorization of phases of matter has been based on symmetryclassification. However, in QH systems all the integer QH states belong to thesame symmetry group and therefore the symmetry classification fails to distinguish5✏kedge states edge statesFigure 1.3: Schematic energy spectrum of a two dimensional electron gasin an integer QH phase when there is a periodic boundary conditionalong one of the directions. The system has two edges in this cylindricalconfiguration. The edge states’ energy spectrum crosses the chemicalpotential for all chemical potentials inside the gap and the two branchesat the two sides of the spectrum correspond to the spatially separatedgapless one dimensional modes localized at the two sides of the system.between different QH states. This also explains the extraordinary robustness of thequantized conductance, since as long as the above invariants remain well-definedthe quantization holds. In fact the invariant remains well-defined as long as thereis a gap in energy between the many-body ground-state and the first excited states.By changing the magnetic field one can close the gap and at the critical point wherethe bands touch, the topological invariant is no longer well defined and that is howthe conductance can change so that we can see different integer plateaus in thetransverse conductivity as a function of the applied magnetic field.Another interesting aspect of a topologically nontrivial QH state is the exis-tence of linearly dispersing chiral one dimensional modes at the boundary. In fact,according to Laughlin’s argument, transverse conductance would not be possiblewithout the existence of such states at the edges because there must be gaplessstates to accommodate the electrons that are being pumped between edges [8]. Theedges are important in fact because they are the interface which separates two topo-logically distinct regions and in fact the existence of linearly dispersing modes canbe associated with this topological mismatch. Remember that a classical vacuumcan be viewed as a topologically trivial gapped state with infinite energy gap. These6chiral one dimensional modes are robust to local perturbations and they cannot begapped out as long as the bulk remains to be in the same topological class [16].The number of such doubly degenerate modes (considering spin degeneracy if oneneglects the Zeeman effect) is the same as the value one gets from calculating theThouless-Kohmoto-Nightingale-Nijs (TKNN) topological invariant, n, defined inEq. (1.4). Indeed, this is another signature of the nontrivial topological state as ithas been seen and confirmed in a number of multichannel transport experiments[7].1.4 Quantum Spin Hall phaseFigure 1.4: Schematic of a two dimensional electron gas in a QSH phase. Ac-cording to bulk-edge correspondence, there is an odd number of pairs ofcounter propagating chiral edge modes. The spin conductance is con-served and therefore well-defined in the absence of Rashba SO coupling.In this case the spin conductance of the edge mode is quantized and lo-cal impurities that respect TR invariance cannot change the conductivityas long as the bulk remains gapped.As we showed in the previous section, integer QH states are TR broken topo-logical phases of matter which can be classified with a single topological invariant[14]. A legitimate question that arises here is whether it is possible or not to havea nontrivial topological state in two dimensions when there is no net uniform ap-plied magnetic field. In other words, whether it is possible or not to get a nonzeroquantized Hall response in the absence of a net uniform magnetic field. Haldane7answered this question in 1988 by introducing a simple two dimensional latticemodel. He showed that in theory this is possible and one can have a QH statewithout a net uniform perpendicular magnetic field [17]. The model he introducedconsists of a honeycomb lattice with modulating magnetic flux going through itshexagons in a way that the net magnetic flux going through each hexagon is zero.Then in a tight-binding approximation he showed that under certain conditions thesystem can in fact form a QH state with n =±1 as its many-body ground-state.A next-nearest neighbour (NNN) tight-binding model used by Haldane for ahoneycomb lattice leads to distinct Dirac fermion modes in the low energy elec-tronic spectrum [18, 19]. In the continuum limit, one can evaluate the TKNN topo-logical invariant for a single massive Dirac mode in two dimensions and see thatthis quantity can only be either +0.5 or−0.5 by considering a proper regularization(see Appendix A.2). The sign depends on the sign of the product of the mass andthe chirality of the corresponding two dimensional Hamiltonian (= sgn(mυxυy)).Since two Dirac modes around the two distinct points in the BZ have opposite chi-rality, inducing equal mass for these Dirac modes always guarantees that the nettransverse conductivity is zero in the absence of the magnetic field. However, Hal-dane came up with the modulating magnetic field configuration on the honeycomblattice and realized that the associated term in the tight-binding Hamiltonian givesrise to opposite masses for these Dirac modes, and therefore, one can consider a sit-uation where the net transverse conductivity is nonzero. In fact, by considering thefull BZ states one can show that the TKNN invariant can be ±1 in the presence ofthe Haldane term. This model was considered as an interesting toy model for morethan two decades and remained unrealistic even after the discovery of Graphene in2004 [20], since inducing magnetic field that modulates at lattice scale is somethingthat is not experimentally plausible. However, this model inspired Kane and Mele[21] to realize that graphene can possibly be driven into a nontrivial topologicalphase in two dimensions if one considers the effect of SO interaction. The trans-verse conductivity is zero for this novel phase, since the TR symmetry is respectedin the system (remember σxy is odd under TR so it can only be zero if the systemhas this symmetry). However, they showed that this phase is totally distinct froman Ordinary Insulator (OI) in a sense that there is no smooth deformation of theHamiltonian to the one that describes an OI as long as TR symmetry is respected8and the system is gapped. The general Hamiltonian they considered can be writtenasH = t∑〈i j〉c†i c j + iλSO ∑〈〈i j〉〉νi jc†i szc j + iλR∑〈i j〉c†i (s× dˆi j)zc j +λv∑iξic†i ci.in which c†i (ci) are the creation (annihilation) operators on site i. λSO is thestrength of the SO interaction, λR is the strength of Rashba interaction, and λv isthe staggered potential strength. dˆi j is the position vector connecting two sites fromsite i to site j. νi j is either +1 or −1 [22].In the most simple case of λR = 0, this nontrivial topological phase can beviewed as two copies of the Haldane state with n = +1 for Sz = h¯/2 states andn = −1 for Sz = −h¯/2 states. In fact the form of the Hamiltonian is the same asthe one studied by Haldane for each spin flavour. Now, since in this case the spinsectors are disconnected, each of them supports chiral edge modes which propagateoppositely for opposite spins. The robustness is guaranteed by the fact that TRsymmetry prevents mixing of states with opposite spins, therefore the counter-propagating spin filtered chiral edge states are protected by a combination of TRsymmetry at the surface and the topological properties of the bulk states (see Fig.(1.4)). Also, the system exhibits quantized spin Hall effect which can be definedas the difference between the transverse charge current in the two spin sectors [21–23]. Therefore, this phase is called the Quantum Spin Hall (QSH) phase.The more interesting situation is when the Hamiltonian does not commute withSz which can happen in the presence of a finite Rashba SO coupling. Kane andMele showed that even in this case, for a wide range of system parameters theground-state can still be a nontrivial topological state with counter propagatingmodes at the boundary [21, 22].One can define a Z2 topological invariant to classify all TR invariant insulatorsin two dimensions which has various formulations [24–27]. This invariant is de-fined as a sum of a surface and a line integral of the Berry curvature that appearedin the TKNN invariant over half of the BZ and its boundary and can be written as90 2/0 2/<101<5 0 5<505 IQSHh  / hRh  / hv SOSOE/tka ka/ /(a) (b)Figure 1.5: Energy bands for a one dimensional “zigzag” strip in the (a) QSHphase λv = .1t and (b) the insulating phase λv = .4t. In both cases λSO =.06t and λR = .05t. The edge states on a given edge cross at ka = pi . Theinset shows the phase diagram as a function of λv and λR for 0 < λSOt. Figure and caption from Ref. [22]ν2D0 =12pi(∮∂EBZ~A(~k) ·d~k−∫EBZ~B(~k) ·d2~k)mod 2. (1.5)When ν2D0 is zero the system would be in a trivial topological phase, and when-ever this invariant is equal to 1, it would be in a topological QSH insulator phasewhich supports counter-propagating modes at its edges (see Fig. (1.5)). Of courseestablishing a physical meaning for this invariant or understanding its connectionwith any physical observable like the edge states is not as easy as it was for theTKNN invariant defined for the integer QH systems. However, one can use analternative picture to see why this phase is non-trivial and cannot be smoothly con-nected to a trivial insulator. If we can prove that no such path exists, then theexistence of the edge modes follows from the same argument we used before forQH states, i.e., they arise due to the topological mismatch between the two sidesof the boundary as can be seen in the most simple case of two copies of Haldanestates with opposite magnetic flux configurations. We come back to this point inthe next section.Clean graphene was not a good candidate for the QSH phase due to its sub10-1.0-0.50.00.51.01.52.0103104105106107IIIIII IVG = 2 e2 /h  R14,23 / Ω(V g- V thr) / VG = 0.3 e2 /hG = 0.01 e2 /hIIT = 30 mKI-1.0-0.50.00.51.005101520G = 2 e2 /hT = 0.03 K  R14,23 / kΩ(V g - V thr) / VT = 1.8 KFigure 1.6: When the film thickness is large enough the HgTe quantum wellundergoes a phase transition to a spin Hall insulator phase which sup-ports robust counter-propagating edge modes. The longitudinal fourchannel resistance, R1,2,3,4 of various normal (d = 5.5mm) (I) and in-verted (d = 7.3) (II, III, IV) QW structures as a function of gate voltageis measured for ~B = 0T at T = 30 mK. Figure and caption from Ref.[28]meV spin-orbit gap despite the fact that it was the main inspiration for the the-oretical development and the introduction of this topological phase. What wasobvious in those theoretical works was that SO plays an important role in driv-ing the system into a topological phase so naturally one had to look for materialswith heavy elements to realize this phase. It was proposed in 2003 by Shou-ChengZhang’s group that Mercury-Telloride (HgTe) might be a good candidate in thatsense [29, 30]. Using a k.p model they showed that thin films of HgTe sandwichedbetween (Hg,Cd)Te barriers can be driven into a QSH phase by increasing thethickness of the thin film, d, and going through a quantum phase transition. Atthe critical point, d = dc, the bands of the electronic spectrum touch and as one11increases the d further, the system enters the topological phase when the bands areinverted. In 2007, the same group reported the discovery of this phase in HgTe inthe experiments done by the group of Laurens Molenkamp [28]. They observedquantized conductance which arise due to the existence of counter-propagatingedge modes in a transport measurement (see Fig. (1.6)). HgTe remains the onlyexample of the QSH phase discovered until now although it has been suggestedrecently that graphene can also become a QSH insulator in the presence of heavyadatoms [31, 32].1.5 Topological insulators in three dimensionsSimilar to two dimensional QSH systems, TR invariant band insulators with non-trivial topological structure exist. It turns out that in fact there are 16 distinct topo-logical phases and 4 invariants are required to fully characterize all band insulatorswith TR symmetry [26, 33, 34]. One of these invariants, ν0 which is called the’strong’ topological invariant is directly related to various physical properties ofthe system and is robust to all local perturbations that preserve TR symmetry.Again, similar to the two dimensional case, there are various formulations ofthis strong invariant. One way to express this invariant is the following [26]ν3D0 =12pi2∫εi jkTr[Ai∂ jAk− i23AiA jAk] mod 2, (1.6)where we haveAαβi =−i < u~kα |∂i|u~kβ >, i = 1,2,3 (1.7)which is the non-Abelian gauge field matrix defined for Bloch states. In theinversion symmetric cases this invariant takes a simpler form [35] and can be ob-tained by evaluating the Pffafian of the inversion operator’s representation in theHilbert space of the occupied states at certain high symmetry momenta in the BZ.In this case this invariant can be expressed as(−1)ν3D0 =8∏i=1δi, δi =N∏m=1ξ2m(Γi), (1.8)12where Γis are the TR invariant momenta and ξ2m = ξ2m−1 is the parity eigen-value of the 2m-th occupied energy band at the TR invariant momenta. Using oneof these two expressions for the different band insulator models, one can find outwhether a system can be in a strong topological phase or not. Several systems havebeen proposed to be in a Topological Insulator (TI) phase based on this characteri-zation. In a ’strong’ TI phase, the quantum states have been formed in a non-trivialfashion so that the system belongs to a different topological class than the classof OIs, meaning that there is no fully gapped deformation path in the phase spaceof the Hamiltonians that connects the Hamiltonian of the system to that of an OI,preserving the TR symmetry [36]. Any phase space path connecting two topologi-cally distinct points in the phase space would contain a point where the associatedHamiltonian is gapless. This intuitively explains the presence of the robust topo-logical surface states at the interfaces between these systems and OIs. It can beunderstood from the fact that one can establish a mapping between any path con-necting a point in the bulk of a TI to a point in the vacuum and the TR symmetrypreserving paths connecting two topologically distinct regions. Therefore thesepaths must contain a critical point where the system is locally gapless and is in themetallic phase. All such critical points span the metallic surface connecting the twotopologically distinct phases. Therefore, any local perturbation on the surface thatrespects the TR symmetry cannot destroy the gapless surface modes. These gap-less surface modes are then protected by the coexistence of TR symmetry on thesurface and the non-trivial topological structure in the bulk of the system [24, 37].In the case of a ‘strong’ TI there is an odd number of Dirac points at anysurface and, importantly, they are not spin degenerate. Furthermore, they exhibit aunique spin-momentum locking [38, 39] as a result of the strong SO interaction inthe underlying ’strong’ TI which is essential for the formation of the topologicallynon-trivial insulator phase with TR symmetry. These properties make electronicsurface states in a ’strong’ TI very distinct from those of graphene with a pair ofspin degenerate Dirac bands near two special points in the Brillouin zone whichexist as a result of the unique arrangement of carbon atoms in the honeycomblattice [18, 19].This novel phase of matter was discovered first in semiconducting alloy Bi1−xSbxby observing these unusual gapless linearly dispersing modes in an Angle Re-13solved Photo Emission Spectroscopy (ARPES) experiment [38, 40]. Soon afterthis discovery, a single Dirac surface mode with unique spin texture was observedin Bi2Se3 with spin-resolved ARPES experiments [41]. In this case it is easier tosee why the system is in a TI phase by calculating the strong topological invariantusing a low-energy Dirac Hamiltonian. We discuss this in more detail in the nextsection.Many of the interesting features of the surface states in a TI also can be cap-tured, at least qualitatively, using a simple non-interacting Dirac HamiltonianH =∑~kΨ†~k [h¯υFσ · (zˆ×~k)+∆0σz]Ψ~k, (1.9)where zˆ is the unit vector perpendicular to the surface, Ψ†~k = (c†↑~k,c†↓~k) and c†↑(↓)~k isthe fermionic creation operator of the spin up(down) states with wave vector~k. ∆0is the intrinsic gap in the surface spectrum which might be nonzero when the TRsymmetry is spontaneously broken due to magnetic ordering. The latter can arisedue to the presence of magnetic dopants with spin S in the proximity of the surfaceexchange-coupled to the electronic spins [42–44] or as a result of electron-electroninteraction [45].The coupling between spin and the momentum in these surface modes hasinteresting consequences in the electromagnetic response. In chapter 3 we considerTI surface modes and we address one interesting feature of these surface modeswhich happens in response to an applied out-of-plane magnetic field and can beused to probe small intrinsic gaps on the surface using an experimental techniquewhich is called β−NMR [46, 47].1.6 A lattice model for topological insulatorsA three dimensional Dirac Hamiltonian can be expressed in terms of two sets ofPauli matrices τ = (τ1,τ2,τ3) and σ = (σ1,σ2,σ3) acting on a two level orbital andspin degrees of system. In momentum space, the gapless Hamiltonian in terms ofthese matrices can be written asH(~k) = τ3(υ1σ1ky−υ2σ2kx)+υ3τ2kz. (1.10)14It is easy to consider a tight-binding cubic lattice model which, near specificpoints in the BZ and at low energies, can be described effectively by the aboveHamiltonian. In a Fourier transformed tight-binding Hamiltonian, the momen-tum dependent Hamiltonian matrix can only depend on crystal wave-vector~k viacos(~ai ·~k) and sin(~ai ·~k) terms therefore one can immediately find the correspond-ing Hamiltonian matrix. This can be expressed by replacing (kx,ky,kz) by theirlattice versions, i.e., (sinkxa1,sinkya2,sinkza3). Assuming that the lattice is cubicwith a1 = a2 = a3 = a we getH(~k) = τ3λ (σ1 sinkya−σ2 sinkxa)+ τ2λz sinkza, (1.11)This Hamiltonian is invariant under the time-reversal meaning that Θ†H(~k)Θ=H(−~k) where the time reversal operator, Θ= iσ2K, in which K is complex conju-gation operator, acts on σ and τ operators in the following wayΘ†(τ1,τ2,τ3)Θ= (τ1,−τ2,τ3), (1.12)Θ†(σ1,σ2,σ3)Θ= (−σ1,−σ2,−σ3). (1.13)There are eight distinct Dirac points at the time-reversal invariant momentagiven as~Γi =pi2(1,1,1)+ pi2(±1,±1,±1). (1.14)In general, there are various ways to open up a gap in the spectrum of theHamiltonian given in Eq. (1.11) by adding combinations of the Pauli matrices.However, there is only one way to open up a gap respecting the time-reversal sym-metry. It is not hard to find this term by forming various possible combinations ofthe Pauli matrices and seeing which one is invariant under time-reversal and at thesame time anti-commutes with all the terms in the Hamiltonian. By this consider-ation the massive time-reversal invariant Dirac Hamiltonian can be written asH(~k) = λτ3(σ1 sinkya−σ2 sinkxa)+λzτ2 sinkza+M(~k)τ1. (1.15)15In a nearest-neighbour tight-binding Hamiltonian, the most general form ofM(~k) can be written asM(~k) = ε−2t coskxa−2t coskya−2t coskza, (1.16)Note that in the above sin~k ·~ai terms cannot exist since TR symmetry requiresthat the function must be even under~k to −~k so the only allowed~k dependenceis via cosine functions. One can compute the strong topological invariant usingthe expression given in Eq. (1.8) since the system is inversion symmetric. It turnsout that the system can be in a topological insulator phase for a wide range ofsystem parameters. Another way to see how the system can be in a topologicalinsulator phase is to calculate the strong topological invariant analytically in thecontinuum limit and considering a proper regularization for a three dimensionalDirac Hamiltonian to resolve the integral divergences arising from the Dirac sea.It turns out that in this case each of the eight existing Dirac modes contribute tothe total value of this invariant which can either be zero or one. This lattice modelwhich is a regularized minimal model for Bi2Se3 has been studied before in thecontext of the magnetic response of the topological insulators [44, 48].For a single Dirac Hamiltonian of mass m and Fermi velocity components(υ1,υ2,υ3), the contribution turns out to beδν3D0 =12sgn(mυ1υ2υ3). (1.17)The ’strong’ invariant, ν3D0 can be obtained by summing the contribution fromall these Dirac points. It turns out that for 2t < ε < 6t the system is in a topologicalphase (ν3D0 = 1).An alternative characterization of a ’strong’ TI follows from its unusual re-sponse to applied electromagnetic fields which is encoded in a bulk ‘axion’ term[26, 37] of the formLaxion = θ(e22pihc)~B ·~E, (1.18)with θ = pi . The axion term (1.18) appears in the electromagnetic Lagrangian inaddition to the standard Maxwell term. Eq. (1.18) underlies the topological mag-netoelectric effect [26, 37] in which electric (magnetic) polarization is induced by16external magnetic (electric) field, as well as the Witten effect [48, 49] that attachesa fractional electric charge to a magnetic monopole.The axion term (1.18) has been introduced in the context of high-energy physicsdecades before the discovery of STIs to resolve the CP non-violation problem inQuantum Chromodynamics (QCD) [50–52]. The corresponding θ(~x, t) field isknown to particle physicists as the axion field [53]. The action of the uniformaxion field can be viewed as a topological term for the θ vacuum in QCD arisingfrom nontrivial topology of such a vacuum.For a generic value of θ the axion term breaks T as well as parity P . This isbecause under time-reversal ~B→−~B, ~E→ ~E, while under spatial inversion ~B→ ~B,~E → −~E. What allows the T - and P-invariant insulators to possess an axionterm with θ = pi is the 2pi-periodicity of the axion action Saxion =∫dtd3xLaxion inparameter θ . Specifically, on periodic space-time (that is used to model an infinitebulk crystal), the integral in the axion action is quantized,(e22pihc)∫dtd3x~B ·~E = Nh¯, (1.19)with N integer. All physical observables depend on exp(iSaxion/h¯) and are thusinvariant under a global transformation θ → θ + 2pi . Consequently, θ = pi andθ = −pi are two equivalent points and describe a T - and P-invariant system.Conversely, in a system invariant under P or T the value of θ is quantized to 0 orpi .The statement regarding the quantization of the expression (1.19) on periodicspace-times has been made in several influential papers [26, 37, 53, 54] but nosimple physical explanation has been given of its validity. In chapter 2, we willpresent our simple proof of quantization using simple arguments based on singleparticle quantum mechanics and classical electrodynamics.1.7 Weyl and Dirac semimetalsThe topological protection is not restricted to just one and two dimensional metallicsystems which arise at the interface of topological insulators. Indeed one can gen-eralize this idea to find a three dimensional metallic system where the band crossing17is protected by the underlying nontrivial topological structure of the quantum statesrather than the discrete symmetries such as the inversion or the TR. Recently, ex-perimentally feasible proposals to realize such 3D metallic phases has been madewhere linear energy crossings (Dirac Points) exist in the band dispersion and areprotected by global topological properties of the electronic states [55–59]. In thiscase there is no discrete symmetry involved in protecting the Dirac point from thegap opening as opposed to the 1D and the 2D TI edge and surface cases wherewe still need to keep the TR symmetry in order to maintain the band crossings[42, 44]. The existence of this nontrivial topological structure associated with theisolated linear touching points would lead to some interesting transport phenom-ena in the bulk and on the surface of such systems. A topological axion term witha non-uniform axion field is predicted to emerge in the effective electromagneticLagrangian which is related to the chiral anomaly in these systems [60–63].The low energy theory of an isolated Weyl point is given by the HamiltonianhW (~k) = b0 +υFσ · (~k−~b), (1.20)where υF is the characteristic velocity, σ a vector of the Pauli matrices, b0 and~b denote the shift in energy and momentum, respectively. Because all three Paulimatrices are used up in hW (~k), small perturbations can renormalize the parameters,b0,~b and υF , but cannot open a gap. This explains why the Weyl semimetal formsa stable phase [36]. Although the phase has yet to be experimentally observed thereare a number of proposed candidate systems, including pyrochlore iridates [58], TImultilayers [55–57, 64], and magnetically doped TIs [65].When such topological protection of band touching exists, the system is in atopological semi-metal phase even in the presence of weak disorder or other typesof interactions that preserve the momentum conservation. These band crossingpoints in the BZ can only exist in pairs [62]. Near an isolated Weyl point the lowlying states can be described by the 2-by-2 Dirac Hamiltonian given in Eq. (1.20).Therefore, any local perturbation that does not violate the momentum conservationcan only shift the position of such a point in the BZ and is not sufficient to gapout the spectrum unless it is strong enough to merge two such points with oppo-site chirality and make the system unstable towards becoming an insulator. These18   Figure 1.7: a) Doubly degenerate massless Dirac cone at the transition froma TI to a band insulator. Weyl semimetals with the individual conesshifted in b) momenta and c) energy. Panel d) illustrates the Weyl insu-lator which can arise when the excitonic instability gaps out the spec-trum indicated in c). In all panels two components of the 3D crystalmomentum~k are shown.points can be thought of as topological defects in the fibre bundle formed by theelectronic states in the BZ as the base manifold and are the magnetic monopoles ofthe pseudo-magnetic field (Berry curvature) associated with the gauge field definedfor the Bloch states. Only the pairwise annihilation of such points with oppositechirality is possible as one can define a well-defined charge for these monopoleswhich is a conserved quantity in the regime where the crystal momentum is stilla good quantum number. The topological charge defined for a Weyl point can beexpressed as the surface integral of the Berry curvature on the Fermi surface sphere.In theory, it is possible to choose a physical parameter to adiabatically drive asystem to a TI phase from an OI or vice versa by tuning such a parameter. Thisphase transition can happen for the lattice model introduced in the previous sec-tion in Eq. (1.15) by changing a single lattice parameter. As one fine tunes such aparameter to the critical point where the gap closes (which is inevitable when wehave TR or inversion symmetry) the system becomes an unstable bulk metal if theTR and inversion symmetry is preserved at the same time. At this critical point thesystem would be in an unstable 3D metallic phase with degenerate bands near thecrossing points. This gapless metallic phase is susceptible to local perturbations19and instabilities that drive the system to either TI or OI phases. Now if anotherparameter in the Hamiltonian is tuned in such a way that it separates the degen-erate bands at the crossing points in the BZ (by breaking the inversion and/or theTR symmetry) then the system would be in a topological semi-metal phase withgapless modes dispersing in three spatial dimensions which are ”robust” againstmomentum conserving perturbations. The topological nature of the phase reflectsitself in the appearance of a nontrivial term in the effective Lagrangian of the elec-tromagnetic fields [63]. This topological term is very similar to what appears inthe effective electromagnetic response of TIs as an abelian axion term, however inthis case the axion field is a non-homogeneous field as opposed to being a constantfield with θ = 0(OI) or pi(TI). This topological term can be expressed asSaxion =C∫θ(~x)F ∧F , (1.21)where C is a constant and θ(x) modulates in space and time with wave-vectorthat depends on the separation of the Weyl nodes in momentum and energy. Fora single pair of Weyl nodes it is given by θ(~x) =~q ·~x− b0t in which ~q is a vectorof the separation of the pair of the Weyl nodes in the crystal momentum and b0 isthe energy offset between two Weyl points [60–63]. This topological term in theeffective electromagnetic Lagrangian is closely related to the topological transportphenomena that is present in these systems [63]. This unusual response is a con-sequence of the chiral anomaly, well known in the quantum field theory of Diracfermions [61, 62, 66], now potentially realized in a Weyl semimetal. The physicalmanifestations of the above topological term can be best understood from the as-sociated equations of motion, which give rise to the following charge density andcurrent response,ρ = e22pi2~b ·~B, (1.22)~j = e22pi2 (~b×~E−b0~B). (1.23)Eq. (1.22) and the first term in Eq. (1.23) encode the anomalous Hall effect thatis expected to occur in a Weyl semimetal with broken T [55, 56, 58]. The sec-ond term in Eq. (1.23) describes the Chiral Magnetic Effect (CME), whereby a20ground-state dissipationless current proportional to the applied magnetic field ~B isgenerated in the bulk of a Weyl semimetal with broken P [67].These observations are based on field theoretical considerations which requiresspecific regularizations, so the important question is whether in a lattice systemthese effects can still survive when two opposite chirality Weyl points are con-nected via high energy modes of the lattice. In chapter 5 we address this importantquestion.1.8 One dimensional topological phase with MajoranafermionsSo far we have only discussed the nontrivial topological phases in two and threedimensional systems. It turns out that even in one dimension nontrivial topologicalphases exist. The most interesting example has been introduced by Alexei Kitaev in2001 [69]. He showed that spin-less electrons in a Nearest-Neighbour (NN) tight-binding model with a p-wave pairing can form an interesting topological phasewhich supports exotic zero-dimensional states near their ends. These states haveexactly the same properties as the elusive Majorana Fermions (MF). Hypothesizedby Ettore Majorana in 1937, MFs are interesting Dirac states which are their ownanti-particles. The creation and annihilation operators of the Majorana state are thesame and they follow non-abelian statistics.Kitaev’s model still remains experimentally unfeasible since electrons are spin-half particles. A p-wave superconductor in two-dimensions also supports Majoranafermions in the vicinity of the magnetic vortex. However, this model did not resolvethe problem since p-wave superconductivity does not occur easily in condensedmatter systems and its existence in Sr2RuO4 is still under debate [70].In 2008, Fu and Kane showed that it is possible to realize a topological phasewith Majorana fermions from s-wave pairing order at the surface of a topologicalinsulator [68, 71]. This was an important step towards realizing Majorana fermionsin condensed matter systems since all the previous models were based on p-wavesuperconducting ordering. The main ingredients in the Fu and Kane two dimen-sional topological superconductor model was a combination of magnetic ordering,spin-orbit coupling and the proximity induced s-wave superconductivity. In 201021(a)s-wave superconductor1D wireBxyz(b) Ekµ∆Esoµ/∆Trivialh/∆1(c) Topolo-gical(d)1D wires-wave SC γ2γ1 BGate γ4γ3 GateFigure 1.8: (a) Basic architecture required to stabilize a topological supercon-ducting state in a 1D spin-orbit-coupled wire. (b) Band structure for thewire when time-reversal symmetry is present (red and blue curves) andbroken by a magnetic field (black curves). When the chemical potentiallies within the field-induced gap at k = 0, the wire appears ‘spinless’. In-corporating the pairing induced by the proximate superconductor leadsto the phase diagram in (c). The endpoints of topological (green) seg-ments of the wire host localized, zero-energy Majorana modes as shownin (d). Figure and caption from Ref. [68]Refael and von Oppen used all these ingredients to propose a quantum wire as atopological superconductor with MFs [72]. Their setup consisted of a semiconduc-tor quantum wire placed on top of an s-wave superconductor under applied in-planemagnetic field to produce a Zeeman field. This was ground breaking proposal sinceall the ingredients of this model are already realizable in experimental setups. Theonly challenge was tuning the chemical potential to the desired mid-gap configu-ration where the gap arises from the Zeeman field which is normally of the orderof 1 meV. In 2012, a group of experimentalists at Delft University of Technology22reported robust zero-bias peak in their experimental setup based on the Refael andvon Oppen proposal [73]. This interesting observation has been under debate andit is now believed that there are other scenarios which can equally explain the ex-istence of zero-bias peaks and a smoking-gun proof of the existence of Majoranafermions in an experimental system still remains to be found. Another interestingproposal is magnetic chain of adatoms placed on top of an s-wave superconduc-tors. Magnetic nuclei on top of superconductors have been shown to bind localizedmid-gap electron modes. Forming a chain of these atoms, which is experimentallyplausible, can potentially lead to a one dimensional topological phase. It turns outthat the topological phase can arise naturally due to the existence of a magneticinstability in the one dimensional electron systems. In chapter 6 we consider thisproposal and we address how this can arise in detail using a simple minimal model.23Chapter 2Axion Angle Periodicity inTopological Insulators2.1 OverviewThe electromagnetic response of TIs is highly affected by the nontrivial topologicalstructure of the underlying electrons. One can see this by integrating out massiveDirac electrons coupled to a classical electromagnetic gauge field. Apart from theusual Maxwell terms in the action, an extra term exists which arises solely due tothe twist in one of the Dirac masses which is essential for driving a system intoa TR symmetric TI. This term in the action modifies the Maxwell equations andexplains the quantized magneto-electric effects in the bulk and at the surface of athree dimensional TI. Since the quantization of the axion term and the related θ -periodicity underlies the essential element of the theory of topological insulators, itis important to have a clear physical understanding of its origin. In the rest of thischapter we provide a direct and simple proof of the axion action quantization onperiodic space-time. We also consider a non-periodic case where the axion actionremains quantized. Our proof is based on the electromagnetic field decomposi-tion into an ‘externally imposed’ uniform constant part which we show can havenon-zero contribution to Saxion and a part generated by space-time periodic chargeand current configurations whose contribution to Saxion vanishes. The quantizationcondition (1.19) follows from the requirement that the underlying vector potential24be consistent with the single-valuedness of the electron wave-functions [74]. Inour proof we assume that no magnetic monopoles are present but we comment onthe situation with monopoles in the end.2.2 Electromagnetic field decompositionIn a covariant formulation with the speed of light c = 1 the axion action can bewritten as [53]1h¯Saxion =θ8Φ20∫d4x εµναβFµν(x)Fαβ (x), (2.1)where Fµν = ∂µAν −∂νAµ is the electromagnetic field tensor and Φ0 = h/e is thequantum of the magnetic flux. In the following we consider a space-time hypercubeof side L with periodic boundary conditions imposed on Fµν(x) in all directions.In the absence of monopoles integration by parts gives1h¯Saxion =θ4Φ20∫d4x εµναβ∂α[Fµν(x)Aβ (x)], (2.2)At first glance, from the periodicity of space and time one might conclude thatthe integral in (2.2) vanishes for a general electromagnetic field tensor since it canbe written as a three dimensional hyper-surface integral of FµνAβ which is zero ifthis function is periodic in space-time coordinates. However, a simple example ofconstant uniform fields ~E‖~B shows this conclusion to be erroneous. The point isthat in general the gauge field 4-vector of a periodic electromagnetic field is notperiodic in space and time. As an example consider a lower dimensional case ofT 2 torus with a magnetic flux through its hole increasing linearly with time (Fig.2.1a). This induces an electric field on the torus which is constant and thereforeperiodic in time and the coordinates that parametrize the torus. However the lineintegral of the gauge field over the non-contractible loop enclosing the magneticflux is nonzero which means that the gauge field cannot be chosen periodic. Forthe field configurations of this type, containing field lines along non-contractibleloops, Saxion will be non-vanishing and we must consider these with special care.A question arises here: in general for what kind of periodic electromagneticfield configurations in 3 spatial dimensions, the gauge field cannot be chosen peri-25Figure 2.1: (a) A magnetic field increasing linearly with time and confinedto a torus hole produces a uniform and constant electric field along thetorus. (b) A closed path on the torus can be thought of as enclosing twoareas, Ω1 and Ω2.odic? A gauge potential Aµ cannot be chosen periodic if∮d` ·A 6= 0 (2.3)where the line integral is over any L× L square located in one of the coordinateplanes xµxν with µ 6= ν on the hyper-cube. We note that the above integral (2.3)is gauge invariant and, due to the periodicity of Fµν(x), its value is independent ofthe position of the L×L square, i.e. it is invariant under any space-time translation.In terms of the field tensor Eq. (2.3) can be written as∫dxµdxνFµν 6= 0, (2.4)with no summation over µ , ν . Physically, this means that the total magnetic fluxthrough one of the spatial faces of the hypercube is non-zero and a similar conditionfor the electric field (see below).The above considerations motivate a decomposition of the gauge potential intotwo pieces,Aµ(x) = A0µ(x)+δAµ(x), (2.5)26such that ∮d` ·δA = 0. (2.6)Hence δAµ can be chosen periodic, while A0µ(x) contains any non-periodic part.Similarly, we writeFµν(x) = F0µν(x)+δFµν(x), (2.7)with F0µν ≡ ∂µA0ν − ∂νA0µ . To make this decomposition unique (up to a gaugetransformation) we furthermore impose a condition that components of F0µν areuniform and constant,F0µν =1L2∫dxµdxνFµν . (2.8)In terms of electric and magnetic field vectors our decomposition corresponds to~E = ~E0 +δ~E, ~B = ~B0 +δ~B, (2.9)where ~E0, ~B0 are constant uniform fields while δ~E, δ~B are space-time varyingfields derived from the periodic gauge potential δAµ(x). The constant fields canonly be produced by magnetic and electric fluxes through the holes in the T 3 torusembedded in a four dimensional space, and it is not possible to devise non-singularcharge and current sources within the periodic three dimensional space to producethem [75].Fields δ~E and δ~B are produced by ordinary charge and current sources. Theyhave the following physical properties: (i) Magnetic fluxes associated with thesemagnetic fields through each spatial face vanish,ε i jk∫dxidx jδBk = 0 i, j,k = 1,2,3. (2.10)There is no summation on indices and the equation holds for all xk and x0. (ii) Theintegral over any space-time face∫dx0dxiδEi = 0, (2.11)with no summation on i = 1,2,3. (i) and (ii) are properties of the fields producedby a general space-time periodic charge and current sources. With this preparation27we can now proceed to evaluate the axion action.It is most convenient to employ Eq. (2.2) where in view of our decomposition(2.6,2.7) the expression FµνAβ is replaced byF0µνA0β +2F0µνδAβ +δFµνδAβ . (2.12)An integration by parts has been performed on δFµνA0β to obtain the factor of 2in the middle term. Now the second and the third term in the above expression(2.12) are explicitly space-time periodic and therefore their contribution to Saxionidentically vanishes. The only contribution to the action comes from the first termwhich represents the uniform constant part of the electromagnetic fields. Thus,1h¯Saxion =θΦ20∫d4x ~E0 ·~B0. (2.13)It remains to be demonstrated that the action is quantized for these constant anduniform fields.2.3 Quantization of the Chern numbersOur arguments thus far have been purely classical. At the level of classical elec-trodynamics, clearly, the integral in Eq. (2.13) can attain any desired value and isnot quantized. To proceed, we must recall that in the present context the axionterm results from integrating out the electron degrees of freedom in a topologicalinsulator. Electron behaviour is inherently quantum mechanical. The axion actionquantization then follows from the requirement that the gauge potential Aµ thatcouples to the electron wave-functions, be consistent with the quantum theory ofelectrons in periodic space-time.In the following we assume for simplicity that our fields are pointed along thex3 direction, ~E0 = E3xˆ3 and ~B0 = B3xˆ3. Other components can be treated in anidentical fashion. For this configuration we may decompose our space-time torusT 4 into a direct product T 212×T 203 and write1h¯Saxion =θΦ20∫dx1dx2B3∫dx0dx3E3. (2.14)28It remains to show that each of these integrals is an integer multiple of the magneticflux quantum Φ0. The first integral represents the total magnetic flux through thex1x2 face of the hypercube. The quantization of this term follows from the standardarguments for the electron motion in applied magnetic field, which we now brieflyreview for completeness.Imagine an arbitrary closed path C on the T 212 torus. As illustrated in Fig.2.1b it encloses area denoted as Ω1. Alternately, it can be viewed as enclosing itscomplement on T 212 denoted as Ω2. Using Stokes’ theorem we may write∫Ω1~B ·dS =∮C~A ·dl (2.15)∫Ω2~B ·dS = −∮C~A′ ·dl, (2.16)where the prime on the vector potential signifies the subtle but important fact thatthe equality is required to hold only up to a gauge transformation Aµ → A′µ =Aµ − ∂µ f , with f (x) a scalar function. Now the line integral of ~A along a closedpath is normally thought of as a gauge invariant quantity in which case adding Eqs.(2.15) and (2.16) immediately implies∫Ω1+Ω2~B ·dS= 0. This suggests that Saxion/h¯is indeed quantized but the only value allowed is 0. However, there exists a class of‘large’ gauge transformations f (x) which change the value of the line integral butleave the wave-function single valued. The latter transforms as Ψ(x)→ Ψ′(x) =eie f (x)Ψ(x) and the relevant f (x) contains a vortex (a Dirac string) at some point ofthe T 212 torus, i.e. eie f (x) ∼ einϕ where ϕ is an angle in x1x2 plane measured fromthe vortex center and n is an integer. Since∮C ∇ f ·dl = 2pin the inclusion of largegauge transformations of this type can be seen from Eqs. (2.15) and (2.16) to allowfor non-zero quantized values∫ dx1dx2B3 = nΦ0.One can advance the same argument to establish the quantization of the secondsurface integral in Eq. (2.14). Consider a closed path, this time on T 203, enclosingΩ1 and Ω2 regions. It is straightforward to check that all steps proceed exactly asbefore. The large gauge transformations now involve space-time vortices in f (x)(i.e. vortices in the x0x3 plane) and lead to the analogous result∫ dx0dx3E3 = mΦ0with m integer.29Combining the above results we find1h¯Saxion = Nθ , (2.17)with N = nm. Eq. (2.17) shows that the axion action for electromagnetic field isquantized on periodic space-time and, consequently, the amplitude exp(iSaxion/h¯)is invariant under the shift of the axion angle θ by any integer multiple of 2pi .A more abstract and rigorous way to think of the quantization of these quan-tities is using fibre bundle mathematics mentioned briefly in the beginning. Thefibre bundles [3] can be classified in terms of the so called Chern characters. Theseare forms whose integral over closed base space of the bundle always return aninteger number. One consequence is that the integral of a first Chern character ofthe Abelian U(1) gauge theory, Fµν/Φ0 on the closed base space T 2µν must alwaysbe an integer. On the other hand the axion action is a second Chern character in-tegral evaluated for this abelian gauge theory. We showed that this can be writtenas a product of the two first Chern character integrals whose quantization, as wediscussed, has a clear physical interpretation.2.4 Open boundaries caseAssuming periodic boundary conditions in all directions is the simplest way toavoid edges and to concentrate on the bulk response. However, in real experimentalsetup one must deal with a situation where the the fields are present in a finiteportion of space and over a finite time duration. The question arises whether theaxion action remains quantized under these non-periodic conditions. The answeris “yes” provided that one additional condition on the gauge potential is satisfied.Specifically, it is possible to show that Eq. (1.19) remains valid if (i) the fields ~B and~E vanish outside a space-time volume V and (ii) the underlying gauge field Aµ issuch that its presence cannot be detected by any Aharonov-Bohm type experimentperformed using charge e particles outside V . For the magnetic field this implies,for example, that the total flux enclosed by any closed trajectory is nΦ0 with ninteger. In that case the Aharonov-Bohm phase acquired by charge e particle is2pin and thus indistinguishable from 0.302.5 Abelian versus non-abelian gauge fieldsTable 2.1: Chern Character for Abelian and non-Abelian gauge fieldsAbelian gauge non-Abelian gaugeBase space Minkowski Space Four-dimensional BZGauge field Aµ = (φ ,~A) Aαβi =−i < α,~k|∂i|β ,~k >Field strength Fµν = ∂ µAν −∂ νAµ F αβi j = ∂iAαβj −∂ jAαβi + i[Ai,A j]αβFirst Chern Φ/Φ0 σi j/σ0Second Chern Saxion/θ C2As we discussed in the beginning, the electromagnetic response of the TIs canbe studies by considering a Dirac field coupled to a classical electromagnetic gaugefield. Chern characters and numbers can be defined for this abelian gauge field asthey are shown in the table. Apart from the abelian gauge field, one can defineChern characters in the same fashion for the non-abelian pseudo magnetic gaugefield of the electronic Bloch states.All the possible configurations for the occupied electronic states can be char-acterized in terms of invariants which are the integral of the so called Chern Char-acters over the closed base space (the BZ). In this case, first Chern numbers canbe defined by integrating the first Chern character over a 2D closed sub-manifold(≡ T 2) as discussed in the quantization of the magnetic flux in the second chapter.CA1 =1Φ0∫dx∫dyFxy =1Φ0∮~B ·da = ΦΦ0, (2.18)One can also define the second Chern number which can take integer valuesand can be factorized in terms of two first Chern characters defined on two 2Dsub-manifolds as we showed in our proof of the quantization of the axion action.Till now all the Chern characters we discussed were defined for the QED’s AbelianU(1) gauge field over real space that couples to the Dirac field via minimal cou-pling.In (3+1) we can define three first Chern numbers (σxy,σyz,σxz) which all van-ish for a time-reversal invariant system. For such a system with zero first Chernnumbers we can have a non-zero second Chern number which is related to the θ31axion field [37]. This quantity is an integral of the second Chern character for anon-Abelian gauge field evaluated over a 4-dimensional BZ. Table (2.1) comparesthe Abelian and non-Abelian cases.2.6 Effect of magnetic monopolesIn passing from Eq. (2.1) to (2.2) we have assumed that a term Aβ εµναβ∂αFµν thatappears in the integration by parts vanishes on the account of partial derivativescommuting and εµναβ being antisymmetric. This assumption would appear to failin the presence of magnetic monopoles. Consider e.g. the β = 0 component ofthe above expression which equals 2A0∇ ·~B. In the presence of the non-vanishingmonopole density ∇ · ~B 6= 0 such term will give non-zero contribution to Saxionwhenever A0 is non-zero. Similarly, β = 1,2,3 terms correspond to monopolecurrents and may be non-vanishing as well. Careful evaluation of Saxion in thepresence of monopoles leads to a contribution that appears to violate the globalsymmetry θ → θ +2pi of Saxion on periodic space-time.One nevertheless expects the axion action to remain a valid description of topo-logical insulators in the presence of monopoles. For instance the occurrence of theWitten effect, [49] i.e. electric charge −e(θ/2pi + n) bound to the fundamentalmonopole, has been recently verified using a model topological insulator [48].We believe that the problem with the apparent violation of the θ → θ + 2pisymmetry lies in the fact that the presence of magnetic monopoles contradicts thedescription of the system in terms of the gauge potential Aµ . It is well known [75]that it is mathematically impossible to find a globally defined vector potential ~Athat would give rise to non-vanishing ∇ ·~B. Vector potentials commonly used todescribe monopole-like field configurations [76] invariably contain a singularity.For example the usual choice of the vector potential ~A =−Φ0(1+cosϑ)∇ϕ , with(ϑ ,ϕ) the spherical angles, is singular along the positive z-axis. Physically, thissingularity can be viewed as an infinitely thin solenoid, a “Dirac string”, that bringsone quantum of the magnetic flux to the location of the monopole. The string caneither extend to infinity or terminate at an anti-monopole. A Dirac string can bethought of as a limiting case of a solenoid with the radius approaching zero. Such32a thin solenoid carrying integer number of flux quanta is invisible to electrons as itimparts Aharonov-Bohm phase that is an integer multiple of 2pi . Nevertheless if weinclude the fields inside the solenoid in the computation of Saxion then the condition∇ ·~B = 0 is upheld, all magnetic field lines are closed, and the term Aβ εµναβ∂αFµνidentically vanishes. Our original argument therefore goes through as before andimplies symmetry under θ → θ + 2pi of Saxion on periodic space-time even in thepresence of Dirac monopoles.2.7 ConclusionsWe have presented a simple and intuitive proof of the quantization of the topolog-ical axion action on periodic space-time. Our considerations show that the theoryis invariant under a global θ → θ +2pi transformation consistent with the Z2 char-acter of the fundamental ‘strong’ invariant describing the physics of time-reversalinvariant band insulators. An important observation was that for U(1) Abeliangauge field the second Chern number is a direct product of two first Chern num-bers and once a first Chern number vanishes the second will vanish too but inthe SU(2) non-abelian case even when the first Chern numbers vanish the secondChern number can be non-zero as it can be the case in a TI [37].33Chapter 3Response of Surface Dirac States3.1 OverviewTopologically protected electronic states residing on the surface of topological in-sulators (TI) [33, 38, 38, 40, 41, 77–79] form a unique 2D metal distinct from thoseso far realized in solid state systems. Similar to the low energy electronic states in asingle layer graphene [19], the robust 2D metal on an TI surface has conic branchestouching (in the absence of the intrinsic gap) at high-symmetry points in the firstBrillouin zone. This resemblance in the energy dispersion is responsible for somecommon properties between these metallic systems, such as the square-root de-pendence of Landau Level (LL)s spacing to the applied magnetic field. However,there are important differences between these systems, as a result of the topologicalnature of the surface states in TIs.One of the interesting aspects in which these 2D metallic systems behaveuniquely is their magnetic response [80]. Here, motivated by the recent progressin the experimental methods and the importance of the spin susceptibility for ourunderstanding of electronic systems, we study the spin response of electrons onthe surface of a TI and show that it exhibits interesting features even in a simplenon-interacting limit. We find that the characteristic spin-momentum locking ofthese electrons leads to a unique spin response that is distinct from spin-degeneratesystems like graphene and 2DEG. Instead of the oscillatory behaviour of the sus-ceptibility as a function of the chemical potential found in spin degenerate systems,34in TIs our work predicts a plateau-type behaviour which arises from the strong cor-relation between spin and orbital degrees of freedom. In addition we find that theexistence of a special LL with full spin polarization leads to a jump in the magne-tization as the chemical potential crosses the energy of this LL. When an intrinsicmagnetic ordering is present there is also a jump in the magnetization as a func-tion of the applied magnetic field. Our results can therefore assist in detectionof such intrinsic magnetization [42, 43] which is known to have profound conse-quences for the nature of the surface state; the magnetized surface state of a TI ispredicted to become a quantum Hall liquid with half-integer quantized Hall con-ductivity [35, 37] and many unusual [54, 81–85] and potentially useful [85, 86]physical properties.There are various ways of measuring the weak magnetization produced by elec-trons on a metallic surface. The SQUID scanning magnetometry is a highly sen-sitive probe which can detect tiny magnetization on the surfaces. However, it isa challenge to use this device to probe magnetization when there is a large ap-plied magnetic field which interferes with the superconducting part of the SQUIDand causes noise. A variant of this method using a superconducting pickup coilhas been developed in order to study deHaas-van Alphen oscillations of the 2Dmetallic systems in a large perpendicular magnetic field [87, 88]. High-sensitivitymicro-mechanical cantilever magnetometry [89] is another way of measuring elec-tronic magnetization, however, this method measures the magnetization of thewhole sample and it is difficult to isolate the contribution from the electrons ona single surface.The nuclear magnetic resonance (NMR) is another powerful experimental tech-nique which can be used to study electronic spin magnetic response in a bulk metal[90]. Through the so-called Knight shift of the nuclear resonance peak, it is pos-sible to probe the spin part of the magnetic susceptibility of the electrons as theyinteract with the resonating nuclei in their proximity. Unfortunately, this method, inits conventional form, fails to be useful in very thin films and 2D metallic systemsdue to the limitation in the number of available nuclei and the resulting weaknessof the signal. Thanks to the progresses made by experimentalists in controlling andimplementing high energy beams of unstable ions, an experimental technique, theso-called β -NMR, has been recently developed to overcome the above limitations.35Briefly, in this exotic variety of NMR, unstable radioactive ions such as 8Li and11Be are implanted in the sample surface. The nuclear spin precession signal isthen detected through the products of the beta decay of the radioactive nucleus.Since the ion implantation depth can be controlled by tuning the beam energy itis possible to acquire information about the behaviour of the electronic spins invery thin metallic films [46, 47]. Experiments are currently underway to study thesurface magnetic response of TI crystals Bi2Se3 and Bi2Te3 using β -NMR [46].This chapter is organized as follows. First we introduce a simple model knownto describe the electrons on the surface of a TI in the presence of a perpendicularmagnetic field. We review the exact solutions of its eigenvalue problem which hasbeen studied previously in various contexts [80, 91]. We then calculate the spinmagnetization and susceptibility assuming that the chemical potential lies insidethe gap between positive and negative energy eigenstates. We discuss the magneticresponse of the surface as one tunes the magnetic field and the chemical potentialfor various values of the intrinsic gap. In the end, we explain how a β -NMR ex-periment might be able to detect these effects through Knight shift measurements.3.2 Dirac modes in out-of-plane applied magnetic fieldTurning on a perpendicular magnetic field adds two terms to the above Hamilto-nian. One is the minimal coupling of the magnetic vector potential, h¯~k→ h¯~k+ e~A(electron charge −e). The other is the coupling of the spins to the magnetic field,the Zeeman effect, expressed as δHZ = −gsµBh¯−1~B ·~s, where gs is the effectiveelectron gyromagnetic constant. In the bulk Bi2Se3 crystal gs ' 30 (Ref. [92]) al-though much smaller values have been reported for electrons near the surface [93].In our calculations below we use two representative values, gs = 8 and gs = 30,which yield qualitatively similar results with some interesting differences.In the continuum limit, the leading order Hamiltonian describing these surfacestates in the presence of the applied magnetic field, ~B = B0zˆ, in the Landau gauge~A =−(B0y,0), can be written in the following formH =∑kx∫dy Ψ†kx(y)H (kx,y)Ψkx(y), (3.1)36where Ψ†kx(y) is the creation operator for the spinor mode extended along the xdirection and localized at y. H (kx,y) for B0 > 0 is defined as(a)(b)B = B0zˆB = 0zˆkx"0kx""(a)(b)B = B0zˆB = 0zˆkx"0kx""Figure 3.1: The surface spectrum of a strong topological insulator in the ab-sence of magnetic dopants (∆0 = 0), (a) in the absence of the externalmagnetic field, (b) in the presence of an applied perpendicular magneticfield.H (kx,y) =(∆ iεcakx−iεca†kx −∆), (3.2)where εc = υF√2eh¯|B0| and∆= ∆0−gsµBB02. (3.3)The term in ∆ proportional to the magnetic field is the Zeeman contribution. akx isthe one-dimensional harmonic oscillator Bosonic operator defined asakx =1√2(ylB+ lB(∂y− kx)), l2B =h¯e|B0|. (3.4)Note that varying kx shifts the position of the localized state, produced by the ap-plication of a†kx on the vacuum state, along the y direction. In the B0 < 0 case,H (kx,y) can be obtained from the one given for B0 > 0 in Eq. (3.2) by exchang-ing the off-diagonal elements and replacing kx by its time reversed counterpart−kx.The eigenstates for B0 > 0 and n > 0 are given by [80, 91]φ+kx,n(y) =cos(δn/2)ϕn−1(y− kxlB2)−isin(δn/2)ϕn(y− kxlB2) , (3.5)37φ−kx,n(y) =sin(δn/2)ϕn−1(y− kxlB2)icos(δn/2)ϕn(y− kxlB2) , (3.6)while for n = 0 we haveφkx,0(y) =0ϕ0(y− kxlB2) . (3.7)In the above cosδn = ∆/ε+n and ϕn are the one-dimensional harmonic oscillatoreigenstates, i.e., a†aϕn = nϕn. We remark that the n = 0 eigenstate in Eq. (3.7) isvery special since it is fully spin polarized. This will have important consequencesfor the magnetic response discussed below.The eigenvalues associated with φ±kx,n(y) eigenstates are given byε±n =±√nε2c +∆2, n > 0, (3.8)and for the fully spin-polarized n = 0 eigenstatesε0 =−sgn(B0)∆. (3.9)Note that the form of the eigenstates for B0 < 0 is different from that given inEq.(3.5-3.7) since the Hamiltonian is different in that case.3.3 Spin susceptibility and magnetizationThe electronic magnetic moment due to spin is proportional to the spin operatorµ e =−γes, (γe =−gsµB/h¯), (3.10)Therefore, to calculate the spin part of the magnetic moment for the eigenstatesgiven in the previous section, we only need to find the expectation value of thespin operator. A straightforward evaluation using Eqs. (3.5-3.7) shows that all theelectronic states within the same LL contribute equally to the magnetization. For38each of them we haveM αx,n =M αy,n = 0, (3.11)M αz,n =gsµB2· ∆εαn, (3.12)Mz,0 =−gsµB2sgn(B0), (3.13)The total magnetization for each Landau level can be obtained by multiplyingthe above quantities by the Landau level degeneracy L2/(2pil2B), representing thetotal number of states with characteristic length lB that the surface area L2 can ac-commodate. Since the magnetization contribution computed above for each eigen-state is an explicit function of its energy, we can perform the following integral tofind the total magnetization density due to the electronic spinsMs =∫dε D(ε)Ms(ε)nF(ε), (3.14)where Ms(ε) = gsµB∆/(2ε) is the magnetization of the eigenstate with energy εand nF(ε) = 1/[e(ε−µ)/kBT +1] is the Fermi-Dirac distribution function. The elec-tronic density of states associated with the surface states is D(ε). For the Hamilto-nian we used in the previous section it takes the following formD(ε) = 12pil2B[δ (ε− ε0)+nc∑n>0,α=±δ (ε− εαn )], (3.15)where nc ≡ (Λ2−∆2)/ε2c is the Landau level index beyond which the energy ex-ceeds the cutoff energy Λ. The cutoff can be chosen to be the energy where the sur-face band becomes degenerate with the bulk bands and here we assume Λ = 300meV. Using this density of states function to perform integration in Eq. (3.14)yieldsMsB0= χ0[−nF(ε0)+ sgn(B0)nc∑n=1nF(ε+n )−nF(ε−n )ε+n /∆], (3.16)where χ0 ≡ (egsµB)/2h.39-2 0-100 -50  0  50  100! (meV)bMs-2 0-100 -50  0  50  100! (meV)bMs-4-2 0-100 -50  0  50  100! (meV)bMsFigure 3.2: Spin magnetization δMs = Ms(T,µ)−Ms(0,0) in units of (χ0·Tesla) as a function of the chemical potential for a nonmagnetic surface(∆0 = 0) at B0 = 3.0 Tesla, gs = 8 (top panel) and 30 (bottom panel).kBT = 0.1,1.0,5.0 meV (red, green, blue).For a constant magnetic field B0 the magnetization of a single TI surface givenby Eq. (3.16) shows an interesting behaviour as a function of chemical potential µillustrated in Fig. 3.2. In order to avoid ambiguity associated with the high-energycutoff we choose to display δMs = Ms(T,µ)−Ms(0,0), i.e. spin magnetizationrelative to the neutrality point at T = 0. For the negative values of µ magnetizationinitially decreases reflecting the fact that the negative-energy surface states exhibitnegative spin polarization, as can be seen from Eq. (3.12). The large jump in δMs40-15 0 15-15 -10 -5  0  5  10  15B0 (Tesla)Ms-10-5 0 5 10-4 -3 -2 -1  0  1  2  3  4B0 (Tesla)MsFigure 3.3: Spin magnetization in units of (χ0· Tesla) as a function of themagnetic field for (from left to right) ∆0 = −2,−1,0,1,2 meV andkBT = 0.1meV. gs = 8 (top panel) and 30 (bottom panel).near µ = 0 results from electrons filling the fully spin-polarized n = 0 Landaulevel. Further increase in µ results in increase of δMs now reflecting the fact thatthe positive-energy surface states exhibit positive spin polarization. To get a bettersense of the order of the magnitude, assuming gs = 30 and B0 = 1Tesla, the jumpin the spin magnetization on the surface area of size 1cm2 is of the order of 107µB.The above behavior is unique to topological insulators for it results from theLandau level structure in a single Dirac point (or more generally an odd numberthereof). In a TI the ‘other’ Dirac point is located on the opposite surface where themagnetic field points in the direction opposite relative to the surface normal, see4160(meV)-10 -5  0  5  10B0(Tesla)-6-4-2 0 2 4 6-4-2 0 2 4Figure 3.4: The color at each point (B0,∆0) represents the magnitude of thespin susceptibility in the linear response regime, i.e., χ = Ms/B0, inunits of χ0. The step (diagonal white line) is given by Eq. (3.18) andresults from n = 0 Landau level crossing the chemical potential. Thediscontinuity evident at B0 = 0 reflects the fact that susceptibility χ di-verges as B0→ 0. We have assumed gs = 8 and kBT = 0.01meV in thisgraph.Fig. 3.1. The contribution of this surface to δMs would be the same. We empha-size that for a relatively thick TI slab β -NMR will be sensitive to a single surfacefacing the beam and the behaviour predicted here is in principle observable, exceptthat continuous tuning of the chemical potential in a crystal might be difficult toachieve.A much more feasible experiment involves varying magnetic field B0 whilekeeping µ constant. We now show that a unique signature of the fully polarizedn = 0 Landau level still exists in samples with intrinsic magnetic ordering, whenthe chemical potential resides in the gap (i.e. both the bulk and the surface are in-sulating in the absence of the field). In his situation we can set µ = 0 in our model.Now consider the effect of the applied magnetic field. Since the gap between theadjacent energy levels in which the Fermi energy is located is fairly large (∼ 100 Kfor a 1T field), at sufficiently low temperatures we can replace the Fermi function42in Eq. (3.16) by the step function nF(ε)→Θ(−ε) and writeMsB0= χ0[−Θ(−ε0)− sgn(B0)nc∑n=1∆√ε2c n+∆2], (3.17)We have assumed that the chemical potential remains pinned at zero energy anddoes not change as we tune the magnetic field. With these assumptions the magne-tization has an interesting discontinuity at a finite magnetic field. The discontinuityin Ms occurs since tuning the magnetic field forces the fully spin polarized Landaulevel energy to evolve according to Eq. (3.3) and cross the chemical potential. Thecritical magnetic field at which the jump happens is given byBc =2gsµB∆0. (3.18)Assuming that the intrinsic gap ∆0 remains independent of B0, we plot the resultingmagnetization in Fig. 3.3. This predicted behaviour could be employed to exper-imentally detect the intrinsic magnetization gap in the surface of a magneticallydoped TI and measure the size of ∆0 through Eq. (3.18). Although this signatureonly occurs when the chemical potential lies very close to the surface state Diracpoint, we note that magnetically doped samples satisfying this requirement havebeen grown and studied [42].It is important to note that ∆0 is in general not independent of B0, since mag-netic moments of dopants will tend to align with the applied field. Thus, like ina ferromagnet, there will be a hysteresis effect whose features will depend on thematerial details [42]. One possible scenario is shown schematically in Fig. 3.5. Inthe case when ∆0 depends on the field, the equation (3.18) continues to hold butmust now be viewed as an implicit equation for the critical field Bc. It is also im-portant to note that the behaviour of the magnetization as a function of B0, will bemodified if we assume that ∆0 depends on the magnetic field. Assuming that ∆0remains uniform with a change in the magnetic field, one can obtain the behaviourof the spin magnetization as a function of the magnetic field by extracting the val-ues of the magnetization on the corresponding hysteresis path in the (∆0,B0) planeusing Eq. (3.16).43B01234B0ε0∆0(a) (b)Figure 3.5: (a) Schematic behaviour of ∆0, i.e., the intrinsic magnetizationgap on the surface in a system with a slab geometry shown in Fig. 3.1versus the applied magnetic field. (b) The energy of the n = 0 Landaulevel versus the magnetic field. Starting at the point 1 and by decreasingthe magnetic field gradually, the energy would follow the path shown bythe blue curve, i.e., 1→ 2→ 3→ 4, this happens if ∆0 is described bythe blue part of the cycle in (a). Now by increasing the magnetic fieldfrom a negative value, corresponding to the point 4, the energy wouldfollow the path given by the red curve, i.e., 4→ 2→ 3→ 1, since thistime ∆0 would be given by the red curve in the hysteresis cycle.Another question that arises has to do with the origin of the electrons that fillthe n = 0 Landau level upon changing the field through Bc. One may wonderwhere the extra electrons come from in a fully gapped isolated system. The answerlies in the side surfaces which under generic conditions remain gapless and act as areservoir of electrons. The model we consider here does not capture these states butthey reflect themselves in solutions of the Hamiltonian which are not normalizablein an infinite system. These are fully spin polarized with opposite energy and spindirection. In fact they are the particle-hole conjugates of the fully spin polarizedLL given in Eq. (3.7). Taking into account these electronic states and the fact that itis more favourable for electrons with higher energy to be transferred to the negativeenergy states the counting problem can be resolved.3.4 Knight shiftWe now outline how a β -NMR experiment can in principle be used to probe someof the physics discussed in the previous section. The valence and conduction elec-44trons in a metal posses magnetic moments arising from both their orbital motionand their spin degrees of freedom. Nuclear magnetic resonance technique can beused as a probe of the spin part of the total magnetization in the presence of themagnetic field by measuring the relative shift in the nuclear resonance peak withrespect to the same resonance peak in a reference insulating system. This effect,which is due to the interaction between electronic spins and those of the nuclei, isknown in the literature as the Knight shift and has been extensively studied in boththeory and experiment [90].The mobile electrons in a metal interact with the nuclei in their proximity andthe Knight shift in the resonance peak of these nuclei can be described by a localFermi contact interaction term given by Hint =−8pi3 µ e ·∑i µ iδ (r−Ri) where Ri isthe position of the ith nucleus and µ i = γNI i is its magnetic moment. The mag-nitude of γN , the gyromagnetic ratio, depends on the nuclear quantum state. Thenucleus total spin, I i, couples to the applied magnetic field and therefore the posi-tion of the peak depends on the magnitude of the total magnetic field experiencedby the nucleus which has a contribution due to the interaction with electrons. Itturns out that this shift is proportional to the spin susceptibility. The constant ofproportionality, known as the hyperfine coupling, can be computed using first prin-ciple calculations for the implanted nuclei. On the other hand, if we assume thatthe presence of the nuclei does not significantly alter the electronic states, then itis possible to approximate the shift for them by taking the expectation value ofthe aforementioned interaction term using the unperturbed electronic states. Thisis the lowest order approximation in the perturbative treatment of the interactionterm. For metallic systems with spin degenerate bands this shift is proportional tothe spin susceptibility, as can been seen from a simple calculation considering thefact that the spatial and spin degrees of freedom are uncorrelated [90].The spin-momentum locking on the surface of a TI along with the energy de-pendence of the penetration depth can in principle change the above simple physics.Since it is not possible anymore to separate spin and orbital degrees of freedom,one might question the validity of the linear relation between the Knight shift andthe spin susceptibility. We devote the rest of this section to addressing this issue byconsidering a very simple model. We assume that the nuclei do not alter the elec-tronic states around them. It is important to note that this assumption may break45down for the implanted nuclei if they modify the electronic states around themsignificantly and computing the Knight shift would then require a first principlecalculations.The field experienced by the ith nucleus due to the interaction with the proxi-mate electrons is given byδ~Bi ≡ −8pi3γe〈σδ (~r−~Ri)〉T , (3.19)where 〈...〉T is the expectation value over electronic states at temperature T . There-fore, the effective Hamiltonian for the ensemble of nuclei takes the formH effN =−h¯γN∑iI i · (~B0 +δ~Bi). (3.20)This way, the ith nucleus would have a resonance peak ωi = γN(B0 + δBiz). TheKnight shift is then defined by comparing the resonance frequency with the fre-quency in a similar material without these electronic statesKi =ωi−ω0ω0= B0 +δBiz−B0B0= δBizB0. (3.21)The shift in the resonance peak of the nuclei ensemble is the average of the Knightshift from each individual nucleus and is given byK = 1N ∑iδBizB0, (3.22)where N is the number of the implanted nuclei. Using the electronic eigenstatesgiven in Eq. (3.5-3.7) we get the following expression for K− 8piγe3NB0∑iocc∑kx,n,α|ψαn (zi)|2(|φαn,↑|2(~R⊥i)−|φαn,↓|2(~R⊥i)). (3.23)Here |ψαn (z)|2 appears as a factor in the realistic 3D electronic wave functions ofthe surface electrons reflecting the fact that the electrons have an energy dependentpenetration depth into the bulk. The nuclei implanted in the system have a spatialprobability distribution PNuc(z,r⊥), which depends on the energy and diameter of46the beam of the ions used in the β -NMR experiment. If the distribution functionis known, we can replace the above summation with a 3D integral over the crystalvolume1N ∑i→∫dzd2r⊥PNuc(z,r⊥). (3.24)Assuming that the distribution is uniform in the plane of the surface, i.e., PNuc(z,r⊥)=P(z), we getK = 8pi3l2BB0occ∑n,αf αn ·M αz,n, (3.25)where we have performed the integration over the in-plane degrees of freedom andreplaced the summation over kx with the LL degeneracy. We have also defined thenth LL weight, f αn , asf αn ≡∫dz|ψαn (z)|2P(z). (3.26)Now if we assume that different LL have the same penetration depth we have f αn =f0 for all n < nc and we obtainK = 8pi f03B01l2Bocc∑n,αMαz,n =8pi f03χse . (3.27)We thus recover the linear proportionality of the Knight shift to the surfaceelectronic spin susceptibility under reasonable assumptions. Note that if we relaxthe assumption that different LLs can now have different penetration depths, thenthe Knight shift would no longer be linearly proportional to the total spin suscep-tibility. Instead, it would be a weighted superposition of contributions from eachindividual LL to the spin susceptibility. Nevertheless, the Knight shift will stilldisplay the interesting behavior discussed in this study as long as f αn is a reason-ably slowly varying function of n. We should emphasize once again that althoughabove considerations elaborate on the differences caused by the spin-momentumlocking and the energy dependent penetration depth, they do not take into accountthe fact that the electronic wave-functions could be altered by the presence of theimplanted nuclei and the hybridization with the adjacent atoms.473.5 ConclusionsThe magnetic response of spins of the Dirac-like electrons on the surface of a topo-logical insulator shows interesting features both in the absence and the presence ofan intrinsic gap ∆0. When the surface states are gapped owing to the time-reversalbreaking perturbation (i.e. due to magnetic doping) the n = 0 LL, which is fullyspin polarized, can have positive or negative energy depending on the sign of theintrinsic gap ∆0 relative to that of the applied magnetic field. It will therefore becompletely filled or empty when the chemical potential potential is tuned to zeroenergy. Our study shows that this structure results in an observable jump in thespin susceptibility, measurable e.g. through the β -NMR Knight shift, as one tunesthe applied magnetic field through the critical value Bc given by Eq. (3.18). Theeffect may be used as a means to measure the magnitude of the intrinsic gap on thesurface of a magnetically doped topological insulator if β -NMR or another surface-sensitive technique (e.g., LE-µsR, CE-Mossbauer, PAD [94–96]) could capture themagnetic response of the electronic spins on the surface. This behaviour is a uniquefeature of the topological insulator exotic surface states closely related to the spe-cial form of the spin-momentum entanglement.48Chapter 4Models for Weyl Semimetals4.1 OverviewAs we discussed in the first chapter, the parent phase in a Weyl semimetal phase isa degenerate Dirac semimetal at the critical point in the transition between a topo-logical and an ordinary insulator where time-reversal and inversion symmetries arepreserved. Starting from this Dirac semimetal phase which has already been re-alized experimentally in Na3Bi [97], one can drive the system to what is calleda Weyl semimetal phase with a pair of isolated linear band-touching points. TheWeyl points’ separation in momentum space or energy has some interesting conse-quences that can potentially be observed in an actual experiment. Anomalous Hallcurrent is one of these interesting consequences in which applying an electric cur-rent in the direction which is perpendicular to the direction of the Weyl separationleads to a current in the transverse direction which is proportional to the length ofthe vector that connects the two Weyl points. Another interesting but questionableprediction is the CME which arises if the form of the topological action predictedbased on subtle field theoretical consideration is correct. It is important to studyvarious lattice models that can be in a Weyl semimetal phase in order to understandthe rich physical phenomena associated with this phase.The original proposal is very simple and it is based on considering a layeredheterostructure that is made by stacking thin films of ordinary magnetic insulatorsand topological insulators on top of each other [55]. This way one can realize a49Dirac Hamiltonian at low energies with proper term to separate Weyl points for arange of system parameters. One can generalize the cubic lattice model describedin the introduction chapter to model a topological insulator and realize a Weylsemimetal by breaking the time-reversal and inversion symmetries at the criticalpoint where the Dirac mass is zero. This system is the core model in our studyof the electromagnetic response of Weyl semimetals on a lattice model as we aregoing to study in the next chapter. However, before that we introduce another toymodel for Weyl semimetals where multiple pairs of Weyl nodes can exist and weshow that the cumulative effect of these pairs on the transport can lead to differentbehaviours in the linear response theory.4.2 Multiple Weyl pairsIt turns out that a system with more than one pair of Weyl fermions might have zeronet anomalous Hall conductivity since pairs can have total cancelling contributionsdue to the exactly opposite separations in momentum space. Here we consider asimple theoretical model which can lead to a topological semi-metal phase withtwo Weyl pairs under certain circumstances. Although there are big challenges onthe way to experimental realization of such a system, it can be used as a platformto study the bulk and the surface properties of a system in a Weyl semi-metal phasewhen there is more than one pair of Weyl nodes.In the rest of this chapter we introduce a lattice model made by arranging par-allel TI nano-wires in a honeycomb fashion with a lattice spacing which is smallenough to allow a significant hopping of the electrons between these wires andthen we discuss how one can achieve a topological semimetal phase with anoma-lous Hall charge and valley current from this arrangement.We show that a Weyl semimetal phase with isolated Weyl nodes in momentumspace can arise in a system of parallel topological insulator nano-wires arranged ina honeycomb fashion where the anomalous Hall response can be absent. This in-troduces another theoretical example of a topological semi-metal phase with morethan one pair of Weyl nodes and due to the simple form of its Hamiltonian it canbe used to study various interesting phenomena associated with this phase. Ourresults indicates that depending on the separation of the Weyl nodes a topological50electromagnetic charge response might or might not be present and two pairs canhave overall cancelling contributions to the net anomalous Hall conductivity. Weshow that when the separation of the two Weyl pairs are exactly opposite, there is anon-zero anomalous Hall valley-current which is proportional to the length of theseparation vector in momentum space.4.3 TI nano-wiresThe surface modes of a cylindrical topological insulator system can be describedby a Dirac Hamiltonian in a curved space [98] (h¯ = 1)Hk =υ2[∇ · nˆ + nˆ · (~p×σ )+(~p×σ ) · nˆ], (4.1)where σ is the Pauli matrix vector which acts on the spin Hilbert space. ~p =−i∇is the momentum operator and nˆ is the unit vector normal to the surface. For acylinder of radius R along zˆ axis we have nˆ = cosϕ xˆ+ sinϕ yˆ. The Hamiltonianthat governs the TI surface modes in the presence of a sufficiently thin magneticflux, φ = ηφ0, can then be written as￿kz > 0kz < 0φ = ηφ0kz2ΛFigure 4.1: The surface spectrum of a wire made from a strong topologicalinsulator in the presence of half integer multiple of the magnetic fluxquantum along the wire. The bandwidth in which the spectrum is non-degenerate is Λ = h¯υ0/R, where R is the radius of the wire. The spintexture is shown schematically for the states dispersing along kz withthe blue arrows.51Hk =12R+(1R(i∂ϕ +η) −ikze−iϕikzeiϕ − 1R(i∂ϕ +η)), (4.2)We can simplify this Hamiltonian by a unitary transformation H˜k =U†(ϕ)HkU(ϕ)in which U(ϕ) is given byU(ϕ) =(1 00 eiϕ), (4.3)using this transformation we can get rid of the phase in the off-diagonal compo-nents of the Hamiltonian matrixHk =(1R(i∂ϕ +η− 12) −ikzikz − 1R(i∂ϕ +η− 12)), (4.4)The eigenstates of this Hamiltonian are given by ψkl(ϕ) = eiϕlψkl where ψklare eigenstates of the Hkl defined asHkl = kzσ2−1Rσ3(l +12−η). (4.5)When there is a magnetic flux through the wire equal to the odd multiple of thehalf quantum of the magnetic flux η = n+ 0.5, we get a non-degenerate gaplessband for l = nH0 = kzσ2, (4.6)We assume that R is small enough in a way that there exists a significant energyrange, Λ, in which the gapless band does not overlap with other bands (See Fig.4.1), then we can use this energy band as a building block of the model to realize aWeyl semimetal phase.52⊥zRR+ δFigure 4.2: Two adjacent parallel wires. The electronic states on one caninteract with those on the other via a finite hopping term t.4.4 Honeycomb lattice of parallel wiresWhen the distance between two parallel nano-wires is sufficiently small, therewould be an overlap between electronic wave-functions and this leads to a hop-ping between adjacent electronic states of the surface modes. This can also arisefrom the inter-wire electron-electron interactions in a mean-field approximation.Here we consider a honeycomb arrangement of these nano-wires by consideringthe nondegenerate gapless modes of these wires that can be achieved by exposingthem to a static magnetic field along the wires. We assume that the spin-momentumlocking that happen at the surface of topological insulators are in opposite direc-tions for wires in the A and B sublattices. We should point out here that all theso far discovered topological insulators happen to have the same direction forthe spin-momentum locking. In theory, one can get a system with an oppositespin-momentum locking by simply changing the sign of the spin-orbit couplingrequired to get the original topological insulator system. We also assume that theDirac points for both A and B wires have the same energy, however, in the endwe relax this assumption, and we discuss how this is essential in order to get aWeyl semimetal phase. On the other hand, these wires are multi-band systems,therefore, they must be sufficiently thin to allow a significant gap, Λ, in which thegapless band does not overlap with other subbands (see Fig. (4.2)). With these as-sumptions, and by considering only the non-degenerate lowest lying band in each53wire in the energy range 2Λ (see Fig. 4.1) we can split the Hamiltonian into twoparts as followsH = H0 +Ht, (4.7)in which H0 sums up the gapless nondegenarate modes of each individual wire ina second quantized notation and is given in terms of the Fourier components asH0 =∫dkz2pi ∑~k⊥Ψ†~k⊥(kz)(υ0kzτ3σ2)Ψ~k⊥(kz), (4.8)where τ3 is the pauli matrix acting on the sublattice Hilbert space, kz is the mo-mentum along the wire (Fig. 4.1) and~k⊥ spans the honeycomb’s reciprocal lattice(similar to what one gets for graphene) and Ψ† = (ψ†A+,ψ†A−,ψ†B+,ψ†B−). The first(last) two components act on the spin space of the wire in the A (B) sub-lattice.ψ†A/Bα creates an electron in the α branch of the A/B wire when it acts on thevacuum state. Using this representation, Ht which represents the direct hoppingbetween nearest-neighbour wires, can be written asHt =∫dkz2pi ∑~k⊥Ψ†~k⊥(kz)[(0 g(~k⊥)g?(~k⊥) 0)⊗1]Ψ~k⊥(kz), (4.9)in which g(~k⊥) =−t∑δi e−i~k⊥·δi and δi are three vectors that connect a site in hon-eycomb lattice to the three adjacent points. The energy spectrum of the total Hamil-tonian would then be doubly degenerateε±(kz,~k⊥) =±√υ20 k2z + |g(~k⊥)|2, (4.10)g(~k⊥) vanishes at two inequivalent points in the two-dimensional reciprocal lattice,i.e., at ~K± and is linear near these points. Therefore, we get two degenerate three-dimensional Dirac points at (kz,~k⊥) = (0, ~K±). In order to get a topologicallyprotected semimetal phase we need to break time-reversal or inversion symmetryin such a way that it separates the Dirac points. Breaking inversion symmety canbe easily realized by relaxing the assumption we had in the beginning, i.e., allowthe Dirac point at two sub-lattices to have different energies. By revising H0 to54account for such an energy difference, V , we getH0 =∫dkz2pi ∑~k⊥Ψ†~k⊥(kz)(υ0kzτ3σ2 +Vτ3)Ψ~k⊥(kz), (4.11)In this case the degeneracy is lifted and we have four bands εsr(kz,~k⊥) (s,r =±)given byεsr(kz,~k⊥) = r√(υ0kz + sV )2 + |g(~k⊥)|2, (4.12)This spectrum has two pairs of Weyl points in the three dimensional BZ atwhich the bands cross linearly. Each pair consists of two Weyl points centred at(kz,~k⊥) = (0, ~K±). They are separated along kz axis by Q = 2V/υ0. Accordingto the argument presented in the previous section, these crossings are topologi-cally protected against various local momentum conserving perturbations as longas they are separated in the BZ. It is important to note that although these isolatedWeyl nodes are robust against local momentum-conserving perturbations, the pe-culiarity in the Hall response that is present in a system with only one pair of Weylnodes with broken time-reversal symmetry does not exist here since the separa-tion of the Weyl points arise from breaking inversion symmetry and the system istime-reversal invariant. The transverse conductivity in the presence of the appliedmagnetic field along the wires, σxy is zero [17, 55] since the two pairs contributeoppositely to the Hall conductivity, as they are separated in the opposite way at twovalleys considering their chirality. On the other hand, the topological nature of thesystem shows itself in the nonzero valley Hall current which can be defined as theanti-symmetric combination of the two valley contributions to the charge currentin the systemσVxy = σ+xy−σ−xy =2Vυ0e2h(4.13)There is also a possibility of a phase transition to an insulator phase due tothe electron-electron interactions. This instability arises in a mean-field treatmentof the inter-rod electron-electron interactions. When the wire radius, R, is muchsmaller than the honeycomb lattice constant, a, the Coulomb interaction between55electrons in adjacent wires can be written asHe−e =U∫ L/2−L/2dz∫ ∞−∞dunˆ(~R,z)nˆ(~R+δ ,z+u)√1+(ua)2, (4.14)where nˆ is the electronic density operator and U = e2/(4piεa). ~R, δ and z havebeen defined in the Fig. (4.2). In a mean-field treatment of the above term andby considering a Kekule type modulating order parameter with a wave-vector ~G =~K+− ~K−, it is possible to connect the Weyl points separated by ~G. This wouldthen open up a gap and the system becomes an insulator. The critical coupling, Ucat which such a phase transition to an insulator phase occurs can be computed forthis system. It is a function of the potential difference, V , as well as the hoppingstrength, t, and is given by∓￿λ · ￿k±￿λ · ￿kK+K−Q−++−K+K−kx kykzkx kykzFigure 4.3: The three dimensional BZ before and after separating the Weylnodes along the kz. We have two inequivalent pairs of Weyl nodes.Uc =4Λ3(1+ ln αV (Λ−V )t)−1, (4.15)in which α = (8/√3pi)1/2. This critical coupling would be of the order of thesingle wire’s non-degenerate bandwidth Λ (see Fig. 4.1) whenever the hoppingstrength is significant enough. For U <Uc the system would remain in the topolog-ical semimetal phase and for U >Uc the system would become a three-dimensionalversion of Kekule insulator which has been discussed previously in two dimensionsfor graphene [99]. It is possible to make the system stable against such a phase tran-56sition for all ranges of interactions at least in this mean-field channel by introducinga term in the Hamiltonian which separates the two Weyl nodes at each valley (~K+and ~K−) by a different amount. In this case the system becomes stable against theperturbation that connects two valleys since all the opposite chirality Weyl pairswould have incommensurate separation in momentum space after addition of sucha term. This term can be induced by considering next nearest-neighbour inter-wirehopping of the electronic states. The hopping amplitude is imaginary consideringa modulating magnetic flux with a zero net average through each hexagon (see Fig.4.4). This can be achieved by applying a modulating magnetic field instead of theuniform magnetic field that is required to induce half-quantum magnetic fluxes inthe wires. Such a term was first introduced for a honeycomb lattice to realize aquantum Hall phase in a system without a net uniform applied magnetic field [17].The Hamiltonian for this next-nearest neighbor hopping in momentum space canbe written asH ′ =−t ′∫dkz2pi ∑~k⊥Ψ†~k⊥(kz)(µ(~k⊥)+V (~k⊥)τ3)Ψ~k⊥(kz), (4.16)which is similar to the term which has been introduced in Eq. (4.11) but here the Vwhich separates the Dirac nodes along kz is a function of the~k⊥ and takes differentvalues at ~K+ and ~K−. The presence of µ(~k⊥) separates the two pairs in energysince it also depends on ~k⊥ and can take different values at two valleys. In theoriginal Haldane’s [17] flux configuration µ(~k⊥) and V (~k⊥) are given byV (~k⊥) =−2t ′ sinφ ∑i=1,2,3sin~k⊥ ·bi, (4.17)andµ(~k⊥) = 2t ′ cosφ ∑i=1,2,3cos~k⊥ ·bi, (4.18)bi and φ have been defined in the caption of Fig. 4 and t ′ is the next-nearestneighbor hopping amplitude.Therefore, the distance between each pair of Weyl nodes is now an incom-mensurate wave-vector since the separations of Weyl nodes at two valleys, i.e.,~q1 = 2V (~K+)zˆ/υ0 and ~q2 = 2V (~K−)zˆ/υ0 are not exactly opposite. This makes57the system stable against charge density perturbations that connect quantum statesof two valleys. The axion field is not zero in this case as the separations are notexactly opposite and instead it modulates in space [63].￿δ2￿δ3 ￿δ1Figure 4.4: The modulations in the magnetic flux with a zero average througheach hexagon introduces a complex next nearest neighbor hopping pa-rameter, t ′eiφ . The vectors that connect adjacent sites in the same sub-lattice are ±(~b1,~b2,~b3) =±(δ 1−δ 2,δ 2−δ 3,δ 3−δ 1)Therefore in the presence of the Haldane term, the system breaks the timereversal symmetry and it would have a nontrivial topological response. In this casethe transverse conductivity in the presence of the magnetic field in the z directioncan be obtained by a summation over the transverse conductivity of the massivetwo-dimensional Dirac fermions for each kz [17, 55] . At ~k⊥ = ~K+ and −q1 <kz < q1 the sign of the masses of two-dimensional Dirac states are in such a waythat they contribute constructively to the transverse Hall conductivity. Similarly, at~k⊥ = ~K− and −q2 < kz < q2 the transverse Hall conductivity is nonzero. The netσxy can be obtained, when the chemical potential is inside the gap in the presence ofan applied magnetic field in the z direction, by a summation over the contributionof all two-dimensional Dirac fermions labeled by kz and the valley index, i.e., 1and 2σxy =∫ Λ−Λ[ν1(kz)−ν2(kz)]dkz2pie2h, (4.19)in which ν1 = Θ(q1−|kz|) and Θ(q2−|kz|). Using Eq. (4.17) the transverse con-ductivity becomes58σxy =6√3t ′ sinφ e2pih , (4.20)This result highlights the fact that although the momentum conserving pertur-bations cannot gap out the system, this does not necessarily imply the existence ofa topological electromagnetic response in the system and two pairs of Weyl nodescan have cancelling contributions to the electromagnetic response in special casesand therefore σxy can be zero even when the Weyl nodes are well separated in theBZ which is the case for our model in the absence of the Haldane term. Adding thenext nearest hopping and introducing the Haldane term would lead to a nonzero nettopological electromagnetic response since two Weyl pairs would now have differ-ent separations and therefore their contributions to the transverse conductivity donot sum up to zero.4.5 ConclusionsTo summarize, in order to realize multiple Weyl pairs, we considered a system ofparallel nano-wires in the time-reversal invariant topological insulator phase andby breaking inversion and time-reversal symmetry, we found that it is possible torealize a topological semimetal phase with two pairs of Weyl nodes in the three di-mensional BZ which are protected against local perturbations that conserve crystalmomentum. A nontrivial topological electromagnetic response might arise undercertain circumstances when the contributions from two pairs do not exactly canceleach other.It is important to note that although experimentalists can grow large-scale ver-tically aligned nano-rods and nanopillars with various arrangements including hon-eycomb [100–102], realizing a system for which one can observe the robust Weylenergy crossings described in this chapter seems not to be experimentally plau-sible. One can name a few obstacles on the way of its experimental realization.First, wires are multi-band systems and the two-band model approximation usedhere requires a significant gap for the other existing subbands in the nano-wires.For the so far discovered topological insulators, the surface Fermi velocity [41], υ0,59is of the order of 105m/s (5× 105m/s for Bi2Se3 [77]), therefore, the wire diam-eter required (2R . 60nm) to get a significant gap (Λ & 10meV) is still out of theexperimentally feasible sub-micron ranges [100–102]. Finally, another challengeto realize such a system is to find two types of topological insulators with oppositespin-momentum lockings. Although in theory it is possible to have systems withopposite chirality, in all the so far discovered topological insulators surface elec-trons’ are directed around the Dirac point have the same handedness. We note thatthe direction of the spin-momentum locking is a material dependent property andin the theoretical lattice models for the topological insulators in three dimensions,it is possible to change it on the surface by varying physical parameters of the bulk.60Chapter 5Electromagnetic Response ofWeyl Semimetals5.1 OverviewThe anomalous Hall effect we discussed in the previous chapter is known to com-monly occur in solids with broken time-reversal symmetry. In the present case ofthe Weyl semimetal its origin and magnitude can be understood from simple phys-ical arguments [55, 56, 58, 103] applied to the bulk system as well as in the limit ofdecoupled 2D layers [104]. Understanding the chiral magnetic effect (CME) in asystem with non-zero energy shift b0 presents a far greater challenge. The issue be-comes particularly intriguing in the case of a Weyl insulator, illustrated in Fig. 1.7d,which will generically arise due to the exciton instability in the presence of repul-sive interactions and nested Fermi surfaces. According to Ref. [63] CME shouldpersist even when the chemical potential resides inside the bulk gap. At the sametime, standard arguments from the band theory of solids dictate that filled bandscannot contribute to the electrical current [105]. We remark that using a differentregularization scheme for the Weyl fermions Ref. [106] found that CME occurs inthe semimetal but is absent in the insulator, while Ref. [104] concluded that it onlyoccurs when~b2− b20 ≥ m2D, where mD denotes the gap magnitude. Semiclassicalconsiderations [107] on the other hand predict a vanishing electrical current in theWeyl semimetal but non-zero ‘valley current’ proportional to ~B.61CME, if present, could have interesting technological applications, as it consti-tutes a dissipationless ground state current, controllable by an external field. Dis-agreements between the various field-theory predictions, however, raise importantquestions about the existence of CME in Weyl semimetals and insulators. Theimplied contradiction with one of the basic results of the band theory calls intoquestion whether the results based on the low-energy Dirac-Weyl Hamiltoniansare applicable to the real solid with electrons properly regularized on the lattice.In this chapter we undertake to resolve these questions by constructing and ana-lyzing a lattice model of a Weyl medium. Using simple physical arguments andexact numerical diagonalization, we confirm the existence of the anomalous Halleffect as implied by Eqs. (1.22,1.23) when~b 6= 0. We find, using the same modelwith b0 6= 0, that CME does not occur in either the Weyl semimetal or insulator,in agreement with arguments from the band theory of solids which we review insome detail.5.2 ModelOur starting point is the standard model describing a 3D TI in the Bi2Se3 fam-ily [108, 109], regularized on a simple cubic lattice which we introduced in theintroduction chapter. If we assume the lattice constant is a = 1 then we haveH0(~k) = 2λσz(sx sinky− sy sinkx)+2λzσy sinkz+ σxM~k, (5.1)with ~σ and ~s the Pauli matrices in orbital and spin space, respectively, and M~k =ε−2t∑α coskα . For λ ,λz > 0 and 2t < ε < 6t the above model describes a strongtopological insulator with the Z2 index (1;000). In the following, we shall focus onthe vicinity of the phase transition to the trivial phase that occurs at ε = 6t, via thegap closing at~k = 0.It is easy to see that Weyl semimetal emerges when we add the following per-turbation to H0,H1(~k) = b0σysz +~b · (−σxsx,σxsy,sz). (5.2)Nonzero b0 breaks P but respects T while ~b has the opposite effect. The two62symmetries are generated as follows,P: σxH(~k)σx = H(−~k) andT : syH∗(~k)sy =H(−~k). For simplicity and concreteness we focus on the case~b = bzzˆ, which yieldsOut[56]=Z G M Z-4-2024Z G M Z-4-2024Z G M Z-4-2024Z G M Z-4-2024kEkEkEk EcbdaFigure 5.1: The band structure of the Weyl semimetal lattice model, dis-played along the path~k : (pi,0,pi)→ (0,0,0)→ (0,0,pi)→ (pi,0,pi).a) Doubly degenerate 3D Dirac point when H1 = 0 and ε = 6t. b)Momentum-shifted Weyl point for b= 0.9 and b0 = 0. c) Energy-shiftedWeyl points for bz = 0 and b0 = 0.7. d) Weyl insulator with bz = 0 andb0 = 0.7 and the exciton gap modeled by taking ε = 5.9t. In all panelswe take λ = λz = 1.0, t = 0.5 and the energy is measured in units of λ .Red circles mark the location of the Dirac/Weyl points.a pair of Weyl points at~k = ±(bz/2λz)zˆ. The band structure of H = H0 +H1 forvarious cases of interest is displayed in Fig. 5.1.5.3 Anomalous Hall currentWe now address the anomalous Hall current by directly testing Eq. (1.22). Tothis end we consider a rectangular sample of the Weyl semimetal with a base of(L×L) sites in the x-y plane and periodic boundary conditions, infinite along thez-direction. The effect of the applied magnetic field is included via the standardPeierls substitution, t→ t exp [2pii/Φ0∫ ji~A ·dl ], where Φ0 = hc/e is the flux quan-63tum, ~A is the magnetic vector potential and the integral is taken along the straightline between sites~ri and~r j of the lattice. For ~B = zˆB(x,y) we retain the transla-tional invariance along the z-direction and the Hamiltonian becomes a matrix ofsize 16L2 for each value of kz. We find the eigenstates φn,kz(x,y) of H by means ofexact numerical diagonalization and use these to calculate the charge densityρ(x,y) = e ∑n∈occ∑kz|φn,kz(x,y)|2. (5.3)        	  		Figure 5.2: a) Charge density δρ(x,y) accumulated in the vicinity of the fluxtubes Φ= 0.01Φ0 in the Weyl semimetal. b) Total accumulated chargeper layer δQ near one of the flux tubes, in units of e/2pi for indicatedvalues of bz. Dashed lines represent the expectation based on Eq (5.4).We use λ = λz = t = 0.5, ε = 3.0, L = 14 and Lz = 160 independentvalues of kz.Figure 5.2a displays ρ for the magnetic field configuration B(x,y) = Φ[δ (x−L/4)−δ (x+L/4)]δ (y), i.e. two flux tubes separated by L/2 along the x direction.In accord with Eq. (1.22) charge accumulates near the flux tubes, although ρ(x,y) issomewhat broadened compared to B(x,y). We expect the total accumulated chargeper layer δQ to be proportional to the total flux,δQ = epi(bz2λz)ΦΦ0, (5.4)where we have restored the physical units. Fig. 5.2b shows that this proportionalityholds very accurately when the flux through an elementary plaquette is small com-pared to Φ0. [When the flux approaches Φ0/2 we no longer expect Eq. (5.4) to hold64because of the lattice effects.] We have also tested the effect of a non-zero Diracmass, mD = ε − 6t, and non-zero b0 on the anomalous Hall effect. These termscompete with bz and for m2D + b20 > b2z one expects the Hall effect to disappear[104, 106]. This is indeed what we observe in Figs. 5.3a,b. We have performedsimilar calculations for other field profiles B(x,y) reaching identical conclusionsfor the anomalous Hall effect.  			   	   	   	    		  		Figure 5.3: Charge accumulations as a function of bz in the presence of non-zero Dirac mass and b0. Parameters as above except b0 = 0.1,0.2,0.3in a) and ε = 3.0,2.9,2.8,2.7 for the curves in b) from left to right.5.4 Can CME exist in an equilibrium state?We now address the chiral magnetic effect, predicted to occur when b0 6= 0. Weconsider the same sample geometry as above, but now with uniform field ~B = zˆB.In order to account for possible contribution of the surface states we study systemswith both periodic and open boundary conditions along x. To find the current re-sponse we introduce a uniform vector potential Az along the z-direction (in additionto Ax and Ay required to encode the applied magnetic field). The second-quantizedHamiltonian then readsH (Az) =∑kzHαβ (kz− eAz)c†kzαckzβ , (5.5)65where α , β represent all the site, orbital and spin indices. The current operator isgiven byJz =∂H (Az)∂Az∣∣∣∣Az→0=−e∑kz∂Hαβ (kz)∂kzc†kzαckzβ . (5.6)This leads to the current expectation valueJz =−e∑n,kz〈φn,kz∣∣∣∣∂H(kz)∂kz∣∣∣∣φn,kz〉nF [εn(kz)], (5.7)where nF indicates the Fermi-Dirac distribution and εn(kz) the energy eigenval-ues of H(kz). We note that Eq. (5.7) remains valid in the presence of the excitoncondensate as long as it is treated in the standard mean field theory.We have evaluated Jz from Eq. (5.7) for various system sizes, boundary condi-tions, field strengths and parameter values corresponding to energy- and momentum-shifted Weyl semimetals and insulators. In all cases we found Jz = 0 to within thenumerical accuracy of our computations, typically 6-8 orders of magnitude smallerthan CME expected on the basis of Eq. (1.23).         		       	  	 Figure 5.4: a) Chiral current Jz as a function of energy offset b0 for variousvalues of the momentum cutoff Λ. The dashed line indicates the fieldtheory prediction Eq. (1.23). b) The slope dJz/db0 in units of eη/2pi asa function of cutoff Λ. Slope 1.0 is expected on the basis of Eq. (1.23).For an insulator, vanishing of Jz comes of course as no surprise. At T = 0 and66using the fact that ∂kz〈φn,kz |φn,kz〉= 0 one can rewrite Eq. (5.7) asJz =−e ∑n∈occ∫BZdkz2pi∂εn(kz)∂kz, (5.8)which vanishes owing to the periodicity of εn(kz) on the Brillouin zone. Moregenerally, for a system at non-zero temperature and when partially filled bands arepresent we can rewrite Eq. (5.7) asJz =−e∑n∫BZdkz2pi∂εn(kz)∂kznF [εn(kz)], (5.9)where the sum over n extends over all bands. By transforming the kz-integral inEq. (5.9) into an integral over the energy it is easy to see that it identically vanishesfor any continuous energy dispersion εn(kz) that is periodic on the Brillouin zoneand for any distribution function that only depends on energy. This reflects thewell-known fact that one must establish a non-equilibrium distribution of electronsto drive current in a metal, e.g. by applying an electric field.5.5 ConclusionsGiven these arguments we conclude that, as a matter of principle, CME cannotoccur in a crystalline solid, at least when interactions are unimportant and the de-scription within the independent electron approximation remains valid. There areseveral notable cases when filled bands do contribute currents. A superconductorcan be thought of as an insulator for Bogoliubov quasiparticles and yet it supportsa supercurrent. This occurs because Bogoliubov quasiparticles, being coherent su-perpositions of electrons and holes, do not carry a definite charge and consequentlythe current cannot be expressed through Eq. (5.7). In quantum Hall insulators non-zero σxy also implies non-vanishing current. In the standard Hall bar geometry,used in transport measurements, it is well known that the physical current is car-ried by the gapless edge modes, not through the gapped bulk. In the Thoulesscharge pump geometry the current indeed flows through the insulating bulk butthis requires a time-dependent Hamiltonian (the magnetic flux through the cylin-der is time dependent). Our considerations leading to Eq. (5.7) are only valid for67time-independent Hamiltonians. Finally, there are known cases [110] when thetransition from Eq. (5.7) to (5.8) fails because the Hamiltonian is not self-adjointon the space of functions that includes derivatives of φ . This can happen whenthe Hamiltonian is a differential operator but in our case H(kz) is a finite-size her-mitian matrix with a smooth dependence on kz, which precludes any such exoticpossibility. In any case, our numerical calculations addressed directly Eq. (5.7) soself-adjointness cannot possibly be an issue.68Chapter 6Self Organized Topological phasewith Majorana Fermions6.1 OverviewTopological phases, quite generally, are difficult to come by. They either occurunder rather extreme conditions (e.g. the quantum Hall liquids [9], which requirehigh sample purity, strong magnetic fields and low temperatures) or demand finetuning of system parameters, as in the majority of known topological insulators[78, 108, 111]. Many perfectly sensible topological phases, such as the Weylsemimetals [58] and topological superconductors [70, 108], remain experimentallyundiscovered.The paucity of easily accessible, stable topological materials has been in a largepart responsible for the relatively slow progress towards the adoption of topologi-cal phases in the mainstream technological applications. A question that naturallyarises is the following: Is there a fundamental principle behind this “topologicalresistance”? Although unable to give a general answer to this question, we providein this chapter a specific counterexample to this conjectured phenomenon of topo-logical resistance. We consider a simple model system which, as we demonstrate,wants to be topological in a precisely defined sense. The key to this “topofilia” isthe existence in the system of a dynamical parameter that adjusts itself in responseto changing external conditions, so that the system self-tunes into the topological69phase.-p p-2-112-p p-2-1122kFaε(q)ε (q)STM tipSC substrateqqµµcbFigure 6.1: a) Schematic depiction of the system with the red spheres repre-senting the adatoms and blue arrows showing their magnetic momentsarranged in a spiral. b) Two spin-degenerate branches of the normal-state spectrum of the system in the absence of magnetic moments mod-elled by the nearest-neighbour tight-binding model Eq. (6.1). c) Withthe magnetic moments, the two branches shift in momentum by ±Gand the gap JS opens at q = 0,pi . Dashed lines show the shifted spectralbranches indicated in panel (b) with no gap for comparison.The specific model system we consider is depicted in Fig. 6.1a and consistsof a chain of magnetic atoms, such as Co, Mn or Fe, deposited on the atomicallyflat surface of an ordinary s-wave superconductor. We note that scanning tunnellingmicroscopy (STM) techniques now enable fairly routine assembly of such and evenmuch more complicated nanostructures [112, 113].It has been pointed out previously [1, 114, 115] that if the magnetic momentsin the chain exhibit a spiral order then the electrons in the chain can form a 1Dtopological superconductor (TSC) with Majorana zero modes localized at its ends70[69]. For a given chemical potential µ , however, the spiral must have the correctpitch in order to support the topological phase. This connection is illustrated inFig. 6.1b,c and will be discussed in more detail below. Exactly how the pitch of thespiral depends on the system parameters and its thermodynamic stability are twokey issues that have not been previously discussed. In this Letter we show that,remarkably, under generic conditions the pitch of the spiral that minimizes the freeenergy of the system coincides with the one required to establish the topologicalphase.The physics behind the self-organization phenomenon outlined above is easyto understand and is similar to that leading to the spiral ordering of nuclear spinsproposed to occur in 1D conducting wires [116, 117] and 2D electron gases [118].Some experimental evidence for such an ordering has been reported [119, 120]. Ifwe for a moment neglect the superconducting order and assume a weak couplingof the adatoms to the substrate then the electrons in the chain can be thought ofas forming a 1D metal. The natural wavevector for the spiral ordering in sucha 1D metal is G = 2kF where kF denotes its Fermi momentum. This is becausethe static spin susceptibility χ0(q) of a 1D metal has a divergence at q = 2kF .Electron scattering off of such a magnetic spiral results in opening of a gap inthe electron excitation spectrum but only for one of the two spin-degenerate bands[116, 117]. In the end, we are left with a single, non-degenerate Fermi crossingat ±2kF , illustrated in Fig. 6.1c. According to the Kitaev criterion [69] this isexactly the condition necessary for a 1D TSC to emerge. In the following we willshow that this reasoning remains valid when we include superconductivity fromthe outset and when we describe the chain by a tight-binding model appropriate fora discrete atomic chain.6.2 ModelWe begin by studying the simplest model of tight-binding electrons coupled tomagnetic moments ~Si described by the HamiltonianH0 =−∑i jσti jc†iσc jσ −µ∑iσc†iσciσ + J∑i~Si · (c†iσσ σσ ′ciσ ′) (6.1)71Here c†jσ creates an electron with spin σ on site j, J stands for the exchange cou-pling constant and σ = (σ x,σ y,σ z) is the vector of Pauli spin matrices. We assumethat the substrate degrees of freedom have been integrated out, leading to a super-conducting order ∆ in the chain described byH =H0 +∑j(∆c†j↑c†j↓+h.c.). (6.2)We consider a co-planar helical arrangement of atomic spins as indicated in Fig.6.1a,~S j = S[cos(Gx j),sin(Gx j),0] (6.3)where G is the corresponding wavevector and the chain is assumed to lie along thex-axis. We note that Hamiltonian (6.2) is invariant under the simultaneous globalSU(2) rotation of the electron and atomic spins so the discussion below in factapplies to any co-planar spiral.To find the spectrum of excitations, it is useful to perform a spin-dependentgauge transformation [1],c j↑→ c j↑ei2 Gx j , c j↓→ c j↓e−i2 Gx j , (6.4)upon which the Hamiltonian becomes translationally invariant and can be writtenin the momentum space asH = ∑q[ξ (q)c†qσcqσ +b(q)c†qσσ zσσ ′cqσ ′ (6.5)+ JSc†qσσ xσσ ′cqσ ′+(∆c†q↑c†−q↓+h.c.)].In the above ξ (q)= 12 [ε0(q−G/2)+ε0(q+G/2)]−µ , and b(q)= 12 [ε0(q−G/2)−ε0(q+G/2)] with ε0(q) = −∑ j t0 jeiqx j the normal-state dispersion in the absenceof the exchange coupling. We note that the Hamiltonian (6.5) is essentially a lat-tice version of the model semiconductor wire studied in Refs. [72, 121] with b(q)playing the role of the spin-orbit coupling and JS standing for the Zeeman field. Its72normal state spectrum is given byε(q) = ξ (q)±√b(q)2 + J2S2, (6.6)and is displayed in Fig. 6.1c for the case of nearest-neighbour hopping with ε0(q)=−2t cosq.If viewed as a rigid band structure then, according to the Kitaev criterion [69],the chain will support topological superconductivity when there is an odd numberof Fermi crossings in the right half of the Brillouin zone. This requires µ such that|µ ± 2t cos(G/2)| < JS. However, in the SU(2) symmetric model under consid-eration, G is a dynamical parameter that will assume a value that minimizes thesystem free energy. Taking ~Si to be classical magnetic moments and working atT = 0, we thus proceed to minimize the ground state energy of the electrons Eg(G)for a given value of µ and ∆. The result of this procedure is shown in Fig. 6.2aand confirms that at minimum G ≈ 2kF , as suggested by the general argumentsadvanced above. More importantly, for almost all relevant values of µ and ∆ theself-consistently determined spiral pitch G is precisely the one required for the for-mation of the topological phase. This fails only close to the half filling (µ = 0)where G = pi indicates an antiferromagnetic ordering. In this case the symmetry ofthe band structure prohibits an odd number of Fermi crossings, so the system mustbe in the trivial phase. Also, it is clear that no value of G can bring about the TSCphase when µ lies outside of the tight-binding band and the system is an insulator.The resulting topological phase diagram is displayed in Fig. 6.2b.These results indicate that, as we argued on general grounds, the pitch of themagnetic spiral self-tunes into the topological phase for nearly all values of thechemical potential µ for which such a tuning is possible. The emergence of Majo-rana zero modes at the two ends of such a topological wire [69, 72, 121] and theirsignificance for the quantum information processing have been amply discussed inthe recent literature [68, 122, 123].We now address the adatom coupling to the substrate in greater detail. We con-sider a more complete Hamiltonian H =H0 +HSC +Hcd , where H0 is defined73in Eq. (6.1), whileHSC =∑~k[ξ0(~k)d†~kσd~kσ +(∆0d†~k↑d†−~k↓+h.c.)](6.7)describes the SC substrate with electron operators d†~kσ . The substrate is character-ized by a three-dimensional normal-state dispersion ξ0(~k) = k2/2m− εF and thebulk gap amplitude ∆0. The coupling is effected throughHcd =−r∑jσ(d†jσc jσ +h.c.), (6.8)where d jσ = 1√N ∑~k e−i~k·~R j d~kσ , N is the number of adatoms in the chain and~R jdenotes their positions.We now wish to integrate out the substrate degrees of freedom and ascertaintheir effect on the magnetic chain. Since the Hamiltonian H is non-interactingthis procedure can be performed exactly. As outlined in the Appendix of Ref. [68],a simple result obtains in the limit of the substrate bandwidth much larger than thechain bandwidth 4t, which we expect to generically be the case. In this limit theGreen’s function of the chain readsG −1eff (iωn,q) = G −10 (iωn,q)−piρ0r2iωn− τx∆0√ω2n +∆20, (6.9)where ωn =(2n+1)piT is the Matsubara frequency, ρ0 =ma2/2pi h¯2 is the substratenormal density of states projected onto the chain, (with a being the atom spacing)andG −10 (iωn,q) =−iωn + τz [ξ (q)+σ zb(q)]+σ xJS (6.10)the bare chain Green’s function. The above Green’s functions are 4× 4 matricesin the combined spin and particle-hole (Nambu) space, the latter represented bya vector of Pauli matrices τ . In the low-frequency limit ω  ∆0, relevant to thephysics close to the Fermi energy, Eq. (6.9) implies two effects. First, the barechain parameters t, µ and J are reduced by a factor of ∆0/(∆0 +pir2ρ0). Second, aSC gap ∆= pir2ρ0∆0/(∆0 +pir2ρ0) is induced in the chain. In the limit of a weak74chain-substrate coupling, r2  ∆0/ρ0, the latter is seen to become ∆ ' pir2ρ0,independent of the substrate gap ∆0.In order to visualize the above effects we display in Fig. 6.3 the relevant spec-tral function, defined asAeff(ω,q) =−12pi Im Tr [(1+ τz)Geff(ω+ iδ ,q)] (6.11)where δ represents a positive infinitesimal. The figure clearly shows how the bandsself-consistently adjust to the changing chemical potential as well as the expectedtopological phase transitions taking place between the trivial and the TSC phases.Our results thus far relied on the mean field theory and ignored interactionsbeyond those giving rise to superconductivity. There are several effects that canin principle destabilize the topological state found above but we now argue thatthe latter remains stable against both interactions and fluctuations. First, one mayworry that the adatom magnetic moments would be screened by the Kondo effectat temperatures T < TK = εFe−1/ρ(εF )J , where ρ(εF) is the density of states in thesubstrate at the Fermi level. For a normal metal, TK can indeed be sizeable – tensof Kelvins – and the ground state is then non-magnetic [124]. In the presenceof superconductivity, however, ρ(εF) = 0 and a more elaborate treatment of theKondo problem in the presence of a gap shows that TK is much reduced [125],possibly to zero when J is sufficiently small. Thus, generically, we expect thesystem to avoid the Kondo fixed point and remain magnetic at most experimentallyrelevant temperatures. Electron-electron interactions in the wire are additionallyexpected to enhance the magnetic gap [116, 118] compared to its non-interactingvalue JS, which will ultimately further improve the stability of the topologicalphase.Second, one must consider thermal and quantum fluctuations which will tend todestroy any ordering present in a 1D wire. Since the SC order in the wire is phase-locked to the substrate we may ignore its fluctuations. However, fluctuations in themagnetic spiral order must be considered. In a realistic system spin-orbit couplingwill induce a Dzyaloshinsky-Moriya interaction of the form Di j · (~Si×~S j) in theeffective spin Hamiltonian. The latter breaks the SU(2) spin symmetry and pinsthe spiral order so that the spins rotate in the direction perpendicular to D. The75remaining low-energy modes of such a spiral are magnons. The relevant spin-wave analysis and the origin of the DM interaction are outlined in the next section.We find a single linearly dispersing gapless magnon ωq = c|q| which will reducethe classical ordered moment according to〈Sx〉 ' S−a∫BZdq2pi1eβ h¯ωq−1 . (6.12)For an infinite wire the integral diverges logarithmically at long wavelengths, sig-naling the expected loss of the magnetic order in the thermodynamic limit. We are,however, interested in a wire of finite length L where the divergence is cut off atq∼ pi/L. A crude estimate of the transition temperature in this case obtains by as-suming β h¯ωq 1 over the Brillouin zone and setting 〈Sx〉= 0 in Eq. (6.12). ThisgiveskBT∗ ≈ piSlnNh¯ca, (6.13)with N =L/a the number of adatoms in the chain. For N = 100, S= 5/2 and typicalmodel parameters t = 10meV, J = 5meV and µ appropriate for the topologicalphase we find T ∗ of tens of Kelvins as it is discussed in the next section. Due tothe lnN factor T ∗ is only weakly dependent on the chain length.6.3 Stability analysis of the spiral ground stateIn this section we supply some of the technical details pertaining to the stabil-ity analysis of the spiral ground state. In this analysis it is useful to consider aspin-only effective theory that arises upon integrating out the electron degrees offreedom in the relevant Hamiltonian given e.g. by Eq. (6.1) in the previous section.More generally, we start from a Hamiltonian of the formH =H0 + J∑i~Si · (c†iσσ σσ ′ciσ ′) (6.14)where H0 describes the electrons in the absence of the exchange coupling to themagnetic moments ~Si. Assuming translational invariance we can integrate out the76electrons to obtain the effective spin-only HamiltonianHS =J22 ∑qχαβ (q)Sαq Sβ−q (6.15)whereχαβ (q) = i∫ ∞0dt∑i, je−iq(xi−x j)〈[sαi (t),sβj (0)]〉0 (6.16)is the static spin susceptibility evaluated in the ensemble specified by H0 and~si = c†iσσ σσ ′ciσ ′ . Given the electron Green’s function G0(iωn,k) the spin suscep-tibility can be evaluated by analytical continuation of the Matsubara frequencysusceptibilityχαβ (iνm,q) = kBT∑n,kTr[G0(iωn,k)σαG0(iωn−νm,k−q)σβ ] (6.17)in the limit ν → 0.In the simplest case of noninteracting electrons withH0 =∑k(h¯2k2/2m)c†kσckσthe T = 0 susceptibility becomes χαβ (q) = δαβχ0(q) with one-dimensional Lind-hard functionχ0(q) =Lpi2mh¯2qln∣∣∣∣2kF −q2kF +q∣∣∣∣ . (6.18)The latter exhibits singularities at q = ±2kF , as already noted in the main text.These peaks constitute the root cause behind the emergent spiral.For the SU(2) symmetric model considered thus far the helical ordering canoccur in any plane leading to a continuous ground-state degeneracy parametrizedby a unit vector nˆ perpendicular to that plane. One possible mechanism that willbreak the global SU(2) spin symmetry and select a specific spiral plane is theDzyaloshinski-Moriya (DM) interaction. It takes the form ∑i j Di j · (~Si×~S j) andarises in the presence of the spin-orbit coupling (SOC) when spatial inversion sym-metry is broken, as is the case in our system of adatoms placed on a substrate. Ifwe assume that the adatom chain lies along the x direction while the surface normalis along y then a Rashba-type SOC of the form λkσ z is permitted in H0 with λparametrizing the SOC strength. DM interaction arises in this situation because asa result of SOC Eq. (6.17) yields antisymmetric components of the susceptibility77χαβ ∼ Tr(σ zσασβ ) = 2iε3αβ . The D-vector points along the z direction and takesthe form Di j = D(yˆ×~ri j) where~ri j is a vector connecting sites i and j. To estimatethe magnitude D we continue modelling our wire by the continuum free electronmodel with Rashba SOC! (SOC!), and evaluate Eq. (6.17) to obtainχxy(q) = 12[χ0(q+Qλ )−χ0(q−Qλ )] (6.19)with Qλ = λm/h¯2 the characteristic SOC wave-vector. At long wavelengths, as-suming Qλ  2kF , one can expandχxy(q)' L12piqQλεFkF. (6.20)This leads to an estimate D ≈ 112Λ(J/εF)2 with Λ = λ/a the SOC strength peradatom site. Taking J/εF ≈ 1 and Λ of few meV we obtain D of the order offew degrees Kelvin. We expect the same result to qualitatively describe the tight-binding electrons when the chemical potential lies near the bottom of the band.The DM interaction discussed above gives rise to a gap ∆DM 'DS towards theout-of-plane spin excitations and for a large adatom spin, e.g. S = 52 , to a charac-teristic temperature TDM ' 10K below which these can be ignored. We emphasizethat the DM interaction represents only one possible mechanism by which the spi-ral plane can be pinned. Another possibility arises from an easy-plane anisotropywhich is also allowed by symmetry.We now address the effect of the remaining spin-wave fluctuations around thehelical ground state pinned to a given plane. These remain gapless as q→ 0 andfor a chain of a finite length L will destroy the helical order above the crossovertemperature T ∗ given in Eq. (13) of the main text. Our goal here is to determinethe spin-wave velocity c and estimate T ∗. To this end we repeat the analysis givenin Eqs. (1-6) but now including small adatom spin fluctuations δ~S j. We thus write~S j = R j(Sxˆ +δ~S j) (6.21)78whereR j =cos(Gx j) sin(Gx j) 0−sin(Gx j) cos(Gx j) 00 0 1 (6.22)is the rotation matrix. After the gauge transformation (4) we obtainH =H0 + J∑jδ~Si · (c†jσσ σσ ′c jσ ′) (6.23)with H0 given by Eq. (5). We now pass to the bosonic magnon variables usingthe standard Holstein-Primakoff transformation for spin along the xˆ direction Sxj =S− a†ja j and S+j = Szj + iSyj =√2S(1− a†ja j/2S)1/2a j. In the linear spin-wavetheory we can neglect terms O(1/S) and the Hamiltonian (6.23) becomesH =H0 + J∑j[−a†ja jsxj +√S/2(a js−j +a†js+j )], (6.24)where s±j = szj± isyj. Once again we may integrate out the electrons to obtain theeffective magnon Hamiltonian which takes the following formHmag =∑q[Ωqa†qaq +12ηq(a†qa†−q +aqa−q)], (6.25)with Ωq = J〈−sx〉0 + 2J2χ+−(q) and ηq = 2J2χ++(q). Here χαβ (q) is the staticspin susceptibility tensor (6.16) evaluated for the Hamiltonian H0. The magnonspectrum follows from diagonalizing Hmag via the Bogoliubov canonical transfor-mation and readsωq =√Ω2q−η2q . (6.26)We have evaluated the relevant components of χαβ (q) for H0 given by Eq. (5)numerically; some representative results are displayed in Fig. 6.4. For simplic-ity and concreteness we focus here on the non-superconducting state. For J = 0the susceptibility shows divergences at q = 0,±2G inherited from the free-electronLindhard function (6.18). For J > 0 the divergence at q = 0 is quenched, reflect-ing the gap that has opened in the electron excitation spectrum at k = 0 (see Fig.6.1c). The remaining peaks at q = ±2G reflect the fact that a state exists with the79opposite helicity (i.e. described by a rotation matrix R j with G replaced by −G)which is also a classical ground state of the system. This second ground state, be-ing in a different topological sector, is separated from our chosen ground state by alarge energy barrier and will be ignored in our subsequent analysis, which by con-struction focuses on small fluctuations in the vicinity of a single classical groundstate.To study the long-wavelength magnons we note that it is possible to evaluatethe spin susceptibility in the vicinity of q = 0 analytically. The dominant contri-bution to the momentum summation in Eq. (6.16) here comes from the vicinityof k = 0 where the Hamiltonian (5) can be well approximated by a 1D Dirac the-ory with mass mD = JS and electron velocity vF . To the leading order in q thesusceptibility evaluates to χαβ (q) = αβχq with α,β =±,χq =aS4pivF[ln(1+ Θ2J2S2)−1− v2Fq22J2S2], (6.27)and Θ ∼ εF representing the high-energy cutoff for the Dirac theory. In additionwe find 〈−sx〉0 = 4Jχ0 as must be the case to satisfy the Goldstone theorem whichrequires a gapless mode as q→ 0. Substituting these results into Eq. (6.26) yieldsthe magnon spectrumωq = 4J2√χ0(χ0−χq) = cq (6.28)with the spin-wave velocityc = Japi h¯K, (6.29)and K =√12 [ln(1+Θ2/J2S2)−1] a dimensionless constant close to unity. Taking,as an example, Θ/JS = 10 we have K = 1.34 and for N = 100 and S = 52 Eq.(13) yields kBT ∗ = 0.728J. For J = 5meV the crossover temperature T ∗ ' 36K.The above analysis suggests that the classical spiral ground state is reasonablystable and the more likely limiting factor in a realistic system will be the substratesuperconducting critical temparature Tc which is below ∼ 10K for most simplemetals.In closing several remarks are in order. In our calculation of the spin suscepti-80bility we have neglected the SC gap ∆ and worked at T = 0. Inclusion of the formerleads to changes in χαβ (q) near q = 0 which are small as long as ∆ < JS, whichis a necessary condition for the topological phase. Such small changes do not sig-nificantly alter our conclusions regarding the spiral stability. Similarly, working atnon-zero temperature has a negligible effect on the electron spin susceptibility aslong as kBT  JS because the electrons are gapped. Since we found both kBTDMand kBT ∗ to be parametrically smaller than JS, our analysis is self-consistent in thisregard. Finally, one may ask if other spin states might compete with the assumedspiral ground state. We have tried several possibilities, including a spiral with aconstant out-of-plane spin component, but all were higher in energy.6.4 ConclusionsOur results provide strong support for the notion of a self-organized topologicalstate. Magnetic moments of atoms assembled into a wire geometry on a super-conducting surface are indeed found to self-organize into a topological state un-der a wide range of experimentally relevant conditions. The emergent Majoranafermions can be probed spectroscopically by the same STM employed in buildingthe structures and will show zero-bias peaks localized near the wire ends. The sys-tem can be tuned out of the topological phase by applying magnetic field ~B which,when strong enough, will destroy the helical order by polarizing the adatom mag-netic moments. An attractive feature of this setup is the possibility of assemblingmore complex structures, such as T-junctions and wire networks that will aid fu-ture efforts to exchange and braid Majorana Fermions with the goal of probingtheir non-Abelian statistics [126]. We also note that the general self-organizationprinciple described above should apply to other 1D structures, most notably quan-tum wires with nuclear spins considered in Ref. [116]; however the energy scalesare expected to be much smaller due to the inherent weakness of the nuclear mag-netism.81Figure 6.2: Panel a shows the spiral wavevector G that minimizes the systemground state energy Eg(G) as a function of µ , the latter in units of t. Theparameters are S = 5/2, J = 0.1t and ∆= 0 (red), ∆= 0.1t (blue). Thedashed line represents G = 2kF while the green band shows the regionin which G must lie for the system to be topological for a given µ . Panelb shows the topological phase diagram in the µ–J plane, both in unitsof t, for ∆= 0.1t. To distinguish the two phases we have calculated theMajorana number M as defined in Ref. [69]. Topological phase (TSC)is indicated when M = −1 while M = +1 indicates the topologicallytrivial phase (N).82ttµ=<2.50µ=<0.90µ=<2.30µ=<0.25µ=<1.50µ=0.00a ced fbqqq qqqt tttFigure 6.3: Aeff(ω,q) defined in Eq. (6.11) is represented as a density plotfor several representative values of the chemical potential µ , with theappropriate self-consistently determined spiral pitch G shown in Fig.6.2a. In panel a µ is below the bottom edge of the chain band andthe system is in the trivial phase. When µ reaches the band edge atopological phase transition occurs through a gap closing shown in panelb. Further increase of µ puts the system into the topological phaseillustrated in c, d until the gap closes again near half filling e placingthe system back into the trivial phase f. A similar sequence of phasesoccurs for positive values of µ for which the spectral function can beobtained by simply flipping the sign of the frequency ω . The dark bandaround the chemical potential reflects the bulk SC gap. In all panelsfrequency ω is in units of t while JS = 0.4t, ∆0 = 0.6t, ρ0r2 = 0.05t andδ = 0.002t is used to give a finite width to the spectral peaks.830.00.20.40.60.81.01.2-2G -G 0 G 2Gr+- (q)qFigure 6.4: Spin susceptibility χ+−(q) in the units of 1/t evaluated for thelattice model Eq. (5) with µ = 1.5t, ∆ = 0 and JS = 0 (JS = 0.1t) fordashed (solid) line. A small temperature T = 0.01t has been used tocut off the Lindhard divergences. q extends over the first Brillouin zone(−pi/a,pi/a).84Chapter 7Conclusions7.1 SummaryIn chapter 1 we gave a brief historical overview of the role of topology in con-densed matter systems. We introduced basic concepts which are important tobetter understand the questions we address in this thesis. In chapter 2, we con-sidered the topological electromagnetic response responsible for interesting quan-tized magneto-electric phenomena in the bulk and at the surface and showed thatusing simple assumptions this topological action must be quantized. We showedhow this topological term can be factorized and written as a product of two firstChern numbers of the abelian electromagnetic gauge field. The quantization of thefirst Chern numbers follows with an argument which is similar to the Dirac proofof the quantization of monopole charges which arises by requiring that the elec-tronic wave-functions of the constituent particles must be well-defined and there-fore single-valued.The same topological response described above also explains the exotic surfaceresponse where the topological phase terminates. This is possible if one considersa profile for the axion field which can be assumed to be constant at the two sides ofthe interface, i.e., zero at one side and pi at the other side. Since θ changes from pito 0 time-reversal symmetry would be broken at the interface unless there is an op-posite contribution which arises from the gapless surface modes. This establishesa connection between the bulk topological structure and the existence of odd num-85ber of gapless Dirac surface modes when TR is respected. Careful considerationof the lattice model in a topological phase shows that there is an interesting spin-momentum locking in these electronic surface modes which can be seen in theirlow energy Dirac Hamiltonian. In chapter 3 we studied one interesting signatureof this spin-momentum entanglement which can be used as a sensitive probe ofthe small intrinsic gaps in their spectrum when the TR symmetry is broken. Aswe showed, there is a jump in the total magnetization arising from the existence offully spin polarized LL. We proposed a Knight shift β -NMR measurement as anexperimental technique to observe this signature.In chapter 4 we discussed a toy model for realizing a Weyl semimetal withmultiple pairs of isolated Weyl points separated in BZ. A interesting aspect ofthis model is that although Weyl points are separated in BZ, the system respectsTR symmetry. This would not be possible in a system with a single pair of weylpoints. As we show, Weyl points contribute oppositely to the anomalous Hall cur-rent because of their opposite separation in momentum space.In chapter 5 we continued studying Weyl semimetals by focusing on their un-usual electromagnetic response which had been predicted to exist by the field the-oretical considerations based on the effective low energy theory for fermion fieldscoupled to the classical electromagnetic gauge field. We considered a simple cubiclattice model for a Weyl semimetal and we ruled out the possibility of a static equi-librium CME. However, we showed that the anomalous Hall effect is consistentwith the predicted value from the effective field theory.Finally in chapter 6 we considered a one dimensional system consisting ofmagnetic adatoms on top of an s-wave superconductor and introduced the notionof self-organized topological phase where the system energetically prefers to be ina topological phase. This phase supports MF at its ends and can be a promisingplatform for realizing these interesting localized modes.7.2 Future worksTopological phases and their exotic boundary modes have opened a new promisingavenue of research which is growing fast by new discoveries and the high volumeof experimental and theoretical works. There are still many missing elements in the86topological table of phases which await discovery [55, 56, 127]. However, recentadvances in experimental techniques promises a good future and more excitingdiscoveries in the field.In all the discovered three dimensional TR symmetric TIs, an important re-maining challenge is the existence of the mid-gap bulk states due to the bulk im-perfections and impurities which are contributing to the conductance along with thesurface modes [128]. Experimentalists are working to resolve these issues by grow-ing higher quality samples and looking for better potential materials which can bein a TI phase. Another important aspect is to control surface states occupancy (thechemical potential) and also their Dirac masses which in the three dimensional caseis very challenging. The magneto-electric response which arises due to existenceof the quantized topological E ·B action in the effective electromagnetic theory haspotential for interesting technological applications [83, 85, 129] by consideringvarious forms of TI interfaces.In the two dimensional QSH TI phase the only example discovered so far isHgTe, which requires sophisticated experimental techniques which makes the ex-perimental realization a formidable task. Experimentalists are investigating othertheoretical proposals based on the deposition of heavy adatoms on top of graphenemonolayers to hopefully induce a large SO gap desired for the QSH phase. [31, 32]Although very recently a three dimensional Dirac phase has been discoveredin Na3Bi [97], a Weyl semimetal phase with isolated modes in momentum has notyet been discovered. However, some of the candidate systems seem experimentallyfeasible enough to make us believe that they will be discovered in the near future[55–57]. Recently, the negative magneto-resistivity observed in Bi1−xSbx near thetopological phase transition has been attributed to the existence of a topological E ·B term due to the isolated Weyl points which in theory can be formed by breakingthe TR symmetry at the critical topological transition point [130]. However, this isstill not a smoking gun proof for the existence of the isolated Weyl nodes.On the other hand, one dimensional topological superconductors with MFs arealso highly pursued. Manipulating Majorana states in a controlled fashion in orderto exploit their interesting non-Abelian statistics is also an important topic and isanother direction which has a good potential for being used in fault-tolerant quan-tum computation devices that might be invented in the future. More theoretical and87experimental works are required to achieve these goals. The model we describedin chapter 6 can be realized with the current experimental techniques and mightprove easier access to Majorana modes at its ends by constructing more sophisti-cated patterns which is possible by moving adatoms using an Scanning TunnellingMicroscopy (STM) tip [112, 113].It is also important to note that all the phases we discussed here have beenbased on the assumption that the interaction does not play a major role and there-fore can only describe non-interacting or weakly interacting systems. One canimagine more exotic possibilities like fractional topological insulators analogousto the fractional counterparts of the quantum Hall insulators in two spatial dimen-sions [131, 132]. As far as I understand no such fermion field model, that can arisefrom a condensed matter system lattice model, has been proposed yet. All the con-siderations so far have been based on the effective topological field theory for theelectromagnetic gauge fields. 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If the process of changing the Hamiltonian is slow enough, thenthe state remains to be the eigenstate of the Hamiltonian without acquiring energyfrom the environment. This is called an adiabatic process. Interestingly, Although tgoes back to itself and therefore the initial and final Hamiltonians are the same aftergoing around the loop in the parameter space, the final eigenstate is not necessarilythe same as the initial one. In the abelian case the final state differ from the initialone by a phase which is called a Berry phase.Let us consider a simple example. If you consider a general spinor state fors = 1/2 electron in the basis of Sz eigenstates we have101|n(φ ,θ)>=(e−iφ cos θ2sin θ2), (A.2)this state is the ground state for the Hamiltonian H =−ζ~B ·~S with ~B = B0nˆ. Now,adiabatic change of the direction of the magnetic field along a loop in the (φ ,θ)space will change the phase of the initial state by ΦB given by the following equa-tionΦB = i∮< n|∇|n > ·dn =∮~A ·dl , (A.3)where we have defined ~A = i < n|∇|n >. According to the stocks theorem onecan rewrite it in terms of a surface integralΦB =∮~A ·dl =∫S∇×~A ·da, (A.4)It is easy to calculate ~A and then ~B from it~A = (cos θ2)2(sinθ)−1φˆ → ~B = ∇×~A = 12rˆ, (A.5)Therefore we haveΦB =∫S~B ·da = 12∫dΩ= Ω2, (A.6)This geometrical phase also shows up in the path integral representation. TheLagrangian describing the path integral contains a Berry phase term∫d2nˆ2pi |nˆ >< nˆ|= 12×2, (A.7)The Green’s function then isG(nˆ, nˆ0, t, t0) = < nˆ|U(t, t0)|nˆ0 > (A.8)= < nˆ|U(t, t−∆t) ·U(t−∆t, t−2∆t) · · ·U(t0 +∆t, t0)|nˆ0 >= 1(2pi)N∫D[nˆ(t)]ei∫ tt0L (nˆ(t), ˙ˆn(t),t)dt ,102where for H = 0 the Lagrangian L isL = i < nˆ(t)| ddt|nˆ(t)>, (A.9)which is surprisingly nonzero, and it is associated with the Berry phase andtherefore is a topological term in the Lagrangian.This geometrical phase reflects the frustration in choosing the same phase forall of the coherent states.A.2 TKNN invariant for massive Dirac statesThe low energy Hamiltonian describing a two dimensional Dirac state with massm can be expressed asH(~k) = υxkxσx +υykyσy +mσz, (A.10)The eigenstates can be found by diagonalizing above two-by-two matrix. Athalf filling the occupied state as a function of kx and ky have the following spinorformχ~k =(e−iφbk cosθ~k2sinθ~k2), (A.11)where φ~k and θ~k are defined in terms of (kx,ky,m) as followscosθ~k =m√m2 +υ2x k2x +υ2y k2y, (A.12)andcosφ~k + isinφ~k =υxkx + iυyky√υ2x k2x +υ2y k2y, (A.13)Using these expressions and the similarity between these states with the spinorstates defined in the previous sections we can obtain the expressions for the pseudo-magnetic field103~A~k = (cosθ~k2)2(sinθ~k)−1φˆ~k → ~B = ∇~k×~A =12rˆ~k, (A.14)where φˆ~k and rˆ~k are the unit vectors defined in the spherical coordinate system of(kx,ky,m) space.Using the expression given for the TKNN invariant in Eq. (1.4) we can evaluatethis invariant. After some careful consideration and by summing the contributionfrom all the occupied states we findn = sgn(mυxυy)2. (A.15)A.3 Electrons in a periodic potentialIn the absence of the magnetic field and any spin orbit interaction, the Hamiltoniandescribing noninteracting electrons in a periodic potential can be written asH = (− h¯22m~∇2 +V (~r))⊗12×2, (A.16)in which the potential satisfies the following periodicity relation for the three latticevectors,~ai i = 1,2,3V (~r+~ai) =V (~r), (A.17)This periodicity implies that the Hamiltonian is invariant under a symmetrygroup of discrete translations defined by the same lattice vectors. The generatorsof this Abelian symmetry group are three translations in each direction given byT~ai f (~r) = f (~r+~ai), (A.18)According to the Bloch theorem this symmetry implies that the eigenstates ofthe Hamiltonian are also eigenstates of these symmetry operators, therefore theysatisfy the following relationψn~q(~r+~ai) = ei~q·~aiψn~q(~r), (A.19)104where n is the band index and~q is called the lattice wave-vector. Therefore, one canwrite these Hamiltonian eigenfunctions in terms of the periodic Bloch functionsψn~q(~r) = ei~q·~run(~q,~r), (A.20)The invariance of the wave-function under translation by the system lengthrequires that the lattice momentum can only take discrete values~q = 2pi(n1L1, n2L2, n3L3), (A.21)where (L1,L2,L3) are the dimensions of the system. Shifting ~q vectors by thereciprocal vectors ~Gi does not change the wavefunction sinceei~Gi.~ai = e2pii = 1, (A.22)The lattice constructed by these ~Gi vectors is called the reciprocal lattice. The~q points in the unit cell of the reciprocal lattice is called Brillouin Zone (BZ).The ~q points in the BZ are the distinct quantum numbers which label the Blochwavefunctions. The energy as a function of ~q for each band n, i.e., εn(~k) is calledthe band spectrum.A.4 Generalized Bloch statesTurning on a perpendicular magnetic field in a two dimensional electron systembreaks the discrete translational symmetry of the two dimensional lattice. If themagnetic field is commensurate meaning that the magnetic flux going through aunit cell is a rational multiple of quantum of magnetic flux, i.e., (m/n) · (h/e),then it is possible to define a generalized version of the Bloch states. For this tohappen one needs to use a Landau gauge, ~A = (0,eBx). The eigenfunctions of theHamiltonian describing such a system can be chosen in a way that they satisfy ageneralized version of the Bloch condition for the translations along x [11]ψk1k2(x+ma,y) = exp(2piimy/b+ ik1na)ψk1k2(x,y), (A.23)One can then define periodic functions with a generalized periodicity given by105uk1k2(x+ma,y) = exp(2piimy/b)uk1,k2(x,y), (A.24)These are the generalized Bloch functions and are eigenfunctions of the fol-lowing HamiltonianH(k1,k2) =12m(−ih¯ ∂∂x + h¯k1)2+ 12m(−ih¯ ∂∂y + h¯k2− eBx)2+V (x,y),(A.25)In fact one uses these generalized Bloch functions to get non-zero Hall con-ductance in Eq. (1.2).A.5 From discrete to continuum limitConsider a 1D lattice system with lattice spacing a and we can define a quantity fon each site, i.e., fn is the f value on nth site. The average of this quantity on thislattice would bef¯ = 1NN∑n=1fn, (A.26)Now if N = L/a is very large we can approximate a continuum theory for theabove expression as followsf¯ = 1L∫ L0dx f (x), (A.27)Note that this has been obtained using the following substitutionN∑n=1→∫ ∞−∞dxρ(x) = 1a∫ L0dx, (A.28)f (x) is the average of fn in a dx segment with a center at x.we can generalize this to a case like the following< f g >= 1NN∑n=0fngn→1N∫ ∞−∞dxρ(x) f (x)g(x) = 1L∫ L0dx f (x)g(x), (A.29)106Note that here we are approximating f (x)g(x) as being the product of the av-erages of the fn and gn over a dx segment centred at x which would be equal to theaverage of the fngn if the quantities does not vary radically over a length scale dxand they can be approximated by linear functions.another form which is a bit trickier is the following quantityFp =1NN∑n=1( fn− fn−1)p, (A.30)which becomesFp =1N∫ ∞−∞dxρ(x)[ f (x)− f (x−a)]p = 1L∫ L0dx(∂x f (x))p, (A.31)Now let see what happens to a function like fn = Vδnl where δnl is the Kro-necker function so fn is only non-zero at lth sitef¯ = 1NN∑n=1Vδnl =VN, (A.32)Now in the continuum limit we still need to find the same valuef¯ = 1L∫ L0f (x)d(x) = VN, (A.33)It turns out that f (x) has all the properties of the Dirac’s delta function as wetake the limit, i.e., f (x) = aVδ (x− xl).Now as we take a to zero if Va = V remains constant we getf (x) = V δ (x− xl), (A.34)This happens when V grows linearly with N this way we havelima→0Va =V0Na =V0L, (A.35)Now we use above arguments to derive the famous integral representation ofDirac delta function, namely,107δ (k− k0) =12pi limL→∞∫ L/2−L/2dxei(k−k0)x, (A.36)we define a lattice with a lattice spacing 2pi/L when we take L to infinity wecan make every kn ≡ k0 +(2pin)/L sufficiently close to any kthis way we haveI = 12pi∫ L/2−L/2dxei2pinx/L = −iL(2pi)2∮zn−1dz = L2pi δknk0 =δknk0a, (A.37)Now according to our continuum conversion we haveI = δknk0a= δ (kn− k0), (A.38)which proves Eq. (A.36). For finite L we need the periodic boundary condition inx (i.e., x = x+L) in order to get the kronecker function form of Eq. (A.36)A.6 Fourier transform of the step functionThe F.T. of a step function, i.e., Θ(t) can be expressed asΘ(ω) =∫ ∞0eiωt−0+t = iω+ i0+ , (A.39)A.7 Fourier transform of the productConsider two functions f (t) and g(t). The F.T. of the product can be written interms of an integral of the product of the F.T.s components as follows[ f .g](ω) =∫ ∞−∞f (t)g(t)eiωtdt (A.40)=∫ ∞−∞dω ′2pi f (ω′)∫ ∞−∞dtg(t)ei(ω−ω ′)t=∫ ∞−∞dω ′ f (ω ′)g(ω−ω ′),108Now by using what we have got in the previous section we find∫ ∞0dt f (t)e−iωt−0+t =∫ ∞−∞dω ′ f (ω′)ω−ω ′+ i0+ . (A.41)A.8 Green’s function for Schrodinger equationih¯∂Ψ(t)∂ t =H Ψ(t), (A.42)ih¯∂tUˆ(t, t0) =H Uˆ Ψ(t) = Uˆ(t, t0)Ψ(t0), (A.43)where Uˆ(t, t0) for time independent Hamiltonian takes the following formUˆ(t, t0) = eiHˆ (t−t0)/h¯, (A.44)for a time-dependent Hamiltonian for which [Hˆ (t),Hˆ (t ′)] = 0 for all t, t ′ wehaveUˆ(t, t0) = ei∫ tt0Hˆ (t ′)dt ′/h¯, (A.45)for the case where [Hˆ (t),Hˆ (t ′)] 6= 0 we get the dyson seriesUˆ(t, t0) = 1+∞∑n=1(− ih¯)n∫ tt0dt1∫ t1t0dt2...∫ tn−1t0dtnHˆ (t1)Hˆ (t2)Hˆ (t3)...Hˆ (tn),(A.46)using the time-ordering operator it can be written asUˆ(t, t0) =T (ei∫ tt0Hˆ (t ′)dt ′/h¯), (A.47)iGˆ(t, t0)≡ Uˆ(t, t0)Θ(t− t0), (A.48)where Θ(t) is the step functioni(ih¯∂t −Hˆ )Gˆ(t, t0) =−h¯δ (t− t0)Uˆ(t, t0) =−h¯δ (t− t0)1, (A.49)109Therefore we get(ih¯∂t −Hˆ )G(t, t0) = ih¯δ (t− t0)1, (A.50)In position basis it becomes(ih¯∂t −Hˆ (~x))G(~x, t;~x0, t0) = ih¯δ (t− t0)δ (~x−~x0), (A.51)If we F.T. the time we getGˆ(ω) =∫ ∞−∞Gˆ(t0 + t, t0)eiωt−ηt , (A.52)and from Eq. (A.50) we obtain the following equation for Gˆ(ω)(h¯ω−Hˆ + iη)Gˆ(ω) = ih¯, (A.53)Gˆ(ω) = ih¯h¯ω−Hˆ + iη. (A.54)A.9 Density of states function from the Green’s functionThe density of the states function can be defined as followsD(E) =∑iδ (E− εi) = tr[δ (E−Hˆ )], (A.55)but we also have the following limit definition of the Dirac delta function1x± i0+ = P.V.(1x)∓piδ (x), (A.56)So from the previous section we find thatD(E) =− 1pi tr[Im(ˆ¯G(ω = Eh¯))], (A.57)Where ih¯G¯ = G110A.10 Green’s function for a perturbed system(h¯ω−Hˆ0−Vˆ + iη)Gˆ(ω) = ih¯ → Gˆ(ω) = Gˆ0(ω)−ih¯Gˆ0(ω)Vˆ Gˆ(ω), (A.58)we can also write it asGˆ(ω) = 11+ ih¯ Gˆ0(ω)VˆGˆ0(ω), (A.59)Note that we can recover more well-known equation for this if we replace G withih¯GGˆ(ω) = 11− Gˆ0(ω)VˆGˆ0(ω) Gˆ(ω) = Gˆ0(ω)+ Gˆ0(ω)Vˆ Gˆ(ω), (A.60)for a time dependent perturbation Vˆ it takes the following formGˆ(ω) = Gˆ0(ω)−ih¯∫ ∞−∞dω ′2pi Gˆ0(ω)Vˆ (ω′)Gˆ(ω−ω ′). (A.61)A.11 Quantum linear response theoryConsider the following HamiltonianH = H0 + Oˆ1 f (t), (A.62)where H0 is the base Hamiltonian where its ground-state is known. One can askwhat would be the effect of this additional term on the expectation value of anotheroperator like Oˆ2 at an arbitrary time tδ < |Oˆ2(t)|> ≡ <Ψ(t)|Oˆ2|Ψ(t)>−<Ψ0(t)|Oˆ2|Ψ0(t)> (A.63)= < δΨ(t)|Oˆ2|Ψ(t)>+<Ψ(t)|Oˆ2|δΨ(t)>,where |Ψ(t) >= |Ψ0(t) > +|δΨ(t) > and it is easy to see that |δΨ(t) > inlinear order is given by111|δΨ(t)>=−ie−iH0(t−t−∞)∫ tt−∞dt ′Oˆ1(t ′) f (t ′), (A.64)where Oˆ1(t ′) = eiH0(t′−t−∞)Oˆ1e−iH0(t′−t−∞) therefore by substituting the aboveinto Eq.(A.63) we getδ < |Oˆ2(t)|>=−i∫ tt−∞dt ′ <Ψ0|[Oˆ2(t), Oˆ1(t ′)]|Ψ0 > f (t ′), (A.65)which can be written asδ < |Oˆ2(t)|>=∫ ∞−∞dt ′D(t, t ′) f (t ′), (A.66)where D(t, t ′) is defined asD(t, t ′) =−iΘ(t− t ′)<Ψ0|[Oˆ2(t), Oˆ1(t ′)]|Ψ0 >, (A.67)So the response function D(t, t ′) is relation to the correlation functionA.12 Coupled systems and the effective theoryConsider two systems described by Hamiltonians H1 and H2 coupled through HintH = H1⊗1+1⊗H2︸ ︷︷ ︸H0+Hint , (A.68)any observable operator A defined in the system 1 Hilbert space has the followingexpectation value< |A|>= tr(ρ ·A⊗1) = tr1tr2(ρ ·A⊗1) = tr1(ρ1A), (A.69)where ρ1 = tr2(ρ). This can be understood from[A⊗B]i jlk =< φl,ϕk|A⊗B|φi,ϕ j >=< φl|A|φi > ·< ϕk|B|ϕ j >= AilB jk, (A.70)tr(A⊗B) = tr1(A)tr2(B), (A.71)112So the system described by ρ1 is an effective theory in which we have integratedout the degrees of freedom we had from the second system.To understand this better, consider it in the position basis in which we have< xb|ρˆ1|xa >= tr2 < xb|ρˆ|xa >=∫< xb,X |ρˆ|xa,X > dX , (A.72)Consider for example the canonic ensemble where we can have a path integralfor its partition function and therefore relate it to the classical action for the system.So from the path integral representation for the effective partition function ob-tained from the effective density operator ρ1 = tr2(ρ) (for example ρ1 can be e−βHwhen you integrate out the system 2 being the thermal bath) we can find the ef-fective action describing the system where we integrate all degrees of freedomavailable for system 2.So we have the following for the partition functionZ =∫D [x(t)]D [X(t)]ei[S1(x)+S2(X)+Sint(x,X)], (A.73)where (x,X)(t) = (x,Xb), (x,X)(t0) = (x,Xa),S1(x) =∫ tt0L1(x, x˙, t)dt, S2(X) =∫ tt0L2(X , X˙ , t)dt, (A.74)andSint(x,X) =∫ tt0Lint(x, x˙,X , X˙ , t)dt, (A.75)We can write Z = Ze f f ZX in which the two systems are decoupled in a senseand the system 1 through Ze f f carries information from its interaction with thesystem 2.Z =∫D [x(t)]D [X(t)]ei[S1(x)+S2(X)+Sint(x,X)] =∫D [x(t)]eiSe f f (x)︸ ︷︷ ︸Ze f f·∫D [X(t)]eiS2(X),(A.76)where Se f f (x) = S1(x)+SI(x) and SI(x) is defined as113eiSI(x) =∫D [X(t)]ei[S2(X)+Sint(x,X)]∫D [X(t)]eiS2(X) , (A.77)The effective quantum theory for system 1 is in fact approximately a theory inwhich the interaction of it with system 2 has been averaged over the canonicalensemble of system 2Ze f f =< |∫D [x(t)]ei[S1(x)+Sint(x,X)]|>2, (A.78)where < ? >2 is< ? >2=∫D [X(t)]? eiS2(X)∫D [X(t)]eiS2(X) . (A.79)A.13 Topologically protected semi-metalAt low energies, the minimal Hamiltonian describing the unstable 3D gapless sys-tem at the TI-OI transition point with a pair of energy crossing points would havethe following formH = ∑α=±∑|~k−~wα |<ΛΨ†α(~k)[~λα .(~k−~wα)]Ψα(~k), (A.80)where~λα are 2-by-2 matrices given by~λ = (υxσx,υyσy,υzσz), (A.81)in which ~w+ = ~w− = ~w0 are the positions of the Weyl points with opposite chirality( = sgn(υxυyυz)) in the BZ. This Hamiltonian consists of doubly degenerate Diraccones with the Dirac points located at ~w0. The two component spinor operatorΨ†± is associated with the modes on the Dirac cone with chirality α = ±. ThisHamiltonian describes the gapless phase at the phase transition point between atopological insulator and an ordinary insulator. When the opposite chirality Weylpoints happen at the same point in the BZ there is no topological protection andeven momentum conserving perturbations that connect the degenerate states candrive the system to an insulator phase. Now if somehow the opposite chirality114points become separated (~w+−~w− =~q 6= 0) then the system would be in a gaplesstopological phase where small local momentum conserving perturbations cannotproduce a gap in the spectrum and would only shift the position of the Weyl pointsin the BZ.Using the mathematical properties of the Clifford algebra it is possible to seehow such protection can occur for a gapless system containing a pair of separatedopposite chirality Weyl points. The local Hamiltonian matrix in the BZ describ-ing the low energy states in the topological semi-metal phase containing a pair ofopposite chirality Weyl points can be written as followsH (~k) = kxγ1 + kyγ2 + kzγ3 +q2γ ′3, (A.82)where for simplicity and without the loss of the generality we have put |υx,y,z|= 1.Note that by writing the Hamiltonian in this fashion we implicitly exclude any per-turbation that connects two Weyl nodes which might arise from the e-e interactionsand scattering sources and can potentially gap out the Weyl nodes. This Hamilto-nian has three terms (4-by-4 hermitian matrices) which are anti-commuting witheach other, i.e., γ1,2,3. It also has a term which is essential to separate the Weylpoints in the BZ. This term commutes with one of the terms and anti-commuteswith all others. So we have {γi,γ j}= {γ1,2,γ ′3}= 0 and [γ3,γ ′3] = 0. The last termseparates the Weyl points along the kz axis by q and the resulting spectrum is givenbyε~k,s =±√k2x + k2y +(kz +sq2)2, s =±1, (A.83)In order to open up a gap by adding a local momentum conserving term to theHamiltonian in a way that doesn’t merge the Weyl points there must be a hermitianmatrix that anti-commutes with all of the terms in the above Hamiltonian. Thestatement we want to prove here is that no such term exist and any 4-by-4 hermitianmatrix which can be written as a superposition of the normalized Clifford algebramatrices, γg, (γg2 = I4×4) would commute with at least one of the terms in thisHamiltonian and therefore cannot produce a gap in the spectrum and only shiftsthe position of the Weyl nodes. In order to prove this statement consider the most115general form of a normalized matrix that anti-commutes with γ1,2,3 and we showthat it cannot at the same time anti-commute with γ ′3. In order to do so we need tochoose a representation of the present matrices. We can choose a basis in whichboth γ3 and γ ′3 are diagonal since they commute with each other. This way we haveγ3 = diag[1,a1,a2,−1−a1−a2] and γ ′3 = diag[1,b1,b2,−1−b1−b2]. It turns outthat only three independent choices can satisfy the commutation condition, i.e.,(a1,a2,b1,b2) = (1,−1,−1,1),(−1,1,1,−1),(−1,1,−1,−1). For each of thesechoices only two independent normalized matrices exist that anti-commute with γ3and γ ′3 and with themselves. This would then imply that no other matrices exist thatanti-commutes with all the existing matrices in the Hamiltonian. To see how thisworks consider the most general normalized matrix satisfying the aforementionedconditions for the first choice of (a1,a2,b1,b2). For this particular representation itwould be in the following formγp =0 0 0 eiφ0 0 eiθ 00 e−iθ 0 0e−iφ 0 0 0, (A.84)Matrices of this form that anti-commute with each other must satisfy |φi−φ j|=|θi−θ j| = pi/2. It is easy to see that only two independent matrices can be foundin this representation to mutually satisfy this equations. These matrices are al-ready exploited in the Hamiltonian therefore no other matrices can be added to theHamiltonian given in Eq. (A.82) that anti-commutes with all the present matrices.The same argument can be established for the other two choices of (a1,a2,b1,b2).Therefore, Weyl nodes cannot be annihilated unless two of them with oppositechirality merge together at the same point in the BZ. This would not be possi-ble without the sufficiently strong local perturbations (∼ |h¯υFq|) and the systemwould be in a topological semimetal phase in at least a region of the phase space.The above proof is a generalization of the argument in which one considers onlyone of the isolated Weyl nodes in a two band model using two-by-two Pauli matri-ces. Here we have considered all the four bands that are present in the low energytheory describing a pair of Weyl fermions; therefore, it is on a more solid ground.116A.14 Mean-field treatment of HintThe interaction term in the Hamiltonian is made up of some quadratic terms likeΨ†1Ψ1Ψ†2Ψ2. There are various ways to decouple these terms. In the following, weconsider one specific decomposition channel.First we use the anti-commutation relations, i.e., {Ψα ,Ψ†β}= δαβ , {Ψα ,Ψβ}={Ψ†α ,Ψ†β}= 0 to getΨ†1Ψ1Ψ†2Ψ2 =12[Ψ†1Ψ1 +Ψ†2Ψ2]+12[Ψ1Ψ†2Ψ†1Ψ2 +Ψ2Ψ†1Ψ†2Ψ1], (A.85)Now one can rewrite Ψ1Ψ†2Ψ†1Ψ2 as followsΨ1Ψ†2Ψ†1Ψ2 = (Ψ1Ψ†2−<Ψ1Ψ†2 >)(Ψ†1Ψ2−<Ψ†1Ψ2 >) (A.86)+ <Ψ1Ψ†2 >Ψ†1Ψ2+<Ψ2Ψ†1 >Ψ†2Ψ1 + |Ψ1Ψ†2|2,Now we assume that the first term is negligible according to the MF approxima-tion so if we do the same thing for the other term and sum up after some simplecommutations we getΨ†1Ψ1Ψ†2Ψ2 =12[Ψ†1Ψ1 +Ψ†2Ψ2]+<Ψ1Ψ†2 >Ψ†1Ψ2 (A.87)+ <Ψ2Ψ†1 >Ψ†2Ψ1 + |Ψ1Ψ†2|2,Where we have chosen < Ψ1Ψ†2 >= M12 as the order parameter. We canalso drop the first two terms in the RHS since they are independent of the orderparameter and they can be considered as an overall shift in the energy. (we cannotdrop the last term though since although it is an overall shift but, it depends on theorder parameter so it plays a role in when the order parameter varies) So this waywe end up withΨ†1Ψ1Ψ†2Ψ2 =M12Ψ†1Ψ2 +M12?Ψ†2Ψ1 +M12M12?. (A.88)117

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