Science, Faculty of
Physics and Astronomy, Department of
DSpace
UBCV
Pathak, Vedangi
Master of Science - MSc
University of British Columbia
Majorana quasi-particles can exist as zero-energy excitations bound to vortices present on the surface of a topological insulator that is proximity-coupled to a type-II superconductor. Such a system finds its natural realisation on the surface of the iron-based superconductor FeTe₀.₅₅Se₀.₄₅ which has been identified as a potential topological superconductor and is expected to host Majorana quasi-particles as zero-energy excitations on vortices. This thesis aims to explain the occurrence of Majorana vortex bound states present on such materials by constructing a model for a vortex lattice on periodic manifolds. A two-dimensional square lattice model is developed to capture the low energy physics of the surface states of a topological insulator proximity-coupled to an s-wave superconductor. Using the Franz-Tesanovic singular gauge transformation, multiple vortices can be incorporated in the system by circumventing the problem of branch cuts. To construct finite vortex lattice on periodic manifolds (torus in two dimensions), we need to account for the fluxes through the non-contractible loops of the torus, leading to certain correction terms appearing in the singular gauge transformation. This thesis provides the details of the construction of vortex lattice on a periodic manifold on the surface of topological insulator proximity-coupled to a superconductor and investigates the Majorana zero modes present on the vortices. The theory developed for periodic vortex lattice on the topological insulator-superconductor heterostructure is successful in replicating the experimental observations of Majorana vortex bound states on the surface of the iron-based superconductor FeTe₀.₅₅Se₀.₄₅.
Majorana bound states on a periodic superconductingvortex latticebyVedangi PathakM.Sc., Birla Institute of Technology and Science, Pilani - Goa, India, 2018B.E., Birla Institute of Technology and Science, Pilani - Goa, India, 2018A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMaster of ScienceinThe Faculty of Graduate and Postdoctoral Studies(Physics)The University of British Columbia(Vancouver)August 2020c© Vedangi Pathak 2020The following individuals certify that they have read, and recommend to theFaculty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled:Majorana bound states on a periodic superconducting vortex latticesubmitted by Vedangi Pathak in partial fulfillment of the requirements for thedegree of Master of Science in Physics.Examining Committee:Marcel Franz, Physics and AstronomySupervisorRobert Raussendorf, Physics and AstronomyExamining Committee MemberiiAbstractMajorana quasi-particles can exist as zero-energy excitations bound to vorticespresent on the surface of a topological insulator that is proximity-coupled to atype-II superconductor. Such a system finds its natural realisation on the sur-face of the iron-based superconductor FeTe0.55Se0.45 which has been identifiedas a potential topological superconductor and is expected to host Majorana quasi-particles as zero-energy excitations on vortices. This thesis aims to explain the oc-currence of Majorana vortex bound states present on such materials by construct-ing a model for a vortex lattice on periodic manifolds. A two-dimensional squarelattice model is developed to capture the low energy physics of the surface states ofa topological insulator proximity-coupled to an s-wave superconductor. Using theFranz-Tesˇanovic´ singular gauge transformation, multiple vortices can be incorpo-rated in the system by circumventing the problem of branch cuts. To construct fi-nite vortex lattice on periodic manifolds (torus in two dimensions), we need to ac-count for the fluxes through the non-contractible loops of the torus, leading to cer-tain correction terms appearing in the singular gauge transformation. This thesisprovides the details of the construction of vortex lattice on a periodic manifold onthe surface of topological insulator proximity-coupled to a superconductor and in-vestigates the Majorana zero modes present on the vortices. The theory developedfor periodic vortex lattice on the topological insulator-superconductor heterostruc-ture is successful in replicating the experimental observations of Majorana vortexbound states on the surface of the iron-based superconductor FeTe0.55Se0.45.iiiLay SummaryMajorana fermions are particles that are their own anti-particles. In the contextof condensed matter physics, Majorana fermions can occur as zero-energy ex-citations on vortices found on the surface of certain superconducting materials.Iron-based superconductor, FeTe0.55Se0.45, is one such candidate material whichcan host Majorana vortex bound states. Motivated by the experiments conductedon iron-based superconductors, we investigate Majorana zero modes on a peri-odic vortex lattice. Modelling a periodic superconducting vortex lattice whichcan host Majorana zero-modes is a non-trivial theoretical problem that we addressin this thesis. We employ the model developed here to explain the occurrence andcharacteristics of Majorana vortex bound states. The theory developed for peri-odic superconducting vortex lattice is successful in explaining certain experimen-tal observations of Majorana vortex bound states on the surface of the iron-basedsuperconductor, FeTe0.55Se0.45.ivPrefaceThis thesis is original, unpublished, independent work by the author, VedangiPathak, under the supervision of Professor Marcel Franz. All figures and theunderlying data have been generated by the author. The analyses in Chapter 2 andSections 3.1-3.2 from Chapter 3 use analytical derivations found in the existingliterature. Section 3.3 onwards is original work performed during the course ofthis project. The analytical derivations of Section 3.4 have been reported for thefirst time in this thesis.vContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vContents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Superconductor-proximitized topological insulator with vortex . . . 42.1 The Fu-Kane model . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Surface of topological insulator . . . . . . . . . . . . . . . . . . . 52.3 Vortex core states under open boundary conditions . . . . . . . . 113 Periodic vortex lattice . . . . . . . . . . . . . . . . . . . . . . . . . . 153.1 Franz-Tesˇanovic´ transformation . . . . . . . . . . . . . . . . . . 153.2 Derivation of phase factors . . . . . . . . . . . . . . . . . . . . . 183.3 Symmetry considerations . . . . . . . . . . . . . . . . . . . . . . 233.3.1 Internal Gauge Symmetry . . . . . . . . . . . . . . . . . 233.3.2 Particle-Hole Symmetry . . . . . . . . . . . . . . . . . . 273.4 Finite vortex lattice with periodic boundary conditions . . . . . . 29vi4 Majorana vortex bound states . . . . . . . . . . . . . . . . . . . . . 364.1 Experiments on FeTe0.55Se0.45 . . . . . . . . . . . . . . . . . . . 364.2 Majorana zero modes on a triangular vortex lattice . . . . . . . . 384.3 Majorana zero modes on a disordered triangular vortex lattice . . . 425 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . 50Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52A Splitting of zero-modes with µ . . . . . . . . . . . . . . . . . . . . . 54viiList of tables4.1 Simulation parameters and their approximate experimental equiv-alent. Here, the time reversal breaking mass term, m= 1, ensuresoptimal gap at all points in the Brillouin zone. . . . . . . . . . . . 39viiiList of figures2.1 Band structure correresponding to the Hamiltonian for square lat-tice in (2.4). Due to the absence of the mass term, there are fourzero energy solutions. Here, the zero energy solutions are shownat Γ= (0,0), X = (pi,0) and M = (pi,pi). The fourth solution is atY = (0,pi). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Band structure corresponding to square lattice TI in (2.6). Themass term breaks the time-reversal symmetry and creates a gap atall points except Γ = (0,0). This emulates the surface of a threedimensional topological insulator. . . . . . . . . . . . . . . . . . 72.3 Band structure for square lattice TI proximitized to a supercon-ductor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Band structure for square lattice TI at finite chemical potential. . . 92.5 Band structure corresponding to square lattice TI proximitized toa superconductor at finite chemical potential. . . . . . . . . . . . 102.6 Energy eigenvalues of Hamiltonian (2.11) without vortices andsubject to periodic boundary conditions. Dotted red lines signifythe superconducting gap at ∆0 =±0.18. Parameters: L= 20, ∆0 =0.18, µ =−0.44, λ = 1, m= 0.5. . . . . . . . . . . . . . . . . . 112.7 In-gap energy eigenvalues of Hamiltonian (2.11) with 1 vortexwith open boundary conditions. There are two zero modes present.Parameters: L= 30, ∆0 = 0.18, µ =−0.44, λ = 1, m= 0.5. . . . 122.8 Two dimensional local density of states. Vortex core located atthe center hosts the Majorana bound state. Parameters: L = 30,∆0 = 0.18, µ =−0.44, λ = 1, m= 0.5. . . . . . . . . . . . . . . 132.9 Local density of states at vortex. Red markers are located at peaks0,−0.1. Parameters: L= 30, ∆0 = 0.18, µ =−0.44, λ = 1, m= 0.5. 143.1 Example of A (red) and B (blue) vortex sublattices with 2 vor-tices per magnetic unit cell (black). The vortices are placed on thediagonal with vortex spacing to be half of the diagonal length. . . 16ix3.2 The lattice curl of the superfluid velocity gives the 2pi fluxes inthe plaquettes with vortices and zero otherwise. The configura-tion here is 2 vortices per 10×10 magnetic unit cell with vorticesplaced apart at half the length of the diagonal. . . . . . . . . . . . 213.3 Eigenvalues for two vortices per 10× 10 magnetic unit cell us-ing the FT-transformed-BdG Hamiltonian with periodic boundaryconditions. (a) Two zero modes are present for µ = 0 without anyother in-gap modes. (b) The zero modes split when µ 6= 0 andthere are other in-gap states. . . . . . . . . . . . . . . . . . . . . 223.4 Two sublattices ABAB and AABB can be created with 4 vortices(A-red, B-blue) per unit cell used to investigate the internal gaugesymmatery of the FT transformation. . . . . . . . . . . . . . . . . 243.5 Energy eigenvalues plotted for a finite lattice at Γ point compris-ing the 4 vortex configurations ABAB (blue) and AABB (red). Ev-idently, the eigenvalues for both configurations are different fromeach other. Parameters; ∆0 = 0.2, µ =−0.45, λ = 1, m= 0.5 . . . 253.6 Bandstructure plotted for the two choices of 4 vortex configurations-AABB (dotted red) and ABAB (solid blue). Parameters: ∆0 = 0.2,µ =−0.45, λ = 1, m= 0.5 . . . . . . . . . . . . . . . . . . . . . 263.7 Density of states plotted for the two choices of 4 vortex configu-rations - AABB (dotted red) and ABAB (solid blue). Clearly, thedensity of states remains same for both the gauge choices. Param-eters: ∆0 = 0.2, µ =−0.45, λ = 1, m= 0.5 . . . . . . . . . . . . 263.8 Configuration with 4 vortices with sublattice A (red) and B (blue)exhibiting particle-hole asymmetry. . . . . . . . . . . . . . . . . 273.9 Eigenvalues plotted for the configuration 3.8. The eigenvalues(blue) and their time-reversed counterpart (red) do not match. Theerror shown in green is of the order 10−2. Parameters: ∆0 = 0.2,µ =−0.45, λ = 1, m= 0.5 . . . . . . . . . . . . . . . . . . . . . 283.10 Hollow torus with two non-contractible loops [Figure adapted fromWikipedia]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.11 Eigenvalues plotted for the configuration 3.8 using the modifiedFT phases. The eigenvalues (blue) and their time-reversed coun-terpart (red) match now with negligible error shown in green. Pa-rameters: ∆0 = 0.2, µ =−0.45, λ = 1, m= 0.5 . . . . . . . . . . 33x3.12 Resolution of the ABAB and AABB internal gauge symmetry issue.Incorporating the modified FT phases gives the same spectrumnow irrspective of the choice of gauge. Parameters: ∆0 = 0.2,µ =−0.45, λ = 1, m= 0.5 . . . . . . . . . . . . . . . . . . . . . 333.13 Local density of states at vortex remains same irrespective of gaugechoice. Parameters: ∆0 = 0.2, µ =−0.45, λ = 1, m= 0.5 . . . . 343.14 Bandspectrum of the AABB (dotted red) and ABAB (solid blue)matches exactly using the mified FT phases. Parameters: ∆0 =0.2, µ =−0.45, λ = 1, m= 0.5 . . . . . . . . . . . . . . . . . . 354.1 Configurations with 8 and 32 vortices for 32×32 physical lattice. 384.2 Local density of states at vortex in the low field and high fieldregime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.3 Local density of states in the low field region. . . . . . . . . . . . 414.4 Local density of states in the low field region. . . . . . . . . . . . 414.5 Random move acceptance ratio dictated by Metropolis algorithmis shown in (a). Disorder characterised by mean-square deviationof vortex positions increases with temperature is shown in (b). . . 424.7 Temperature evolution of the zero-bias peak at vortex core locatedwithin ±∆r of [17,17] in the low field limit. The peak heightdecreases with temperature for this vortex. . . . . . . . . . . . . . 444.8 L=32, v = 8, ∆0 = 0.4, µ = −0.9. LDOS at zero-energy withincreasing disorder in vortex configuration from (a)-(f). . . . . . . 464.9 L=32, v = 8, ∆0 = 0.2, µ = −0.9. LDOS at zero-energy withincreasing disorder in vortex configuration from (a)-(f). . . . . . . 464.10 L=32, v = 32, ∆0 = 0.4, µ = −0.9. LDOS at zero-energy withincreasing disorder in vortex configuration from (a)-(f). . . . . . . 474.11 L=32, v = 32, ∆0 = 0.2, µ = −0.9. LDOS at zero-energy withincreasing disorder in vortex configuration from (a)-(f). . . . . . . 474.12 Local density of states at representative vortex positions for disor-der ∆r ≈ 1.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.13 Histograms of lowest energy peak for disordered vortices. . . . . . 494.14 Histograms of next to lowest energy peaks for 32 disordered vor-tices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49A.1 Zero mode splitting with change in chemical potential µ for thelow field case with 8 vortices and the high field case with 32 vortices. 54xiAcknowledgementsI would like to express my gratitude to Professor Marcel Franz, my research super-visor, for giving me an opportunity to work in the field of theoretical condensedmatter physics. His advice and vision have been the key driving points behindthis work. His patience and willingness for discussions helped me understand theintricacies involved in this project.I am extremely grateful for the exchange of ideas and enriching discussionsI have had with Stephan Plugge as they played a major role in shaping my workas it stands today. I am thankful for the helpful and motivating discussions withPedro Lopes. I would like to thank Tianyu Liu and Rafael Haenel for their sug-gestions at the beginning of the project. I am also grateful for the help, advice andmotivation provided by my research group members at all times.I am also thankful to Professor Robert Raussendorf for being the second ex-aminer of this thesis and providing his extremely valuable inputs.I thank all my friends from UBC and BITS for helping me get through themost demanding times in the last two years.Lastly, I will like to thank my family for helping me stay calm in the most hec-tic of times. Their proud faces provided me with the required inspiration duringthe thesis.xiiTo my sister-whose interest in physics is elusive like the Majorana,but who always keeps me grounded.xiiiChapter 1IntroductionMajorana fermions are particles that are their own anti-particles. In the contextof condensed matter physics, where electrons and holes serve the purpose of theparticles and anti-particles, Majorana fermions occur as non-trivial emergent ex-citations [1]. The Majorana quasi-particles provide an insight into exotic funda-mental physics along with exciting technological applications such as topologicalquantum computation [2].The pertinent question then is to find out what kind of condensed matter sys-tems can host Majorana quasi-particles. In ordinary metals, any electron and holeexcitations annihilate due to their opposing charges and cannot coexist as Ma-jorana fermions. This is not an issue in platforms where charge conservationis violated such as that in Cooper pair condensation in a superconductor due towhich a quasi-particle can exist as an electron-hole superposition. However, thesequasi-particle operators are still distinguishable from quasi-hole operators due tothe presence of spin. Kitaev proposed [2], that a spinless p-wave superconduc-tor in one dimension and correspondingly a spinless px+ ipy in two dimensionswill host Majorana quasiparticles. The one dimensional p-wave superconductorexhibits a topological phase in its bulk and hosts Majorana quasi-particles as zeroenergy excitations at its boundaries. In two dimensional spinless px+ ipy, Majo-rana zero mode excitations occur at topological defects called vortices. However,the main issue is that electrons are not spinless and such low-dimensional p-wavesuperconductivity is extremely rare in nature which would make the detection andsubsequent control of Majorana quasi-particles in such systems even harder.The seminal Fu-Kane proposal [3] shows that the proximity effect between anordinary s-wave superconductor and the surface states of a strong topological in-sulator leads to the formation of a state equivalent to the spinless two-dimensionalpx+ ipy superconductor. This model provides the practical realisation of a systemthat is the two-dimensional equivalent of Kitaev’s toy lattice model. This two-dimensional state hosts the zero energy Majorana bound states at vortices.1Recent experimental findings suggest that such a system is naturally realisedin the iron based superconductor FeTe0.55Se0.45 [4]. The spin helical surface stateswith superconducting gap induced by the bulk allow one to potentially observe thezero energy Majorana bound states at the vortices on the surface. Moreover, due toits low Fermi energy, zero energy Majorana bound state (if present) should remainrelatively isolated from the higher energy, in-gap, Caroli-de Gennes-Matriconbound states, thereby allowing for a clean detection of a zero-bias peak (ZBP)in the STM conductance measurements. Recently, a major research effort wasdirected to the search for the zero energy Majorana bound states with varying de-grees of sophistication in the experimental techniques as found in [5], [6], [7],[8]. Although in theory, the platform should host robust Majorana zero modes,one group [6] reported the absence of Majorana zero-bias peak and observationof non-zero Caroli-de Gennes-Matricon bound states while another group [5] re-ported the presence of Majorana zero modes but which decrease in intensity withtemperature. These seemingly contradictory observations of a zero-bias peakwere investigated further at a remarkably higher STM resolution in [7]. In thishigh-resolution STM experiment, conducted on FeTe0.55Se0.45, zero-bias peaksare clearly observed. However, all vortices do not uniformly exhibit the presenceor absence of a zero-bias peak. The probability of observing a ZBP monotonicallydecreases with increasing magnetic field. The motivation for this project was todevelop an effective model for such a system at low energies and investigate thepresence/absence and characteristics of the Majorana zero energy modes in a vor-tex lattice such as that found on the surface of FeTe0.55Se0.45.A singular vortex can be included in the superconducting order parameterof the topological insulator-superconductor heterostructure under open boundaryconditions. To study quasi-particles in a vortex lattice, we need the physical ob-servables of the problem to exhibit the periodicity of the vortex lattice. However,if we include the phase of the vortex lattice in the superconducting order param-eter, it is impossible to impose periodic boundary conditions on the model understudy. On encircling a vortex, an electron acquires a phase pi (unlike a Cooper pairwhich acquires a phase 2pi), leading to branch cuts at each vortex which preventthe direct imposition of periodic boundary conditions. One way to deal with theproblem of branch cuts is to divide the vortices into two subgroups. If particlesand holes are constrained to see only one subgroup of vortices each, both particlesand holes now acquire a phase of 2pi on encircling a vortex in order to compensatefor the fact that they see only half the vortices in the system. This would allow us2to circumvent the issue of branch cuts and make the Hamiltonian periodic. Thisis the essence of the Franz-Tesˇanovic´ singular gauge transformation [9, 10].In order to study Majorana quasi-particles, we build the model of a topolog-ical insulator proximitized to an s-wave superconductor and employ the Franz-Tesˇanovic´ singular gauge transformation for the inclusion of vortices. However,construction of a finite vortex lattice with periodic boundary conditions leads tocomplications such as the breaking of the particle-hole symmetry. In this thesis,we propose a modification term to the original theory of the singular gauge trans-formation by taking into account the fluxes arising from non-contractible loops ofthe torus manifold. This leads to the formation of a physical vortex lattice on aperiodic manifold. We use this model to understand the experimental observationspertaining to Majorana vortex bound states on FeTe0.55Se0.45.Following the Fu-Kane proposal, an equivalent square lattice model has beendeveloped in Chapter 2 of this thesis in order to capture the low energy physics ofthe surface states of a topological insulator proximity-coupled to an s-wave super-conductor. The local density of states for a single vortex incorporated with openboundary conditions on this system is studied.Chapter 3 provides detailed calculations for the construction of a vortex latticeon the superconductor-proximitized topological insulator surface model devel-oped in Chapter 2. Using the Franz-Tesˇanovic´ singular gauge transformation [9],multiple vortices can be incorporated in the system by circumventing the prob-lem of branch cuts. However, certain modification terms arising from the fluxesthrough the non-contractible loops of torus (two-dimensional periodic manifold)need to be introduced in the original singular gauge transformation to create a fi-nite vortex lattice with periodic boundary conditions.The theory and techniques developed in Chapters 2 and 3 are applied to inves-tigate the observations of Majorana vortex bound states on periodic vortex latticesin Chapter 4. A periodic triangular lattice, such as that found on FeTe0.55Se0.45,is taken as a reference. Majorana vortex bound states are explored for differentparameters and magnetic field strengths. The effect of disorder in the vortex lat-tice due to thermal fluctuations is studied. We prove the merit of the model bysuccessfully replicating the experimental observations of Majorana vortex boundstates on FeTe0.55Se0.45.3Chapter 2Superconductor-proximitizedtopological insulator with vortexKitaev’s model for obtaining Majorana fermions in two dimensions requires aspinless px + ipy superconductor. This Chapter briefly discusses the Fu-Kanemodel [3] which shows that the proximity effect between an s-wave supercon-ductor and the surface of a strong topological insulator is equivalent to a spinlesspx+ ipy superconductor, hosting Majorana bound states at the vortices. The in-troduction to the Fu-Kane Hamiltonian is followed by a detailed developmentof a two-dimensional lattice model for the surface of the topological insulatorproximity-coupled to a superconductor [11]. Majorana zero modes are obtainedby exact diagonalisation of the effective low energy model Hamiltonian with openboundary conditions.2.1 The Fu-Kane modelStrong three-dimensional topological insulator (TI) such as Bi2Se3, Bi2Te3 andSb2Te3, exhibit fully gapped bulk states and undergo a band inversion at oddnumber of points in the Brillouin zone due to strong spin-orbit interactions. Asa result, an odd number of topologically protected, massless Dirac fermions arepresent on its surface. If an ordinary s-wave superconductor is deposited on thethree-dimensional TI, Cooper pairs may tunnel from the superconductor to thesurface of the TI due to the superconducting proximity effect. This induces a su-perconducting energy gap in the surface states of the topological insulator. Theresultant state resembles a spinless px+ ipy topological superconductor but doesnot break TR symmetry.The Fu-Kane Hamiltonian [3, 11] describing the surface of a topological insu-4lator proximity-coupled with an s-wave superconductor isHFK =∫d2rΨˆ†rHFK(r)Ψˆr. (2.1)Here, Ψˆr =(cr↑,cr↓,c†r↓,−c†r↑) is the Nambu space spinor andHFK is the Bogoliubov-de Gennes (BdG) Hamiltonian matrix for the Fu-Kane Hamiltonian given byHFK = τz[vFσ .(p− τz ecA)−µ]+ τx∆(r), (2.2)where vF is the Fermi velocity, p = −i~∇ is the momentum operator, ∆(r) is thesuperconducting order parameter, σ are the Pauli matrices in the spin space andτ are the Pauli matrices in the Nambu space. In the absence of magnetic field,B = ∇×A, and real ∆(r), the Hamiltonian in (2.2) exhibits a particle-hole (PH)symmetry, Ξ= σ yτyκ (Ξ2 = 1), and a physical time-reversal (TR) symmetry, Θ=iσ yκ (Θ2 =−1), where κ is the complex conjugation operator. The time-reversalsymmetry is lifted in the presence of a Zeeman field and vortices. In the presenceof vortices, superconducting order parameter acquires a phase, ∆(r) = ∆0(r)eiθ(r).∆0(r) is the magnitude of superconducting order parameter and θ(r) is the phasethat encapsulates the information about the phase-winding around the vortices.Majorana fermions exist as vortex bound states at zero energy in such a setup.2.2 Surface of topological insulatorA strong three-dimensional topological insulator has a fully gapped bulk and anodd number of massless Dirac fermions on its surface. Since, the band structureof a strong topological insulator is time-reversal invariant, perturbations obeyingtime-reversal symmetry cannot create a gap on the surface state. For a materialwith a single Dirac cone located the center of the surface Brillouin zone, an effec-tive, low-energy Hamiltonian in the continuum limit is given by,heff = λ (σ xkx+σ yky), (2.3)where σ i are the Pauli matrices in the spin space and λ is a model parameterequivalent to the hopping parameter in a tight-binding model. Next, we need tochoose a lattice system for which we get the above Hamiltonian in the low energycontinuum limit. Therefore, on regularising this Hamiltonian on a simple square5Figure 2.1: Band structure correresponding to the Hamiltonian for square lattice in(2.4). Due to the absence of the mass term, there are four zero energy solutions. Here,the zero energy solutions are shown at Γ= (0,0), X = (pi,0) and M = (pi,pi). The fourthsolution is at Y = (0,pi).lattice model, we obtain,h0(k) = λ (σ xsinkx+σ ysinky). (2.4)The number of zero energy solutions increase from one to four on going from(2.3) to (2.4) as shown in the band structure in Figure 2.1. This is in accordancewith the Nielsen-Ninomyia theorem which states that it is impossible to constructa purely two-dimensional, TR invariant lattice model with an odd number of mass-less Dirac fermions. The two-dimensional surface states of a strong TI with oddnumber of Dirac cones are holographic since they cannot exist without the topo-logically non-trivial three-dimensional bulk.One way to overcome the constraint imposed by the Nielsen-Ninomyia theo-rem while respecting time-reversal symmetry is to construct a pair of strong topo-logical insulator surfaces as shown in [12]. If we forgo the necessity to preservetime-reversal symmetry then we can add a mass term proportional to σ z given byMk = m[(2− coskx− cosky)− 14(2− cos2kx− cos2ky)]. (2.5)6Figure 2.2: Band structure corresponding to square lattice TI in (2.6). The mass termbreaks the time-reversal symmetry and creates a gap at all points except Γ= (0,0). Thisemulates the surface of a three dimensional topological insulator.The additional mass term makes the Hamiltonian,h0(k) = λ (σ xsinkx+σ ysinky)+σ zMk−µ, (2.6)where µ is the chemical potential. This time-reversal symmetry breaking massterm gaps out all the Dirac fermions except the one present at Γ ≡ (0,0) as de-picted in Figure 2.2. Adding such a TR symmetry breaking term will not compro-mise the physics of the interface between a TI and a superconductor that we wantto study because the presence of vortices will ultimately break the time-reversalsymmetry anyway [11]. Moreover, the effect of the time-reversal breaking is smallaround the Γ point as Mk ∝ k4.To incorporate the superconducting proximity effect in the system, we use theBogoliubov-de Gennes Hamiltonian given byHBdG =(h0(k) ∆∆∗ −σ yh∗0(−k)σ y), (2.7)where ∆ is the superconducting order parameter proportional to the identity matrixin the spin space. The term h0(k) = hk(λ ,m,µ) is the Hamiltonian of particles7and −σ yh∗0(−k)σ y = −hk(λ ,−m,µ) is understood as the Hamiltonian of holesin this basis. The energy eigenvalues for the the Hamiltonian (2.7) areε(k) =±[λ 2(sin2kx+ sin2ky)+M2k +∆2+µ2±2√λ 2µ2(sin2kx+ sin2ky)+M2k (∆2+µ2)] 12. (2.8)Figures 2.1 and 2.2 depict the energy spectrum of the normal state Hamiltonianin (2.7) when the mass term is zero and non-zero respectively. In both the cases,the spectrum is particle-hole symmetric as ensured by the BdG Hamiltonian andis degenerate in spin space. Figure 2.3 shows the effect of adding a non-zero su-perconducting order parameter where the spectrum is still particle-hole symmetricbut loses spin space degeneracy. Changing the chemical potential to a non-zerovalue results in the formation of Dirac cones at ±µ centered at the Γ point in thek-space as shown in Figure 2.4. Figure 2.5 depicts the band structure with a finiteµ and ∆. Here, we focus only on the low energy physics around the Dirac cones.In Chapter 4, we will be taking the material FeTe0.55Se0.45 (with ∆= 1.8 meVand µ =−4.4 meV) as a reference for which the effective band structure based onthe model developed here is depicted in Figure 2.5.On performing a Fourier transform on the Hamiltonian in (2.6), the model forthe surface of a topological insulator isH0 = iλ2 ∑r,α(ψ†rσαψr+α −h.c.)+∑rψ†r(32mσ z−µ)ψr− m8 ∑r,α(4ψ†rσzψr+α −ψ†rσ zψr+2α +h.c.),(2.9)where α = x,y and ψr = (cr↑,cr↓)T is a two component spinor at position r on thelattice. λ and m are parameters. The matrix corresponding to this Hamiltonianwill have the dimensions of 2L2× 2L2 (L2 lattice sites and two spin degrees offreedom), making the basis ψ = (c1↑,c1↓, ...,cr↑,cr↓, ...,cL2↑,cL2↓)T .On proximity-coupling the TI with a superconductor, the Cooper pairs tunnelinto the interface between them. The BdG Hamiltonian for this interface is written8Figure 2.3: Band structure corresponding to square lattice TI proximitized to a super-conductor. Due to proximity effect, a superconducting gap (red) develops in the spec-trum. Spin degeneracy is broken (except at Γ point).Figure 2.4: Band structure corresponding to square lattice TI proximitized to a super-conductor with finite chemical potential (grey) without superconductivity.9Figure 2.5: Band structure corresponding to square lattice TI proximitized to a super-conductor at finite chemical potential.asH = H0+∑r(∆rc†r↑c†r↓+h.c.), (2.10)where ∆r is the superconducting order parameter at site r. The basis now ex-pands to include both particle and hole states given by the Nambu spinor ψ =(c1↑,c1↓, ...,cr↑,cr↓, ...,cL2↑,cL2↓,c†1↓,−c†1↑, ...,c†r↓,−c†r↑, ...,c†L2↓,−c†L2↑)T . In thisbasis, the matrix ∆(r) of dimensions 2L2×2L2 becomes proportional to the iden-tity matrix. We can derive the following 4L2× 4L2 dimensional matrix form forthe BdG Hamiltonian,(H0(λ ,µ,m) ∆(r)∆∗(r) −H0(λ ,µ,−m))(un(r)vn(r))= En(un(r)vn(r)), (2.11)where H0(λ ,µ,m) is the Hamiltonian of particles given by (2.9), −H0(λ ,µ,−m)is the Hamiltonian of holes which is the time-reversed counterpart of H0(λ ,µ,m).∆(r) is the position dependent superconducting order parameter, un(r) and vn(r)are the particle and hole eigenstates respectively and En are the corresponding en-ergy eigenvalues. It is important to note here that the particle-hole symmetry ofthe BdG Hamiltonian is a constraint that must be obeyed.10100 75 50 25 0 25 50 75 100n0.60.40.20.00.20.40.6E nFigure 2.6: Energy eigenvalues of Hamiltonian (2.11) without vortices and subject toperiodic boundary conditions. Dotted red lines signify the superconducting gap at ∆0 =±0.18. Parameters: L= 20, ∆0 = 0.18, µ =−0.44, λ = 1, m= 0.5.Figure 2.6 depicts the energy eigenvalues after diagonalisation of (2.11) underperiodic boundary conditions for a uniform superconducting order parameter. Ev-idently, there are no in-gap modes present.2.3 Vortex core states under open boundaryconditionsFor a uniform system, the order parameter is given by ∆(r) = ∆0. In the pres-ence of vortices, the system becomes non-uniform for which we need to considerposition space Hamiltonian with open-boundary conditions. In the presence of avortices located at R j = (x j,y j) on an L×L square lattice, the off-diagonal order1115 10 5 0 5 10 15n0.200.150.100.050.000.050.100.150.20E nFigure 2.7: In-gap energy eigenvalues of Hamiltonian (2.11) with 1 vortex with openboundary conditions. There are two zero modes present. Parameters: L= 30, ∆0 = 0.18,µ =−0.44, λ = 1, m= 0.5.parameter terms in the BdG Hamiltonian, are modelled as∆(r) = ∆0∏jtanh( |r−R j|ξ0)eiφ j , (2.12)where |r−R j| is the distance from the center of the jth vortex, φ(r) is thephase with tan(φ j) =|y−y j||x−x j| and ξ0 =λpi∆0 is the superconducting coherence length.Figure 2.7 depicts the in-gap eigenvalues calculated for the 30×30 square latticewith a single vortex at the center. The zero energy solutions correspond to theMajorana bound states. One of the zero modes is present in the vortex while theother one presents itself as an edge mode. The other states at non-zero energyvalues are Caroli-de Gennes-Matricon bound states [13].We can now evaluate the local density of states. Local density of states is givenby,ρ(ε,r) = ∑En≥0[|un(r)|2δ (ε−En)+ |vn(r)|2δ (ε+En)], (2.13)where En are the energy eigenvalues and the delta function is calculated numeri-120 5 10 15 20 25 300510152025300.0020.0040.0060.0080.0100.0120.014Figure 2.8: Two dimensional local density of states. Vortex core located at the centerhosts the Majorana bound state. Parameters: L = 30, ∆0 = 0.18, µ = −0.44, λ = 1,m= 0.5.cally using the Gaussian function δ (βE) = 1/cosh(βE) with β chosen such thatthe FWHM is approximately 0.005 in the energy units used here.Figure 2.8 shows the local density of states on the surface evaluated at zero en-ergy where it is clear that the zero energy peak is bound to the vortex. Figure 2.9shows the local density of states within the superconducting gap at the vortex. Thedominant central peak is the Majorana zero mode. The in-gap smaller peaks, suchas that in Figure 2.9, are the Caroli-de Gennes-Matricon (CdGM) bound states. Inthe limit ∆ µ , these in-gap states are expected to be of the order of ±n∆2µ , withinteger n, in the presence of a zero-bias peak. In Figure 2.9, the next dominantpeak is at −0.1 which is of the same order as ±∆2µ ≈ 0.08. There are other lessdominant peaks due to spurious edge modes. In the absence of a zero-bias peak,the CdGM modes are expected to be located at ±(n+ 12) ∆2µ .The method described in this section can be used to construct an arbitrary num-ber of vortices. However, as evident from Figures 2.9 and 2.8, there are spuriousedge modes present. In an experimental setup, typically, a very large number ofvortices forming a periodic lattice are studied. In a theoretical model, it is nearly130.15 0.10 0.05 0.00 0.05 0.10 0.15Energy 0.0000.0020.0040.0060.0080.0100.0120.014LDOS(x=15, y=15)Figure 2.9: Local density of states at vortex. Red markers are located at peaks 0, −0.1.Parameters: L= 30, ∆0 = 0.18, µ =−0.44, λ = 1, m= 0.5.impossible to study such a large system with the existing computational resources.However, to effectively model such a realistic lattice, we can study a finite vortexlattice that is repeated periodically in space. The following Chapter discusses theconstruction of multiple vortices forming a periodic Abrikosov lattice.14Chapter 3Periodic vortex latticeTo describe a realistic vortex lattice we need to incorporate periodic boundaryconditions. Moreover, magnetic field must be included in the theory. Quasi-particle dynamics in the vortex phase where superconductivity coexists with mag-netic field is a highly evolved problem that is tackled by the Franz-Tesˇanovic´ (FT)transformation, outlined in Section 3.1. This enables us to circumvent the prob-lem of branch cuts which prevent the inclusion of periodic boundary condition inthe Hamiltonian. The calculative details of this singular gauge transformation asimplemented in the model are discussed and derived in Section 3.2. Section 3.3highlights an important issue related to the internal gauge symmetry and particle-hole symmetry of the system. Section 3.4 handles this issue for a finite vortexlattice on a torus and proposes a correction factor to the original FT phase factors.3.1 Franz-Tesˇanovic´ transformationMagnetic field can be included in (2.11) using Peierls’ substitution, according towhich all hopping terms ψ†rψr+α transform into ψ†rψr+αexp[−i∫ r+αr (e/}c)~A.~dr]where~A is the magnetic vector potential. This gives the magnetic field ~B accordingto ~B= ~∇×~A. The BdG Hamiltonian matrix from (2.11) now reads(∑r,α hr,r+α(λ ,m)e−i∫ r+αre}c~A.~dr ∆(r)∆∗(r) −∑r,α hr,r+α(λ ,−m)ei∫ r+αre}c~A.~dr)+(∑r hr,r(m,µ) 00 −∑r hr,r(−m,µ)),(3.1)where H0(λ ,µ,m) is expressed as a sum of hopping terms, hr,r+α(λ ,m), and on-site terms, hr,r(m,µ).Assuming there are 2n vortices on the lattice under consideration, we can di-vide the vortices into two groups, A and B, of n vortices each. Figure 3.1 shows15(a) Vortices forming two squaresublattices- A (red) and B (blue)0 2 4 6 802468(b) Magnetic unit cell with 2 vortices -A (red) and B (blue)Figure 3.1: Example of A (red) and B (blue) vortex sublattices with 2 vortices permagnetic unit cell (black). The vortices are placed on the diagonal with vortex spacingto be half of the diagonal length.the A and B vortex sub-lattice for a system with two vortices per magnetic unitcell.The total phase contributions to the superconducitng order parameter due tovortices in groups A and B are φA(r) and φB(r) respectively such that they obey~∇×~∇φg = 2pi zˆ∑jδ (r−R jg), (3.2)where R jg is the position of jth vortex of type g= A,B. With this, we can define aunitary transformation,U =(eiφA(r) 00 e−iφB(r)), (3.3)where φA(r)+φB(r) = φ(r) andU has the same dimensions as the BdG Hamilto-nian (2.10). Applying this on the BdG Hamiltonian,H →U−1H U , removes thephase factors from the superconducting order parameter terms in the off-diagonal16blocks of (2.10). The modified BdG Hamiltonian matrix is(∑r,α hr,r+α(λ ,m)eiVαA (r) ∆0∆∗0 −∑r,α hr,r+α(λ ,−m)e−iVαB (r))+(∑r hr,r(m,µ) 00 −∑r hr,r(−m,µ)),(3.4)where VA and VB are the new phase factors applied to the hopping terms. Thesephase factors are given byV αg (r) =∫ r+αr(~∇φg(r)− e}c~A).~dr=2piΦ0∫ r+αr(Φ02pi~∇φg(r)−~A).~dr(3.5)where g = A,B and Φ0 = hc/e is the flux quantum. From equation (3.2), we candeduce that the phases in the unitary transformation (3.3) are not gauge invariant.However, the unitary transformation in combination with the external magneticfield included via Pierls’ substitution gives the modified phases in the hoppingterms as provided in (3.5) which are manifestly gauge invariant. Hence, (3.3) is asingular gauge transformation. The term in the brackets in (3.5) can be treated asan effective vector potential resulting in an effective magnetic field,~Bg = ~∇×(~A− Φ02pi~∇φg(r))= ~B−Φ0zˆ∑jδ (r−R jg).(3.6)One can interpret that quasi-particles experience an effective magnetic field ~BAwhich comprises a uniform physical magnetic field B and delta function spikesof opposing polarity at vortices of type A. Similarly, quasi-holes experience aneffective magnetic field ~BB. The reason for choosing an equal number of vor-tices (n) of type A and B is so that ~Bg is zero when averaged over the lattice. Inother words, there is one flux spike arising due to each vortex of type A or typeB that compensates for each flux quantum of the physical magnetic field. Thecondition 〈 ~Bg〉 = 0 also allows us incorporate periodic boundary conditions inthe transformed BdG Hamiltonian (3.4) which had only allowed open boundaryconditions prior to the application of the unitary transformation (3.3). Therefore,17we can create a periodic vortex lattice with sub-lattices of type A and B.As a practical matter, we just need to calculate the phase factors in (3.5) in aconvenient manner followed by diagonalization of the Hamiltonian (3.4) to findthe spectrum.3.2 Derivation of phase factorsMaxwell’s equation for magnetic field in a superconductor with super-current ~j is~∇×~B= µ0~j= µ0nse~vs,(3.7)where µ0 is the permeability, ns is the carrier density. The super-fluid velocity,~vsis~vs =}m(12~∇φ(r)− e}c~A). (3.8)Taking curl of equation (3.7) results in~∇× (~∇×~B) = µ0nse(~∇×~vs)⇒ ~∇(~∇.~B)−∇2~B= µ0nse}2m(2pi zˆ∑jδ (r−R j)− 2e}c~B).(3.9)Here, the j summation runs over all vortices (of type A and B both). Using ~∇.~B=0 and rearranging terms, we get the London’s equation in the presence of vortices,~B−λ 2s ∇2~B=Φ02zˆ∑jδ (r−R j), (3.10)where λs=√mcµ0nse2is the London penetration depth andΦ0 = hce is the flux quan-tum as before.18Transferring to Fourier space using ~B=∫ d2k(2pi)2 eik.r~Bk, we get,~Bk =Φ02zˆ(∑ j e−ik.Rj1+λ 2s k2)=Φ02zˆ∑ j e−ik.R jA +∑ j˜ e−ik.R j˜B1+λ 2s k2 . (3.11)We define superfluid velocities for the A and B vortex groups as~vg= }m(~∇φg− e}c~A)with g= A,B such that~vs = ~vA+~vB2 . The phase factors in (3.5) can be expressed interms of superfluid velocities asV αg (r) =m}∫ r+αr~vg.~dr. (3.12)Taking curl of~vg and going over to the Fourier space gives,ik×~vgk = 2pi}m(zˆ∑je−ik.Rjg−~BkΦ0)=2pi}mzˆ∑je−ik.Rjg− 12∑ j e−ik.R jA +∑ j˜ e−ik.R j˜B1+λ 2s k2 . (3.13)A vector product with ik on both sides of the equation above and using k.~vgk = 0results in,~vgk =2pi}m(ik× zˆk2)∑je−ik.Rjg− 12∑ j e−ik.R jA +∑ j˜ e−ik.R j˜B1+λ 2s k2 . (3.14)In the limit λ 2s k2 1, the equation above can be approximated as,~vgk =2pi}m(ik× zˆ)[∑ j e−ik.Rjgλ−2s + k2]. (3.15)19Converting back to position space, we get~vg =2pi}m∫ d2k(2pi)2(ik× zˆ)[∑ j eik.(r−Rjg)λ−2s + k2]. (3.16)In this equation, it is implicit that the sum runs over all vortex positions R jg oftype g for an infinite lattice. The infinite lattice comprises magnetic unit cells ofdimensions L×L repeating periodically in space. The centres of these magneticunit cells are given by ~Om and the position of vortices of type g inside these unitcells are given by Rng. The sum in the equation (3.25) above can be split into twodistinct summations as∑je−ik.Rjg =∑me−ik. ~Om∑ne−ik.Rng, (3.17)where m runs over all magnetic unit cells on the infinite lattice and n runs overvortices of type g in one unit cell. The sum over magnetic unit cells is simplifiedas∑me−ik. ~Om= N∑~Gδk,~G, (3.18)where ~G = 2piL (nx,ny) is the reciprocal lattice vector of ~Om and N are the num-ber of magnetic unit cells. Now, we can discretize the intergral over momentumspace using∫ d2k(2pi)2 → 1NL2 ∑k. Using equations (3.25)-(3.18), we can write theexpression for the superfluid velocity as~vg =2pi}mL2∑~G(i~G× zˆ)[∑n ei~G.(r−Rng)λ−2s +G2]. (3.19)For a magnetic unit cell comprising two vortices on a 10×10 physical lattice,taking the lattice curl of the superfluid velocity gives the 2pi fluxes in the plaquetteswith vortices and zero otherwise as shown in Figure 3.2. The phase factors from(3.12) are then calculated as,V αg (r) =2piL2 ∑~G{(i~G× zˆλ−2s +G2).(∑n∫ r+αrei~G.(r−rng)~dr)}. (3.20)In principle, the summation in (3.20) is over an infinite number of reciprocal lat-200 2 4 6 8 1002468100123456Figure 3.2: The lattice curl of the superfluid velocity gives the 2pi fluxes in the plaque-ttes with vortices and zero otherwise. The configuration here is 2 vortices per 10× 10magnetic unit cell with vortices placed apart at half the length of the diagonal.tice vectors ~G. In practice, we employ a Gaussian soft cutoff of the type e−α |G|2G2max ,where G2max is the upper limit and α is some constant. The soft cutoff representsthe vortex core size and is necessary because the London model used to calculatethe superfluid velocity, assumes constant SC order parameter amplitude which isno longer true close to the vortex center. Additionally, these phase factors arepurely real ensuring the hermiticity of the Hamiltonian.Upon inclusion of the phase factors from (3.20) on the hopping terms of theBdG Hamiltonian (2.11) to create two vortices per magnetic unit cell with peri-odic boundary conditions, we get two zero modes corresponding to the Majoranabound states at the vortices as shown in Figure 3.3. For µ = 0, the only in-gapstates present are the Majorana zero modes because of the inclusion of periodicboundary conditions. For µ 6= 0, the degeneracy at zero may lift due to inter-vortextunneling which depends on the inter-vortex distance, µ and the superconductingcoherence length [14, 15].Evidently, the theory developed so far works well for a periodic vortex lattice2110.0 7.5 5.0 2.5 0.0 2.5 5.0 7.5n0.80.60.40.20.00.20.40.60.8E n(a) ∆0 = 0.2, µ = 0, λ = 1, m= 0.510.0 7.5 5.0 2.5 0.0 2.5 5.0 7.5n0.40.20.00.20.4E n(b) ∆0 = 0.2, µ =−0.45, λ = 1, m= 0.5Figure 3.3: Eigenvalues for two vortices per 10× 10 magnetic unit cell using the FT-transformed-BdG Hamiltonian with periodic boundary conditions. (a) Two zero modesare present for µ = 0 without any other in-gap modes. (b) The zero modes split whenµ 6= 0 and there are other in-gap states.22with two vortices per unit cell. Extending this to incorporate more vortices permagnetic unit cell requires a careful consideration of the internal gauge symmetryand the particle-hole symmetry as discussed in the following sections.3.3 Symmetry considerations3.3.1 Internal Gauge SymmetryIn the formulation of the singular gauge transformation, the choice of A and Bvortex sub-lattice was entirely arbitrary. For a magnetic unit cell with two vor-tices, upon exchanging the A and B assignment of the vortices, we obtain thesame spectrum as shown in Figure 3.3. Hence, irrespective of the choice of A-Bconfiguration, the physical observables should remain the same. To check thisinternal gauge symmetry explicitly for higher number of vortices, we choose asystem with four vortices per magnetic unit cell. There are two distinct choices ofA-B sub-lattices, ABAB and AABB, as illustrated in Figure 3.4.It was shown in [10] that for a linearized BdG Hamiltonian in momentumspace, the internal gauge symmetry of the ABAB and AABB configurations isbroken due to the interplay between strong Aharonov-Bohm scattering and theunbounded Dirac spectrum of the continuum Hamiltonian. However, it is ex-pected that the internal gauge symmetry should be preserved in the lattice versionof the BdG Hamiltonian of the type in (2.11), since the spectrum is bounded andthe discrete lattice spacing regularises the behaviour of wave functions near thevortex core.To test this, we plot the energy eigenvalues for the ABAB and AABB config-urations using the phase factors derived in Section 3.2. The two configurations donot exhibit the same eigenvalues, as shown in Figure 3.5. The difference betweenthe eigenvalues in the two configurations is of the order 10−2 which is quite high.This contradicts the expectation that the internal gauge symmetry is valid at allpoints for the lattice BdG Hamiltonian. In other words, for a finite lattice systemwith a periodic geometry, the internal gauge symmetry assumption is invalid.Probing this issue further, we calculate the bandspectrum of an infinite pe-riodic lattice with four vortices per unit cell. An infinite periodic lattice can begenerated by using the Bloch Hamiltonian Hk→ e−ik.rU−1H Ueik.r, which has23ABAB(a) ABAB latticeAABB(b) AABB lattice0 2 4 6 802468ABAB(c) ABAB lattice in one magnetic unitcell0 2 4 6 802468AABB(d) AABB lattice in one magnetic unitcellFigure 3.4: Two sublattices ABAB and AABB can be created with 4 vortices (A-red,B-blue) per unit cell used to investigate the internal gauge symmatery of the FT trans-formation.2420 15 10 5 0 5 10 15 20n0.40.20.00.20.4E nABABAABBFigure 3.5: Energy eigenvalues plotted for a finite lattice at Γ point comprising the 4vortex configurations ABAB (blue) and AABB (red). Evidently, the eigenvalues for bothconfigurations are different from each other. Parameters; ∆0 = 0.2, µ = −0.45, λ = 1,m= 0.5the effect of modifying the hopping terms of the BdG matrix (3.4) as follows,(∑r,α hr,r+α(λ ,m)eiVαA (r)eik.α ∆0∆∗0 −∑r,α hr,r+α(λ ,−m)e−iVαB (r)eik.α)+(∑r hr,r(m,µ) 00 −∑r hr,r(−m,µ)).(3.21)The band structures for the ABAB and AABB configurations are plotted inFigure 3.6. Clearly, the band structures are not aligned, thereby corroborating thefact that the eigenvalues plotted at the Γ = (0,0) point in Figure 3.5 are indeeddifferent. However, the density of states plotted from the band spectra of the twoconfigurations matches exactly, as depicted in Figure 3.7. Therefore, all physicalobservables for the infinite periodic lattice remain the same for both the config-urations, making the two configurations equivalent. The fact that the density ofstates matches perfectly implies that the band structures of the two configurationsare simply related by a rigid shift in the momentum space.25X Mk0.40.20.00.20.4E kFigure 3.6: Bandstructure plotted for the two choices of 4 vortex configurations- AABB(dotted red) and ABAB (solid blue). Parameters: ∆0 = 0.2, µ =−0.45, λ = 1, m= 0.50.0 0.5 1.0 1.5 2.0 2.50.0000.0050.0100.0150.020DOSABABAABBNo fieldFigure 3.7: Density of states plotted for the two choices of 4 vortex configurations -AABB (dotted red) and ABAB (solid blue). Clearly, the density of states remains samefor both the gauge choices. Parameters: ∆0 = 0.2, µ =−0.45, λ = 1, m= 0.5260 2 4 6 802468Figure 3.8: Configuration with 4 vortices with sublattice A (red) and B (blue) exhibitingparticle-hole asymmetry.Although the matching of physical observables (density of states) for an infi-nite periodic lattice proves that the inclusion of the phase factors is independentof the choice of gauge with respect to either the ABAB or the AABB configura-tions, it still does not provide resolution for different eigenvalues and observables(local density of states) for different A/B choices in a finite lattice with periodicboundary conditions.3.3.2 Particle-Hole SymmetryBy construction, the Bogoliubov-de Gennes Hamiltonian is particle-hole symmet-ric. Upon inclusion of vortices using the FT transformation, the spectrum for afinite or an infinite lattice should remain particle-hole symmetric for any arbitraryconfiguration and assignment of A/B vortices. In the two-vortex configuration, thespectrum is always PH symmetric as long as the vortices are placed at the centerof the plaquette of the underlying lattice. For more than two vortices per unit cell,the PH symmetry considerations become more complicated.For the four-vortex configurations (both ABAB and AABB) illustrated in Fig-2710.0 7.5 5.0 2.5 0.0 2.5 5.0 7.5n0.40.20.00.20.4EEnE n[En] [ E n]Figure 3.9: Eigenvalues plotted for the configuration 3.8. The eigenvalues (blue) andtheir time-reversed counterpart (red) do not match. The error shown in green is of theorder 10−2. Parameters: ∆0 = 0.2, µ =−0.45, λ = 1, m= 0.5ure 3.4, the spectrum preserves PH symmetry, as seen in Figure 3.5 for a finite pe-riodic lattice or Figure 3.6 for an infinite periodic lattice, irrespective of whetherthe spectra match for ABAB and AABB configurations. However, if there is dis-order introduced by moving the vortices away from the original positions as illus-trated in Figure 3.8, the particle-hole symmetry breaks down. The in-gap energyeigenvalues for such a configuration are shown in Figure 3.9. For higher numberof vortices per unit cell, more spatially symmetric vortex configurations were alsofound to obey the particle-hole symmetry while disordered non-Bravais latticeconfigurations without any inversion or reflection symmetry broke the PH sym-metry.For a finite vortex system subjected to periodic boundary conditions, we can-not write the conventional BdG Hamiltonian where the phase of the vortex isencoded in the off-diagonal order parameter terms. However, if we start withan FT-transformed BdG Hamiltonian with some specific choice of A/B, periodicboundary conditions can be imposed. The spectrum then will be PH symmetricif the Hamiltonian of holes is the time-reversed counterpart of the Hamiltonianof particles. However, for a random vortex arrangement, if the Hamiltonian of28Figure 3.10: Hollow torus with two non-contractible loops [Figure adapted fromWikipedia].particles and Hamiltonian of holes are not related by an anti-unitary operator, thePH symmetry breaks. Hence, the PH symmetry breaking is related to the choiceof gauge in the theory developed until now.This section highlighted two important issues which were heretofore unex-pected with the formalism developed so far. Firstly, the choice of A/B sub-latticeshould not affect the spectrum for a finite lattice with periodic boundary condi-tions. Additionally, the spectrum should obey particle-hole symmetry irrespectiveof the configuration or the assignment of A/B sub-lattice in that configuration.3.4 Finite vortex lattice with periodic boundaryconditionsTo work with a finite vortex lattice under periodic boundary conditions, we needto ensure that all the observables are independent of the choice of the A/B sub-lattice. Moreover, the resulting Hamiltonian should respect PH symmetry whichis in turn related to the choice of A/B sub-lattice. Hence, it is pertinent to under-stand what selects a specific A/B configuration on a real physical system.The physical constraint governing the choice of A/B sub-lattice is that theelectrons and holes are subjected to the same magnetic field. As a result, theflux ~∇×~vA/B experienced by the electrons and holes on the underlying lattice29differs by a multiple of 2pi across any loop that they encircle. The property ofthe Franz-Tesˇanovic´ transform is that it can change the flux in a loop by ±2pi onthe lattice. However, when periodic boundary conditions are imposed on the FT-transformed-BdG Hamiltonian, the fluxes through two additional non-contractibleloops present on the torus, illustrated in Figure 3.10, need to be accounted for. Ifthe flux through each non-contractible loop present on the torus experienced byelectron and hole differs by a multiple of 2pi , then we can obtain a real physicallattice. The only true choice of A/B assignment is the one for which this principleis valid.Equation (3.21) gives the Hamiltonian at a wave vector k. The modified FTphase factors for the A and B sublattice are given byV˜ αA (r) = VαA (r)+ kα .α, (3.22a)V˜ αB (r) = VαB (r)− kα .α, (3.22b)where α = δx,δy. The problem now translates into the calculation of a wavevector k such that the electrons and holes are subjected to the same flux acrossthe non-contractible loops C1 and C2 on the torus. Recapitulating that the FTphase factor is related to the superfluid velocity as V αg (r) = m}∫ r+αr ~vg(r).~dr withg= A,B, we obtain the condition,∮C(m}~vA(r)+k).~dr =∮C(m}~vB(r)−k).~dr. (3.23)Taking r0 = (0,0) as the origin, the integral over loopC1 runs from r0 = (0,0)to r1 = (L,0) and the integral over loop C2 runs from r0 = (0,0) to r2 = (0,L).This results in the following equations for the wave vector k = (kx,ky),2kxL=m}∫ r1r0(~vB(r)−~vA(r)).~dr, (3.24a)2kyL=m}∫ r2r0(~vB(r)−~vA(r)).~dr. (3.24b)Using the expression of the superfluid velocity in the continuum limit with q =(qx,qy), rewritten below for convenience, is used in the calculation of the integrals30in (3.24).~vg(r) =2pi}m∫ d2q(2pi)2(iq× zˆ)[∑ j eiq.(r−Rjg)λ−2s +q2]. (3.25)The integral in (3.24a) can be carried out with qx → 0. This is followed by thediscretisation procedure described in Section 3.2. A similar procedure is followedfor the integral in (3.24b). The resultant expressions for kx and ky arekx =piL2 ∑Gy 6=0iGy∑j(e−iGyRjBy − e−iGyRjBy)(3.26a)ky =piL2 ∑Gx 6=0−iGx∑j(e−iGxRjBx − e−iGxR jBx)(3.26b)where ~G = 2piL (nx,ny) is the reciprocal lattice vector of the magnetic unit cell asbefore with integer nx and ny and the sum j runs over all the vortex positions oftype A/B within a magnetic unit cell. Since kx and ky are purely real, the equations(3.28) get simplified to,kx =piL2 ∑Gy 6=01Gy∑j(sin(GyRjBy)− sin(GyRjAy)), (3.27a)ky =piL2 ∑Gx 6=0−1Gx∑j(sin(GxRjBy)− sin(GxRjAx)). (3.27b)Equations (3.28) are further simplified to yield a simple formula,kx =piL2∑j(R jAy−RjBy), (3.28a)ky =− piL2∑j(R jAx−RjBx). (3.28b)Taking ∆R = ∑ j(R jAx−RjBx ,RjAy−RjBy)can be reexpressed ask =piL.∆R× zˆL. (3.29)The corresponding flux that must be threaded through the two non-contractibleloops can be evaluated by φl =∫ L0 kldl, where l = x,y which gives31φl = pi∆RlL. (3.30)Using equations (3.22) and (3.29), the FT phase factors including the correc-tion terms emerging from the fluxes in the two non contractible loops, are evalu-ated asV˜ δxA (r) = VδxA (r)+piL2∑j(R jAy−RjBy).δx, (3.31a)V˜δyA (r) = VδyA (r)−piL2∑j(R jAx−RjBx).δy, (3.31b)V˜ δxB (r) = VδxB (r)−piL2∑j(R jAy−RjBy).δx, (3.31c)V˜δyB (r) = VδyB (r)+piL2∑j(R jAx−RjBx).δy, (3.31d)where Vδx/yA/B are calculated from (3.20).Going back to the configuration in Figure 3.8 which was found to be exhibitPH asymmetry, incorporating the correction terms in the FT phases, gives us aPH symmetric spectrum as shown in Figure 3.11. Therefore, the spectrum nowremains PH symmetric, irrespective of the configuration itself or the A/B assign-ment in the configuration.The modified phase factors (3.31) also resolve the issue of differing eigenval-ues and band structures for the ABAB/AABB configurations depicted in Figure3.4. Figure 3.12 now shows exactly same eigenvalues for both ABAB/AABB con-figurations, in contrast with differing eigenvalues for the finite lattice ABAB/AABBconfigurations shown in Figure 3.5. As expected, the local density of states shownin Figure 3.13 is the same for both configurations implying that the physical ob-servables remain the same irrespective of the choice of A/B assignment. More-over, unlike the band spectrum in Figure 3.6, inclusion of the modified FT phasefactors gives the same band structure for both ABAB and AABB configurationsas shown in Figure 3.14. However, the PH symmetry is assured to remain pre-served only at the Γ point in the modified FT-transformed-BdG Hamiltonian forthe configurations that exhibit PH asymmetry for FT-transformed-BdG Hamilto-nian without the correction terms.3210.0 7.5 5.0 2.5 0.0 2.5 5.0 7.5n0.40.20.00.20.4EEnE n[En] [ E n]Figure 3.11: Eigenvalues plotted for the configuration 3.8 using the modified FT phases.The eigenvalues (blue) and their time-reversed counterpart (red) match now with negli-gible error shown in green. Parameters: ∆0 = 0.2, µ =−0.45, λ = 1, m= 0.520 15 10 5 0 5 10 15 20n0.40.20.00.20.4E nABABAABBFigure 3.12: Resolution of the ABAB and AABB internal gauge symmetry issue. Incor-porating the modified FT phases gives the same spectrum now irrspective of the choiceof gauge. Parameters: ∆0 = 0.2, µ =−0.45, λ = 1, m= 0.5330.20 0.15 0.10 0.05 0.00 0.05 0.10 0.15 0.20Energy 0.0000.0050.0100.0150.0200.025LDOS(x=2, y=2)Figure 3.13: Local density of states at vortex remains same irrespective of gauge choice.Parameters: ∆0 = 0.2, µ =−0.45, λ = 1, m= 0.5We have now built a model that captures the low energy physics of a topo-logical insulator proximity coupled to an s-wave superconductor with a periodicvortex lattice. We focus our attention to the quasi-particles residing on such vor-tices and investigate the Majorana zero modes expected to be found on iron-basedsuperconductors.34X Mk0.40.20.00.20.4E kFigure 3.14: Bandspectrum of the AABB (dotted red) and ABAB (solid blue) matchesexactly using the mified FT phases. Parameters: ∆0 = 0.2, µ =−0.45, λ = 1, m= 0.535Chapter 4Majorana vortex bound statesAfter developing the technique for modelling a periodic vortex lattice at the inter-face between a topological insulator and superconductor, we can now study theMajorana quasi-particles residing on the vortices. With the theory developed inChapters 2 and 3, the experimental observations regarding the presence/absenceof Majorana zero modes such as those present on the vortices of the iron-basedsuperconductor, FeTe0.55Se0.45, are explained here.4.1 Experiments on FeTe0.55Se0.45FeTe0.55Se0.45 has spin helical surface states with superconducting gap induced bythe bulk [4]. These features pave the way to potentially observe the zero-energyMajorana bound states at the vortices on the surface. Hence, this material is anatural realisation of the Fu-Kane type model discussed in Chapter 2, without theneed for creating a TI/SC heterostructure. A significant motivation to use this su-perconductor is that its low Fermi energy ensures that the zero-energy Majoranabound state remains relatively isolated from the higher energy, in-gap, Caroli-deGennes-Matricon bound state located at energies of the order ∆2µ as discussed ear-lier.Several recent experiments have investigated the presence of zero-bias peaksand other in-gap CdGM modes in FeTe0.55Se0.45. Chen et al. [6] report the ob-servations of discrete CdGM bound states at the vortices in FeTe0.55Se0.45 but noMajorana zero mode. Additionally, they observe a peak near zero energy in somevortices which cannot be attributed to either the discretised CdGM peaks or thezero-energy Majorana bound state. Other STM experiments conducted on thismaterial [5, 7, 8] have revealed the presence of zero-bias peaks in some vorticesthat are attributed to the emergent Majorana quasi-particles on the surface. Here,we first explain this apparent disparity of the results regarding the observation (or36the lack thereof) of a Majorana zero-mode by simulating a periodic vortex latticewith two magnetic field strengths and different coherence lengths.Wang et al. [5] report a definite observation of a zero-bias peak that can beattributed to a Majorana quasi-particle. However, the observed zero-bias peaksexhibit non-uniform broadening across different vortices. Moreover, the zero-biaspeak intensity decreases with increase in temperature indicating some additionalsuppression mechanisms. In the high-resolution STM experiment [7] conductedon FeTe0.55Se0.45, zero-bias peaks are observed. However, all vortices do notuniformly exhibit the presence or absence of a zero-bias peak. The probabilityof observing a zero-bias peak decreases monotonically with increasing magneticfield. Here, we argue that the experimental results of both [5, 7] can be explainedsuccessfully by considering disorder effects in the periodic vortex lattice due tothermal fluctuations at different magnetic fields.In the STM experiment [8], two distinct classes of coexisting vortices are ob-served to be present on this system. The topologically nontrivial vortices hostthe Majorana zero-bias peaks whereas topologically trivial vortices do not exhibitthese peaks. There is a half-integer level shift between the in-gap vortex boundstates of the two types of vortices. We probe into this bi-partite nature of vorticesusing the same principles.Majorana hybridisation and vortex disorder effects in a three-dimensional tight-binding model, specific to FeTe0.55Se0.45, with vortices under open boundary con-ditions, have been previously investigated by Chiu et al. [16]. Here, we studyhybridisation and thermally induced disorder effects using a vortex lattice on aperiodic manifold on a two-dimensional surface. The local disorder effects in vor-tex configurations repeated periodically in space provide a unique perspective intothe long range physics of Majorana quasi-particles.Here, the simulations are conducted with parameters extracted from the ex-periments on FeTe0.55Se0.45. A periodic, triangular vortex lattice configuration ischosen for more realistic estimates in the low field and high field regimes. Disor-der effects in the vortex lattice due to thermal fluctuations are studied.370 5 10 15 20 25 30051015202530T:0.0, r:0(a) v=8, d¯ = 12.2a00 5 10 15 20 25 30051015202530T:0.0, r:0(b) v=32, d¯ = 6.1a0Figure 4.1: Configurations with 8 vortices (low magnetic field) and 32 vortices (highmagnetic field) for 32×32 physical lattice.4.2 Majorana zero modes on a triangular vortexlatticeA triangular vortex lattice is constructed for the low magnetic field with v = 8vortices and the high magnetic field for v= 32 vortices in a 32×32 system as il-lustrated in Figure 4.1. In the units of lattice spacing, a0, the average inter-vortexdistance, d¯, for the low-field regime is half of that for the high-field regime chosenhere. The finite lattice thus constructed is subjected to periodic boundary condi-tions using the BdG Hamiltonian (3.4) with modified FT phase factors (3.31). Wechoose one set of realistic physical parameters with direct correspondence withthe experimentally observed values, as illustrated in Table 4.1, for our simulationsin the low field and high field. With this setup, the Majorana modes on the super-conducting vortices with two different coherence lengths (ξ = vF∆0 ), correspondingto ∆0 = 0.4 and ∆0 = 0.2, are also studied for better conceptual understanding.Figure A.1 in Appendix A shows the dependence of zero modes on the chemicalpotential µ in the low-field and high-field regime for the two different coherencelengths.In the low field-low coherence length regime with 8 vortices, there is a dis-tinct zero-bias peak, shown in Figure 4.2a, that is attributed to Majorana quasi-38Simulation values Experiment values∆0 0.4 2 meVµ -0.9 -4.5 meV∆20µ 0.17 0.89 meVλ 1 5 meVvF = λa0 1 24 meVξ = vF∆0 2.5 12 nmk−1F =vFµ 1.11 5.33 nma0 1 4.8 nmTable 4.1: Simulation parameters and their approximate experimental equivalent. Here,the time reversal breaking mass term, m = 1, ensures optimal gap at all points in theBrillouin zone.particles. Here, ∆0/µ ≈ 0.4 is the same as the ∆0/µ ratio for FeTe0.55Se0.45.The Majorana coherence length here is ξ = vF∆0 ≈ 2.5, (vF = 1). The two dimen-sional local density plot for the zero mode and in-gap CdGM states located at0.27 and 0.32, shown in Figures 4.3, clarifies that these are indeed vortex boundstates. Since the CdGM peaks are located away from zero, it prevents any quasi-particle poisoning effects, as observed even in the actual experiments conductedon FeTe0.55Se0.45.Increasing the magnetic field in a real system translates to increasing the num-ber of vortices per magnetic unit cell. This, in turn, reduces the inter-vortex dis-tance which increases the hybridisation of vortex core states. The resultant in-crease in inter-vortex tunnelling leads to the splitting of Majorana bound states.This is seen clearly in Figure 4.2c where the zero-bias peak is split for the 32 vor-tex configuration, in sharp contrast with the clear zero-bias peak observed for the8 vortex configuration in Figure 4.2a, for the same set of parameters. Correspond-ingly, even the other in-gap modes split due to inter-vortex tunnelling.Decreasing the value of the superconducting gap results in the increase of theMajorana coherence length and the number of vortex core states. This results in asplit zero-bias peak for the low field 8 vortex configuration as seen in Figure 4.2band a complete absence of a zero-bias peak for the high field, 32 vortex configura-tion in Figure 4.2d. The two dimensional local density plots for the in-gap statesof the 8 vortex configuration with high coherence length are shown in Figures 4.4.390.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4Energy 0.000.010.020.030.040.050.060.07LDOS(x=17, y=17)(a) v= 8, ∆0 = 0.4, µ =−0.90.20 0.15 0.10 0.05 0.00 0.05 0.10 0.15 0.20Energy 0.0000.0050.0100.0150.0200.025LDOS(x=17, y=17)(b) v= 8, ∆0 = 0.2, µ =−0.90.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4Energy 0.0000.0050.0100.0150.0200.0250.0300.035LDOS(x=17, y=17)(c) v= 32, ∆0 = 0.4, µ =−0.90.20 0.15 0.10 0.05 0.00 0.05 0.10 0.15 0.20Energy 0.0000.0020.0040.0060.0080.0100.0120.014LDOS(x=17, y=17)(d) v= 32, ∆0 = 0.2, µ =−0.9Figure 4.2: Local density of states at vortex in the low field (v=8) and high field (v=32)regime. Two different coherence lengths corresponding to ∆0 = 0.4 and ∆0 = 0.2 arechosen here for comparison. Figure (a) shows a prominent zero-bias peak in the lowfield, low coherence length regime. Figure (b) and (c) show split zero-bias peaks corre-sponding to low field-high coherence and high field-low coherence regimes respectively.Figure (d) shows no zero-bias peaks in the high field-high coherence length regime.400 5 10 15 20 25 30051015202530e=0, r=0.00.010.020.030.040.050.060.07(a) v = 8, ∆0 = 0.4, µ =−0.90 5 10 15 20 25 30051015202530e=0.26550.00250.00500.00750.01000.01250.01500.01750.0200(b) v = 8, ∆0 = 0.4, µ =−0.90 5 10 15 20 25 30051015202530e=0.32250.0020.0040.0060.0080.010(c) v = 8, ∆0 = 0.4, µ =−0.9Figure 4.3: Two dimensional local density of states in the low field region for the lowestthree peaks in the superconducting gap.0 5 10 15 20 25 30051015202530e=0.0030.0050.0100.0150.0200.025(a) v = 8, ∆0 = 0.2, µ =−0.90 5 10 15 20 25 30051015202530e=0.11150.0010.0020.0030.0040.0050.0060.0070.008(b) v = 8, ∆0 = 0.2, µ =−0.90 5 10 15 20 25 30051015202530e=0.14550.0010.0020.0030.0040.005(c) v = 8, ∆0 = 0.2, µ =−0.9Figure 4.4: Two dimensional local density of states in the low field region for the lowestthree peaks in the superconducting gap.The disparity between the observations of zero-bias peaks, as observed in theexperiments [6, 5] with similar STM resolution, can be attributed to the hybridi-sation effects due to increase in magnetic field or, alternatively, increase in co-herence length. The absence of a zero-bias peak in [6] is quite possibly due to ahigher magnetic field (∼ 5 T) taken into consideration, in contrast with the obser-vation of zero-bias peak in [5] which takes place at a much lower magnetic field(∼ 0.5 T). Moreover, the peak near zero-energy observed in [6], can be attributedto the split Majorana modes as seen in Figures 4.2b and 4.2c.410 1 2 3 4 5T0.00.10.20.30.40.50.60.7Acceptance ratio(a) Acceptance probability0 1 2 3 4 5T0.00.51.01.52.02.5rv=8v=32(b) Disorder ∆r increasing with TFigure 4.5: Random move acceptance ratio dictated by Metropolis algorithm is shownin (a). Disorder characterised by mean-square deviation of vortex positions increaseswith temperature is shown in (b).However, it is important to note that hybridisation effects alone cannot ex-plain all the experimental observations. In the simulation results produced in thissection, all vortices are identical but in actual experiments there are prominentvariations in the vortex spectra [7], implying that vortex disorder effects (positionfluctuations) need to be taken into account.4.3 Majorana zero modes on a disorderedtriangular vortex latticeThe simplest way to get a more realistic vortex lattice disordered by thermal fluc-tuations is to let the individual vortices move around in a quadratic potential wellcentered at each vortex lattice point. In the periodic triangular lattice shown inFigures 4.1, we allow individual vortices to fluctuate around their equilibrium po-sitions in a harmonic potential. A random move consists of moving one vortexaway from its initial position by one lattice spacing in x or y directions. The en-ergy change is then computed, ∆E = Enew−Eold. A random move is accepted rrejected using the Metropolis Monte-Carlo algorithm at temperature T. The ran-dom move is accepted if ∆E < 0 or if e−∆E/T > n for ∆E > 0, where n is a ran-dom number between 0 an 1. The disorder in configuration is characterised bythe mean-square deviation of vortex positions, r, from their equilibrium positions,420.0 0.35 0.79 0.94 1.12 1.32 1.46 1.62 2.0 2.76r0.40.30.20.10.00.10.20.30.4Energy Eigenvalues(a) v= 8, ∆0 = 0.4, µ =−0.90.0 0.35 0.79 0.94 1.12 1.32 1.46 1.62 2.0 2.76r0.20.10.00.10.2Energy Eigenvalues(b) v= 8, ∆0 = 0.2, µ =−0.90.0 0.18 0.5 0.77 1.08 1.17 1.41 1.76 2.03 2.49r0.40.20.00.20.4Energy Eigenvalues(c) v= 32, ∆0 = 0.4, µ =−0.90.0 0.18 0.5 0.77 1.08 1.17 1.41 1.76 2.03 2.49r0.20.10.00.10.2Energy Eigenvalues(d) v= 32, ∆0 = 0.2, µ =−0.9Figure 4.6: Splitting of Majorana modes with increasing disorder. The modes in greencorrespond to the v= 8 or v= 32 Majorana modes which split from zero with increasingdisorder.req, given by∆r =√〈|r− req|〉. (4.1)Figure 4.5b shows the disorder parameter, ∆r, as a function of the temperatureparameter. For both configurations, with 8 and 32 vortices, the disorder increaseswith temperature. The acceptance probability of Metropolis algorithm, shown inFigure 4.5a, increases with increasing temperature, thereby creating an increas-ingly disordered lattice with temperature.On increasing the disorder in configurations, the splitting on Majorana modesincreases on average. This can be seen from Figures 4.6. For the low field-low430.020 0.015 0.010 0.005 0.000 0.005 0.010 0.015 0.020Energy 0.000.010.020.030.040.050.060.07LDOST=0.0T=0.4T=0.8T=2.0T=4.0T=5.0(a) v= 8, ∆0 = 0.4, µ =−0.90.020 0.015 0.010 0.005 0.000 0.005 0.010 0.015 0.020Energy 0.0000.0050.0100.0150.0200.025LDOST=0.0T=0.4T=0.8T=2.0T=4.0T=5.0(b) v= 8, ∆0 = 0.2, µ =−0.9Figure 4.7: Temperature evolution of the zero-bias peak at vortex core located within±∆r of [17,17] in the low field limit. The peak height decreases with temperature forthis vortex.coherence length regime, the splitting is not very prominent (Figure 4.6a) but onincreasing the cohrence length, the splitting becomes apparent even in the lowfield regime (Figure 4.6b). The Majorana modes are spread out even at low disor-ders for the high field case but there could be zero modes present upon increasingdisorder (Figures 4.6c). The in-gap states come to prominence upon increasingdisorder in the high field-high coherence length case (Figure 4.6d).Since the degree of disorder is related to the change in the thermal parameter,we investigate the evolution of the zero-bias peak in the low field limit, as depictedin Figures 4.7. For the specific vortex location probed, the zero-bias peak strengthdecreases with increase in temperature for the low coherence length limit shownin Figure 4.7a. In the high coherence length limit shown in Figure 4.7b, the splitpeak evolves in the same manner with increase in the temperature parameter. Thisbehaviour resembles very closely to the low field temperature evolution experi-ments conducted by Wang et al. [5]. Since both of these simulations are in thelimit where quasi-particle poisoning is negligible, thermal disorder effects can bethe cause of the of the decreasing peak strength with increase in temperature.The two-dimensional local density of states at different values of disorders for8 vortices are shown in Figures 4.8 and 4.9. All 8 zero modes are prominent for∆ = 0.4 case but there is a large variation in the peaks for the high coherence44length ∆ = 0.2 case because of increased inter-vortex tunnelling in the presenceof disorder. In the high field case, for both ∆ = 0.4 and ∆ = 0.2, the zero-biaspeak intensity varies for different vortices and changes drastically with increasein disorder, as seen from Figures 4.10 and 4.11. In the vortex lattice without dis-order, there were split or no zero-bias peaks present on vortices except in the lowfield-low coherence length case with 8 vortices and ∆ = 0.4. However, upon in-creasing disorder, there could be vortices that host Majorana quasi-particles whileothers may not. This is in accordance with the experimental results by Machida etal. [7] who reported non-uniformity in the observations of zero-bias peak of thetype simulated here. Local density of states near zero energy for a representativevortex core for the four regimes studied, are plotted in Figure 4.12.The probability of observing zero-bias peaks on vortices at ∆r ≈ 1.1 is shownin the histograms of minimum peaks shown in Figure 4.13. The probability ofobeservation of a zero-bias peak on the vortex decreases from 100% to 21% uponincreasing the magnetic field in the low coherence length limit as shown in 4.13aand 4.13c. Similarly, the probability of obeservation of a zero-bias peak on thevortex decreases from 50% to 13% upon increasing the magnetic field in the lowcoherence length limit as shown in 4.13a and 4.13c. The histograms for the nextlowest peaks plotted in Figures 4.14 for high field case shows that there could bepeaks present in the same energy range as the minimum energy peaks resulting inthe observations of multiple near-zero-bias peaks.Although the non-zero CdGM peaks observed for all our simulations occur inthe range of ∼ ∆2µ , we do not observe any specific quantisation of energy modes.Moreover, the vortices that do not exhibit a zero-bias peak, show the occurrenceof other near-zero peaks that need not necessarily be of the order ∼ ∆22µ . While thebi-partite classification of vortices based on the presence or absence of Majoranazero modes can be conducted with our simulations, we note here that quantisationreported in [8] is not evident with the simulation parameters in our model due tothe observation of split peaks near zero-bias.Thus, our model is successful in replicating the behaviour of Majorana boundstates when subjected to varying degrees of intervortex tunnelling and disordereffects on a vortex lattice on a periodic manifold.450 5 10 15 20 25 30051015202530e=0, r=0.353550.010.020.030.040.050.060.070.08(a)0 5 10 15 20 25 30051015202530e=0, r=0.790570.010.020.030.040.050.060.070.08(b)0 5 10 15 20 25 30051015202530e=0, r=0.935410.010.020.030.040.050.060.07(c)0 5 10 15 20 25 30051015202530e=0, r=1.118030.010.020.030.040.050.060.070.08(d)0 5 10 15 20 25 30051015202530e=0, r=1.322880.010.020.030.040.050.060.07(e)0 5 10 15 20 25 30051015202530e=0, r=1.457740.010.020.030.040.050.060.070.08(f)Figure 4.8: L=32, v = 8, ∆0 = 0.4, µ = −0.9. LDOS at zero-energy with increasingdisorder in vortex configuration from (a)-(f).0 5 10 15 20 25 30051015202530e=0, r=0.353550.0050.0100.0150.0200.0250.030(a)0 5 10 15 20 25 30051015202530e=0, r=0.790570.0050.0100.0150.0200.0250.0300.035(b)0 5 10 15 20 25 30051015202530e=0, r=0.935410.0050.0100.0150.0200.0250.0300.035(c)0 5 10 15 20 25 30051015202530e=0, r=1.118030.0050.0100.0150.0200.0250.0300.035(d)0 5 10 15 20 25 30051015202530e=0, r=1.322880.0050.0100.0150.0200.0250.0300.0350.040(e)0 5 10 15 20 25 30051015202530e=0, r=1.457740.0050.0100.0150.0200.0250.0300.0350.040(f)Figure 4.9: L=32, v = 8, ∆0 = 0.2, µ = −0.9. LDOS at zero-energy with increasingdisorder in vortex configuration from (a)-(f).460 5 10 15 20 25 30051015202530e=0, r=0.176780.0010.0020.0030.0040.005(a)0 5 10 15 20 25 30051015202530e=0, r=0.50.0020.0040.0060.0080.0100.0120.014(b)0 5 10 15 20 25 30051015202530e=0, r=0.770550.0020.0040.0060.0080.0100.0120.014(c)0 5 10 15 20 25 30051015202530e=0, r=1.075290.010.020.030.040.05(d)0 5 10 15 20 25 30051015202530e=0, r=1.17260.010.020.030.040.05(e)0 5 10 15 20 25 30051015202530e=0, r=1.414210.010.020.030.040.05(f)Figure 4.10: L=32, v= 32, ∆0 = 0.4, µ =−0.9. LDOS at zero-energy with increasingdisorder in vortex configuration from (a)-(f).0 5 10 15 20 25 30051015202530e=0, r=0.176780.00010.00020.00030.0004(a)0 5 10 15 20 25 30051015202530e=0, r=0.50.00050.00100.00150.0020(b)0 5 10 15 20 25 30051015202530e=0, r=0.770550.00050.00100.00150.00200.00250.00300.0035(c)0 5 10 15 20 25 30051015202530e=0, r=1.075290.0020.0040.0060.0080.0100.0120.0140.016(d)0 5 10 15 20 25 30051015202530e=0, r=1.17260.0020.0040.0060.0080.0100.0120.0140.016(e)0 5 10 15 20 25 30051015202530e=0, r=1.414210.0050.0100.0150.0200.0250.030(f)Figure 4.11: L=32, v= 32, ∆0 = 0.2, µ =−0.9. LDOS at zero-energy with increasingdisorder in vortex configuration from (a)-(f).470.04 0.03 0.02 0.01 0.00 0.01 0.02 0.03 0.04Energy 0.0000.0010.0020.0030.004LDOS(x=9, y=23)(a) v= 8, ∆0 = 0.4, µ =−0.90.04 0.03 0.02 0.01 0.00 0.01 0.02 0.03 0.04Energy 0.00000.00050.00100.00150.00200.00250.0030LDOS(x=6, y=12)(b) v= 8, ∆0 = 0.2, µ =−0.90.04 0.03 0.02 0.01 0.00 0.01 0.02 0.03 0.04Energy 0.0000.0050.0100.0150.020LDOS(x=13, y=20)(c) v= 32, ∆0 = 0.4, µ =−0.90.04 0.03 0.02 0.01 0.00 0.01 0.02 0.03 0.04Energy 0.0000.0020.0040.0060.0080.010LDOS(x=13, y=20)(d) v= 32, ∆0 = 0.2, µ =−0.9Figure 4.12: Local density of states at representative vortex positions for disorder ∆r≈1.1.480.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040|E|012345678Countsr=1.11803, p=100.0%(a) v= 8, ∆0 = 0.4, µ =−0.90.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040|E|01234567Countsr=1.11803, p=50.0%(b) v= 8, ∆0 = 0.2, µ =−0.90.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040|E|0246810Countsr=1.1726, p=21.875%(c) v= 32, ∆0 = 0.4, µ =−0.90.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040|E|0246810Countsr=1.1726, p=12.5%(d) v= 32, ∆0 = 0.2, µ =−0.9Figure 4.13: Histograms of lowest energy peak for disordered vortices.0.00 0.01 0.02 0.03 0.04 0.05|E|01234567Countsr=1.1726(a) v= 32, ∆0 = 0.4, µ =−0.90.00 0.01 0.02 0.03 0.04 0.05|E|0123456Countsr=1.1726(b) v= 32, ∆0 = 0.2, µ =−0.9Figure 4.14: Histograms of next to lowest energy peaks for 32 disordered vortices.49Chapter 5Summary and ConclusionsMotivated by the recent studies pertaining to Majorana vortex bound states iniron-based superconductor FeTe0.55Se0.45, this thesis investigates the Majoranazero modes present on a periodic vortex lattice on a superconductor-proximitizedtopological insulator.Based on the Fu-Kane proposal, a square lattice model is developed to capturethe low energy physics of the surface states of a topological insulator proximity-coupled to an s-wave superconductor. Upon incorporating a single vortex withopen boundary conditions in the system, Majorana quasi-particle was found to bepresent as a zero-energy excitation at the vortex on the lattice.Using the Franz-Tesˇanovic´ singular gauge transformation [9], multiple vor-tices can be incorporated on the superconductor-proximitized topological insula-tor surface model by circumventing the problem of branch cuts. However, uponapplication of this singular gauge transformation to create a finite vortex latticewith periodic boundary conditions (torus manifold), the internal gauge symme-try and particle-hole symmetry of the Hamiltonian get violated. This root causeof this issue originates from the fluxes through the two non-contractible loopspresent on the torus which come to the fore upon application of periodic bound-ary conditions. On further investigation of this issue, a modification term to theoriginal Franz-Tesˇanovic´ singular gauge transformation is proposed, thereby al-lowing the creation of a finite vortex lattice on periodic manifolds.The model with periodic vortex lattice on the proximitized topological insula-tor is applied to investigate the observations of Majorana vortex bound states onFeTe0.55Se0.45. A periodic triangular vortex lattice is constructed on the latticemodel of the surface of proximitized topological insulator. The inter-vortex dis-tance controlled by change in magnetic field and the coherence length of the vor-tices dependent on the superconducting gap controls the presence or absence of azero-bias peak corresponding to the Majorana mode. Majorana bound states exist50at low fields for the parameters observed in the experiments. However, an orderedvortex lattice cannot explain all experimental observations since only a fraction ofvortices host Majorana zero-bias peaks in the STM measurements conducted onFeTe0.55Se0.45. In the presence of disorder introduced by thermal fluctuations, thetranslation symmetry is broken and there is a large variation in the vortex spec-tra which is in accordance with the observations in the experiments. A vortexlattice disordered by thermal fluctuations helps in understanding the temperatureevolution of the zero-bias peaks. It also clarifies the observations of increasedor decreased fraction of vortices hosting Majorana zero modes with change inmagnetic field. Although the other in-gap states are located away from the zeros,the multiple inequivalent vortices can host multiple states near zero due to the un-equal splitting of Majorana zero modes which, in turn, is dependent on intervortextunnelling. In conclusion, our model successfully replicates the observations ofMajorana zero modes in iron based superconductor FeTe0.55Se0.45.51Bibliography[1] J. Alicea, “New directions in the pursuit of majorana fermions in solid statesystems,” Reports on progress in physics. Physical Society (Great Britain),vol. 75, p. 076501, 07 2012.[2] A. Y. Kitaev, “Unpaired majorana fermions in quantum wires,”Physics-Uspekhi, vol. 44, pp. 131–136, oct 2001.[3] L. Fu and C. L. Kane, “Superconducting proximity effect and majoranafermions at the surface of a topological insulator,” Phys. Rev. Lett., vol. 100,p. 096407, Mar 2008.[4] P. Zhang, K. Yaji, T. Hashimoto, Y. Ota, T. Kondo, K. Okazaki, Z. Wang,J. Wen, G. D. Gu, H. Ding, and S. Shin, “Observation of topologicalsuperconductivity on the surface of an iron-based superconductor,” Science,vol. 360, no. 6385, pp. 182–186, 2018.[5] D. Wang, L. Kong, P. Fan, H. Chen, S. Zhu, W. Liu, L. Cao, Y. Sun, S. Du,J. Schneeloch, R. Zhong, G. Gu, L. Fu, H. Ding, and H.-J. Gao, “Evidencefor majorana bound states in an iron-based superconductor,” Science,vol. 362, no. 6412, pp. 333–335, 2018.[6] M. Chen, X. Chen, H. Yang, Z. Du, X. Zhu, E. Wang, and H.-H. Wen,“Discrete energy levels of Caroli-de Gennes-Matricon states in quantumlimit in FeTe0.55Se0.45,” Nature Communications, vol. 9, no. 1, p. 970,2018.[7] T. Machida, Y. Sun, S. Pyon, S. Takeda, Y. Kohsaka, T. Hanaguri,T. Sasagawa, and T. Tamegai, “Zero-energy vortex bound state in thesuperconducting topological surface state of Fe(Se,Te),” Nature Materials,vol. 18, no. 8, pp. 811–815, 2019.[8] L. Kong, S. Zhu, M. Papaj, H. Chen, L. Cao, H. Isobe, Y. Xing, W. Liu,D. Wang, P. Fan, Y. Sun, S. Du, J. Schneeloch, R. Zhong, G. Gu, L. Fu,52H.-J. Gao, and H. Ding, “Half-integer level shift of vortex bound states inan iron-based superconductor,” Nature Physics, 2019.[9] M. Franz and Z. 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Zhang, “Scalablemajorana vortex modes in iron-based superconductors,” Science Advances,vol. 6, no. 9, 2020.53Appendix ASplitting of zero-modes with µ-0.0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.90.040.020.000.020.04E 00=0.40=0.2(a) v= 8-0.0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.90.100.050.000.050.10E 00=0.40=0.2(b) v= 32Figure A.1: Zero mode splitting with change in chemical potential µ for the low fieldcase with 8 vortices and the high field case with 32 vortices.The maximum split in the zero modes for the low field and the high field caseis depicted in Figure A.1. On an average, the splitting increases with increase inmagnetic field and coherence length. References [15, 14] discuss the exact phe-nomenological and analytical expressions for the zero-mode splitting observed.54
Thesis/Dissertation
eng
Physics
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Majorana bound states on a periodic superconducting vortex lattice
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