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Majorana bands in topological superconductors Liu, Tianyu 2015

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Majorana Bands in TopologicalSuperconductorsbyTianyu LiuB.Sc., University of Science and Technology of China, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)June 2015c© Tianyu Liu 2015AbstractMajorana fermions can exist in condensed matter systems as quasi-particleexcitations called Majorana bands. The details of Majorana bands will bethe central concern of this thesis. In the thesis, Majorana bands are stud-ied analytically and numerically in two square lattice systems with vortices.The p+ip superconductor, containing two vortices in each magnetic unit cell,exhibits slightly dispersing Majorana bands in the middle of the supercon-ducting gap. With the same vortex geometry, the Fu-Kane model showssimilar Majorana bands, which, however, can become completely flat whenchemical potential is tuned to coincide with the Dirac point. By comparisonto a tight binding model of vortex lattice, it is clear that the dispersion ismainly contributed by first and second nearest neighbor hoppings of Majo-rana fermions bound in vortices. The hoppings, which are extracted fromnumerical diagonalization, are not quite identical to the existing analyticalprediction. Therefore, we built two simple equations that show the phe-nomenologically correct trends of the hoppings.iiPrefaceThis thesis is a summary of the author’s M.Sc. project, formulated by Prof.Marcel Franz, focusing on the details of Majorana bands with the presenceof vortex lattice. The author is in charge of generating all numerical dataand figures as well as some of the analytical derivation.In chapter 2, section 2.1 and section 2.2 are reproductions of the ana-lytical results in the work [1] that Prof. Franz used to participate in. Fig-ure 2.3 and Figure 2.6 are also reproductions, though the parameters maybe slightly different. In chapter 3, the analytical derivation in section 3.1 iswholly developed by Prof. Franz. In chapter 4, Figure 4.1 and Figure 4.2are reproductions of the work by D. J. J. Marchand and M. Franz [2] withsame parameter settings. In appendix, Figure A.1 and Figure A.3 are bothreproductions of the work [1] with same parameter settings.All the other analytical and numerical work is original and done by theauthor under supervision of Prof. Franz. A rough draft containing analyt-ical derivation and numerical results with details based on chapter 3 andchapter 4 has been prepared by the author and is being modified by Prof.Franz for future submission.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Majorana fermion . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Theoretical proposals . . . . . . . . . . . . . . . . . . 21.1.3 Experimental realization . . . . . . . . . . . . . . . . 31.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 The p+ip superconductor . . . . . . . . . . . . . . . . . . . . . 52.1 Hamiltonian of p+ip superconductor . . . . . . . . . . . . . . 52.2 Superfluid velocity . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Majorana band and spectrum . . . . . . . . . . . . . . . . . . 103 Majorana lattice . . . . . . . . . . . . . . . . . . . . . . . . . . 163.1 Tight binding Hamiltonian . . . . . . . . . . . . . . . . . . . 163.2 Hoppings of p+ip superconductor . . . . . . . . . . . . . . . 204 The Fu-Kane model . . . . . . . . . . . . . . . . . . . . . . . . 264.1 The Fu-Kane Hamiltonian . . . . . . . . . . . . . . . . . . . 264.2 Majorana band and spectrum . . . . . . . . . . . . . . . . . . 324.3 Hoppings in Fu-Kane model . . . . . . . . . . . . . . . . . . 39ivTable of Contents5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43AppendixA The s-wave superconductor . . . . . . . . . . . . . . . . . . . . 47vList of Figures1.1 Kitaev’s toy model. Upper panel: trivial phase with ∆ =t = 0 and µ 6= 0. Majorana fermions are paired to form aregular Dirac fermion chain. Lower panel: topological phasewith ∆ = t and µ = 0. Majorana fermions are paired in apattern that two unpaired Majorana fermions are left on theends of the chain. . . . . . . . . . . . . . . . . . . . . . . . . 22.1 Two vortex sublattices. Each magnetic unit cell (orange square)encloses two different vortices on the diagonal with vortexspacing to be half of the diagonal length. . . . . . . . . . . . . 62.2 Magnetic unit cell with 10δ × 10δ geometry. The order j ofcrystalline site is assigned row by row so that rj is explicitlydefined. The vortices are located in the center of correspond-ing plaquettes on the diagonal of the unit cell. The spectrumdoes not rely on the specific positions of the vortices as longas they are on the diagonal with spacing to be (5δ, 5δ). . . . . 112.3 Density of states (DOS) for p+ip superconductor. The mag-netic unit cell is chosen to be 10δ×10δ, with δ being the latticeconstant of the underlying crystalline lattice. The supercon-ductor order parameter is ∆0 = 0.5034 and the chemical po-tential is εF = −2.2. The blue line is the DOS of magneticfield B = 0 so that no vortices will appear. The sudden dropsat 1.8 and 6.2 result from band edges corresponding to thebottom and the top of the band respectively. The Van Hovesingularity at 2.2 results from the saddle points of the band.The red line shows the DOS of magnetic field B 6= 0 and thusa vortex lattice appears. Landau levels show at the top of thespectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Magnetic Brillouin zone for square vortex lattice with the no-tation of high-symmetry points used in plotting band struc-ture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12viList of Figures2.5 Band structure of p+ip superconductor with no vortices. Themagnetic unit cell is chosen to be 10δ × 10δ, with δ beingthe lattice constant of the underlying crystalline lattice. Thesuperconductor order parameter is ∆0 = 0.5034 with chemicalpotential εF = −2.2. . . . . . . . . . . . . . . . . . . . . . . . 132.6 Band structure of p+ip superconductor with square vortexlattice. The magnetic unit cell is chosen to be 10δ × 10δ,with δ being the lattice constant of the underlying crystallinelattice. The superconductor order parameter is ∆0 = 0.5034with chemical potential εF = −2.2. . . . . . . . . . . . . . . 142.7 Low-energy bands of p+ip superconductor in 10δ × 10δ mag-netic unit cell with ∆0 = 0.5034 and εF = −2.2. The red linesshow the Majorana bands with slight dispersion. . . . . . . . 153.1 Gauge of tight binding model of square vortex lattice. Eachtriangular plaquette has a phase factor of pi/2. . . . . . . . . 173.2 Diamond-shape reduced Brillouin zone (solid lines) in tightbinding model with Q = (pi/2,−pi/2). . . . . . . . . . . . . . 183.3 Majorana bands of p+ip superconductor in 10δ×10δ magneticunit cell with ∆0 = 0.5034 and εF = −2.2. The square dotsare Majorana bands given by numerical diagonalization, whilethe red lines are (k) given by tight binding model (Equa-tion 3.17) with gaps ∆Γ = 0.00155 and ∆X = 0.02434 ex-tracted from numerical diagonalization. . . . . . . . . . . . . . 203.4 First nearest neighbor hopping t in p+ip superconductor with50δ × 50δ magnetic unit cell and ∆0 = 0.1. Black dots showhow t varies with chemical potential εF numerically. Thered line (Equation 3.26) is the phenomenological fit with theoscillation characterized by cos (kR+ pi/4) as is suggested byCheng et al. [3, 4]. The linearly varying amplitude is A+BkRwith parameters A = 2.438 × 10−4, B = 3.981 × 10−6, andR = 25√2. The blue line (Equation 3.24) is analytical resultgiven by Cheng et al. [3, 4]. . . . . . . . . . . . . . . . . . . 23viiList of Figures3.5 Second nearest neighbor hopping t′ in p+ip superconductorwith 50δ × 50δ magnetic unit cell and ∆0 = 0.1. Black dotsshow how t′ varies with chemical potential εF numerically.The red line (Equation 3.31) is the phenomenological fit withthe oscillation characterized by cos (kR′ + pi/4) as is suggestedby Cheng et al. [3, 4]. The linearly varying amplitude isA+BkR′ with parameters A = 5.91×10−5, B = 3.904×10−8,and R′ = 50. The blue line (Equation 3.30) is analytical resultgiven by Cheng et al. [3, 4]. . . . . . . . . . . . . . . . . . . . 254.1 Spectrum of the simplified topological insulator model witht¯ = 0.5. The spectrum is doubly degenerate with a singleDirac point at Γ. . . . . . . . . . . . . . . . . . . . . . . . . . 274.2 Spectrum of the simplified topological insulator model witht¯ = 0.5. One of the surfaces is gapped by mass m = 0.2. . . . 284.3 Spectrum of the simplified topological insulator model witht¯ = 0.5. The mass is m = 0.2, the superconductor orderparameter is ∆ = 0.5, and the chemical potential is µ = 0. . 294.4 Majorana bands in the Fu-Kane model in a 30δ×30δ magneticunit cell with εF = 0.25, m = 0.5, ∆0 = 0.4, t¯ = 0.5, andδ0 = 0.23. Black dots represent numerical data while redlines are (k) given by tight binding model (Equation 3.17)with gaps ∆Γ = 2.76×10−4 and ∆X = 1.336×10−2 extractedfrom numerical data. . . . . . . . . . . . . . . . . . . . . . . 334.5 Low-energy bands in the Fu-Kane model in a 30δ × 30δ mag-netic unit cell with εF = 0.25, m = 0.5, ∆0 = 0.4, t¯ = 0.5,and δ0 = 0.23. The red lines are dispersing Majorana bands. 344.6 Low-energy bands in the Fu-Kane model in a 30δ × 30δ mag-netic unit cell with εF = 0, m = 0.5, ∆0 = 0.4, t¯ = 0.5, andδ0 = 0.23. The spacing of two vortices in the magnetic unitcell is d = (15δ, 15δ). The bandwidth of Majorana band (redline) is 5.7× 10−5. . . . . . . . . . . . . . . . . . . . . . . . . 354.7 Low-energy bands in the Fu-Kane model in a 30δ × 30δ mag-netic unit cell with εF = 0, m = 0.5, ∆0 = 0.4, t¯ = 0.5, andδ0 = 0.23. The geometry is changed by bringing two vorticescloser to d = (10δ, 10δ) along the diagonal. The bandwidthof Majorana band (red line) is 5.6× 10−5. . . . . . . . . . . . 36viiiList of Figures4.8 Low-energy bands in the Fu-Kane model in a 30δ × 30δ mag-netic unit cell with εF = 0, m = 0.5, ∆0 = 0.4, t¯ = 0.5, andδ0 = 0.23. The geometry is changed by bringing two vor-tices further closer to d = (5δ, 5δ) along the diagonal. Thebandwidth of Majorana band (red line) is 4.9× 10−5. . . . . . 374.9 Low-energy bands in the Fu-Kane model in a 30δ × 30δ mag-netic unit cell with εF = 0, m = 0.5, ∆0 = 0.4, t¯ = 0.5, andδ0 = 0.23. The geometry is changed by merging two vorticesalong the diagonal into a double vortex. The bandwidth ofMajorana band (red line) is 4.9× 10−5. . . . . . . . . . . . . 384.10 First nearest neighbor hopping t of the Fu-Kane model in a30δ× 30δ magnetic unit cell with m = 0.5, ∆0 = 0.4, t¯ = 0.5,and δ0 = 0.23. Black dots are numerical data and red line isphenomenological fit. Fitting parameters are A = 7.85×10−3,B = 1.79, D = 0.341, and R = 7.87. . . . . . . . . . . . . . . 404.11 Second nearest neighbor hopping t′ of the Fu-Kane model in a30δ× 30δ magnetic unit cell with m = 0.5, ∆0 = 0.4, t¯ = 0.5,and δ0 = 0.23. Black dots are numerical data and red line isphenomenological fit. Fitting parameters are A = 5.87×10−4,B = 2.05, D = 0.26, and R = 12.06. . . . . . . . . . . . . . . 41A.1 Density of states for an s-wave superconductor. The magneticunit cell is chosen to be 10δ × 10δ, with δ being the latticeconstant of the underlying crystalline lattice. The supercon-ductor order parameter is ∆0 = 1 and the chemical potentialis εF = −2.2. The blue line is the DOS with magnetic fieldB = 0 so that no vortices will appear. The sudden drops at2.1 and 6.3 result from the edges of the band correspondingto the bottom and the top of the band respectively. The VanHove singularity at 2.4 results from the saddle points of theband. The red line shows the DOS with non-zero magneticfield B 6= 0 and thus a vortex lattice appears. Landau levelsshow at the top of the spectrum. . . . . . . . . . . . . . . . . 48A.2 Band structure of s-wave superconductor with no vortices.The magnetic unit cell is chosen to be 10δ × 10δ, with δ be-ing the lattice constant of the underlying crystalline lattice.The superconductor order parameter is ∆0 = 1 with chemicalpotential εF = −2.2. . . . . . . . . . . . . . . . . . . . . . . . 49ixList of FiguresA.3 Band structure of s-wave superconductor with square vortexlattice. The magnetic unit cell is chosen to be 10δ×10δ, with δbeing the lattice constant of the underlying crystalline lattice.The superconductor order parameter is ∆0 = 1 with chemicalpotential εF = −2.2. . . . . . . . . . . . . . . . . . . . . . . . 50xAcknowledgementsThe author extraordinarily acknowledges Prof. Marcel Franz, who is a con-siderate supervisor offering the cutting-edge project on topological supercon-ductors and timely help throughout the project. His rigorous scholarship,outstanding professionalism, and modest character are really impressive andpromote the success of the project. The author is indebted to Dr. Ching-KaiChiu for insightful discussions and help at the beginning of the project. Theauthor is also grateful to Dr. Dominic Marchand for his guidance on com-putation facilities. At last, the author thanks to Anffany Chen, Xiaoyu Zhu,Dr. Dmitry Pikulin, and Dr. Armin Rahmani for their inspiring questionsand support.xiTo My FamilyxiiChapter 1IntroductionOur central concern in this thesis is the Majorana bands existing as quasi-particle excitations in p+ip superconductor and the Fu-Kane model. Inthe first section of this chapter, we will review some critical literatures oncreating Majorana fermions theoretically and experimentally. In the secondsection, we will state our motivation and summarize our main results.1.1 Majorana fermion1.1.1 BackgroundIn 1928, Dirac built his relativistic wave equation [5] for spin-1/2 particlespredicting the existence of antimatter, which was confirmed later by thediscovery of positrons [6]. In 1937, Ettore Majorana separated Dirac equa-tion into a pair of real wave equations [7] named after him. The Majoranaequation is satisfied by fermionic particle named Majorana fermion, which istheoretically defined as its own anti-particle. Thus, Majorana fermions are ina sense "real-valued" particles, while Dirac fermions are "complex-valued"particles. Mathematically, a Majorana fermion is understood as the real(imaginary) half of a Dirac fermion. And in return, a Dirac fermion can al-ways be written as a superposition of two Majorana fermions. When the twoMajorana fermions are spatially separated, their state is robust against localperturbations. However, the adiabatic exchange of two Majorana fermionsdoes change the state up to a phase other than 0 or pi by their non-Abeliananyonic nature[8], and leads to the idea of topological quantum computation[9–11] resorting to braiding of Majorana fermions. Also, with no intrinsicelectric and magnetic moments [12, 13], a Majorana fermion has little re-sponse to electromagnetic field, making it a potential candidate for darkmatter [14, 15].11.1. Majorana fermion1.1.2 Theoretical proposalsTo realize the appealing properties of Majorana fermions, we shall figureout a way to create Majorana fermions first. Majorana fermions are neverreally observed as elementary particles in high energy physics. In condensedmatter, however, many theoretical models have been developed to realize Ma-jorana fermions as quasi-particle excitations, which are naturally expectedto appear in superconductors where particle-hole symmetry is automaticallysatisfied.For spin triplet superconductor, the pioneering work was done by Kitaev[16] with a toy model for 1D fermionic chain.H =N∑j=1[−t(c†jcj+1 + c†j+1cj)− µ(c†jcj −12) + (∆c†j+1c†j + ∆∗cjcj+1)] (1.1)Kitaev’s model considers nearest neighbor hoppings and p-wave pairing forN fermionic particles and can be rewritten with 2N Majorana operators{γ1, . . . , γ2N}. In a special case when hopping and pairing are identical andchemical potential is zero, there are two operators, γ1 and γ2N , not enteringinto the Hamiltonian. Therefore, Kitaev chain is a realization of spatiallyseparated Majorana fermion pair, which is promising to be used as a qubit.Majorana fermions can also be realized in 2D p+ip superconductor (andsuperfluid) with vortices [17–19] or with electric defects [20].Figure 1.1: Kitaev’s toy model. Upper panel: trivial phase with ∆ = t = 0and µ 6= 0. Majorana fermions are paired to form a regular Dirac fermionchain. Lower panel: topological phase with ∆ = t and µ = 0. Majoranafermions are paired in a pattern that two unpaired Majorana fermions areleft on the ends of the chain.For spin singlet superconductor, the breakthrough in searching Majoranafermions was made by Fu and Kane [21]. They patterned an s-wave super-conductor to 3D strong topological insulator surface (the Fu-Kane model)such that both spin singlet pairing and strong spin orbital coupling can existon the interface. The former will be deformed by the latter giving a p+ip21.1. Majorana fermiontype low-energy spectrum potentially holding Majorana fermions. Generally,a Zeeman splitting is expected to break time reversal symmetry so that theKramers’ degeneracy is relieved and the low-energy physics is recovered tobe "spinless". The Zeeman splitting can enter the system by doping mag-net in the Sc-TI interface [22], by patterning a magnetic insulator [9], or byexternal field [23]. After Fu and Kane’s groundbreaking development, manysimilar proposals were reported to realize Majorana fermions with s-wavesuperconductor [24–29]. It is worth noting that topological insulator is nota must in the Fu-Kane model. What really matters is the strong spin orbitalcoupling [9, 23, 30].1.1.3 Experimental realizationAs theoretical models come out prosperously, experimentalists have been at-tempting to confirm the correctness of the theories. The Fu-Kane modelargues that we no longer need a p-wave superconductor, which is generallyhard to fabricate. And further tests show that topological insulator is notnecessary either, provided that there is strong spin orbital coupling. There-fore, it is feasible to assemble topological superconductors manually. Mouriket al. [31] put InSb semiconductor nanowire, which is proved to have strongspin-orbital coupling, in proximity to NbTiN s-wave superconductor, andfound zero-bias peaks possibly to be Majorana fermions due to the fact thatthese zero-bias peaks disappear when any necessary ingredient of Majoranaproposal is taken out and is robust when varying chemical potential and ap-plied field in a large range. This is a nice check to models [28, 29] based onsemiconductor. However, it is recently pointed out that some other trivialbound states [32] can highly mimic those zero-bias peaks.Three similar works on semiconductor-superconductor structure weredone in the same year with InSb/Nb [33, 34] and InAs/Al [35], respec-tively. The original Fu-Kane model was also realized later by consideringPb/Bi2Se3/Pb, which showed striking departure from the common Joseph-son junction behaviors. Such deviation can be easily explained by assuminga 1D Majorana chain along the width of the junction [36].In late 2014, Nadj-Perge et al. [37] proposed a slightly different schemeby fabricating ferromagnetic iron atomic chains on the surface of supercon-ducting lead. The major difference to semiconductor-superconductor modelis that the strong spin orbital coupling is provided by the conventional su-perconductor instead of semiconductor nanowire. Low temperature scanningtunneling microscopy (STM) observation of the Fe/Pb system showed sim-ilar zero-bias peaks and zero-energy end states, which should be a strong31.2. Motivationevidence for the formation of topologically non-trivial phase and Majoranamodes in the iron chains.1.2 MotivationIn the preceding section, we have grasped basics on Majorana fermions in-cluding its origin, application, and realization. It is worth noting that Ma-jorana fermions in those condensed matter systems are quasi-particles sat-isfying non-Abelian statistics instead of Fermi-Dirac statistics. Therefore,it is more rigorous to call the corresponding excitations as Majorana bandsor Majorana zero modes. Although many proposals have been brought upto realize Majorana bands, the dispersion of these bands has not yet beenstudied so far. When vortex density is large, the inter-vortex tunnelings areinevitable. Therefore, the Majorana bands have to be dispersing. What thedispersion looks like is the central question worthy of further study sinceunderstanding and characterizing the electronic structure will be a prereq-uisite of manipulating these Majorana modes for topological quantum com-putation. Our motivation in this thesis will be to systematically study thedispersion of Majorana bands.With this motivation in mind, we have, both numerically and analyti-cally, studied two specific models – the p+ip superconductor and the Fu-Kane model – in the presence of vortex lattice. By a singular gauge trans-formation, we avoided the difficulty in figuring out the phase field φ(r) ofsuperconductor order parameter. We found dispersing Majorana bands forboth models, which is contrary to the flat band prediction [38] in whichmagnetic field is simply neglected. We further found that a tight bindingmodel of vortex lattice is capable to capture the dispersion of Majoranaband provided that only first and second nearest neighbor hoppings are con-sidered and matches highly with numerical results. Therefore, the hoppingscan be extracted from numerical diagonalization by examining gaps at highsymmetry points. These hoppings show oscillating behavior, with respect tochemical potential, whose period is well predicted by Cheng et al. [3, 4] butwith different amplitude. Specifically, for the Fu-Kane model, both hoppingsare zero when chemical potential εF is tuned to coincide with Dirac point,leaving a completely flat Majorana band. We numerically tested the flatMajorana band under different vortex configuration and showed that suchflat band possesses robustness, which is protected by chiral symmetry [39].4Chapter 2The p+ip superconductorIn this chapter, we will reproduce some key results for the 2D spinless p+ipsuperconductor. We shall see that such chiral superconductor [40] can bedescribed by a simple BCS type lattice Hamiltonian. A unitary transforma-tion can move the phase of superconductor order parameter to the crystallineblocks, whose total phase then can be rewritten as a linear integral of thesuperfluid velocity along the crystalline lattice. The superfluid velocity isexpressed as an infinite integral with integrand’s decay following the k−1rule. Thus, the integral can be cut off softly and well approximated by asummation, making numerical diagonalization feasible. Most results in thechapter have been developed by Vafek et al. [1] and Franz and Tešanović[41]. However, it is worth reproducing them for the completeness and coher-ence of the thesis. The techniques in this chapter will be used again in thefollowing chapters.2.1 Hamiltonian of p+ip superconductorThe 2D spinless p+ip superconductor can be described by the following BCStype [42] lattice Hamiltonian in the basis of Φr = (cr, c†r)T with r being thecoordinates of lattice sites of the underlying crystalline lattice.H =(hˆ ∆ˆ∆ˆ∗ −hˆ∗)(2.1)where the Hamiltonian for the crystalline lattice ishˆ = −τ∑δe−i(e/h¯c)∫ r+δr A(r)·dlsˆδ − εF (2.2)and superconductor order parameter [43] is∆ˆ = ∆0∑δeiφ(r)/2ηˆδeiφ(r)/2 (2.3)52.1. Hamiltonian of p+ip superconductorFrom now on, we will set the hopping τ on crystalline lattice to be unityand measure all energies in this p+ip superconductor in unit of τ . Specifi-cally, for p+ip superconductor, the operator ηˆδ is defined asηˆδ ={∓isˆδ if δ = ±xˆ±sˆδ if δ = ±yˆ(2.4)where the operator sˆδ works as shift operator sˆδu(r) = u(r+δ). Applying aproper magnetic field, a square vortex lattice {ri} is formed with two sets ofsublattices {rAi } and {rBi }. Square vortex lattice [44, 45] has been confirmedin Sr2RuO4, which shows p-wave pairing [46] similar to 3He-A [47–49]. Thesuperconductor order parameter phase field φ(r) is constrained by topologyto wind a phase of 2pi around each vortex. Explicitly,∇×∇φ(r) = 2pizˆ∑iδ(r − ri) (2.5)The summation is over both types of sublattices. As we have two sets ofsublattices {rAi } and {rBi }, we can separate the phase into two parts.∇×∇φµ(r) = 2pizˆ∑iδ(r − rµi ) µ = A,B (2.6)where the summation is now over a certain type of sublattice.Figure 2.1: Two vortex sublattices. Each magnetic unit cell (orange square)encloses two different vortices on the diagonal with vortex spacing to be halfof the diagonal length.With the definition of the two phase fields φA(r) and φB(r), we can62.2. Superfluid velocitydefine a unitary matrixU =(eiφA(r) 00 e−iφB(r))(2.7)and a singular gauge transformation H → U−1HU under which the eigenval-ues of the Hamiltonian remain same. Such transformation may help removethe phase of ∆(r) and brings mathematical simplicity, because of the factthat φA(r)+φB(r) = φ(r). After the transformation, the Hamiltonian reads,U−1HU =−∑δeiVAδ (r)sˆδ − εF ∆0∑δeiAδ(r)ηˆδ∆0∑δeiAδ(r)ηˆ∗δ∑δe−iVBδ (r)sˆδ + εF (2.8)with the phase factors defined asVµδ (r) =∫ r+δr(∇φµ −eh¯cA)· dl µ = A,B (2.9)andAδ(r) =12∫ r+δr(∇φA −∇φB) · dl =12[VAδ (r)− VBδ (r)] (2.10)2.2 Superfluid velocityTo calculate the phase factors, we may rewrite the phase factor Vµδ in termsof superfluid velocity vµs .Vµδ (r) =mh¯∫ r+δrvµs (r) · dl (2.11)wherevµs (r) =h¯m(∇φµ −eh¯cA)(2.12)Without knowing φµ and A(r), we are not able to tell the value of vµs (r).Therefore, we will need to develop a more practical form of superfluid veloc-ity. We first take the curl of the superfluid velocity∇× vµs (r) =emc[zˆφ0∑iδ(r − rµi )−B](2.13)72.2. Superfluid velocitywhere the flux quantum φ0 = hc/e. We combine two velocities together tosuppress index µ,2B +2mce∇× vs = zˆφ0∑iδ(r − ri) (2.14)The Maxwell equation−∇2B = ∇× (∇×B) = ∇×4piJc= ∇×4pinsecvs (2.15)Plugging this back to the curl of vs, we get the conventional London equation,B − λ2∇2B =12φ0zˆ∑iδ(r − ri) (2.16)where London penetration depth λ2 = mc24pinse2. Perform Fourier transformB(r) =∫d2k(2pi)2eik·rBk (2.17)to the London equation , we would getBk + λ2k2Bk =12zˆφ0∑ie−ik·ri (2.18)and equivalently,Bk =12 zˆφ0∑ie−ik·ri1 + λ2k2(2.19)Perform Fourier transform to the curl of vµs and plug in Bkik × vµsk(r) =2pih¯m[zˆ∑ie−ik·rµi −12zˆ∑ie−ik·ri1 + λ2k2](2.20)We then perform a cross productik × (ik × vµsk) =2pih¯mik × zˆ[λ2k21 + λ2k2∑ie−ik·rµi+12(1 + λ2k2)(∑ie−ik·rµi −∑ie−ik·rνi)](2.21)82.2. Superfluid velocityThe second term can be safely dropped. We further apply transverse gaugek · vµsk = 0, so thatvµsk =2pih¯λ2mik × zˆ1 + λ2k2∑ie−ik·rµi (2.22)Then the superfluid velocity in real spacevµs (r) =2pih¯m∫d2k(2pi)2ik × zˆk2∑ieik·(r−rµi ) (2.23)where we assume λ→∞. As rµi represents the position of the type µ vortexof the i-th magnetic unit cell, it can be rewrite as rµi = Ri + δµ, with Ribeing the center of the i-th magnetic unit cell and δµ being the position oftype µ vortex with respect to the center. By this separation of index i andindex µ, we can first conduct the summation and getvµs (r) =2pih¯mN∑G∫d2k(2pi)2ik × zˆk2eik·(r−δµ)δk,G (2.24)where N is the number of magnetic unit cells and we use the relation∑ie−ik·Ri =∑Gδk,G (2.25)with G being the reciprocal lattice vector of lattice {Ri}. For an L × Lmagnetic cell, where G = 2piL (nx, ny), we have∫d2k(2pi)2→1L2N∑kThis will help us further simplify the superfluid velocity tovµs (r) =2pih¯m1L2∑GiG× zˆG2eiG·(r−δµ) (2.26)And finally, the phase factor will beVµδ (r) =2piL2∑G∫ r+δreiG·(r−δµ) iG× zˆG2· dl (2.27)Numerically, we cannot sum over all possible reciprocal vector G. The sum-mand decays as G−1 so that we may perform a soft cut-off to the summation.92.3. Majorana band and spectrumFor a lattice problem, by plugging in e±ik·r, we may write the Bloch Hamil-tonian Hk = e−ik·rUHU−1eik·r .Hk =−∑δeiVAδ (r)eik·δ sˆδ − εF ∆0∑δeiAδ(r)eik·δηˆδ∆0∑δeiAδ(r)eik·δηˆ∗δ∑δe−iVBδ (r)eik·δ sˆδ + εF (2.28)in which we use the fact thate−ik·rSˆδeik·rf(r) = eik·δf(r + δ) = eik·δSˆδf(r) (2.29)Here Sˆδ is a general shift operator that can be chosen among sˆδ, ηˆδ, andηˆ∗δ. And f(r) is an arbitrary function defined on the crystalline lattice.We write the Schrödinger equation for such Bloch Hamiltonian HkΦnk =εnkΦnk with Φnk = [unk(rj), vnk(rj)]T , where rj is the coordinates of the j-th crystalline site. Here we will choose a magnetic unit cell contains 100 siteswith 10δ×10δ geometry and periodic boundary conditions in both directions.For j = 1, 2, . . . , 100, when we plug Φnk(rj) in Schrödinger equation, we willhave 200 equations, which in return can be written as a 200× 200 matrix ofHamiltonian. Such Hamiltonian is our final version that can be numericallydiagonalized directly.2.3 Majorana band and spectrumNumeric diagonalization of the Bloch Hamiltonian will show us DOS andspectra. It can be seen that in zero magnetic field, which means no vorticeswill appear, the spectrum is fully gapped. However, a pair of slightly dispers-ing midgap states appear provided that a square vortex lattice is presentedby some nonzero magnetic field. We denote the states as ψ± that satisfySchrödinger equation Hψ± = ±Eψ±. Due to the particle-hole symmetry,we have ψ− = σxψ∗+. By linear combination, we can separate two linearlyindependent statesψ1 =12(1− i)ψ+ +12(1 + i)ψ− (2.30)ψ2 =12(1 + i)ψ+ +12(1− i)ψ− (2.31)with particle-hole symmetry ψ1,2 = σxψ∗1,2, which means the particle in suchstate is equal to its own antiparticle. Hence, the new states characterizeMajorana fermions. Such midgap states hold non-Abelian statistics [50].102.3. Majorana band and spectrumFigure 2.2: Magnetic unit cell with 10δ × 10δ geometry. The order j ofcrystalline site is assigned row by row so that rj is explicitly defined. Thevortices are located in the center of corresponding plaquettes on the diagonalof the unit cell. The spectrum does not rely on the specific positions of thevortices as long as they are on the diagonal with spacing to be (5δ, 5δ).112.3. Majorana band and spectrum0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 4 . 5 5 . 0 5 . 5 6 . 0 6 . 50 . 0 00 . 0 50 . 1 00 . 1 50 . 2 00 . 2 5   DOS E n e r g yFigure 2.3: Density of states (DOS) for p+ip superconductor. The magneticunit cell is chosen to be 10δ × 10δ, with δ being the lattice constant ofthe underlying crystalline lattice. The superconductor order parameter is∆0 = 0.5034 and the chemical potential is εF = −2.2. The blue line is theDOS of magnetic field B = 0 so that no vortices will appear. The suddendrops at 1.8 and 6.2 result from band edges corresponding to the bottom andthe top of the band respectively. The Van Hove singularity at 2.2 results fromthe saddle points of the band. The red line shows the DOS of magnetic fieldB 6= 0 and thus a vortex lattice appears. Landau levels show at the top ofthe spectrum.Figure 2.4: Magnetic Brillouin zone for square vortex lattice with the no-tation of high-symmetry points used in plotting band structure.122.3. Majorana band and spectrum0123456Γ  EnergyΓ M MX YFigure 2.5: Band structure of p+ip superconductor with no vortices. Themagnetic unit cell is chosen to be 10δ×10δ, with δ being the lattice constantof the underlying crystalline lattice. The superconductor order parameter is∆0 = 0.5034 with chemical potential εF = −2.2.132.3. Majorana band and spectrum0123456Γ  Energy Γ M MX YFigure 2.6: Band structure of p+ip superconductor with square vortexlattice. The magnetic unit cell is chosen to be 10δ × 10δ, with δ being thelattice constant of the underlying crystalline lattice. The superconductororder parameter is ∆0 = 0.5034 with chemical potential εF = −2.2.142.3. Majorana band and spectrum- 0 . 8- 0 . 6- 0 . 4- 0 . 20 . 00 . 20 . 40 . 60 . 8  ΓEnergy Γ M MX YFigure 2.7: Low-energy bands of p+ip superconductor in 10δ×10δ magneticunit cell with ∆0 = 0.5034 and εF = −2.2. The red lines show the Majoranabands with slight dispersion.15Chapter 3Majorana latticeAs Majorana fermions are trapped in the cores of vortices, the vortex latticewill present us a lattice of Majorana fermions. Thus a tight binding modelcan be used here to find Majorana band analytically. In this chapter, we willfirst develop the tight binding Hamiltonian by considering first and secondnearest neighbor hoppings. The hoppings are the origins of the dispersingMajorana bands and open gaps at high-symmetry points. Therefore, we canextract both hoppings from these points to plot the analytical dispersion,which is actually a good match with the numerical dispersion. We foundthat the hoppings oscillate with chemical potential, which is consistent withanalytical prediction [3, 4] by Cheng et al.. However, the dispersion devel-oped by Cheng et al. can only capture the period of the oscillation. Thismay be because Cheng et al. obtained the dispersion from continuous modelof p+ip superconductor. It seems that a linearly varying amplitude is goodenough to fit our numerical data.3.1 Tight binding HamiltonianWe first consider a generic case about vortex lattice with each vortex carryinga flux quantum Φ0 = pih¯c/e and a Majorana fermion in the core. TheHamiltonian, due to its own hermicity, can be written asH = it∑i,jsijγiγjwith sij = −sji = ±1. The elements of s matrix can only be determinedup to a sign, because a local Z2 transformation γi → −γi will not affectthe Majorana commutation relation. Although sij is gauge dependent, theproduct of sij along a closed loop is not. It has been proved [51] that forany lattice whose plaquette is a polygon of n vortices, the flux enclosed inthe plaquette is∑polygonφij =pi2(n− 2)163.1. Tight binding Hamiltonianindicating the product of hopping terms around the plaquette to be tn exp[i(n− 2)pi/2] = −intn. Therefore, the product of the sij around the plaquette is-1. Specifically, in our vortex lattice (Figure 2.1), a possible gauge is shownin Figure 3.1. The dispersion can then be found analytically by diagonalizingthe following tight binding HamiltonianH = Hfn +Hsn,A +Hsn,B (3.1)with the first nearest neighbor hopping Hfn and second nearest neighborhopping Hsn,A(B) to beHfn = it∑RγR,A(γR,B − γR−xˆ−yˆ,B + γR−xˆ,B + γR−yˆ,B) (3.2)Hsn,A = it′∑RγR,A(−γR+xˆ,A + γR+yˆ,A) (3.3)Hsn,B = it′∑RγR,B(γR+xˆ,B − γR+yˆ,B) (3.4)with t and t′ to be the hoppings. Perform Fourier transform,Figure 3.1: Gauge of tight binding model of square vortex lattice. Eachtriangular plaquette has a phase factor of pi/2.173.1. Tight binding HamiltonianFigure 3.2: Diamond-shape reduced Brillouin zone (solid lines) in tight bind-ing model with Q = (pi/2,−pi/2).γR,A(B) =√2N∑kei(k+Q)·Rγk,A(B) Q = (pi2,−pi2) (3.5)The first nearest neighbor hopping Hamiltonian isHfn = 4t∑kei(kx+ky)/2[sin(kx + ky2)− i sin(kx − ky2)]γk,Aγ†k,B (3.6)The two second nearest neighbor hoppings areHsn,A = −2t′∑k(e−ikx + e−iky)γk,Aγ†k,A (3.7)Hsn,B = 2t′∑k(e−ikx + e−iky)γk,Bγ†k,B (3.8)By using Q, we can rewrite all three Hamiltonians in a diamond-shape re-duced Brillouin zone (Figure 3.2) such that there will be a single Dirac conecoinciding with the Γ point. Then the nearest neighbor hopping isHfn = −∑k∈RBZh∗kγ†k,Aγk,B −∑k∈RBZhkγ†k,Bγk,A (3.9)183.1. Tight binding Hamiltonianwith the parameterhk = 4tei(kx+ky)/2(sinkx + ky2− i sinkx − ky2)(3.10)And the second nearest neighbor hoppings areHsn,A = 2t′∑k∈RBZ(e−ikx + e−iky)−∑k∈RBZh′kγ†k,Aγk,A (3.11)Hsn,B = −2t′∑k∈RBZ(e−ikx + e−iky) +∑k∈RBZh′kγ†k,Bγk,B (3.12)with the parameterh′k = −4t′(cos kx + cos ky) (3.13)Therefore, we can write the total tight binding Hamiltonian in the space ofΓk = (γk,A, γk,B)T asH = Hfn +Hsn,A +Hsn,B =∑k∈RBZΓ†kHkΓk (3.14)whereHk =(−h′k −h∗k−hk h′k)(3.15)Diagonalization gives the analytical solution of dispersion of Majorana band(k) = ±√|hk|2 + |h′k|2 (3.16)Therefore, at Dirac point Γ the Hamiltonian opens a gap ∆Γ = 8t′. At pointX, it opens a gap ∆X = 4√2t. We can rewrite the dispersion in terms of∆Γ and ∆X .(k) = ±√∆2X2(sin2kx + ky2+ sin2kx − ky2)+∆2Γ4(cos kx + cos ky)2(3.17)As long as we extract the gap parameters ∆Γ and ∆X from the numericaldiagonalization, we can plot the whole dispersion of Majorana bands. Fora 10δ × 10δ magnetic unit cell with superconductor order parameter ∆0 =0.5034, the match between the analytical results and numerical results isgood (Figure 3.3).193.2. Hoppings of p+ip superconductor- 0 . 0 3- 0 . 0 2- 0 . 0 10 . 0 00 . 0 10 . 0 20 . 0 3EnergyΓ X M Y Γ MFigure 3.3: Majorana bands of p+ip superconductor in 10δ × 10δ magneticunit cell with ∆0 = 0.5034 and εF = −2.2. The square dots are Majoranabands given by numerical diagonalization, while the red lines are (k) givenby tight binding model (Equation 3.17) with gaps ∆Γ = 0.00155 and ∆X =0.02434 extracted from numerical diagonalization.3.2 Hoppings of p+ip superconductorIn the section, we are going to study the hoppings in p+ip superconduc-tor. For simplicity, we will set h¯ = 1 and crystalline lattice constant δ = 1.The hoppings in p+ip superconductor will show exponentially decaying be-havior in the regime ∆20 > 2mεF v2F and oscillating behavior in the regime∆20 < 2mεF v2F . The chemical potential εF is counted from the bottomof the band and therefore is always positive. We are more interested inthe latter regime with large vortex spacing R  max(k−1, ξ), where thesuperconducting coherence length is ξ = vF /∆0 and the momentum is203.2. Hoppings of p+ip superconductork2 = 2mε2F −∆20/v2F . Analytically, the energy splitting is predicted [3, 4] asE =√2pi∆0| cos (kFR+ pi4 )|√kFRexp(−Rξ)(3.18)if we temporarily ignore the sign. Such energy splitting is obtained by eval-uating an overlap integral associated with zero modes at two vortices [3, 4],which are separated by a distance of R. In the language of second quanti-zation, this energy splitting E is identical to the hopping t(t′) in the tightbinding model in the preceding section 3.1, when first(second) nearest neigh-bor spacing is substituted. By plotting this hopping, we will be able to tellhow well such hopping, which is calculated from continuous p+ip model, fitsour numerical data. But the first task will be figuring out all correspondingparameters. The dispersion for the underlying crystalline lattice that we areusing is = −2(cos kx + cos ky) (3.19)with the energy scaled in unit of τ and momentum scaled in unit of δ−1.The first nearest neighbor hopping happens along the diagonal in magneticunit cell between different sublattices. Therefore, we takekx = ky =kF√2(3.20)when calculating chemical potential εF . We may write the dispersion asεF = −2[2− 4 sin2( kF2√2)](3.21)Equivalently,kF = 2√2 arcsin√εF + 48(3.22)The Fermi velocity isvF =∂εF∂kF= 4√2 sinkF2√2coskF2√2=√16− ε2F2(3.23)By plugging in all parameters, we can plotE(εF ) =√2pi∆0∣∣∣ cos(2√2 arcsin√εF+48 R+pi4)∣∣∣√2√2 arcsin√εF+48 Rexp(−√2R∆0√16− ε2F)(3.24)213.2. Hoppings of p+ip superconductorwith the parameter ∆0 = 0.1 and R = 25√2. It looks that such analyticalequation can capture the period of our numerical data when εF is not sosmall, while at the bottom of the band the fitting is not very good. Thesecond problem is that E cannot fit the amplitude of our numerical data(Figure 3.4, blue line). To improve the fitting at low energy, we can use thereal momentum k, but in a more practical form,k =√k2F −∆20v2F(3.25)At low energy, this approximated expression of momentum will fix the fitof period. At high energy, this expression reduces to the Fermi momentumkF , which is proved to fit the period very well. Phenomenologically, byusing such momentum, a linearly varying amplitude A + BkR combinedwith cos (kR+ pi/4) is good enough to fit our numerical data. The explicitrelation ist(εF ) =(A+B√8R2 arcsin2√εF + 48−2R2∆2016− ε2F)× cos(√8R2 arcsin2√εF + 48−2R2∆2016− ε2F+pi4)(3.26)with the parameter ∆0 = 0.1 and R = 25√2. Such function will fit bothperiod and amplitude very well for all chemical potential εF > −3.95,where oscillating behavior is expected (Figure 3.4, red line). For the regimeεF < −3.95, the exponentially decaying behavior is expected other thanoscillation.223.2. Hoppings of p+ip superconductor- 4 - 3 - 2 - 1 00 . 0 0 00 . 0 0 10 . 0 0 20 . 0 0 30 . 0 0 4t  εF  Figure 3.4: First nearest neighbor hopping t in p+ip superconductor with50δ × 50δ magnetic unit cell and ∆0 = 0.1. Black dots show how t varieswith chemical potential εF numerically. The red line (Equation 3.26) is thephenomenological fit with the oscillation characterized by cos (kR+ pi/4) asis suggested by Cheng et al. [3, 4]. The linearly varying amplitude is A+BkRwith parameters A = 2.438× 10−4, B = 3.981× 10−6, and R = 25√2. Theblue line (Equation 3.24) is analytical result given by Cheng et al. [3, 4].For second nearest neighbor hopping, the hopping is along either x or y,giving a slightly different dispersionεF = −2(1 + cos kF ) = −4 + 4 sin2(kF2)(3.27)And the Fermi momentum is changed tokF = 2 arcsin√εF + 44(3.28)The Fermi velocity is then given asvF =∂εF∂kF= 2 sin kF =√−ε2F − 4εF (3.29)233.2. Hoppings of p+ip superconductorBy plugging in all parameters, we can plotE(εF ) =√2pi∆0∣∣∣ cos(2 arcsin√εF+44 R′ + pi4)∣∣∣√2 arcsin√εF+44 R′exp(−R′∆0√−ε2F − 4εF)(3.30)where superconducting order parameter ∆0 = 0.1 and the spacing betweensecond nearest neighbors R′ = 50. Still, this function cannot capture theamplitude (Figure 3.5, blue line). Phenomenologically, by using the realmomentum k, the second nearest neighbor hopping can be written as aproduct of a linear function and an oscillating functiont′(εF ) =(A+B√4R′2 arcsin2√εF + 44−R′2∆20−ε2F − 4εF)× cos(√4R′2 arcsin2√εF + 44−R′2∆20−ε2F − 4εF+pi4)(3.31)Such phenomenological fit is good (Figure 3.5, red line) for εF < −1. Whenwe keep going up to the corner of the reduced Brillouin zone, the numericaldata becomes slightly disordered. And the fit shows a little deviation.243.2. Hoppings of p+ip superconductor- 4 - 3 - 2 - 1 00 . 0 0 0 0 00 . 0 0 0 2 50 . 0 0 0 5 00 . 0 0 0 7 50 . 0 0 1 0 0t '  εF  Figure 3.5: Second nearest neighbor hopping t′ in p+ip superconductor with50δ × 50δ magnetic unit cell and ∆0 = 0.1. Black dots show how t′ varieswith chemical potential εF numerically. The red line (Equation 3.31) is thephenomenological fit with the oscillation characterized by cos (kR′ + pi/4)as is suggested by Cheng et al. [3, 4]. The linearly varying amplitude isA+BkR′ with parameters A = 5.91×10−5, B = 3.904×10−8, and R′ = 50.The blue line (Equation 3.30) is analytical result given by Cheng et al. [3, 4].25Chapter 4The Fu-Kane modelThe Fu-Kane model [21] may be realized by patterning a topologically trivials-wave superconductor (see Appendix) to a surface of a strong topological in-sulator. The Cooper pairs can tunnel to the interface of superconductor andtopological insulator, which will modify the surface Hamiltonian to one thatresembles the p+ip superconductor Hamiltonian. Thus, Majorana fermionsare possible to appear. In this chapter, we will start from a simplified Hamil-tonian [2] with all bulk degrees of freedom being integrated out. Physically,such a Hamiltonian represents two parallel surfaces of the topological insula-tor coupled by some interaction. It is proven that such Hamiltonian capturesthe low-energy physics of a realistic strong topological insulator. By addingin an s-wave superconducting term and a mass term, one builds a latticeHamiltonian realizing Fu-Kane model. Diagonalization of such Hamiltonianin the same vortex geometry (Figure 2.1) shows midgap states indicatingMajorana fermions. For non-zero chemical potential, such Majorana bandsshow dispersion that is consistent with what we expect from the tight bindingMajorana lattice. For zero chemical potential, the Majorana bands are flatto high numerical accuracy. The flat bands are protected by chiral symmetry[39] and therefore are robust when bringing two vortices closer.4.1 The Fu-Kane HamiltonianThe simplified Hamiltonian [2] for 3D strong topological insulator isHk,T I =(hk M¯kM¯k −hk)(4.1)with blocks defined ashk = 2λ(sy sin kx − sx sin ky) (4.2)M¯k = 2t¯(2− cos kx − cos ky) (4.3)The Hamiltonian is written in basis Ψk = (ck,1,↑, ck,1,↓, ck,2,↑, ck,2,↓)T . Heresx and sy are Pauli matrices in spin space. Hk,T I considers the two parallel264.1. The Fu-Kane Hamiltoniansurfaces denoted by hk and −hk coupled by matrix M¯k. From now on, wewill take λ to be unity and measure all energies in terms of λ. The spectrumof Hk,T I is(k) = ±√4(sin2 kx + sin2 ky) + M¯2k (4.4)which is doubly degenerate because of the fact that each surface will have agapless Dirac cone.- 4- 2024   EnergyΓM MXFigure 4.1: Spectrum of the simplified topological insulator model with t¯ =0.5. The spectrum is doubly degenerate with a single Dirac point at Γ.An interesting modification is to open a gap at one of the surface byadding a mass termHk,Mag = m(c†k,1,↑ck,1,↑ − c†k,1,↓ck,1,↓) (4.5)such that the low-energy physics is completely determined by the ungappedsurface. Due to the additional mass term, the Hamiltonian is nowHk,T I +Hk,Mag =(hk +msz M¯kM¯k −hk)(4.6)274.1. The Fu-Kane HamiltonianThe non-degenerate spectrum is(k) = ±√4(sin2 kx + sin2 ky) + M¯2k +12m2 ±√14m4 +m2M¯2k (4.7)- 4- 2024   EnergyΓ X MMFigure 4.2: Spectrum of the simplified topological insulator model with t¯ =0.5. One of the surfaces is gapped by mass m = 0.2.We now pattern a slab of s-wave superconductor to the ungapped sur-face. The tunneling of Copper pair will introduce another term to the TIHamiltonianHk,Sc = ∆c†k,2,↑c†−k,2,↓ + ∆∗c−k,2,↓ck,2,↑ (4.8)This term will open a gap and is the last piece of the Fu-Kane model. Thefull Hamiltonian isHFK =12∑kΦ†kHk,FKΦk (4.9)284.1. The Fu-Kane HamiltonianwithHk,FK =hk +msz M¯k 0 0M¯k −hk − µ 0 i∆σy0 0 h∗k −msz −M¯k0 −i∆∗σy −M¯k −h∗k + µ (4.10)in the basis Φk = (ck,1,↑, ck,1,↓, ck,2,↑, ck,2,↓, c†−k,1,↑, c†−k,1,↓, c†−k,2,↑, c†−k,2,↓)T .- 4- 2024   EnergyΓ X MMFigure 4.3: Spectrum of the simplified topological insulator model with t¯ =0.5. The mass is m = 0.2, the superconductor order parameter is ∆ = 0.5,and the chemical potential is µ = 0.We would expect Majorana bands to appear provided that there is a non-zero magnetic field to create a set of vortex lattice (Figure 2.1). The vorticeswill break the translational symmetry of the underlying crystalline lattice.Thus, the momentum kx(y) is not a good quantum number, indicating thatwe will need to solve this in real space. Perform Fourier transform,ck,α =1√N∑re−ik·rcr,α (4.11)294.1. The Fu-Kane Hamiltonianwhere α specifies surface and spin. In an 8-component Nambu spinor Φr =(cr,1,↑, cr,1,↓, cr,2,↑, cr,2,↓, c†r,1,↑, c†r,1,↓, c†r,2,↑, c†r,2,↓)T , the Hamiltonian readsHFK =12∑rΦ†r(hˆr ∆ˆr∆ˆ†r −hˆ∗r)Φr (4.12)with the blockshˆr =m i∑δηˆ∗δ 4t¯− t¯∑δsˆδ 0i∑δηˆδ −m 0 4t¯− t¯∑δsˆδ4t¯− t¯∑δsˆδ 0 −εF −i∑δηˆ∗δ0 4t¯− t¯∑δsˆδ −i∑δηˆδ −εF(4.13)and∆ˆr =0 0 0 00 0 0 00 0 0 ∆0 0 −∆ 0 (4.14)where ∆ = ∆0eiφ. When a magnetic field is applied, a vortex lattice isformed, and the hoppings, which are characterized by shift operators, shouldbe modified by Peierls phase θr = eh¯c∫ r+δr A(r) · dl. The diagonal blockhˆr(θr) is nowm i∑δe−iθr ηˆ∗δ 4t¯− t¯∑δe−iθr sˆδ 0i∑δe−iθr ηˆδ −m 0 4t¯− t¯∑δe−iθr sˆδ4t¯− t¯∑δe−iθr sˆδ 0 −εF −i∑δe−iθr ηˆ∗δ0 4t¯− t¯∑δe−iθr sˆδ −i∑δe−iθr ηˆδ −εFWorking in same vortices geometry (Figure 2.1), the similar singular gaugetransformation will remove the phase of superconductor order parameter andmake the Hamiltonian easy to be diagonalized numerically. A natural choiceof such transformation may beU = diag(eiφA,1,↑ , eiφA,1,↓ , eiφA,2,↑ , eiφA,2,↓ ,e−iφB,1,↑ , e−iφB,1,↓ , e−iφB,2,↑ , e−iφB,2,↓) (4.15)304.1. The Fu-Kane HamiltonianAs for our topological insulator, only one surface is connected to supercon-ductor, φA(B),1,↑(↓) does not have real physical meaning, because for a zerosuperconductor order parameter its phase can be arbitrary. We further as-sume phase is independent on spin orientation. By setting φA,1(2),↑(↓) = φAand φB,1(2),↑(↓) = φB, where φA + φB = φ, we may get a more convenientunitary transformationU =(eiφA 00 e−iφB)(4.16)with each block being a 4×4 matrix. After the unitary gauge transformation,we further plug in e−ik·r and eik·r, arriving at an 8× 8 Bloch HamiltonianHk,FK = e−ik·rU−1(hˆr(θr) ∆ˆr∆ˆ†r −hˆ∗r(θr))Ueik·r (4.17)where the diagonal blocks H(1,1)k,FK and H(2,2)k,FK are defined asm i∑δeiVAδ eik·δ ηˆ∗δ 4t¯−t¯∑δeiVAδ eik·δ sˆδ 0i∑δeiVAδ eik·δ ηˆδ −m 0 4t¯−t¯∑δeiVAδ eik·δ sˆδ4t¯−t¯∑δeiVAδ eik·δ sˆδ 0 −εF −i∑δeiVAδ eik·δ ηˆ∗δ0 4t¯−t¯∑δeiVAδ eik·δ sˆδ −i∑δeiVAδ eik·δ ηˆδ −εFand−m i∑δe−iVBδ eik·δ ηˆδ t¯∑δe−iVBδ eik·δ sˆδ−4t¯ 0i∑δe−iVBδ eik·δ ηˆ∗δ m 0 t¯∑δe−iVBδ eik·δ sˆδ−4t¯t¯∑δe−iVBδ eik·δ sˆδ−4t¯ 0 εF −i∑δe−iVBδ eik·δ ηˆδ0 t¯∑δe−iVBδ eik·δ sˆδ−4t¯ −i∑δe−iVBδ eik·δ ηˆ∗δ εFrespectively. And off-diagonal blocks areH(1,2)k,FK = −H(2,1)k,FK =0 0 0 00 0 0 00 0 0 ∆00 0 −∆0 0 (4.18)This is our final version of Fu-Kane Hamiltonian that can be numericallydiagonalized.314.2. Majorana band and spectrum4.2 Majorana band and spectrumWe will focus on the regime δ0 < ∆0 < λ, where δ0 is the energy differencebetween the lowest two Landau levels. In the regime δ0 < ∆0, the low-energy bands are mainly determined by superconductor instead of magneticfield. On the other hand, for δ0 > ∆0, the magnetic field dominates. Andone will see flat Landau levels. Accordingly, δ0 is obtained by diagonalizingthe Bloch Hamiltonian Hk,FK with ∆0 = 0 and finding the gap between twolowest flat bands. As we are interested in the low-energy physics, we shallrestrict both δ0 and ∆0 within the bulk gap of topological insulator. Thebulk gap is roughly measured by energy unit λ (Figure 4.1). In this regime,the low-energy bands should resemble those of p+ip superconductor.By diagonalizing the Bloch Hamiltonian (Equation 4.17) with some non-zero chemical potential, one can confirm that Majorana bands (Figure 4.4, 4.5)have the same dispersion as the one in p+ip superconductor (Figure 3.3),which can be explained by first and second nearest neighbor hoppings inthe tight binding model. Specifically, when the chemical potential is tunedto coincide with Dirac point, namely, εF = 0, the Majorana bands becomeflat to high numeric accuracy (Figure 4.6), which is protected by chiral sym-metry. Therefore, such flat band is robust when one brings two vortices ina magnetic unit cell closer until they merge into a double vortex. It canbe shown numerically that the bandwidth of Majorana band does not havequalitative change in the merging process (Figure 4.6, 4.7, 4.8, 4.9).324.2. Majorana band and spectrum- 0 . 0 2 0- 0 . 0 1 5- 0 . 0 1 0- 0 . 0 0 50 . 0 0 00 . 0 0 50 . 0 1 00 . 0 1 50 . 0 2 0EnergyΓ X M Y Γ MFigure 4.4: Majorana bands in the Fu-Kane model in a 30δ × 30δ magneticunit cell with εF = 0.25, m = 0.5, ∆0 = 0.4, t¯ = 0.5, and δ0 = 0.23. Blackdots represent numerical data while red lines are (k) given by tight bindingmodel (Equation 3.17) with gaps ∆Γ = 2.76× 10−4 and ∆X = 1.336× 10−2extracted from numerical data.334.2. Majorana band and spectrum- 0 . 5- 0 . 4- 0 . 3- 0 . 2- 0 . 10 . 00 . 10 . 20 . 30 . 40 . 5MΓ  EnergyΓ MX YFigure 4.5: Low-energy bands in the Fu-Kane model in a 30δ×30δ magneticunit cell with εF = 0.25, m = 0.5, ∆0 = 0.4, t¯ = 0.5, and δ0 = 0.23. Thered lines are dispersing Majorana bands.344.2. Majorana band and spectrum- 0 . 5- 0 . 4- 0 . 3- 0 . 2- 0 . 10 . 00 . 10 . 20 . 30 . 40 . 5Γ  EnergyΓ M MX YFigure 4.6: Low-energy bands in the Fu-Kane model in a 30δ×30δ magneticunit cell with εF = 0, m = 0.5, ∆0 = 0.4, t¯ = 0.5, and δ0 = 0.23. Thespacing of two vortices in the magnetic unit cell is d = (15δ, 15δ). Thebandwidth of Majorana band (red line) is 5.7× 10−5.354.2. Majorana band and spectrum- 0 . 5- 0 . 4- 0 . 3- 0 . 2- 0 . 10 . 00 . 10 . 20 . 30 . 40 . 5   MΓEnergyΓ MX YFigure 4.7: Low-energy bands in the Fu-Kane model in a 30δ×30δ magneticunit cell with εF = 0, m = 0.5, ∆0 = 0.4, t¯ = 0.5, and δ0 = 0.23. Thegeometry is changed by bringing two vortices closer to d = (10δ, 10δ) alongthe diagonal. The bandwidth of Majorana band (red line) is 5.6× 10−5.364.2. Majorana band and spectrum- 0 . 5- 0 . 4- 0 . 3- 0 . 2- 0 . 10 . 00 . 10 . 20 . 30 . 40 . 5MΓ  EnergyΓ MX YFigure 4.8: Low-energy bands in the Fu-Kane model in a 30δ×30δ magneticunit cell with εF = 0, m = 0.5, ∆0 = 0.4, t¯ = 0.5, and δ0 = 0.23. Thegeometry is changed by bringing two vortices further closer to d = (5δ, 5δ)along the diagonal. The bandwidth of Majorana band (red line) is 4.9×10−5.374.2. Majorana band and spectrum- 0 . 5- 0 . 4- 0 . 3- 0 . 2- 0 . 10 . 00 . 10 . 20 . 30 . 40 . 5  ΓEnergyΓ MX Y MFigure 4.9: Low-energy bands in the Fu-Kane model in a 30δ×30δ magneticunit cell with εF = 0, m = 0.5, ∆0 = 0.4, t¯ = 0.5, and δ0 = 0.23. Thegeometry is changed by merging two vortices along the diagonal into a doublevortex. The bandwidth of Majorana band (red line) is 4.9× 10−5.384.3. Hoppings in Fu-Kane model4.3 Hoppings in Fu-Kane modelAs working in the same vortex geometry (Figure 2.1), the analytical dis-persion (Equation 3.17) of Majorana band will still be unchanged. Thefirst nearest neighbor hopping is t = ∆X/4√2 and second nearest neighborhopping is t′ = ∆Γ/8. It is easy to test that such hoppings still hold os-cillating behaviors that can be characterized by simple sine functions. Phe-nomenologically, we will use an exponentially decaying amplitude, whichexplicitly is e−B|k(εF )|, with momentum k = εF + Dε3F . The s-wave super-conductor is patterned to the surface characterized by surface Hamiltonianhk = −2(sy sin kx − sx sin ky), which only depends on sine functions. Thus,only odd powers should enter k(εF ). Therefore, the explicit fitting equationfor our hopping will be = |Ae−B(|εF |+D|εF |3) sin[R(|εF |+D|εF |3)]| (4.19)with  representing either t or t′.394.3. Hoppings in Fu-Kane model- 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 00 . 0 0 0 00 . 0 0 0 50 . 0 0 1 00 . 0 0 1 50 . 0 0 2 00 . 0 0 2 50 . 0 0 3 0  tεFFigure 4.10: First nearest neighbor hopping t of the Fu-Kane model in a30δ×30δ magnetic unit cell with m = 0.5, ∆0 = 0.4, t¯ = 0.5, and δ0 = 0.23.Black dots are numerical data and red line is phenomenological fit. Fittingparameters are A = 7.85× 10−3, B = 1.79, D = 0.341, and R = 7.87.404.3. Hoppings in Fu-Kane model- 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 00 . 0 0 0 00 . 0 0 0 10 . 0 0 0 20 . 0 0 0 30 . 0 0 0 40 . 0 0 0 5  t 'εFFigure 4.11: Second nearest neighbor hopping t′ of the Fu-Kane model in a30δ×30δ magnetic unit cell with m = 0.5, ∆0 = 0.4, t¯ = 0.5, and δ0 = 0.23.Black dots are numerical data and red line is phenomenological fit. Fittingparameters are A = 5.87× 10−4, B = 2.05, D = 0.26, and R = 12.06.41Chapter 5ConclusionIn this thesis, our research question is successfully accomplished. We sys-tematically studied the dispersion of Majorana bands in topological super-conductors. For both p+ip superconductor and the Fu-Kane model withvortices, we found the dispersion by numerical diagonalization of the BlochHamiltonian and by analytical diagonalization of the tight binding Hamil-tonian. The two methods showed good consistency. The dispersion impliesinter-vortex tunneling of Majorana fermions and is mainly contributed byfirst and second nearest neighbor hoppings of Majorana fermions. The hop-pings are not identical to the previous analytical prediction [3, 4]. Thus,we used simplified equations to phenomenologically show the trends of thehoppings.Although showing dispersing Majorana bands similar to p+ip supercon-ductor at nonzero chemical potentials, the Fu-Kane model distinguishes itselfby exhibiting flat Majorana bands at neutrality point where chemical poten-tial is zero, coinciding with Dirac point. We also found that this additionalfeature is robust under different vortex configuration due to the extra chiralsymmetry. This finding is consistent with previous theoretical prediction [39]on topological classification of the zero mode.Our work may contribute to a better understanding of electronic struc-tures of p+ip superconductor and the Fu-Kane model and should be benefi-cial to manipulating Majorana modes for topological quantum computation.Most analytical and numerical techniques in our work may be transplantedto study other "Fu-Kane model like" lattice systems, i.e., interface between a3D strong topological insulator and a d-wave superconductor. A further stepmay be to consider a realistic model for 3D strong topological insulator byadding bulk layer Hamiltonians and corresponding couplings to the simplifiedHamiltonian (Equation 4.1), which will allow us to study the bulk physicsof the topological insulator in proximity to a conventional superconductor.42Bibliography[1] O. Vafek, A. Melikyan, M. Franz, and Z. Tešanović. Phys. Rev. B,63:134509, 2001.[2] D. J. J. Marchand and M. Franz. Phys. Rev. B, 86:155146, 2012.[3] M. Cheng, R. M. Lutchyn, V. Galitski, and S. Das Sarma. Phys. Rev.Lett., 103:107001, 2009.[4] M. Cheng, R. M. Lutchyn, V. Galitski, and S. Das Sarma. Phys. Rev.B, 82:094504, 2010.[5] P. A. M. Dirac. In Proc. Roy. Soc. Lon. A: Math, volume 117, page610. The Royal Society, 1928.[6] C. D. Anderson. Phys. Rev., 43:491, 1933.[7] E. Majorana. Nuovo Cim., 14:50, 1937.[8] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma. Rev.Mod. Phys., 80:1083, 2008.[9] J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma. Phys. Rev.Lett., 104:040502, 2010.[10] J. Alicea, Y. Oreg, G. Refael, F. von Oppen, and M. P. A. Fisher. Nat.Phys., 7:412, 2011.[11] S. Tewari, S. Das Sarma, C. Nayak, C. Zhang, and P. Zoller. Phys. Rev.Lett., 98:010506, 2007.[12] B. Kayser and A. S. Goldhaber. Phys. Rev. D, 28:2341, 1983.[13] E. E. Radescu. Phys. Rev. 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Hashimoto, K. Yoshida, S. Nishizaki, T. Fujita, J. G.Bednorz, and F. Lichtenberg. Nature, 372:532, 1994.[48] T. M. Rice and M. Sigrist. J. Phys. Condens. Matter, 7:L643, 1995.[49] A. P. Mackenzie, S. R. Julian, A. J. Diver, G. J. McMullan, M. P. Ray,G. G. Lonzarich, Y. Maeno, S. Nishizaki, and T. Fujita. Phys. Rev.Lett., 76:3786, 1996.45[50] D. A. Ivanov. Phys. Rev. Lett., 86:268, 2001.[51] E. Grosfeld and A. Stern. Phys. Rev. B, 73:201303(R), 2006.46BibliographyAppendix AThe s-wave superconductorIn this appendix, we will show that s-wave superconductor is topologicallytrivial. Namely, there are no midgap states when vortex lattice is provided.The s-wave superconductor can also be described by the BCS type latticeHamiltonian (Equation 2.1) that we used for p+ip superconductor. The onlydifference is that the off-diagonal shift operator is now a constant ηˆδ = 14 .The lattice Bloch Hamiltonian is now,Hk,s =−t∑δeiVAδ (r)eik·δ sˆδ − εF ∆0∆0 t∑δe−iVBδ (r)eik·δ sˆδ + εF (A.1)The definitions of VAδ and VBδ keep unchanged (Equation 2.9). By usingthe same technique (Equation 2.27) to calculate these two phase factors,the Hamiltonian Hk,s can be diagonalized numerically so that both DOSand spectrum are able to be checked. We can see that the ground stateis lowered if the s-wave superconductor possesses a vortex lattice, which isreflected in DOS as several discrete peaks below the superconducting gap∆0 = 1. However, the spectrum is gapped regardless of the appearance ofvortices.47Appendix A. The s-wave superconductor0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 4 . 5 5 . 0 5 . 5 6 . 0 6 . 50 . 00 . 10 . 20 . 30 . 40 . 5  DOSE n e r g yFigure A.1: Density of states for an s-wave superconductor. The magneticunit cell is chosen to be 10δ × 10δ, with δ being the lattice constant ofthe underlying crystalline lattice. The superconductor order parameter is∆0 = 1 and the chemical potential is εF = −2.2. The blue line is the DOSwith magnetic field B = 0 so that no vortices will appear. The sudden dropsat 2.1 and 6.3 result from the edges of the band corresponding to the bottomand the top of the band respectively. The Van Hove singularity at 2.4 resultsfrom the saddle points of the band. The red line shows the DOS with non-zero magnetic field B 6= 0 and thus a vortex lattice appears. Landau levelsshow at the top of the spectrum.48Appendix A. The s-wave superconductor0123456  EnergyΓ ΓM MX YFigure A.2: Band structure of s-wave superconductor with no vortices. Themagnetic unit cell is chosen to be 10δ×10δ, with δ being the lattice constantof the underlying crystalline lattice. The superconductor order parameter is∆0 = 1 with chemical potential εF = −2.2.49Appendix A. The s-wave superconductor0123456

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