{"@context":{"@language":"en","Affiliation":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","AggregatedSourceRepository":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","Campus":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","Creator":"http:\/\/purl.org\/dc\/terms\/creator","DateAvailable":"http:\/\/purl.org\/dc\/terms\/issued","DateIssued":"http:\/\/purl.org\/dc\/terms\/issued","Degree":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","DegreeGrantor":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","Description":"http:\/\/purl.org\/dc\/terms\/description","DigitalResourceOriginalRecord":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","FullText":"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note","Genre":"http:\/\/www.europeana.eu\/schemas\/edm\/hasType","GraduationDate":"http:\/\/vivoweb.org\/ontology\/core#dateIssued","IsShownAt":"http:\/\/www.europeana.eu\/schemas\/edm\/isShownAt","Language":"http:\/\/purl.org\/dc\/terms\/language","Program":"https:\/\/open.library.ubc.ca\/terms#degreeDiscipline","Provider":"http:\/\/www.europeana.eu\/schemas\/edm\/provider","Publisher":"http:\/\/purl.org\/dc\/terms\/publisher","Rights":"http:\/\/purl.org\/dc\/terms\/rights","RightsURI":"https:\/\/open.library.ubc.ca\/terms#rightsURI","ScholarlyLevel":"https:\/\/open.library.ubc.ca\/terms#scholarLevel","Title":"http:\/\/purl.org\/dc\/terms\/title","Type":"http:\/\/purl.org\/dc\/terms\/type","URI":"https:\/\/open.library.ubc.ca\/terms#identifierURI","SortDate":"http:\/\/purl.org\/dc\/terms\/date"},"Affiliation":[{"@value":"Science, Faculty of","@language":"en"},{"@value":"Physics and Astronomy, Department of","@language":"en"}],"AggregatedSourceRepository":[{"@value":"DSpace","@language":"en"}],"Campus":[{"@value":"UBCV","@language":"en"}],"Creator":[{"@value":"Liu, Tianyu","@language":"en"}],"DateAvailable":[{"@value":"2015-06-24T18:57:51Z","@language":"en"}],"DateIssued":[{"@value":"2015","@language":"en"}],"Degree":[{"@value":"Master of Science - MSc","@language":"en"}],"DegreeGrantor":[{"@value":"University of British Columbia","@language":"en"}],"Description":[{"@value":"Majorana fermions can exist in condensed matter systems as quasi-particle excitations called Majorana bands. The details of Majorana bands will be the central concern of this thesis. In the thesis, Majorana bands are studied\nanalytically 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 superconducting gap. With the same vortex geometry, the Fu-Kane model shows similar Majorana bands, which, however, can become completely flat when chemical potential is tuned to coincide with the Dirac point. By comparison to a tight binding model of vortex lattice, it is clear that the dispersion is mainly contributed by first and second nearest neighbor hoppings of Majorana fermions bound in vortices. The hoppings, which are extracted from numerical diagonalization, are not quite identical to the existing analytical prediction. Therefore, we built two simple equations that show the phenomenologically correct trends of the hoppings.","@language":"en"}],"DigitalResourceOriginalRecord":[{"@value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/53962?expand=metadata","@language":"en"}],"FullText":[{"@value":"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\u00a9 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\u2019s 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\u2019s toy model. Upper panel: trivial phase with \u2206 =t = 0 and \u00b5 6= 0. Majorana fermions are paired to form aregular Dirac fermion chain. Lower panel: topological phasewith \u2206 = t and \u00b5 = 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 10\u03b4 \u00d7 10\u03b4 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 (5\u03b4, 5\u03b4). . . . . 112.3 Density of states (DOS) for p+ip superconductor. The mag-netic unit cell is chosen to be 10\u03b4\u00d710\u03b4, with \u03b4 being the latticeconstant of the underlying crystalline lattice. The supercon-ductor order parameter is \u22060 = 0.5034 and the chemical po-tential is \u03b5F = \u22122.2. 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 10\u03b4 \u00d7 10\u03b4, with \u03b4 beingthe lattice constant of the underlying crystalline lattice. Thesuperconductor order parameter is \u22060 = 0.5034 with chemicalpotential \u03b5F = \u22122.2. . . . . . . . . . . . . . . . . . . . . . . . 132.6 Band structure of p+ip superconductor with square vortexlattice. The magnetic unit cell is chosen to be 10\u03b4 \u00d7 10\u03b4,with \u03b4 being the lattice constant of the underlying crystallinelattice. The superconductor order parameter is \u22060 = 0.5034with chemical potential \u03b5F = \u22122.2. . . . . . . . . . . . . . . 142.7 Low-energy bands of p+ip superconductor in 10\u03b4 \u00d7 10\u03b4 mag-netic unit cell with \u22060 = 0.5034 and \u03b5F = \u22122.2. 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,\u2212pi\/2). . . . . . . . . . . . . . 183.3 Majorana bands of p+ip superconductor in 10\u03b4\u00d710\u03b4 magneticunit cell with \u22060 = 0.5034 and \u03b5F = \u22122.2. The square dotsare Majorana bands given by numerical diagonalization, whilethe red lines are \u000f(k) given by tight binding model (Equa-tion 3.17) with gaps \u2206\u0393 = 0.00155 and \u2206X = 0.02434 ex-tracted from numerical diagonalization. . . . . . . . . . . . . . 203.4 First nearest neighbor hopping t in p+ip superconductor with50\u03b4 \u00d7 50\u03b4 magnetic unit cell and \u22060 = 0.1. Black dots showhow t varies with chemical potential \u03b5F 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 \u00d7 10\u22124, B = 3.981 \u00d7 10\u22126, andR = 25\u221a2. The blue line (Equation 3.24) is analytical resultgiven by Cheng et al. [3, 4]. . . . . . . . . . . . . . . . . . . 23viiList of Figures3.5 Second nearest neighbor hopping t\u2032 in p+ip superconductorwith 50\u03b4 \u00d7 50\u03b4 magnetic unit cell and \u22060 = 0.1. Black dotsshow how t\u2032 varies with chemical potential \u03b5F numerically.The red line (Equation 3.31) is the phenomenological fit withthe oscillation characterized by cos (kR\u2032 + pi\/4) as is suggestedby Cheng et al. [3, 4]. The linearly varying amplitude isA+BkR\u2032 with parameters A = 5.91\u00d710\u22125, B = 3.904\u00d710\u22128,and R\u2032 = 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\u00af = 0.5. The spectrum is doubly degenerate with a singleDirac point at \u0393. . . . . . . . . . . . . . . . . . . . . . . . . . 274.2 Spectrum of the simplified topological insulator model witht\u00af = 0.5. One of the surfaces is gapped by mass m = 0.2. . . . 284.3 Spectrum of the simplified topological insulator model witht\u00af = 0.5. The mass is m = 0.2, the superconductor orderparameter is \u2206 = 0.5, and the chemical potential is \u00b5 = 0. . 294.4 Majorana bands in the Fu-Kane model in a 30\u03b4\u00d730\u03b4 magneticunit cell with \u03b5F = 0.25, m = 0.5, \u22060 = 0.4, t\u00af = 0.5, and\u03b4\u000f0 = 0.23. Black dots represent numerical data while redlines are \u000f(k) given by tight binding model (Equation 3.17)with gaps \u2206\u0393 = 2.76\u00d710\u22124 and \u2206X = 1.336\u00d710\u22122 extractedfrom numerical data. . . . . . . . . . . . . . . . . . . . . . . 334.5 Low-energy bands in the Fu-Kane model in a 30\u03b4 \u00d7 30\u03b4 mag-netic unit cell with \u03b5F = 0.25, m = 0.5, \u22060 = 0.4, t\u00af = 0.5,and \u03b4\u000f0 = 0.23. The red lines are dispersing Majorana bands. 344.6 Low-energy bands in the Fu-Kane model in a 30\u03b4 \u00d7 30\u03b4 mag-netic unit cell with \u03b5F = 0, m = 0.5, \u22060 = 0.4, t\u00af = 0.5, and\u03b4\u000f0 = 0.23. The spacing of two vortices in the magnetic unitcell is d = (15\u03b4, 15\u03b4). The bandwidth of Majorana band (redline) is 5.7\u00d7 10\u22125. . . . . . . . . . . . . . . . . . . . . . . . . 354.7 Low-energy bands in the Fu-Kane model in a 30\u03b4 \u00d7 30\u03b4 mag-netic unit cell with \u03b5F = 0, m = 0.5, \u22060 = 0.4, t\u00af = 0.5, and\u03b4\u000f0 = 0.23. The geometry is changed by bringing two vorticescloser to d = (10\u03b4, 10\u03b4) along the diagonal. The bandwidthof Majorana band (red line) is 5.6\u00d7 10\u22125. . . . . . . . . . . . 36viiiList of Figures4.8 Low-energy bands in the Fu-Kane model in a 30\u03b4 \u00d7 30\u03b4 mag-netic unit cell with \u03b5F = 0, m = 0.5, \u22060 = 0.4, t\u00af = 0.5, and\u03b4\u000f0 = 0.23. The geometry is changed by bringing two vor-tices further closer to d = (5\u03b4, 5\u03b4) along the diagonal. Thebandwidth of Majorana band (red line) is 4.9\u00d7 10\u22125. . . . . . 374.9 Low-energy bands in the Fu-Kane model in a 30\u03b4 \u00d7 30\u03b4 mag-netic unit cell with \u03b5F = 0, m = 0.5, \u22060 = 0.4, t\u00af = 0.5, and\u03b4\u000f0 = 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\u00d7 10\u22125. . . . . . . . . . . . . 384.10 First nearest neighbor hopping t of the Fu-Kane model in a30\u03b4\u00d7 30\u03b4 magnetic unit cell with m = 0.5, \u22060 = 0.4, t\u00af = 0.5,and \u03b4\u000f0 = 0.23. Black dots are numerical data and red line isphenomenological fit. Fitting parameters are A = 7.85\u00d710\u22123,B = 1.79, D = 0.341, and R = 7.87. . . . . . . . . . . . . . . 404.11 Second nearest neighbor hopping t\u2032 of the Fu-Kane model in a30\u03b4\u00d7 30\u03b4 magnetic unit cell with m = 0.5, \u22060 = 0.4, t\u00af = 0.5,and \u03b4\u000f0 = 0.23. Black dots are numerical data and red line isphenomenological fit. Fitting parameters are A = 5.87\u00d710\u22124,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 10\u03b4 \u00d7 10\u03b4, with \u03b4 being the latticeconstant of the underlying crystalline lattice. The supercon-ductor order parameter is \u22060 = 1 and the chemical potentialis \u03b5F = \u22122.2. 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 10\u03b4 \u00d7 10\u03b4, with \u03b4 be-ing the lattice constant of the underlying crystalline lattice.The superconductor order parameter is \u22060 = 1 with chemicalpotential \u03b5F = \u22122.2. . . . . . . . . . . . . . . . . . . . . . . . 49ixList of FiguresA.3 Band structure of s-wave superconductor with square vortexlattice. The magnetic unit cell is chosen to be 10\u03b4\u00d710\u03b4, with \u03b4being the lattice constant of the underlying crystalline lattice.The superconductor order parameter is \u22060 = 1 with chemicalpotential \u03b5F = \u22122.2. . . . . . . . . . . . . . . . . . . . . . . . 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\u201311] 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\u2211j=1[\u2212t(c\u2020jcj+1 + c\u2020j+1cj)\u2212 \u00b5(c\u2020jcj \u221212) + (\u2206c\u2020j+1c\u2020j + \u2206\u2217cjcj+1)] (1.1)Kitaev\u2019s model considers nearest neighbor hoppings and p-wave pairing forN fermionic particles and can be rewritten with 2N Majorana operators{\u03b31, . . . , \u03b32N}. In a special case when hopping and pairing are identical andchemical potential is zero, there are two operators, \u03b31 and \u03b32N , 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\u201319] or with electric defects [20].Figure 1.1: Kitaev\u2019s toy model. Upper panel: trivial phase with \u2206 = t = 0and \u00b5 6= 0. Majorana fermions are paired to form a regular Dirac fermionchain. Lower panel: topological phase with \u2206 = t and \u00b5 = 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\u2019 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\u2019s groundbreaking development, manysimilar proposals were reported to realize Majorana fermions with s-wavesuperconductor [24\u201329]. 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 \u2013 the p+ip superconductor and the Fu-Kane model \u2013 in the presence of vortex lattice. By a singular gauge trans-formation, we avoided the difficulty in figuring out the phase field \u03c6(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 \u03b5F 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\u2019s decay following the k\u22121rule. 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 Te\u0161anovi\u0107[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 \u03a6r = (cr, c\u2020r)T with r being thecoordinates of lattice sites of the underlying crystalline lattice.H =(h\u02c6 \u2206\u02c6\u2206\u02c6\u2217 \u2212h\u02c6\u2217)(2.1)where the Hamiltonian for the crystalline lattice ish\u02c6 = \u2212\u03c4\u2211\u03b4e\u2212i(e\/h\u00afc)\u222b r+\u03b4r A(r)\u00b7dls\u02c6\u03b4 \u2212 \u03b5F (2.2)and superconductor order parameter [43] is\u2206\u02c6 = \u22060\u2211\u03b4ei\u03c6(r)\/2\u03b7\u02c6\u03b4ei\u03c6(r)\/2 (2.3)52.1. Hamiltonian of p+ip superconductorFrom now on, we will set the hopping \u03c4 on crystalline lattice to be unityand measure all energies in this p+ip superconductor in unit of \u03c4 . Specifi-cally, for p+ip superconductor, the operator \u03b7\u02c6\u03b4 is defined as\u03b7\u02c6\u03b4 ={\u2213is\u02c6\u03b4 if \u03b4 = \u00b1x\u02c6\u00b1s\u02c6\u03b4 if \u03b4 = \u00b1y\u02c6(2.4)where the operator s\u02c6\u03b4 works as shift operator s\u02c6\u03b4u(r) = u(r+\u03b4). 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\u201349]. Thesuperconductor order parameter phase field \u03c6(r) is constrained by topologyto wind a phase of 2pi around each vortex. Explicitly,\u2207\u00d7\u2207\u03c6(r) = 2piz\u02c6\u2211i\u03b4(r \u2212 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.\u2207\u00d7\u2207\u03c6\u00b5(r) = 2piz\u02c6\u2211i\u03b4(r \u2212 r\u00b5i ) \u00b5 = 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 \u03c6A(r) and \u03c6B(r), we can62.2. Superfluid velocitydefine a unitary matrixU =(ei\u03c6A(r) 00 e\u2212i\u03c6B(r))(2.7)and a singular gauge transformation H \u2192 U\u22121HU under which the eigenval-ues of the Hamiltonian remain same. Such transformation may help removethe phase of \u2206(r) and brings mathematical simplicity, because of the factthat \u03c6A(r)+\u03c6B(r) = \u03c6(r). After the transformation, the Hamiltonian reads,U\u22121HU =\uf8eb\uf8ec\uf8ed\u2212\u2211\u03b4eiVA\u03b4 (r)s\u02c6\u03b4 \u2212 \u03b5F \u22060\u2211\u03b4eiA\u03b4(r)\u03b7\u02c6\u03b4\u22060\u2211\u03b4eiA\u03b4(r)\u03b7\u02c6\u2217\u03b4\u2211\u03b4e\u2212iVB\u03b4 (r)s\u02c6\u03b4 + \u03b5F\uf8f6\uf8f7\uf8f8 (2.8)with the phase factors defined asV\u00b5\u03b4 (r) =\u222b r+\u03b4r(\u2207\u03c6\u00b5 \u2212eh\u00afcA)\u00b7 dl \u00b5 = A,B (2.9)andA\u03b4(r) =12\u222b r+\u03b4r(\u2207\u03c6A \u2212\u2207\u03c6B) \u00b7 dl =12[VA\u03b4 (r)\u2212 VB\u03b4 (r)] (2.10)2.2 Superfluid velocityTo calculate the phase factors, we may rewrite the phase factor V\u00b5\u03b4 in termsof superfluid velocity v\u00b5s .V\u00b5\u03b4 (r) =mh\u00af\u222b r+\u03b4rv\u00b5s (r) \u00b7 dl (2.11)wherev\u00b5s (r) =h\u00afm(\u2207\u03c6\u00b5 \u2212eh\u00afcA)(2.12)Without knowing \u03c6\u00b5 and A(r), we are not able to tell the value of v\u00b5s (r).Therefore, we will need to develop a more practical form of superfluid veloc-ity. We first take the curl of the superfluid velocity\u2207\u00d7 v\u00b5s (r) =emc[z\u02c6\u03c60\u2211i\u03b4(r \u2212 r\u00b5i )\u2212B](2.13)72.2. Superfluid velocitywhere the flux quantum \u03c60 = hc\/e. We combine two velocities together tosuppress index \u00b5,2B +2mce\u2207\u00d7 vs = z\u02c6\u03c60\u2211i\u03b4(r \u2212 ri) (2.14)The Maxwell equation\u2212\u22072B = \u2207\u00d7 (\u2207\u00d7B) = \u2207\u00d74piJc= \u2207\u00d74pinsecvs (2.15)Plugging this back to the curl of vs, we get the conventional London equation,B \u2212 \u03bb2\u22072B =12\u03c60z\u02c6\u2211i\u03b4(r \u2212 ri) (2.16)where London penetration depth \u03bb2 = mc24pinse2. Perform Fourier transformB(r) =\u222bd2k(2pi)2eik\u00b7rBk (2.17)to the London equation , we would getBk + \u03bb2k2Bk =12z\u02c6\u03c60\u2211ie\u2212ik\u00b7ri (2.18)and equivalently,Bk =12 z\u02c6\u03c60\u2211ie\u2212ik\u00b7ri1 + \u03bb2k2(2.19)Perform Fourier transform to the curl of v\u00b5s and plug in Bkik \u00d7 v\u00b5sk(r) =2pih\u00afm[z\u02c6\u2211ie\u2212ik\u00b7r\u00b5i \u221212z\u02c6\u2211ie\u2212ik\u00b7ri1 + \u03bb2k2](2.20)We then perform a cross productik \u00d7 (ik \u00d7 v\u00b5sk) =2pih\u00afmik \u00d7 z\u02c6[\u03bb2k21 + \u03bb2k2\u2211ie\u2212ik\u00b7r\u00b5i+12(1 + \u03bb2k2)(\u2211ie\u2212ik\u00b7r\u00b5i \u2212\u2211ie\u2212ik\u00b7r\u03bdi)](2.21)82.2. Superfluid velocityThe second term can be safely dropped. We further apply transverse gaugek \u00b7 v\u00b5sk = 0, so thatv\u00b5sk =2pih\u00af\u03bb2mik \u00d7 z\u02c61 + \u03bb2k2\u2211ie\u2212ik\u00b7r\u00b5i (2.22)Then the superfluid velocity in real spacev\u00b5s (r) =2pih\u00afm\u222bd2k(2pi)2ik \u00d7 z\u02c6k2\u2211ieik\u00b7(r\u2212r\u00b5i ) (2.23)where we assume \u03bb\u2192\u221e. As r\u00b5i represents the position of the type \u00b5 vortexof the i-th magnetic unit cell, it can be rewrite as r\u00b5i = Ri + \u03b4\u00b5, with Ribeing the center of the i-th magnetic unit cell and \u03b4\u00b5 being the position oftype \u00b5 vortex with respect to the center. By this separation of index i andindex \u00b5, we can first conduct the summation and getv\u00b5s (r) =2pih\u00afmN\u2211G\u222bd2k(2pi)2ik \u00d7 z\u02c6k2eik\u00b7(r\u2212\u03b4\u00b5)\u03b4k,G (2.24)where N is the number of magnetic unit cells and we use the relation\u2211ie\u2212ik\u00b7Ri =\u2211G\u03b4k,G (2.25)with G being the reciprocal lattice vector of lattice {Ri}. For an L \u00d7 Lmagnetic cell, where G = 2piL (nx, ny), we have\u222bd2k(2pi)2\u21921L2N\u2211kThis will help us further simplify the superfluid velocity tov\u00b5s (r) =2pih\u00afm1L2\u2211GiG\u00d7 z\u02c6G2eiG\u00b7(r\u2212\u03b4\u00b5) (2.26)And finally, the phase factor will beV\u00b5\u03b4 (r) =2piL2\u2211G\u222b r+\u03b4reiG\u00b7(r\u2212\u03b4\u00b5) iG\u00d7 z\u02c6G2\u00b7 dl (2.27)Numerically, we cannot sum over all possible reciprocal vector G. The sum-mand decays as G\u22121 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\u00b1ik\u00b7r, we may write the Bloch Hamil-tonian Hk = e\u2212ik\u00b7rUHU\u22121eik\u00b7r .Hk =\uf8eb\uf8ec\uf8ed\u2212\u2211\u03b4eiVA\u03b4 (r)eik\u00b7\u03b4 s\u02c6\u03b4 \u2212 \u03b5F \u22060\u2211\u03b4eiA\u03b4(r)eik\u00b7\u03b4\u03b7\u02c6\u03b4\u22060\u2211\u03b4eiA\u03b4(r)eik\u00b7\u03b4\u03b7\u02c6\u2217\u03b4\u2211\u03b4e\u2212iVB\u03b4 (r)eik\u00b7\u03b4 s\u02c6\u03b4 + \u03b5F\uf8f6\uf8f7\uf8f8 (2.28)in which we use the fact thate\u2212ik\u00b7rS\u02c6\u03b4eik\u00b7rf(r) = eik\u00b7\u03b4f(r + \u03b4) = eik\u00b7\u03b4S\u02c6\u03b4f(r) (2.29)Here S\u02c6\u03b4 is a general shift operator that can be chosen among s\u02c6\u03b4, \u03b7\u02c6\u03b4, and\u03b7\u02c6\u2217\u03b4. And f(r) is an arbitrary function defined on the crystalline lattice.We write the Schr\u00f6dinger equation for such Bloch Hamiltonian Hk\u03a6nk =\u03b5nk\u03a6nk with \u03a6nk = [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 10\u03b4\u00d710\u03b4 geometry and periodic boundary conditions in both directions.For j = 1, 2, . . . , 100, when we plug \u03a6nk(rj) in Schr\u00f6dinger equation, we willhave 200 equations, which in return can be written as a 200\u00d7 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 \u03c8\u00b1 that satisfySchr\u00f6dinger equation H\u03c8\u00b1 = \u00b1E\u03c8\u00b1. Due to the particle-hole symmetry,we have \u03c8\u2212 = \u03c3x\u03c8\u2217+. By linear combination, we can separate two linearlyindependent states\u03c81 =12(1\u2212 i)\u03c8+ +12(1 + i)\u03c8\u2212 (2.30)\u03c82 =12(1 + i)\u03c8+ +12(1\u2212 i)\u03c8\u2212 (2.31)with particle-hole symmetry \u03c81,2 = \u03c3x\u03c8\u22171,2, 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 10\u03b4 \u00d7 10\u03b4 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 (5\u03b4, 5\u03b4).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 10\u03b4 \u00d7 10\u03b4, with \u03b4 being the lattice constant ofthe underlying crystalline lattice. The superconductor order parameter is\u22060 = 0.5034 and the chemical potential is \u03b5F = \u22122.2. 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 spectrum0123456\u0393 Energy\u0393 M MX YFigure 2.5: Band structure of p+ip superconductor with no vortices. Themagnetic unit cell is chosen to be 10\u03b4\u00d710\u03b4, with \u03b4 being the lattice constantof the underlying crystalline lattice. The superconductor order parameter is\u22060 = 0.5034 with chemical potential \u03b5F = \u22122.2.132.3. Majorana band and spectrum0123456\u0393 Energy \u0393 M MX YFigure 2.6: Band structure of p+ip superconductor with square vortexlattice. The magnetic unit cell is chosen to be 10\u03b4 \u00d7 10\u03b4, with \u03b4 being thelattice constant of the underlying crystalline lattice. The superconductororder parameter is \u22060 = 0.5034 with chemical potential \u03b5F = \u22122.2.142.3. Majorana band and spectrum- 0 . 8- 0 . 6- 0 . 4- 0 . 20 . 00 . 20 . 40 . 60 . 8 \u0393Energy \u0393 M MX YFigure 2.7: Low-energy bands of p+ip superconductor in 10\u03b4\u00d710\u03b4 magneticunit cell with \u22060 = 0.5034 and \u03b5F = \u22122.2. 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 \u03a60 = pih\u00afc\/e and a Majorana fermion in the core. TheHamiltonian, due to its own hermicity, can be written asH = it\u2211i,jsij\u03b3i\u03b3jwith sij = \u2212sji = \u00b11. The elements of s matrix can only be determinedup to a sign, because a local Z2 transformation \u03b3i \u2192 \u2212\u03b3i 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\u2211polygon\u03c6ij =pi2(n\u2212 2)163.1. Tight binding Hamiltonianindicating the product of hopping terms around the plaquette to be tn exp[i(n\u2212 2)pi\/2] = \u2212intn. 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\u2211R\u03b3R,A(\u03b3R,B \u2212 \u03b3R\u2212x\u02c6\u2212y\u02c6,B + \u03b3R\u2212x\u02c6,B + \u03b3R\u2212y\u02c6,B) (3.2)Hsn,A = it\u2032\u2211R\u03b3R,A(\u2212\u03b3R+x\u02c6,A + \u03b3R+y\u02c6,A) (3.3)Hsn,B = it\u2032\u2211R\u03b3R,B(\u03b3R+x\u02c6,B \u2212 \u03b3R+y\u02c6,B) (3.4)with t and t\u2032 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,\u2212pi\/2).\u03b3R,A(B) =\u221a2N\u2211kei(k+Q)\u00b7R\u03b3k,A(B) Q = (pi2,\u2212pi2) (3.5)The first nearest neighbor hopping Hamiltonian isHfn = 4t\u2211kei(kx+ky)\/2[sin(kx + ky2)\u2212 i sin(kx \u2212 ky2)]\u03b3k,A\u03b3\u2020k,B (3.6)The two second nearest neighbor hoppings areHsn,A = \u22122t\u2032\u2211k(e\u2212ikx + e\u2212iky)\u03b3k,A\u03b3\u2020k,A (3.7)Hsn,B = 2t\u2032\u2211k(e\u2212ikx + e\u2212iky)\u03b3k,B\u03b3\u2020k,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 \u0393 point. Then the nearest neighbor hopping isHfn = \u2212\u2211k\u2208RBZh\u2217k\u03b3\u2020k,A\u03b3k,B \u2212\u2211k\u2208RBZhk\u03b3\u2020k,B\u03b3k,A (3.9)183.1. Tight binding Hamiltonianwith the parameterhk = 4tei(kx+ky)\/2(sinkx + ky2\u2212 i sinkx \u2212 ky2)(3.10)And the second nearest neighbor hoppings areHsn,A = 2t\u2032\u2211k\u2208RBZ(e\u2212ikx + e\u2212iky)\u2212\u2211k\u2208RBZh\u2032k\u03b3\u2020k,A\u03b3k,A (3.11)Hsn,B = \u22122t\u2032\u2211k\u2208RBZ(e\u2212ikx + e\u2212iky) +\u2211k\u2208RBZh\u2032k\u03b3\u2020k,B\u03b3k,B (3.12)with the parameterh\u2032k = \u22124t\u2032(cos kx + cos ky) (3.13)Therefore, we can write the total tight binding Hamiltonian in the space of\u0393k = (\u03b3k,A, \u03b3k,B)T asH = Hfn +Hsn,A +Hsn,B =\u2211k\u2208RBZ\u0393\u2020kHk\u0393k (3.14)whereHk =(\u2212h\u2032k \u2212h\u2217k\u2212hk h\u2032k)(3.15)Diagonalization gives the analytical solution of dispersion of Majorana band\u000f(k) = \u00b1\u221a|hk|2 + |h\u2032k|2 (3.16)Therefore, at Dirac point \u0393 the Hamiltonian opens a gap \u2206\u0393 = 8t\u2032. At pointX, it opens a gap \u2206X = 4\u221a2t. We can rewrite the dispersion in terms of\u2206\u0393 and \u2206X .\u000f(k) = \u00b1\u221a\u22062X2(sin2kx + ky2+ sin2kx \u2212 ky2)+\u22062\u03934(cos kx + cos ky)2(3.17)As long as we extract the gap parameters \u2206\u0393 and \u2206X from the numericaldiagonalization, we can plot the whole dispersion of Majorana bands. Fora 10\u03b4 \u00d7 10\u03b4 magnetic unit cell with superconductor order parameter \u22060 =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 3Energy\u0393 X M Y \u0393 MFigure 3.3: Majorana bands of p+ip superconductor in 10\u03b4 \u00d7 10\u03b4 magneticunit cell with \u22060 = 0.5034 and \u03b5F = \u22122.2. The square dots are Majoranabands given by numerical diagonalization, while the red lines are \u000f(k) givenby tight binding model (Equation 3.17) with gaps \u2206\u0393 = 0.00155 and \u2206X =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\u00af = 1 and crystalline lattice constant \u03b4 = 1.The hoppings in p+ip superconductor will show exponentially decaying be-havior in the regime \u220620 > 2m\u03b5F v2F and oscillating behavior in the regime\u220620 < 2m\u03b5F v2F . The chemical potential \u03b5F is counted from the bottomof the band and therefore is always positive. We are more interested inthe latter regime with large vortex spacing R \u001d max(k\u22121, \u03be), where thesuperconducting coherence length is \u03be = vF \/\u22060 and the momentum is203.2. Hoppings of p+ip superconductork2 = 2m\u03b52F \u2212\u220620\/v2F . Analytically, the energy splitting is predicted [3, 4] asE =\u221a2pi\u22060| cos (kFR+ pi4 )|\u221akFRexp(\u2212R\u03be)(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\u2032) 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\u000f = \u22122(cos kx + cos ky) (3.19)with the energy scaled in unit of \u03c4 and momentum scaled in unit of \u03b4\u22121.The first nearest neighbor hopping happens along the diagonal in magneticunit cell between different sublattices. Therefore, we takekx = ky =kF\u221a2(3.20)when calculating chemical potential \u03b5F . We may write the dispersion as\u03b5F = \u22122[2\u2212 4 sin2( kF2\u221a2)](3.21)Equivalently,kF = 2\u221a2 arcsin\u221a\u03b5F + 48(3.22)The Fermi velocity isvF =\u2202\u03b5F\u2202kF= 4\u221a2 sinkF2\u221a2coskF2\u221a2=\u221a16\u2212 \u03b52F2(3.23)By plugging in all parameters, we can plotE(\u03b5F ) =\u221a2pi\u22060\u2223\u2223\u2223 cos(2\u221a2 arcsin\u221a\u03b5F+48 R+pi4)\u2223\u2223\u2223\u221a2\u221a2 arcsin\u221a\u03b5F+48 Rexp(\u2212\u221a2R\u22060\u221a16\u2212 \u03b52F)(3.24)213.2. Hoppings of p+ip superconductorwith the parameter \u22060 = 0.1 and R = 25\u221a2. It looks that such analyticalequation can capture the period of our numerical data when \u03b5F 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 =\u221ak2F \u2212\u220620v2F(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(\u03b5F ) =(A+B\u221a8R2 arcsin2\u221a\u03b5F + 48\u22122R2\u22062016\u2212 \u03b52F)\u00d7 cos(\u221a8R2 arcsin2\u221a\u03b5F + 48\u22122R2\u22062016\u2212 \u03b52F+pi4)(3.26)with the parameter \u22060 = 0.1 and R = 25\u221a2. Such function will fit bothperiod and amplitude very well for all chemical potential \u03b5F > \u22123.95,where oscillating behavior is expected (Figure 3.4, red line). For the regime\u03b5F < \u22123.95, 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 \u03b5F Figure 3.4: First nearest neighbor hopping t in p+ip superconductor with50\u03b4 \u00d7 50\u03b4 magnetic unit cell and \u22060 = 0.1. Black dots show how t varieswith chemical potential \u03b5F 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\u00d7 10\u22124, B = 3.981\u00d7 10\u22126, and R = 25\u221a2. 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 dispersion\u03b5F = \u22122(1 + cos kF ) = \u22124 + 4 sin2(kF2)(3.27)And the Fermi momentum is changed tokF = 2 arcsin\u221a\u03b5F + 44(3.28)The Fermi velocity is then given asvF =\u2202\u03b5F\u2202kF= 2 sin kF =\u221a\u2212\u03b52F \u2212 4\u03b5F (3.29)233.2. Hoppings of p+ip superconductorBy plugging in all parameters, we can plotE(\u03b5F ) =\u221a2pi\u22060\u2223\u2223\u2223 cos(2 arcsin\u221a\u03b5F+44 R\u2032 + pi4)\u2223\u2223\u2223\u221a2 arcsin\u221a\u03b5F+44 R\u2032exp(\u2212R\u2032\u22060\u221a\u2212\u03b52F \u2212 4\u03b5F)(3.30)where superconducting order parameter \u22060 = 0.1 and the spacing betweensecond nearest neighbors R\u2032 = 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\u2032(\u03b5F ) =(A+B\u221a4R\u20322 arcsin2\u221a\u03b5F + 44\u2212R\u20322\u220620\u2212\u03b52F \u2212 4\u03b5F)\u00d7 cos(\u221a4R\u20322 arcsin2\u221a\u03b5F + 44\u2212R\u20322\u220620\u2212\u03b52F \u2212 4\u03b5F+pi4)(3.31)Such phenomenological fit is good (Figure 3.5, red line) for \u03b5F < \u22121. 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 ' \u03b5F Figure 3.5: Second nearest neighbor hopping t\u2032 in p+ip superconductor with50\u03b4 \u00d7 50\u03b4 magnetic unit cell and \u22060 = 0.1. Black dots show how t\u2032 varieswith chemical potential \u03b5F numerically. The red line (Equation 3.31) is thephenomenological fit with the oscillation characterized by cos (kR\u2032 + pi\/4)as is suggested by Cheng et al. [3, 4]. The linearly varying amplitude isA+BkR\u2032 with parameters A = 5.91\u00d710\u22125, B = 3.904\u00d710\u22128, and R\u2032 = 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\u00afkM\u00afk \u2212hk)(4.1)with blocks defined ashk = 2\u03bb(sy sin kx \u2212 sx sin ky) (4.2)M\u00afk = 2t\u00af(2\u2212 cos kx \u2212 cos ky) (4.3)The Hamiltonian is written in basis \u03a8k = (ck,1,\u2191, ck,1,\u2193, ck,2,\u2191, ck,2,\u2193)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 \u2212hk coupled by matrix M\u00afk. From now on, wewill take \u03bb to be unity and measure all energies in terms of \u03bb. The spectrumof Hk,T I is\u000f(k) = \u00b1\u221a4(sin2 kx + sin2 ky) + M\u00af2k (4.4)which is doubly degenerate because of the fact that each surface will have agapless Dirac cone.- 4- 2024 Energy\u0393M MXFigure 4.1: Spectrum of the simplified topological insulator model with t\u00af =0.5. The spectrum is doubly degenerate with a single Dirac point at \u0393.An interesting modification is to open a gap at one of the surface byadding a mass termHk,Mag = m(c\u2020k,1,\u2191ck,1,\u2191 \u2212 c\u2020k,1,\u2193ck,1,\u2193) (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\u00afkM\u00afk \u2212hk)(4.6)274.1. The Fu-Kane HamiltonianThe non-degenerate spectrum is\u000f(k) = \u00b1\u221a4(sin2 kx + sin2 ky) + M\u00af2k +12m2 \u00b1\u221a14m4 +m2M\u00af2k (4.7)- 4- 2024 Energy\u0393 X MMFigure 4.2: Spectrum of the simplified topological insulator model with t\u00af =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 = \u2206c\u2020k,2,\u2191c\u2020\u2212k,2,\u2193 + \u2206\u2217c\u2212k,2,\u2193ck,2,\u2191 (4.8)This term will open a gap and is the last piece of the Fu-Kane model. Thefull Hamiltonian isHFK =12\u2211k\u03a6\u2020kHk,FK\u03a6k (4.9)284.1. The Fu-Kane HamiltonianwithHk,FK =\uf8eb\uf8ec\uf8ec\uf8edhk +msz M\u00afk 0 0M\u00afk \u2212hk \u2212 \u00b5 0 i\u2206\u03c3y0 0 h\u2217k \u2212msz \u2212M\u00afk0 \u2212i\u2206\u2217\u03c3y \u2212M\u00afk \u2212h\u2217k + \u00b5\uf8f6\uf8f7\uf8f7\uf8f8 (4.10)in the basis \u03a6k = (ck,1,\u2191, ck,1,\u2193, ck,2,\u2191, ck,2,\u2193, c\u2020\u2212k,1,\u2191, c\u2020\u2212k,1,\u2193, c\u2020\u2212k,2,\u2191, c\u2020\u2212k,2,\u2193)T .- 4- 2024 Energy\u0393 X MMFigure 4.3: Spectrum of the simplified topological insulator model with t\u00af =0.5. The mass is m = 0.2, the superconductor order parameter is \u2206 = 0.5,and the chemical potential is \u00b5 = 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,\u03b1 =1\u221aN\u2211re\u2212ik\u00b7rcr,\u03b1 (4.11)294.1. The Fu-Kane Hamiltonianwhere \u03b1 specifies surface and spin. In an 8-component Nambu spinor \u03a6r =(cr,1,\u2191, cr,1,\u2193, cr,2,\u2191, cr,2,\u2193, c\u2020r,1,\u2191, c\u2020r,1,\u2193, c\u2020r,2,\u2191, c\u2020r,2,\u2193)T , the Hamiltonian readsHFK =12\u2211r\u03a6\u2020r(h\u02c6r \u2206\u02c6r\u2206\u02c6\u2020r \u2212h\u02c6\u2217r)\u03a6r (4.12)with the blocksh\u02c6r =\uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8edm i\u2211\u03b4\u03b7\u02c6\u2217\u03b4 4t\u00af\u2212 t\u00af\u2211\u03b4s\u02c6\u03b4 0i\u2211\u03b4\u03b7\u02c6\u03b4 \u2212m 0 4t\u00af\u2212 t\u00af\u2211\u03b4s\u02c6\u03b44t\u00af\u2212 t\u00af\u2211\u03b4s\u02c6\u03b4 0 \u2212\u03b5F \u2212i\u2211\u03b4\u03b7\u02c6\u2217\u03b40 4t\u00af\u2212 t\u00af\u2211\u03b4s\u02c6\u03b4 \u2212i\u2211\u03b4\u03b7\u02c6\u03b4 \u2212\u03b5F\uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8(4.13)and\u2206\u02c6r =\uf8eb\uf8ec\uf8ec\uf8ed0 0 0 00 0 0 00 0 0 \u22060 0 \u2212\u2206 0\uf8f6\uf8f7\uf8f7\uf8f8 (4.14)where \u2206 = \u22060ei\u03c6. When a magnetic field is applied, a vortex lattice isformed, and the hoppings, which are characterized by shift operators, shouldbe modified by Peierls phase \u03b8r = eh\u00afc\u222b r+\u03b4r A(r) \u00b7 dl. The diagonal blockh\u02c6r(\u03b8r) is now\uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8edm i\u2211\u03b4e\u2212i\u03b8r \u03b7\u02c6\u2217\u03b4 4t\u00af\u2212 t\u00af\u2211\u03b4e\u2212i\u03b8r s\u02c6\u03b4 0i\u2211\u03b4e\u2212i\u03b8r \u03b7\u02c6\u03b4 \u2212m 0 4t\u00af\u2212 t\u00af\u2211\u03b4e\u2212i\u03b8r s\u02c6\u03b44t\u00af\u2212 t\u00af\u2211\u03b4e\u2212i\u03b8r s\u02c6\u03b4 0 \u2212\u03b5F \u2212i\u2211\u03b4e\u2212i\u03b8r \u03b7\u02c6\u2217\u03b40 4t\u00af\u2212 t\u00af\u2211\u03b4e\u2212i\u03b8r s\u02c6\u03b4 \u2212i\u2211\u03b4e\u2212i\u03b8r \u03b7\u02c6\u03b4 \u2212\u03b5F\uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8Working 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(ei\u03c6A,1,\u2191 , ei\u03c6A,1,\u2193 , ei\u03c6A,2,\u2191 , ei\u03c6A,2,\u2193 ,e\u2212i\u03c6B,1,\u2191 , e\u2212i\u03c6B,1,\u2193 , e\u2212i\u03c6B,2,\u2191 , e\u2212i\u03c6B,2,\u2193) (4.15)304.1. The Fu-Kane HamiltonianAs for our topological insulator, only one surface is connected to supercon-ductor, \u03c6A(B),1,\u2191(\u2193) 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 \u03c6A,1(2),\u2191(\u2193) = \u03c6Aand \u03c6B,1(2),\u2191(\u2193) = \u03c6B, where \u03c6A + \u03c6B = \u03c6, we may get a more convenientunitary transformationU =(ei\u03c6A 00 e\u2212i\u03c6B)(4.16)with each block being a 4\u00d74 matrix. After the unitary gauge transformation,we further plug in e\u2212ik\u00b7r and eik\u00b7r, arriving at an 8\u00d7 8 Bloch HamiltonianHk,FK = e\u2212ik\u00b7rU\u22121(h\u02c6r(\u03b8r) \u2206\u02c6r\u2206\u02c6\u2020r \u2212h\u02c6\u2217r(\u03b8r))Ueik\u00b7r (4.17)where the diagonal blocks H(1,1)k,FK and H(2,2)k,FK are defined as\uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8edm i\u2211\u03b4eiVA\u03b4 eik\u00b7\u03b4 \u03b7\u02c6\u2217\u03b4 4t\u00af\u2212t\u00af\u2211\u03b4eiVA\u03b4 eik\u00b7\u03b4 s\u02c6\u03b4 0i\u2211\u03b4eiVA\u03b4 eik\u00b7\u03b4 \u03b7\u02c6\u03b4 \u2212m 0 4t\u00af\u2212t\u00af\u2211\u03b4eiVA\u03b4 eik\u00b7\u03b4 s\u02c6\u03b44t\u00af\u2212t\u00af\u2211\u03b4eiVA\u03b4 eik\u00b7\u03b4 s\u02c6\u03b4 0 \u2212\u03b5F \u2212i\u2211\u03b4eiVA\u03b4 eik\u00b7\u03b4 \u03b7\u02c6\u2217\u03b40 4t\u00af\u2212t\u00af\u2211\u03b4eiVA\u03b4 eik\u00b7\u03b4 s\u02c6\u03b4 \u2212i\u2211\u03b4eiVA\u03b4 eik\u00b7\u03b4 \u03b7\u02c6\u03b4 \u2212\u03b5F\uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8and\uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed\u2212m i\u2211\u03b4e\u2212iVB\u03b4 eik\u00b7\u03b4 \u03b7\u02c6\u03b4 t\u00af\u2211\u03b4e\u2212iVB\u03b4 eik\u00b7\u03b4 s\u02c6\u03b4\u22124t\u00af 0i\u2211\u03b4e\u2212iVB\u03b4 eik\u00b7\u03b4 \u03b7\u02c6\u2217\u03b4 m 0 t\u00af\u2211\u03b4e\u2212iVB\u03b4 eik\u00b7\u03b4 s\u02c6\u03b4\u22124t\u00aft\u00af\u2211\u03b4e\u2212iVB\u03b4 eik\u00b7\u03b4 s\u02c6\u03b4\u22124t\u00af 0 \u03b5F \u2212i\u2211\u03b4e\u2212iVB\u03b4 eik\u00b7\u03b4 \u03b7\u02c6\u03b40 t\u00af\u2211\u03b4e\u2212iVB\u03b4 eik\u00b7\u03b4 s\u02c6\u03b4\u22124t\u00af \u2212i\u2211\u03b4e\u2212iVB\u03b4 eik\u00b7\u03b4 \u03b7\u02c6\u2217\u03b4 \u03b5F\uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8respectively. And off-diagonal blocks areH(1,2)k,FK = \u2212H(2,1)k,FK =\uf8eb\uf8ec\uf8ec\uf8ed0 0 0 00 0 0 00 0 0 \u220600 0 \u2212\u22060 0\uf8f6\uf8f7\uf8f7\uf8f8 (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 \u03b4\u000f0 < \u22060 < \u03bb, where \u03b4\u000f0 is the energy differencebetween the lowest two Landau levels. In the regime \u03b4\u000f0 < \u22060, the low-energy bands are mainly determined by superconductor instead of magneticfield. On the other hand, for \u03b4\u000f0 > \u22060, the magnetic field dominates. Andone will see flat Landau levels. Accordingly, \u03b4\u000f0 is obtained by diagonalizingthe Bloch Hamiltonian Hk,FK with \u22060 = 0 and finding the gap between twolowest flat bands. As we are interested in the low-energy physics, we shallrestrict both \u03b4\u000f0 and \u22060 within the bulk gap of topological insulator. Thebulk gap is roughly measured by energy unit \u03bb (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, \u03b5F = 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 0Energy\u0393 X M Y \u0393 MFigure 4.4: Majorana bands in the Fu-Kane model in a 30\u03b4 \u00d7 30\u03b4 magneticunit cell with \u03b5F = 0.25, m = 0.5, \u22060 = 0.4, t\u00af = 0.5, and \u03b4\u000f0 = 0.23. Blackdots represent numerical data while red lines are \u000f(k) given by tight bindingmodel (Equation 3.17) with gaps \u2206\u0393 = 2.76\u00d7 10\u22124 and \u2206X = 1.336\u00d7 10\u22122extracted from numerical data.334.2. Majorana band and spectrum- 0 . 5- 0 . 4- 0 . 3- 0 . 2- 0 . 10 . 00 . 10 . 20 . 30 . 40 . 5M\u0393 Energy\u0393 MX YFigure 4.5: Low-energy bands in the Fu-Kane model in a 30\u03b4\u00d730\u03b4 magneticunit cell with \u03b5F = 0.25, m = 0.5, \u22060 = 0.4, t\u00af = 0.5, and \u03b4\u000f0 = 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 . 5\u0393 Energy\u0393 M MX YFigure 4.6: Low-energy bands in the Fu-Kane model in a 30\u03b4\u00d730\u03b4 magneticunit cell with \u03b5F = 0, m = 0.5, \u22060 = 0.4, t\u00af = 0.5, and \u03b4\u000f0 = 0.23. Thespacing of two vortices in the magnetic unit cell is d = (15\u03b4, 15\u03b4). Thebandwidth of Majorana band (red line) is 5.7\u00d7 10\u22125.354.2. Majorana band and spectrum- 0 . 5- 0 . 4- 0 . 3- 0 . 2- 0 . 10 . 00 . 10 . 20 . 30 . 40 . 5 M\u0393Energy\u0393 MX YFigure 4.7: Low-energy bands in the Fu-Kane model in a 30\u03b4\u00d730\u03b4 magneticunit cell with \u03b5F = 0, m = 0.5, \u22060 = 0.4, t\u00af = 0.5, and \u03b4\u000f0 = 0.23. Thegeometry is changed by bringing two vortices closer to d = (10\u03b4, 10\u03b4) alongthe diagonal. The bandwidth of Majorana band (red line) is 5.6\u00d7 10\u22125.364.2. Majorana band and spectrum- 0 . 5- 0 . 4- 0 . 3- 0 . 2- 0 . 10 . 00 . 10 . 20 . 30 . 40 . 5M\u0393 Energy\u0393 MX YFigure 4.8: Low-energy bands in the Fu-Kane model in a 30\u03b4\u00d730\u03b4 magneticunit cell with \u03b5F = 0, m = 0.5, \u22060 = 0.4, t\u00af = 0.5, and \u03b4\u000f0 = 0.23. Thegeometry is changed by bringing two vortices further closer to d = (5\u03b4, 5\u03b4)along the diagonal. The bandwidth of Majorana band (red line) is 4.9\u00d710\u22125.374.2. Majorana band and spectrum- 0 . 5- 0 . 4- 0 . 3- 0 . 2- 0 . 10 . 00 . 10 . 20 . 30 . 40 . 5 \u0393Energy\u0393 MX Y MFigure 4.9: Low-energy bands in the Fu-Kane model in a 30\u03b4\u00d730\u03b4 magneticunit cell with \u03b5F = 0, m = 0.5, \u22060 = 0.4, t\u00af = 0.5, and \u03b4\u000f0 = 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\u00d7 10\u22125.384.3. 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 = \u2206X\/4\u221a2 and second nearest neighborhopping is t\u2032 = \u2206\u0393\/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\u2212B|k(\u03b5F )|, with momentum k = \u03b5F + D\u03b53F . The s-wave super-conductor is patterned to the surface characterized by surface Hamiltonianhk = \u22122(sy sin kx \u2212 sx sin ky), which only depends on sine functions. Thus,only odd powers should enter k(\u03b5F ). Therefore, the explicit fitting equationfor our hopping will be\u000f = |Ae\u2212B(|\u03b5F |+D|\u03b5F |3) sin[R(|\u03b5F |+D|\u03b5F |3)]| (4.19)with \u000f representing either t or t\u2032.394.3. 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 t\u03b5FFigure 4.10: First nearest neighbor hopping t of the Fu-Kane model in a30\u03b4\u00d730\u03b4 magnetic unit cell with m = 0.5, \u22060 = 0.4, t\u00af = 0.5, and \u03b4\u000f0 = 0.23.Black dots are numerical data and red line is phenomenological fit. Fittingparameters are A = 7.85\u00d7 10\u22123, 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 '\u03b5FFigure 4.11: Second nearest neighbor hopping t\u2032 of the Fu-Kane model in a30\u03b4\u00d730\u03b4 magnetic unit cell with m = 0.5, \u22060 = 0.4, t\u00af = 0.5, and \u03b4\u000f0 = 0.23.Black dots are numerical data and red line is phenomenological fit. Fittingparameters are A = 5.87\u00d7 10\u22124, 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. 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The onlydifference is that the off-diagonal shift operator is now a constant \u03b7\u02c6\u03b4 = 14 .The lattice Bloch Hamiltonian is now,Hk,s =\uf8eb\uf8ec\uf8ed\u2212t\u2211\u03b4eiVA\u03b4 (r)eik\u00b7\u03b4 s\u02c6\u03b4 \u2212 \u03b5F \u22060\u22060 t\u2211\u03b4e\u2212iVB\u03b4 (r)eik\u00b7\u03b4 s\u02c6\u03b4 + \u03b5F\uf8f6\uf8f7\uf8f8 (A.1)The definitions of VA\u03b4 and VB\u03b4 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\u22060 = 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 10\u03b4 \u00d7 10\u03b4, with \u03b4 being the lattice constant ofthe underlying crystalline lattice. The superconductor order parameter is\u22060 = 1 and the chemical potential is \u03b5F = \u22122.2. 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 Energy\u0393 \u0393M MX YFigure A.2: Band structure of s-wave superconductor with no vortices. Themagnetic unit cell is chosen to be 10\u03b4\u00d710\u03b4, with \u03b4 being the lattice constantof the underlying crystalline lattice. The superconductor order parameter is\u22060 = 1 with chemical potential \u03b5F = \u22122.2.49Appendix A. The s-wave superconductor0123456","@language":"en"}],"Genre":[{"@value":"Thesis\/Dissertation","@language":"en"}],"GraduationDate":[{"@value":"2015-09","@language":"en"}],"IsShownAt":[{"@value":"10.14288\/1.0166331","@language":"en"}],"Language":[{"@value":"eng","@language":"en"}],"Program":[{"@value":"Physics","@language":"en"}],"Provider":[{"@value":"Vancouver : University of British Columbia Library","@language":"en"}],"Publisher":[{"@value":"University of British Columbia","@language":"en"}],"Rights":[{"@value":"Attribution-NonCommercial-NoDerivs 2.5 Canada","@language":"en"}],"RightsURI":[{"@value":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/2.5\/ca\/","@language":"en"}],"ScholarlyLevel":[{"@value":"Graduate","@language":"en"}],"Title":[{"@value":"Majorana bands in topological superconductors","@language":"en"}],"Type":[{"@value":"Text","@language":"en"}],"URI":[{"@value":"http:\/\/hdl.handle.net\/2429\/53962","@language":"en"}],"SortDate":[{"@value":"2015-12-31 AD","@language":"en"}],"@id":"doi:10.14288\/1.0166331"}