Doru C. Sticleț

Edge States in Chern Insulators andMajoranaFermions in Topological Superconductors

Gilles MontambauxDavid Carpentier

GilRefaelJulia S. MeyerKaryn Le HurPascal Simon

Frederic Piéchon

Président du juryRapporteurRapporteurExaminatriceExaminatriceDirecteur de thèseMembre invité

Discipline: Physique Théorique

Université Paris-Sud

soutenue le 27 Novembre 2012

par

Thèse de doctorat

École doctorale: Physique de la région parisienne

Laboratoire de Physique des Solides

Composition du jury :

Contents

Acknowledgments v

Introduction 1

I Topological insulators 5

1 uantum anomalous Hall phases in a two-dimensional Chern insulator 71.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Bulk characterization of a Chern insulator . . . . . . . . . . . . . . . . 8

1.2.1 Topological invariant . . . . . . . . . . . . . . . . . . . . . . . 81.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.3 Model building of a topological insulator with large Chern phases . . 181.3.1 General discussion . . . . . . . . . . . . . . . . . . . . . . . . 181.3.2 A model with ve Chern phases . . . . . . . . . . . . . . . . . 20

1.4 Large Chern phases in the Haldane model . . . . . . . . . . . . . . . . 261.4.1 Distant-neighbor hopping in graphene . . . . . . . . . . . . . 271.4.2 Phases of the Haldane model . . . . . . . . . . . . . . . . . . 34

1.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2 Edge states in a Chern two insulator 412.1 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 422.2 Analytical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.2.1 Edge states from bulk Hamiltonian . . . . . . . . . . . . . . . 432.2.2 Edge states from the Schrödinger equation . . . . . . . . . . . 44

2.3 Extension: Edges in a Z2 insulator . . . . . . . . . . . . . . . . . . . . 492.4 Extension: Stripes of a Z insulator . . . . . . . . . . . . . . . . . . . . 512.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

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II Majorana fermions 55

3 Introduction 573.1 Majorana fermion primer . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.1.1 Dirac equation and the Majorana condition . . . . . . . . . . 573.1.2 Particle-hole symmetry . . . . . . . . . . . . . . . . . . . . . . 59

3.2 Condensed matter realizations . . . . . . . . . . . . . . . . . . . . . . 623.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.2.2 Kitaev model . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.2.3 1D spin-coupled semiconducting wire in proximity to an s-

wave superconductor . . . . . . . . . . . . . . . . . . . . . . . 663.3 Majorana polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4 Topological semiconducting wire with Rashba and Dresselhaus spin-orbitcoupling 734.1 Model Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.2 Topological invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.3 Majorana wave function solutions . . . . . . . . . . . . . . . . . . . . 764.4 Numerical study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5 Extended Majorana states in Josephson junctions 855.1 Superconductor-normal junctions . . . . . . . . . . . . . . . . . . . . 865.2 Fractional Josephson effect . . . . . . . . . . . . . . . . . . . . . . . . 885.3 Superconductor-normal-superconductor junction . . . . . . . . . . . 915.4 Ring with a uniform phase gradient . . . . . . . . . . . . . . . . . . . . 93

5.4.1 Constant phase gradient in a wire . . . . . . . . . . . . . . . . 945.4.2 Ring with a uniform phase gradient . . . . . . . . . . . . . . . 97

5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6 Multiple Majorana fermions in a two-band model 1016.1 Topological properties of a two-band BDI topological superconductor 102

6.1.1 Symmetry constraints . . . . . . . . . . . . . . . . . . . . . . 1026.1.2 Winding number of a circuit and role of distant site couplings 103

6.2 Model Hamiltonian and phase diagram . . . . . . . . . . . . . . . . . 1046.3 Transport in SN junctions . . . . . . . . . . . . . . . . . . . . . . . . . 1056.4 Josephson junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.4.1 e 1− 1 Josephson junction . . . . . . . . . . . . . . . . . . 1066.4.2 e 2− 2 Josephson junction . . . . . . . . . . . . . . . . . . 1086.4.3 e 1− 2 Josephson junction . . . . . . . . . . . . . . . . . . 110

6.5 Wires with an inhomogeneous superconducting phase . . . . . . . . . 1126.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

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Conclusion and Perspectives 117

Bibliography 121

iii

Acknowledgments

Here you have a thesis with an author on its covers. Nevertheless, it is not the contribu-tion of one voice. It conveys the results of a continuous dialog. Sometimes it involveda mute relation with articles and books, but, at its best, it was an exchange of ideas withthe people in the scienti c community. Here are some of them...

I thank my thesis advisor, Pascal Simon, for his unrelenting drive and his ability tospot open directions for research. e work on Majorana fermions is particularly in-debted to the fruitful collaboration withCristina Bena andDenis Chevallier. Our jointwork deserves future cooperation.

A special thanks is owed to the unofficial thesis advisor, Fred Piéchon. Countlesshours spent circling around the subject of topological insulators taughtme themeaningof research in physics. His availability for intelligent discussionwas a fuel formany ideasdeveloped in this thesis.

anks to my colleagues, LPS has become in three years a welcoming environmentfor studying the topological matter. Hopefully, the in uence of the clear and sharpmind of Jean-Noël Fuchs transpires in the following pages. His scienti c generosity hasbene ted all people around him. e same can be said about the good-humored MarcGoerbig, who made coffee breaks an occasion to learn physics wrapped in anecdotes.

It is hard to match the fortuitous encounters with Pavel Kalugin. e few momentsin which I shared his mathematical perspective on the subject of topological insulatorswere transcending.

I also bene ted fromdiscussionwithGillesMontambaux, PierreDelplace, Rafaël deGail, Lih-King Lim, Guangquan Wang and Clément Dutreix. e “graphene-people”are naturally close to the topics presented inside this thesis and, even unknowingly, haveshaped the formulation of ideas presented here.

It was a pleasure to collaborate with François Crépin, who made discussions aboutbosonization and renormalization fun. I cannot forget several discussionswith the ever-inquisitive Nicolas iébaut. I hope our topics will meet in the future and the black-board discussions can turn in a collaboration. In the same vein, I thank to EmilioWino-grad, Jean-René Souquet, Yi Liu,MattiasAlbert,NicolasCharpentier, and IlyaBelopol-ski for introducingme into their respective elds of research. It is true that science is just

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A

a big system of communicating vessels. It was a pleasure to share ideas with JasonAlicea.Talking with him was as illuminating as reading his clear articles. Misha Polianski isa lost friend; his presence inspired my rst year of PhD and his sudden disappearancecannot delete his memory.

Whenmathematics becamedisorienting,AdrianViorel, IoanMărcuţ, andOriYudile-vich were always available for discussions. ank you!

All physicists that crossedpathwithme in the institute, in courses, in summer schoolsetc. deserve a mention. Alas, only some of them could be named in this brief acknowl-edgment. Furthermoremy gratitude is extended to all the people at the LPS, whomadethe institute such a comfortable place for research.

Finally, nothing would have been possible without my family. When the long daysin a foreign land turned bleak, they always brightened them up.

vi

Introduction

e recent years have witnessed a revolution in condensed matter physics. It consistedin discovering a wealth of new phases of matter that are not distinguished by brokensymmetries, but from the topology of the wave-functions describing the system [1–3].

e tremendous work from recent years has a starting point in the theoretical proposalof two-dimensional Z2 topological insulators (the quantum spin Hall insulators) [4, 5]and the rapid experimental discovery in HgTe/CdTe quantum wells [6]. Soon a er, itwas understood that the 2D quantum spin Hall insulator is but an instance of possibletopological insulators and superconductors. Venerable phenomena such as the Shock-ley surface states [7], the topological excitations in the Su-Schrieffer-Heeger polymerchain [8], the integer quantumHall effect [9], andmany others, were all put together aspieces of a greater picture; these phenomena crucially depend on the presence of edgestates that are manifestations of topological phases in a gapped bulk.

e topological insulators and superconductors share the same intriguing feature:the presence of robust edge states in the bulk gap. For example, the edge states are not re-moved under the action of weak disorder which does not close the bulk gap. Hence thepersistence of these states cannot be attributed to thepoint group symmetries, which canbe destroyed in the presence of disorder. In the absence of point group symmetries, oneis le with at least two basic discrete symmetries: time-reversal symmetry (TRS) andcharge conjugation or particle-hole symmetry (PHS). ey form the basis for the clas-si cation of non-interacting gapped Hamiltonians in arbitrary dimensions [10]. isallowed to group the Hamiltonians into ten classes in each spatial dimension.

More precisely, the time-reversal and charge conjugation acting on a Hamiltonianmatrix, in a basis of creation and annihilation operators, are represented by the anti-unitary operators,T and C, that can square to±1 [11]. en, counting also the possibil-ity that the system is not invariant under these symmetries, there are 9 possible classes. Athird chiral or sublattice symmetry (SLS) is represented by a unitary operatorS = TC.

e value of S is entirely determined by the behavior of C and T, except in the partic-ular case where SLS is a symmetry of the system when both TRS and PHS are broken.

is case raises the number of symmetry classes to 10 [11]. e resulting classi cationis represented in Tab. 1.

1

I

System Cartan nomenclature TRS PHS SLS d = 1 d = 2 d = 3

standard A (unitary) 0 0 0 - Z -(Wigner-Dyson) AI (orthogonal) +1 0 0 - - -

AII (symplectic) −1 0 0 - Z2 Z2

chiral AIII (chiral unit.) 0 0 1 Z - Z(sublattice) BDI (chiral orthog.) +1 +1 1 Z - -

CII (chiral sympl.) −1 −1 1 Z - Z2

BdG D 0 +1 0 Z2 Z -C 0 −1 0 - Z -

DIII −1 +1 1 Z2 Z2 ZCI +1 −1 1 - - Z

Table 1: Classi cation of free fermionic gapped Hamiltonians as a function of sym-metries: TRS, PHS and SLS. e dimensionality is denoted by d. e present thesistouches almost exclusively models belonging to the classes marked in red. (Table takenfrom Ref. [13].)

Furthermore, not all gapped ground states for a given class are identical. Five classesin every dimension have gapped ground states that are divided into topological sectors.

e Hamiltonians in these classes form the topological insulators and superconduc-tors [11–14]. Note from Tab. 1 that they have associated a descriptor, Z or Z2. isdenotes the topological invariant that describes the ground state of the system and isused throughout to identify the type of topological insulator (superconductor). e in-variant counts how many distinct gapped phases are in a speci c class: only two, Z2, ora numerable in nity of phases, Z. Always one of the gapped phases is trivial in the sensethat it can be adiabatically connected to the vacuum. Passing from a gapped phase to an-other requires closing the bulk gap [12]. Connecting two topologically distinct gappedphases of topological insulators (superconductors) creates an interfacewhere edge statesappear. is general result is called the bulk-boundary correspondence [15, 16] and isthe basis of the physics grown around the subject of topological insulators (supercon-ductors). e robust edge states are a consequence of connecting ground states withdifferent values of the topological invariant. Adding a weak perturbation that obeys thesymmetries speci c to a class, cannot destroy these states.

From a practical perspective the robustness of edge states constitutes a central moti-vation to study the topological materials. For example in 2D and 3D, the edges harborquantized metallic states that are robust with respect to disorder and could carry cur-rent without dissipation. A prime example are the integer quantum Hall edge states.

ey belong to a 2D Z insulator in class A that is described by an integer topologicalinvariant, the rst Chern number C. Weak disorder does not destroy the edge statesand, furthermore, they are quantized; they carry current and have a Hall conductanceproportional to C [17–19].

ere are ve non-interacting topological insulators and superconductors in each

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I

dimension. Nevertheless, not all of them have found an experimental realization. isposes a continuous challenge to condensedmatter physics; from the experimental pointof view, it is to nd or engineer systems that will support topological phases of matterand, of course, to detect these exotic states of matter. e theoretician needs to pro-pose possible candidates, imagine detection schemes and good quantities to measure.Furthermore, the road ahead is not entirely mapped. At the moment, it still remains tohave a systematical view of the topological matter in the context of interacting systems.Already the classi cation schemes for free Hamiltonians need to be revised in this newlight [20–22].

In this thesise present thesis will not dwell on the abstract matters concerning the classi cation

of topological insulators and superconductors. It is applied entirely to non-interactingtopological insulators and superconductors as the ones classi ed in Tab. 1. e tableshould be used to pinpoint the object of the present study in a more general context.

e thesis is divided into two parts. Each one will receive a more detailed introduc-tion at its respective beginning. It suffices here to draw the main directions of research.

e rst part of the thesis is focused mostly on 2D topological insulators in classA. e most famous inhabitant was already named: the integer quantum Hall effect(IQHE). Here TRS is broken through an external magnetic eld. However, a differ-ent possibility exists, that was rst illustrated byHaldane [23]. eoretically the IQHEphysics can arise in the absence of an external eld, by adding uxes at the scale smallerthat the cell size, but which cancel overall. us TRS is still broken, and one could ex-pect dissipationless current at zero magnetic eld. Such a model did not have an exper-imental ful llment, but was central in imagining the rst Z2 insulator in graphene [4].Due to its connection to IQHE, the Z insulator was named quantum anomalous Hall(QAH) insulator. e topological invariant characterizing the 2D QAH insulators re-mains a Chern number.

e present study focuses mostly on minimal two-band models of QAH insulatorsand investigates the conditions for the production of bands with high Chern number.

e nontrivial aspect of the research is that the high Chern number is not obtained bymultiplying the bands, but by creating a single band with a high Chern number. is isre ected as usual in a multiplication of the edge channels.

Chap. 1 is concerned entirely with the bulk characterization of QAH insulators ina tight-binding formulation. It is shown how one can simplify the treatment of thesemodels to the studyof systemswithDiracpoints. Subsequently, producinghigherChernnumber reduces in this case to the requirement that the nodes in the dispersion for gap-lessmodels aremultiplied through addition of hopping terms between distant sites. etheory is rst put to test in the context of an endogenous arti cialmodelwith veChernphases, and, secondly, by modifying the QAH Haldane model.

Chap. 2 contendsmostlywith numerical and analytical solutions for edge-statewave

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I

functions in a model with Chern |C| = 2. Moreover, a few extensions to four-bandmodels are analyzed using the methods developed in the previous chapter, and theirtopological phase diagram is determined.

e second part of the thesis is centered around the study of Majorana fermions ina spin-orbit coupled semiconducting wire in the proximity of an s-wave superconduc-tor [24, 25]. is physical system realizes a one-dimensional topological superconduc-tor in class D (see Tab. 1) that supports particular edge states: the Majorana fermions.An introduction to these fascinating (quasi)particles is offered in Chap. 3.

InChap. 4 the system is reconsidered in the presence ofDresselhaus spin-orbit inter-action. It is shown that the spin of the electronic degrees of freedom of the zero-energyedgemodes responds to ratio between the Rashba andDresselhaus spin-orbit coupling.

ere is an opposite spin-polarization of the edge modes, in a direction transverse to amagnetic eld, which could be accessible through tunneling spectroscopy. e follow-ing chapter (5) considers hybrid structures of the types: superconductor-normal andsuperconductor-normal metal-superconductor built upon the aforementioned system.Furthermore, it investigates ring geometries, under the action of a uniform supercon-ducting phase gradient, which in certain condition canbemapped to a SNSheterostruc-ture. e central interest lies in the extended nature of Majorana fermions that developin the normal part of the heterostructures.

Finally, Chap. 6 changes gears and touches upon Majorana fermions in a two-bandtight-binding BDI class superconductor. e interest lies in the fact that the system isdescribedby aZ topological invariant and supports severalMajorana fermions at its edge(see Tab. 1). It is important to note the connection with the Z insulators treated in therst part. e same mechanism, i.e. addition of coupling terms between distant sites, is

responsible for creating multiple zero modes. Conclusion and possible perspectives arecontained in the nal chapter.

4

Part I

Topological insulators

5

Chapter 1uantum anomalousHall phases in a

two-dimensional Chern insulator

1.1 Introduction

e quantum spin Hall (QSH) insulator that debuted the recent excitement in theeld of topological insulators (and superconductors) [1–3] has one important precur-

sor which is the subject of the rst part of this thesis. It is the quantum anomalous Hall(QAH) insulator that received a rst theoretical realization in theworkofHaldane [23].It is a Z insulator that has chiral edge states similar to those of integer quantum Hall ef-fect (IQHE) with the crucial difference that they exist in zero magnetic eld. A QAHinsulator in two dimensions exists only in class A, which lacks TRS, PHS or the sublat-tice symmetry (see Tab. 1). It will be also referred in the following as a Chern insulatordue to the fact that it is described by aChern topological invariant C. Because it requiresno particular discrete symmetry, one expects that edge state of this system are extremelyrobust, similarly to the ones in IQHE [9]. Furthermore, it acquires a quantized Hallconductivity proportional to the Chern index, σH = C × e2/h.

e Haldane tight-binding model is treated in more detail in Sec. 1.4. It suffices tosay that it describes spinless electrons and the TRS is broken through currents inducedat a scale smaller that the cell size. More generally, in a spinful system such physics couldbe caused bymagnetic ordering in the presence of spin-orbit interaction [23, 26]. eo-retical proposals to realizeQAH invoke effects of disorder inmetallic ferromagnets [26]ormagnetic doping ofQSH insulators [27]. However, to thismoment, there is no indis-putable physical realization of the QAH insulator, and only recently experimentalistsclaim detection in magnetic topological insulators [28]. By contrast, the QSH effectwas ostensibly investigated in HgTe/CdTe quantum wells [6]. e birth of the QSHinsulator is related to the insight that combining aQAH insulator with its time reversalcopy produces a time-reversal invariant (TRI)Z2 insulator that can have spin-polarizedchiral edge states [4].

7

1.2 B C

e question that drives the rst part of the thesis is how to determine efficiently thetopological invariant for the case of a two-band non-interacting Chern insulator (seeSec. 1.2). Moreover, how can one multiply the topological phases in the system whilekeeping only twobands? Equivalently, howcan there be a single bandwith a largeChernnumber? Note that without the two-band constraint, the answer must follow the samelines of the IQHE; the number of edge channels ismultiplied by havingmore bands [29,30]. e search for bands with high Chern numbers in QAH insulators has recentlyintersected with the study of at band topological models in which it is expected toencounter fractional quantum Hall effect [31–33]. In this context, the search for highChern numbers wasmotivated by the need to discover physics above the lowest Landaulevel [34–37].

e short general answer that is elaborated in the present thesis claims: a single bandcan increase its Chern number by adding distant-site hoppings in the system [38]. ( esame question was answered in the context of at-band topological models by showingthat a multi-band system can be projected to a two-band model with effective distant-neighbor hoppings [37].) In particular, these questions are given a more concrete an-swer by building a model with ve Chern phases in Sec. 1.3 and by creating high Chernphases in the Haldane model in Sec. 1.4. is question can be seen as a bridge to thesecond part of the thesis, where couplings between distant sites can produce severalMajorana modes at a single edge. ere the time-reversal and chiral symmetries ensurethat they do not hybridize to form regular electronic states (see Chap. 6). e secondchapter treats mostly the edge physics in a model QAH insulator. ere the question ishow to determine analytically and numerically the edge states in a model with |C| = 2.Finally, the chapter contains two extensions that show ways in which the analytical de-termination of the Chern number can be used in cases of four-band models. Sec. 2.3treats a four-band TRI Z2 model. Here the edge states from |C| = 2 are gapped by TRIone-particle perturbation. In Sec. 2.4 theChern number is used to predict newmetallicstates that appear in a “striped” topological insulator.

1.2 Bulk characterization of a Chern insulator

1.2.1 Topological invariantChern number for a class A insulator in 2D

is subsection contains a description of the topological invariant that characterizes atwo-dimensionalZ topological insulator in class A. is insulating system has no chiralsymmetry or any of the anti-unitary symmetries: TRS andPHS. Subsequently it is char-acterized by a Chern number. e main argument advanced here is that the topologicalinvariant can receive a discrete formulation that allows to draw a direct parallel with thephysics of Dirac fermions. e discrete formulation of the topological invariant has theadded interest that gives an efficient way to compute and discriminate the topologically

8

1. QAH 2D C

insulating phases of a two-band insulators.e tight-binding Hamiltonian for a free fermion theory is written in a site basis

H =∑i

εi|i⟩⟨i|+∑ij

tij|i⟩⟨j|. (1.1)

e on-site energy is given by εi and the hopping integrals between different neighborsites are given by tij . In the following, only systems without disorder are considered.

erefore the system is invariant under a translation with a Bravais lattice vector. AFourier transform allows to express the Hamiltonian in reciprocal space

H =∑k∈BZ

H(k)|k⟩⟨k|, (1.2)

where |k⟩ are Bloch states. Because there are no anti-unitary symmetries imposed, theHamiltonian H is generally a matrix inHerm(n), the set of n × nHermitian matriceswith complex coefficients. ere are n bands that can be due to the presence of differ-ent orbitals per site and nonequivalent atoms in a unit cell. Note that the variable k iscontinuous and the Brillouin zone is a manifold, a torus T 2. To each point in the BZ,H(k) associates a value, and therefore it is a mapping from the torus to the parameterspace ofH

H : T 2 → Herm(n). (1.3)

Topology enters the discussion with the following question: when are two Hamil-tonians H equivalent? ey are equivalent if the functions can be smoothly deformedinto each other. From a topological point of view they are homotopically equivalent.

eChernnumber indexes the classes of homotopically equivalent functionsH. Forthe multiband system it can be de ned using the notion of projector on the occupiedbands. If there is a gap between the valence and conduction band, then the phases areindexed by

C =i

2π

∫BZ

Tr(dP ∧ PdP ), (1.4)

where P is the projector on the occupied bands [39].A non-zero Chern number can be understood as an obstruction to a global gauge

choice for the wave function on the BZ. Physically this phenomenon is directly relatedto the quantization of theHall conductivity σH = C×e2/h, where C corresponds to thenumber of edge states [17, 40].

In the following, the focus is almost entirely on two-band band translation invariantHamiltonians (n = 2). Hence the momentum space Hamiltonian can generally bedecomposed in a basis of Pauli matrices

H =3∑

µ=0

hµ(k) · σµ. (1.5)

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1.2 B C

e σ Paulimatrices are not given a physical interpretation for themoment. eymightrefer to a space of two orbitals on a site or two nonequivalent sites in the unit cell. eterm h0σ0 just shi s identically the energy bands and does not modify the topology ofthe Hamiltonian. It is neglected in the following, and therefore theHamiltonian reads

H(k) = h(k) · σ. (1.6)

is form is reminiscent of a Zeeman Hamiltonian for a spin 1/2 particle in a magneticeld, except that the “ eld” h(k) is de ned in momentum space. ere are two energy

bands given by

E± = ±|h|. (1.7)

For an insulating system, there is always a gap between the bands. erefore thethree vector components of h(k) never vanish simultaneously, when k varies in the BZ.

en it follows that the unit vector Hamiltonian h

h = h/|h| (1.8)

is well de ned. Under the simpli cation of having only two bands, the Chern numberwill index phases of h

h : T 2 → S2, (1.9)

from the BZ to the Bloch sphere. e bands of h correspond to a attening of the bandspertaining to the original Hamiltonian. is adiabatic attening preserves the gap andthus preserves the topology for the Hamiltonian [12]. is observation amounts to saythat the target space of h, i.e. R\(0, 0, 0), is homotopic equivalent to S2. en h canbe thought of as a composition h = proj h, where

proj : R3\(0, 0, 0) → S2 (1.10)

is the central projection to the unit sphere.All functions h that can be smoothly transformed into each other, while preserving

the spectral gap, form a homotopy class. In Z insulators there is a integer index dis-tinguishing the different phases (i.e. the different homotopy classes) of h, the Chernnumber C. A different way to express this idea is to say that the homotopy group ofthe mapping h is the group of integers Z. Without elaborating on these issues that gobeyond the scope of the present section, it is noteworthy to point out that Chern num-bers index mappings between d-spheres, here S2 → S2, and the homotopy group isπ(Sd, Sd) ≡ πd(S

d) = Z. However, the homotopy group of the torus is equivalent tothat of the sphere,

π(T 2, S2) = π2(S2) = Z. (1.11)

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1. QAH 2D C

en TKNN numbers used to index the integer Hall phases [17, 41] are indeed Chernnumbers [42].

ese considerations extend to the case of the n-band systems with nondegeneratebands where the topological invariant is given by a vector of Chern number of dimen-sion n− 1

π2(Herm(n)) =n−1⊕i=1

Z. (1.12)

Eachbandhas an associated invariant, such that the sumofChernnumbers for the entiresystem is zero. e n− 1 dimension of the vector follows because the invariant for anyone of the bands is entirely determined by the sum of the other invariants [42].

Chern number as a nite sum

Let us come back to the two-band case in order to nd a workable formula for the topo-logical invariant. e projector on the occupied band in terms of h reads

P =1

2(σ0 − h · σ). (1.13)

e substitution of P in Eq. (1.4) yields immediately the expression for the rst Chernnumber for the occupied band

C =1

4π

∫BZd2k h · (∂kxh× ∂ky h). (1.14)

e above formula shows that the Chern number is a winding number that countshow many times does the surface traced by h wrap around the origin (0, 0, 0) when kvaries in the BZ [43]. e only practical difficulty in determining the Chern numberlies in performing the integration in Eq. (1.14). e main point of this section is thatC can be computed using a discrete summation by determining directly the Brouwerdegree of the map h [44, 45] (de ned below).

e condition to calculate the degree of h are met: T 2 and S2 are orientable man-ifolds without boundary and have the same dimension, T 2 is compact and S2 is con-nected. In the general case, for a point k in T 2 one de nes the derivative map betweentangent vector spaces

dh(k) : TkT 2 → TzS2. (1.15)

Let sgn dh(k) stand for the sign of the corresponding Jacobian atk. enChernnumberC is equal to the Brouwer degree of h at a regular point z on the unit sphere

C =∑

k∈h−1(z)

sgn dh(k). (1.16)

11

1.2 B C

e Chern number can be computed also in terms of h in the following way. Forconvenience, letM denote the image of the BZ through h

h(T 2) = M ⊂ R3\(0, 0, 0). (1.17)

Consider the set Y = M∩ proj−1(z). en one has

C =∑y∈Y

∑k∈h−1(y)

sgn[(∂kxh× ∂kyh) · n], (1.18)

where the n is the unit vector towards z. is expression is just the generalization of thecalculation of the winding number for a closed curve in 2D wrapping around a pointp [45, 46].

e formula in Eq. (1.18) can be further simpli ed by an appropriate choice of thepoint z. e central projection proj maps any intersection point between M and a rayoriginating at (0, 0, 0) to z. If this ray does not cross the surface M traced by h, then itfollows that the surface does not wrap around the origin and the Chern number is zero.For a point z on S2, one can immediately obtain a set of points on M that project to zthrough proj. Since the expression (1.16) does not depend on z, the choice of the lattercan be guided by convenience. For instance, one can consider z lying at a coordinateaxis. Let us choose for example the σ3-axis which intersects M in a set of points. isis equivalent to say that the components on the other axes are zero. Said differently, theintersection of M with the σ3-axis are images of the band touchings originating fromthe simpli ed Hamiltonian σ1h1 + σ2h2. Consequently, instead of integrating over theentireBZ, oneonly needs to consider the band touchings of the simpli edHamiltonian.Note that if h3 is also zero at these points, then the system is not in a gapped phase.

It only remains to account for the orientation of the surface at the intersection pointswith σ3-axis. is can be done by studying the projection of the surface normal vectoron the σ3-axis (∂kxh× ∂kyh)3. When h3 > 0, assume that the orientation is (+1) whenthe sign of the projection is positive and, (−1), when the sign is negative; the converseis true when h3 < 0. en nding the Chern number amounts to a computation of anite sum. Since the entire σ3-axis was considered, instead of a ray, the sum yields twice

the value of the Chern number.e above argument can be generalized and summarized in the following formula

for Chern numbers describing 2-band systems

C =1

2

∑k∈Di

sgn(∂kxh× ∂kyh

)isgn(hi). (1.19)

where i is an arbitrary axis chosen in (pseudo-)spin space. erefore the integral overmomenta k in the BZ becomes a nite sum over k in the set of Dirac points Di forHamiltoniansH[hi = 0] (whereH is the originalHamiltonianh·σ). Note that divisionby two is required because the entire axis was considered, instead of a ray originating in(0, 0, 0).

12

1. QAH 2D C

Fig. 1.1: An example of a discrete calculation of aChern number. e topological invari-ant is calculated at single point z on the Bloch sphere. Here it has two preimage pointson the BZ, P1 and P2. Under h, a basis at P2 changes orientation when going to z. enat z there are two topological charges canceling to give a zero Chern number.

Discussion

e topological invariant for an insulating system was connected to a simpler analy-sis for gapless two-band system that possess Dirac points. Once the Dirac points wereidenti ed, it is immediate to compute their chirality χ

χi(κ) = sgn[(∂kxh× ∂kyh

)i]|κ. (1.20)

e quantityχ indicates if theBerry phase gainedby thewave function around theDiracpoint is ±π. Subsequently the system is gapped by hi, which is the so called mass term.

en mass sign hi and the chirality χi are sufficient to determine the Chern number Cwhich indexes the insulating phase of the Hamiltonian. Note that due to the generalarguments made above, there is no intrinsic meaning for the mass term and any compo-nent of h can play the mass role.

ere is an important caveat to the above formula thatneeds tobe addressed: it is notalways true that preimage points for the intersections of axis with the Bloch sphere areDirac points for the gapless systems. ey can correspond to nonlinear band touchingssuch that the sign of the Jacobian at the band touching κ is not de ned. is situationarises when the band touching is due to amerging ofDirac points. en the rst deriva-tives of h vanish, leading to χ = sgn[0]. Note that these cases correspond to a foldingof the manifold M exactly on the spin axes (see Fig. 1.2). However the Eq. (1.19) re-mains useful. e rst solution to maintaining its pertinence, is to choose a differentaxis in order to avoid the diabolical points whereM folds. e second solution, whichis illustrated in Sec. 1.4, consists in determining C from an analysis of chiralities pertain-ing to converging Dirac points. e idea rests on the fact that the merging solutionsrequires tuning the convergence of multiple Dirac points and it is generally unstable toperturbations. en a small perturbation in the parameters of the Hamiltonian (a per-turbation that does not cause a topological transition) splits themerging point into a set

13

1.2 B C

Fig. 1.2: Examples of situations that arisewhen computing theChernnumberC as a nitesum. Here the privileged axis is σ3 and therefore h1σ1 + h2σ2 is a gapless Hamiltonian.

e surface traced by the fullh,M, has a fold atσ3-axis, such that the preimage of z givesa band touching with quadratic dispersion for the gapless model. is is an example ofa diabolical point on the Bloch sphere that must be avoided in order to determine theinsulating phases of the full model, using Eq. (1.19). At the other pole, there is the usualcase of a well-behaved point w, that has a Dirac point as a preimage. When there areonly Dirac points as preimages of z andw, the topological invariant is resolved as a sumover their chiralities using the nite sum formula.

of Dirac points. Furthermore, it was shown that the topological charge associated to aband touching with higher dispersion is conserved and equals the sum over the chirali-ties of the Dirac points [47, 48]. en the diabolical points can be treated as limit casesfor the same system close to the merging point and with multiple Dirac fermions.

Let us suppose for the moment that the privileged axis is σ3 and the poles have onlyDirac points as preimages in BZ. ere are a few interesting consequences that ensue.

e sum over the chiralities of Dirac points is always zero. is is seen by adding alarge constant mass term such that there is zero Chern number. Equivalently the Blochsphere is translated on the σ3 axis such that the origin (0, 0, 0) is no longer included inthe sphere. en the sign of the mass can be factored out, and under the constraintC = 0, the sum over chiralities must yield zero.

Also note that in order to get a nonzero Chern number, the mass term must changeits sign at least once. is gives a meaning to the requirement of “inverted gap” in orderto have nontrivial phases.

14

1. QAH 2D C

ere is an equal number of Dirac fermions with χ = 1 as those with χ = −1,because the sum over chiralities is zero,. It follows that there is always an even numberof Dirac fermions. is affirmation agrees with the predicted doubling of fermions ona lattice [49]. However, an odd number of Dirac fermions are permitted when the fullHamiltonian h is considered. is is the case at an interface between two Z insulatorswhere the Chern number changes by an odd integer.

Finally, for a given model there is always a limit to the largest possible Chern phaseand it manifestly depends on the minimum number of Dirac points. Let us supposethat the number is 2n because there is an even number of Dirac fermions (n ∈ Z). enthe largest Chern number phase has |C| = n and it corresponds to gapping all Diracfermions with a mass term that changes sign between the Dirac points. More precisely,the product between the chirality of a Dirac fermions and the sign of the mass thatgaps must remain constant. is observation opens the road to the present chapter thatessentially explores the notion of creating large Chern topological phases in model Zinsulators in class A.

1.2.2 Examples

e efficiency of Eq. (1.19) to discriminate the topological phases is exempli ed here ona couple of popular models of topological insulators: the Haldane model [23] and theBernevig-Hughes-Zhang [5] (BHZ) “spin up” Hamiltonian for the HgTe/CdTe quan-tum wells. ey will be treated at a formal level, only as an illustration of the technique.

Haldane model

Let us start by considering the paradigmaticHaldanemodel [23]. Amore detailed anal-ysis is undertaken in Sec. 1.4. Here suffices to observe that it can be seen as a modi ca-tion on the graphene tight-binding system. e latter lives on a hexagonal lattice builtout of two inter-penetrating triangular sub-lattices with A and B atoms. It is usually ap-proximated as having only nearest-neighbor (NN) electron hopping with the hoppingintegral t1. e Haldane model contains also next-nearest-neighbor (NNN) hoppingt2, such that when the hopping is performed clockwise in the unit cell the electron gainsa phase ϕ. However, the overall phase on the unit cell is zero; there is no net magneticux. Let the vectors (a1, a2, a3)describe the displacements fromBatoms toNNAatoms

and bi =12ϵijk(aj−ak) vectors relatingNNN sites (see Fig. 1.3). e time-reversal sym-

metry and particle-hole symmetry is broken by the presence of the ux ϕ. It also hasan on-site energy ±M that has a sign alternating between the A and B sites. is termdestroys the chiral symmetry of the lattice and therefore according to the classi cationof non-interacting, gapped, fermion systems, the Haldane model has insulating phasesdescribed by a Chern number.

15

1.2 B C

e Bloch Hamiltonian reads

H(k) =3∑

i=1

2t2 cos(ϕ) cos(k · bi)σ0 + t1[cos(k · ai)σ1 + sin(k · ai)σ2]

+[M3

− 2t2 sin(ϕ) sin(k · bi)]σ3. (1.21)

a1a2

a3

b3

b1 b2

Fig. 1.3: Schematic representation of theHaldane model. e A (B) sites are repre-sented bywhite (black) bullets, (•). Alongthe dashed lines an electron gains a phase ϕin the direction of the red arrows. e vec-tors a and b are the NN and, respectively,NNN displacement vectors.

e term σ3h3 is chosen as the massterm, and therefore the gapless submodelis the graphene Hamiltonian (M → 0and t2 → 0). e Dirac points ingraphene are positioned at time-reversedpointsK andK′ = −K onBZ,whereK =(

4π3√3, 0). e chirality of theDirac points

is readily determined from Eq. (1.20)

χ(±K) = ±1. (1.22)

In order to calculate the Chern num-ber, one must also consider the mass signat the Dirac points,M± := h3(±K),

M± =M ∓ 3√3t2 sin(ϕ). (1.23)

erefore, using Eq. (1.19) one recoversHaldane result for the Chern number

C =1

2(sgnM− − sgnM+). (1.24)

e transition from a topological in-sulator to a normal insulator is marked bya semi-metal state, where the gap closesat least at one Dirac point when M =±3

√3t2 sin(ϕ). In the normal insulating phase Dirac points have identical associated

mass sign, such that C vanishes.

BHZ model

e Chern number calculation can be exempli ed in the case of the recently discoveredZ2 insulators such as the 2D HgTe/CdTe quantum wells [5]. e low energy BlochHamiltonian is written in a basis of four states |E1,mJ = 1/2⟩, |H1,mJ = 3/2⟩,|E1,mJ = −1/2⟩, |H1,mJ = −3/2⟩ and it has the form

H =

(H(k) 00 H∗(−k)

). (1.25)

16

1. QAH 2D C

Dirac points (0,0) (π,0) (0,π) (π,π) Cmass h3 M M − 4B M − 4B M − 8Bchirality + − − +M < 0 − + + − 0M ∈ (0, 4B) + + + − +M ∈ (4B, 8B) + − − − −M > 8B + − − + 0

Table 1.1: Chern phases for a “spin-up” BHZ model as a function of system parameters,withB > 0.

e systemrespects time reversal symmetry and realizes aZ2 topological insulator. How-ever it is assembled out of two Chern insulators: H(k) and its time reversed copy, in-dexed by C and−C. If there are no interblockmatrix elements, the topological invariantcan be determined as a spin Chern number, expressed as C mod 2 [50].

erefore it is enough to pick one Chern insulator, and illustrate the computationof C[H(k)].

H(k) = A sin(kx)σ1 + A sin(ky)σ2 + [M − 2B(2− cos(kx)− cos(ky))]σ3,(1.26)

where A,B,M are material parameters. Let us consider again the surface traced by hand choose σ3 as a special axis. e points where the σ3-axis pierces the surface are givenby the condition that h1 and h2 vanish simultaneously. at determines four “Diracpoints” (qx, qy) ∈ (0, 0), (0, π), (π, 0), (π, π).

e chirality of each Dirac point is given by Eq. (1.20)

χ(q) = sgn[cos(qx) cos(qy)] (1.27)

evaluated at all Dirac points. e mass term h3 has the following expression at theDiracpoints, h3(0, 0) =M , h3(0, π) = h3(π, 0) =M − 2B and h3(π, π) =M − 4B.

e Chern number can then be easily computed for different values of M and Bby summing over the Dirac points. e results for the case B > 0 are summarized inTab. 1.1.

From Tab. 1.1 it follows that as M varies between 0 and 8B, the Chern insulatorH(k) exhibits two topologically nontrivial phases with C = ±1. WhenM is outside the(0, 8B) region there is only a trivial insulator phase. erefore nontrivial Chern phaseswill yield also nontrivial Z2 (QSH) phases.

17

1.2 M …

1.3 Model building of a topological insulator with largeChern phases

1.3.1 General discussione objective of the present section is to explore in more detail the question of how to

imagine two-band topological insulators with highChern number. It is argued that un-derstanding the topological invariant as a nite sum in addition to the symmetry con-straints is sufficient to produce Hamiltonians with large Chern phases. In particular,following theoretical considerations, a simple model which can be tuned through veinsulating phases, C ∈ 0,±1,±2 is produced step by step.

Note that when more than two bands are allowed, high Chern phases can ensuereadily by having each occupied band contributing to the overall conductance. is isindeed the case of the integer quantum Hall effect where the larges Hall conductanceis given by the number of bands. However the question posed here is how to create asingle band with a high Chern number. Arguably, there is a gain in theoretical controlof this situation, with all the insulating phases determined analytically.

e answer to the central question was already mentioned brie y in the previoussection. It comes down to the way aZ insulator was understood by decomposing it intoa gapless model with Dirac fermions and a mass term that gaps them [38].

For a system with translation symmetry, the application of Bloch theorem allows towrite a general Hamiltonian in k-space

H = h · σ, (1.28)

where the identity h0σ0 was dropped out because it does not weigh on the topologicalproperties of the model. e Pauli matrices represent a pseudo-spin degree of freedomdue to the presence of two orbitals on a site or two sites in the unit cell.

To x ideas, the h3(k)σ3 is chosen as the mass term throughout the section. Hencethe gapless modelH12 reads

H12 = h1(k)σ1 + h2(k)σ2. (1.29)

e Chern number from Eq. (1.19) reads

C =1

2

∑k∈kerH12

sgn(∂kxh× ∂kyh

)3sgn(h3). (1.30)

e kernel ofH12 contains all the values of k for which h1 and h2 vanish simultaneously.ey are band touchings of the the gapless model H12 and, in general, they are Dirac

points with chirality given by χ = sgn(∂kxh× ∂kyh

)3.

en a necessary condition to obtain a Chern insulator with large Chern phases isto create multiple Dirac points. e largest Chern phase can be obtained by tuning a

18

1. QAH 2D C

mass term such that the product of the chirality and the gapping term is constant for allDirac points. erefore, if there are 2nDirac points the largest Chern number is n.

Because h is periodic on BZ, the components hi can be Fourier analyzed to give

hi(k) = c(i)00 +

∑m,n

c(i)m,n cos(k · (ma1 +na2))+∑m,n

s(i)m,n sin(k · (ma1 +na2)), (1.31)

wherem and n are integers and never both equal to zero. e vectors a1 and a2 are theBravais lattice vectors are . Adding coefficients s and c corresponds to adding hoppingterms between orbitals. en multiplying the Dirac points for H12 requires produc-ing distant-neighbor hopping termswhich contain sinusoidal components that oscillatefaster and faster as m and n grow. at results in an energy dispersion which acquiresmore nodes when distant-neighbor hoppings are included.

e requirement of distant-neighbor hopping terms to produce large Chern phasesposes a problem from a physical point of view. e wave functions in the tight-bindingmodel are localized and presence of distant-neighbor interaction can be usually neglect-ed. However it was already shown that low-energy models of multi-band system can bemapped to two-band systems with large Chern number [37]. Regardless of the physicalrealization, the inclusion of higher harmonics (or distant-neighbor hopping terms) isthe unavoidable requirement to producing a large topological invariant.

To complete the discussion it is necessary to examine the symmetries, or lack thereof,for the Z topological insulator in two dimensions. In order to have a Chern insulatorin class A it is necessary to break the time reversal, the particle-hole and the chiral (orsublattice) symmetries (see Tab. 1).

e symmetries can impose gneral constraints on the components of h to the effectof a vanishing Chern number. For example, the TRS and the PHS are represented byanti-unitary operators that will relate opposite momenta on the BZ. erefore they willimpose parity constraints on the components of h. A rule of thumb to eliminate thesesymmetries is to mix odd and even functions in an arbitrary component of h. e sub-lattice symmetry is represented by an unitary Hermitian operator S and it is brokenby adding hopping terms between the equivalent sites (orbitals). is corresponds todestroying the bipartite nature of the system [51].

Let us elaborate on the example of theTRS. For spinless electronmodels consideredhere, the time-reversal operator does not act on orbital or site space an hence it is simplythe complex conjugationK required to reverse the momentum direction. Demandingtime-reversal invariance and Hermiticity yields

h1(k) = h1(−k),h2(k) = −h2(−k), (1.32)h3(k) = h3(−k).

Hence in a TRI system the Chern number vanishes because the integrand in equa-tion (1.14) is an odd function of k,

h · (∂kxh× ∂ky h) = 0. (1.33)

19

1.3 M …

Generally, if a symmetry imposes that either exactly one of the components of h isan odd function, or that all components are odd, then the Chern vanishes under thesame argument as above. e PHS will not be of concern in the following because byconstruction there are no superconducting pairing terms.

1.3.2 Amodel with ve Chern phasese objective is to create a toy model of a spinless Chern insulator with |C| = 2. It is

shownhere that themodel can be createdwith atmost nearest-neighbor hopping terms.Such amodel serves as a proof of principle for the formof an insulatorwith a largeChernnumber and it will be of use later, when the interest shi s to the investigation of edgesolutions in a Z insulator.

First a gapless model with four Dirac points is proposed as a prerequisite. Secondly,theDirac points are gappedby amass term that canbe locally tuned so that theHamilto-nian passes through all possibleChern phases 0,±1,±2. e geometrical engineeringof the insulators takes place in momentum space. At the end of the section, a real-spaceimplementation of the model on a triangular lattice is discussed.

Topological phases for momentum space Hamiltonian

Let us start with the gapless model that serves as a template for the construction of theinsulator. To have |C| = 2, it is necessary to have four Dirac points. One of the mostsimple model has only NN hopping (m < 1 or n < 1) in the Fourier series (1.31).Choosing even functions h1 and h2, it follows that

H12 = h1(k)σ1 + h2(k)σ2= 2t1[cos(kx)σ1 + cos(ky)σ2]. (1.34)

e energy dispersion reads

E = ±2t1

√cos2(kx) + cos2(ky), (1.35)

with four Dirac points at q = (±π/2,±π/2).Note that to have time reversal invariance in a spinless model, h2 must be an odd

function. Hence the choice cos(ky) ensures that the TRS is broken.e chirality coresponding to Dirac points is given by the sign of the Jacobian

χ(q) = sgn[sin(qx) sin(qy)]. (1.36)

is determines the chirality χ of the four Dirac points as summarized in Tab. 1.2.Note that the points at q and −q have the same chirality. ey will be referred in

the following as a pair of Dirac points.To obtain a Chern insulator one needs to add a mass term. As one can see from Eq.

(1.30), the Chern index depends on both the chiralities of theDirac points and the sign

20

1. QAH 2D C

Dirac points (π2, π2) (π

2,−π

2) (−π

2, π2) (−π

2,−π

2)

χ + − − +

Table 1.2: Chirality χ of the four different Dirac points.

of the mass term in their vicinity. Let us add a mass term of the form h3(q)σ3. Since h3is a periodic function on the BZ, in the general case, its zeros form a set of closed lineson the two-dimensional torus. e rst condition in order to gap the initial system isthat these lines must not pass through the Dirac points. us the lines of zeros delimitregions, R1 and R2, where the mass term has the same sign. For a proper choice of thegap term, it is somewhat simpler to see the problem from a geometrical point of view.In order to maximize |C|, one needs that the lines of zeros separate the pairs of Diracpoints such that a pair of points of a given chirality are contained in a region of positivemass, while pair of points of opposite chirality are contained in a region of negativemassregion. In short, each pair ofDirac points are placed in regions where themass term hasdifferent values. On the contrary, if a pair of Dirac points is “broken” (such that one isin region R1 and the other in region R2), then the topological charges will cancel outas can be directly inferred from Eq. (1.30).

Let us rst realize topological phases with C = ±2, where each pair of Dirac pointsis in a different region R. e chirality is an even function; hence keeping the productof chirality and mass sign constant is accomplished by an even function h3. A simplesolution is the periodic function

h3 ∝ cos(kx + ky) (1.37)

with lines of zeros given by ky = −kx + 2n+12π. e regionsR1 andR2 for this term are

represented in Fig. 1.4(a). e mass term is negative in regionR1 and positive in regionR2. e lowest Chern number C = −2 is obtained for this model. Let us consider thatthe term corresponds to a hopping term with an amplitude t2. Hence, the insulatingHamiltonian with |C| = 2 reads

H(1)(k) = 2t1 cos(kx)σ1 + 2t1 cos(ky)σ2 + 2t2 cos(kx + ky)σ3. (1.38)

e Chern number is readily determined

C(1) = −2sgn[t2]. (1.39)

Intuitively this is understood as a double covering of the Bloch sphere by h(1) due to thefact that cos(kx + ky) has twice the frequency of cos(kx,y).

e Hamiltonian in Eq. (1.38) lacks trivial insulating phases or Chern phases withC = ±1. To produce a trivial phase with C = 0, it suffices to add a large, “staggeredpotential”mσ3, such that all Dirac points are gapped identically. en because the sumover chiralities is zero, C vanishes. Note that since Pauli matrices can act on any degree

21

1.3 M …

(a) (b)

Fig. 1.4: Differentmass terms, h3, gap differently theDirac points on the BZ to the effectof changing the Chern number. e solid circles represent Dirac points of the initialgapless system; the corresponding chiralities are marked by the signs within. Dashedlines are the zeroes of the mass term, while R1 and R2 denote regions with oppositesign for the mass term. (a) h3 = cos(kx + ky). e mass term changes its sign an oddnumber of times between Dirac points of different chirality, therefore C = ±2. (b)h3 = sin(kx) + sin(ky). Phases with C = ±1 are possible a er the remaining Diracpoints,⊖, are identically gapped.

of freedom, the term “staggered” could for instance refer to the fact that two sites in acell or two orbitals on a site have an associated±m constant energy.

erefore the Hamiltonian changes from H(1) to H(2) = H(1) + mσ3, and, conse-quently, the Chern number becomes

C(2) = sgn[−m− 2t2] + sgn[m− 2t2]. (1.40)

Hence, for a large energym, |m| > 2|t2|, the system enters a trivial phase and a transitionbetween a C = ±2 phase to a C = 0 phase takes place.

To create Chern phases with C = ±1, it is necessary to gap, with the same gap sign,a pair of Dirac points, while a second pair has its respective mass changing sign betweenthe Dirac points. Because the chirality is an even function, the mass term has to be anodd function in order to get an odd function for the product between chirality andmasssign. e simplest choice would be to add the term proportional to sin(kx) + sin(ky).

e mass for one pair of Dirac points is unchanged, while, for the pair (q,−q)with q =(π/2, π/2), the mass changes. If only sin(kx) + sin(ky) is present in the mass term, thenthe system is not an insulator, because there are band touchings at q = ±(π/2,−π/2)(see Fig. 1.4(b)). However adding small even functions in themass gaps identically theseDirac points and C = ±1 phases follow.

22

1. QAH 2D C

−2 −1 0 1 2t3

−4

−2

0

2

4

m −2

0

0

−1 −1

1 1

Fig. 1.5: Phase diagram of the system for t2 = 1. Each region is denoted by the corre-sponding Chern number and represents an insulating topological phase. e bound-aries of the regions represent topological transitions where the system becomes gapless.

Adding all terms together gives the following complete Bloch Hamiltonian for amodel that has Chern phases in the set 0,±1,±2

H = 2t1 cos(kx)σ1+2t1 cos(ky)σ2+[m+2t2 cos(kx+ky)+2t3(sin(kx)+sin(ky))]σ3.(1.41)

ere are four free parameters (m, t1, t2, t3) in the model. Let us assume that allparameters are real. All the phases can be reached by varying (m, t2, t3)while keeping t1xed. e Chern number for the nal model reads

C = sgn(−m− 2t2) +1

2[sgn(m− 2t2 + 4t3) + sgn(m− 2t2 − 4t3)]. (1.42)

e expression for C yields immediately the phase diagram associated with the sys-tem described by the Hamiltonian in Eq. (1.41). Note that the parameter t1 does notenter in the determination of the phases, but manifestly needs to be nite in order tohave non-vanishing σ1 and σ2 components in the model. e formula (1.42) can be il-lustrated by the phase diagram in the Fig. (1.5). is diagram contains only four phasesof the model; the phase with C = 2 needs t2 < 0.

Direct space realization

Until now the system was described abstractly in momentum space. However it will beof future interest to investigate the structure of its edge and therefore it is necessary topropose a lattice implementation for the model.

23

1.3 M …

(i+ 1, j)

(i+ 1, j + 1)

(i, j)

E

a1

a2

m

−m

Fig. 1.6: Direct space realization of themodel (1.41), see Eq. (1.43). (i, j) denotes latticesites. e Bravais lattice vectors are denoted by a1/2, and • () represents orbitals withenergym (−m). e vertical axis represents the on-site energy difference between thetwo nonequivalent orbitals. Black lines represent t1 hoppings, blue lines, t2 hoppings,and red lines, t3 hoppings. An arrow on a link indicates that an electron hopping in thecorresponding direction gains a π/2 phase. Similarly a double line indicates a π phasegain.

e system could be realized on a triangular lattice with two orbitals on each site.e parameters are interpreted as kx,y = k · a1,2 with a1 and a2, the two Bravais vectors.en the Pauli matrices are operating in the space of two orbitals A and B with energy

±m.e Hamiltonian (1.41) can be rewritten in direct space

H =∑ij

[c†ijm

2σ3cij + c†i+1j(t1σ1 + it3σ3)cij + c†ij+1(t1σ2 + it3σ3)cij

+ c†i+1j+1t2σ3cij +H.c.], (1.43)

where the Fourier transformation for the annihilation operator reads

ck =1√N

∑rij

cije−ik·rij . (1.44)

24

1. QAH 2D C

e position of sites on the lattice is given by rij = ia1 + ja2, with a1 and a2 as Bravaisvectors making an angle 2π/3 between them (see Fig (1.6)). e lattice constant was setto one, a = 1.

e model is represented in Fig. 1.6. On each site there are two different orbitalswith energy ±m. e Hamiltonian (1.43) describes through t-terms the overlap be-tween (non)equivalent orbitals.

Note that there is no net ux perpendicular to the two dimensional plane. But theTRS is broken, because for certain closed paths involving the hopping term t1σ2, for theelectron moving between two nonequivalent orbitals there is a gain of a phase ±π/2.However for each path with a phase gain of π/2 there is one with −π/2 and thereforethere is no net magnetic ux over the entire cell. Such a system realizes a IQHE in theabsence of absence of an external magnetic eld. erefore it realizes a Chern insulatorin the same class with the Haldane model [23].

25

1.4 L C H

1.4 Large Chern phases in the Haldane modele rst theorized Chern insulator is the Haldane model. It consists of a simple tight-

binding model for spinless electrons on a hexagonal lattice.e model can be understood as an underlying NN (nearest-neighbor) graphene

with additional mass terms that gap the system. First there is an on-site energy ±Mon the two nonequivalent atoms in the unit cell, A and B. is term breaks inversionsymmetry and creates a gap. Secondly, there are N2 (next-nearest-neighbor) hoppingswhere an electron gains a phase ϕ. is term breaks TRS and is also called the Haldanemass term. e Hamiltonian of the system reads

H =∑m

Mmc†mcm +

∑⟨m,n⟩

t1c†mcn +

∑⟨⟨m,n⟩⟩

eiϕmnt2c†mcn. (1.45)

e sumruns over all the sites on thehexagonal lattice, and ⟨. . .⟩ represent nearest neigh-bors and ⟨⟨. . .⟩⟩ next-nearest neighbors. e hopping integrals are denoted by tj . eindex of the hopping integral signi es the order of the (non)equivalent neighbors. ephase for a clockwise hopping from n tom is ϕmn = ϕ and, −ϕ, for anticlockwise hop-ping. e on-site energy isMm =M , whenm denotes an A atom, and−M , in case of aB atom.

Due to themass terms (M and t2), the system is generally a band insulator. Note thatthroughout the chapter, a system is considered an insulator if its bands can be attenedsuch that the Fermi surface rests in the gap (and not as usually, i.e. if the Fermi energyis between the conductance and valence band). e essential feature of the model restsin the fact that there is no net magnetic ux per unit cell. erefore such a model canbe envisaged in zero external magnetic eld. However, one can nd paths in real spacewhere the electron gains a phase.

Even though there is no net magnetic eld, Haldane showed that there are Chernnumbers that index the insulating phases of the Hamiltonian. When the system is im-plemented on a nite geometry, there will be charge transported on the edge channels.More precisely there will be a quantized conductance that is proportional to the Chernnumber C. e conductivity is given similarly to the integer quantum Hall effect byσH = e2/h× C. In this way, the model is a realization of a Hall effect without an exter-nal magnetic eld.

Although it was only a theoretical model, its importance for the evolution of thesubject of topological insulators cannot be understated. e rst proposal forQSHE ingraphene is inspired directly from theHaldanemodel [4, 52]. e spin-orbit interactionopens a gap in graphene and each spin realizes a copy Haldane with opposite Chernnumber. erefore it was predicted that the gapped graphene could host chiral andspin resolved edge states.

Returning from QSH insulator to the original Haldane model, note that the maxi-mumChern number is |C| = 1, and thus there can be atmost one edge state. In contrast,

26

1. QAH 2D C

in the IQHE, one can increase the number of edge states by lling more Landau levels.en a natural way to increase the edge conductance in the Haldane model, would be

to create a multi-band system from a stack of two dimensional Haldane models, suchthat the edge states of every layer have the same chirality.

However, there is an alternative route to augment the number of possible edge chan-nels. e system can be decomposed in a gapless model and a mass term. Generally theband touchings of the gapless model are Dirac fermions. en a necessary condition tohave a Chern number |C| = n is to have 2n Dirac points. e Chern number will bedetermined entirely from the chirality of the Dirac fermions and the sign of the massterm that gaps them.

e Haldane model is a particularly interesting platform on which to test this ideabecause it is decomposed in anunderlying graphenemodel and amass term that containsthe phase ϕ dependence.

First subsection focuses on the studyof a graphene-likemodelwithdistant-neighbor-hopping integrals included. e solution for all Dirac points are studied up to N7(next×6-nearest-neighbor-hopping) graphene. eir evolution in parameter space hasparticular points that are called super-mergings where all Dirac points meet to createa band touching with higher than linear dispersion relation. ese are unstable pointsthat are characterized however by a small topological charge.

Second subsection considers gapping the Dirac fermions by the mass term. iswill have the effect of creating topological insulating phases with large Chern num-ber. It is equivalent to say that more edge channels can be created for the nite sys-tem. At the topological transitions between the phases the system becomes metallicand the exchange in Chern number can be seen in the number of Dirac fermions thatare formed.[53]

1.4.1 Distant-neighbor hopping in grapheneTo analyze the topological properties of the model it is advantageous to go in momen-tum space where the topological index is readily de ned. e two-band Bloch Hamil-tonian is developed in a basis of Pauli matrices σ in the AB site space

H =3∑

µ=0

hµσµ. (1.46)

For addressing the topological properties, one can throw away the term h0σ0 that breaksparticle-hole symmetry and shi s (topologically) trivially the bands. Let us also choosethe Bravais lattice vectors a1 and a2, with a1 =

(√32, 32

)a and a2 =

(−

√32, 32

)a; a is the

lattice constant and it is set to one in the following. en the Hamiltonian reads

H = t1[1 + cos(k · a1) + cos(k · a2)]σ1 + t1[sin(k · a1) + sin(k · a2)]σ2+ [M − 2t2 sinϕ[sin(k · a2)− sin(k · a1) + sin(k · (a1 − a2))]σ3. (1.47)

27

1.4 L C H

hopping physical chemical neighbors gndistance distancet1 1 1 3 1 + e−ik·a1 + e−ik·a2

t3 2 2 3 2 cos(a1 − a2) + e−ik·(a1+a2)

t4√7 2 6 eik·a1 + eik·a2 + e−2ik·a1 + e−2ik·a2 + eik·(a1−2a2) + eik·(a2−2a1)

t7√13 3 6 cos(2k · (a1 − a2)) + e−ik·a1 cos(2k · a2) + e−ik·a2 cos(2k · a1)

Table 1.3: Hopping terms between AB atoms. e physical distance is counted in unitsof lattice constant a. e chemical distance is the smallest number of bonds passedwhenhopping between two sites. In the column “neighbors” it is counted the number of sitessituated at the same physical distance from the central site.

e underlying graphene-like model contains only the h1(k) and h2(k) terms. eDirac points of themodel are obtained from the zeros of the function f(k) = h1(k)σ1−ih2(k)σ2. e Dirac points are positioned in the BZ at K(′) = (± 4π

3√3, 0). e interest is

to keep the system two-band and to simultaneously increase the number ofDirac points.e claim is that this can be realized by including distant-neighbor-hopping terms.Let us generalize the graphenemodel by includingNn (next×(n−1) nearest neigh-

bor) hopping terms. e AA and BB terms will contribute only to the identity termh0σ0. Because they will not have an impact on the topological index, they are droppedout from the Hamiltonian. By contrast, the hopping between different sites, AB, con-tributes to h1(k) and h2(k) terms.

e Hamiltonian reads

H =

(0 f(k)

f ∗(k) 0

)(1.48)

with

f(k) =∑n

tngn(k). (1.49)

e contributions gn from the distant-neighbor hoppings are tabulated to rst or-ders in Tab. 1.3. Here are treated cases up to N7 graphene for which one can obtainfairly straightforward analytical solutions for the Dirac points. erefore the only hop-ping terms that appear in f are t1, t3, t4 and t7. In the following the hopping integralsare considered in units of t1, such that there are 3 free parameters le for determiningthe position of Dirac points.

To determine the position of the zeroes, one can keep on the high symmetry linebetween K and K′. Under the TRS of the model and C3 symmetry of the Γ point, allthe other solutions readily follow. One of the linesKK′ lies at ky = 0. en the equationf(k) = 0 depends only on the parameter kx. If cos(

√3kx/2) is denoted by x, it follows

that the band touchings are given by the equation

(1 + 2x)(8t7x3 + 4t4x

2 + 2(t3 − 4t7)x+ 1− t3 − 2t4 + 2t7) = 0. (1.50)

28

1. QAH 2D C

t1

t2t3

t4t5

t6 t7

t8t9

Fig. 1.7: e possible hoppings in graphene N9 model. From a central B atom, theneighbors are arranged in concentric circles. Hopping integrals between distant sitesis denoted by ti where i grows with the distance to the neighbor.

Eq. (1.50) has always a solution at x = −1/2. It corresponds to the regularK andK′

points in the NN graphene. is solution due to the symmetry of the hexagonal latticeis general and exists for any distant-neighbor-hopping model as long as it respects theoriginal symmetries of the lattice. erefore there will always be band touchings at thesepoints in the graphene-like model.

If all solutions are real and distinct, then they correspond toDirac points. However,if a solution has amultiplicity higher than one, then they describe points with nonlineardispersion. In fact one can think of them as mergings of Dirac fermions. ey will posesome conceptual problem for our method as the chirality

χ(κ) = sgn(∂kxh× ∂kyh)3∣∣κ, (1.51)

is not de ned at the merging points. ese are points where the rst derivatives vanishand χ = sgn(0) is not well-de ned. However the topological charge of the mergingpoints is just the sum of the chiralities for the Dirac points that are converging to it.

is fast calculation of charge associated to a band touching will be referred to as thesum rule [48].

Note that equation (1.50) has the number of free parameters equal to the order ofthe equation. is indicates that there might be a solution maximally degenerate. isunique point in parameter space is called a super-merging.

Let us suppose for the moment that all solutions are distinct, such that they repre-sent Dirac points. e spectrum is symmetric under rotations with 2π/3 around thecenter Γ of the Brillouin zone. erefore for each solution near one of the points, two

29

1.4 L C H

more follow around the central one. Under the TRS, a mirror triplet is sure to exist atthe opposite corner of the BZ. In conclusion, each x = −1/2 solution carries 6 bandtouchings. Let us treat in detail the cases with more distant hoppings.

N3 Graphene

e N3 graphene was already investigated in Ref. [54]. Here the presence of a strongt3 hopping integral was shown to produce three more satellite points “orbiting” aroundeach regular Dirac point. Solving Eq. (1.50) gives a possible band touching near K:

κ =2√3

3arccos

(t3 − 1

2t3

)× (1, 0). (1.52)

e solution is real only when t3 obeys

t3 − 1

2t3∈ (−1, 1). (1.53)

en for a large hopping t3 ∈ (−∞,−1) ∪ (1/3,∞), these satellites appear and movealong the high-symmetry lines between the regular Dirac points with the variation oft3. Due to the TRI (time reversal invariance) and C3 symmetry of the spectrum it isenough to follow the motion of a single satellite Dirac point in the BZ. Choosing theone in Eq. (1.52), it is seen in Fig. 1.8 that when t3 varies from −∞ to −1 the satellitepoints appear at mid distance (point Σ) between K(K′) points and the center Γ of theBZ, and move to annihilate at Γ. When t3 varies from 1

3to∞, the satellite Dirac points

appear atM point and vanish at Σ.It is noteworthy that during the evolution of a satellite point, there is a particular

value of t3 for which the satellite points merge with the regular Dirac points at K(K′).At t3 = 1/2, there are three Dirac points merging with the associated regular Diracpoints to form a point with quadratic dispersion. A merging point, where all satellitescollide into the central Dirac point, is called a super-merging. It will be shown that upto N7 graphene the super-merging is unique.

e remaining satellites can be obtained by applying rotations of 2π/3 aroundK andK′. For example, there are two satellites around K, denoted for notation simplicity alsoby κ

κ = arccos(t3 − 1

2t3

)×(−

√3

3,±1

). (1.54)

Under time-reversal symmetry there are additional Dirac points at κ′ = −κ.e chirality χ of a Dirac point placed at k on BZ is readily determined by comput-

ing the sign of the Jacobian

χ(κ) = sgn(∂kxh× ∂kyh)3∣∣κ. (1.55)

30

1. QAH 2D C

Γ K MK′

Σ

Fig. 1.8: In blue we represent the evolution of a satellite point in BZ, when t3 varies from−∞ to−1 and in red, when it varies from 1/3 to∞.

Dirac points K 3× κ K′ 3× κ′

χ 1 −1 −1 1

Table 1.4: Chirality of the Dirac points in N3 graphene.

e symmetries of the system allow one to readily reduce the problem to that of ndingthe chirality of only twoDirac points: a regular one and a satelliteDirac point associatedto it. If one is placed at κ, then its time reversed partner exists at −κ and has oppositechirality. Satellite Dirac points that are obtained under a rotation by 2π/3 around Γshare the same chirality.

As mentioned before, there are notable exceptions where formula (1.51) fails fort3 tuned at the merging of satellite points at K(′). is is because the merging pointhas a quadratic dispersion and the formula is unde ned. However, near the mergingpoint three satellite points of a given chirality converge to a regular Dirac point of op-posite chirality. Such a scenario was treated before [47, 48] and was shown to yield aband touching point with quadratic dispersion in both directions. e chirality sum-mation rule ±(1 − 3) is still applicable and indicates that the merging point at K(′)

is characterized by a chirality ±2. Comparable cases are also encountered in bilayergraphene [47, 55, 56].

e topological charge−2 of the band touching atK can be equally read by expand-ing in small momenta (qx, qy) around the merging point

k = K+ q. (1.56)

In this case, the function f in the effective Hamiltonian at in K-valley is proportionalto z2, with z = qx + iqy. e power indicates a Berry phase of−2π picked by a fermionmaking one counter-clockwise revolution around themerging pointK. e othermerg-ing point is at K′ and will have a topological charge 2.

31

1.4 L C H

graphene Super-mergingf(K+ q) charge

t1 t3 t4 t7NN 1 0 0 0 z 1N3 1 1/2 0 0 z2 −2N4 1 2/5 1/5 0 zz2 1N7 1 7/12 1/4 1/12 z3z −2

Table 1.5: Super-merging characteristics at K. e function f from the effective low-energy Hamiltonian Heff = 1

2fσ+ + H.c. is written as a function of small momenta

z = qx + iqy and up to a multiplicative constant which is neglected.

N4 Graphene

In the case of N4 graphene, there are two additional solutions possible at

x± = − 1

4t4[t3 ± (t23 + 8t24 + 4t3t4 − 4t4)

1/2]. (1.57)

erefore the maximum number of satellites around K can grow up to six. e depen-dence on two parameters makes it harder to nd the existence conditions for the satel-lites. It is however easy to determine the parameters for which there is a super-mergingsolution. Imposing a triply degenerate solution, x = −1/2 in Eq. 1.50, it follows thatthe super-mergings develop at

t3 =2

5, t4 =

1

5. (1.58)

By perturbing the parameters around the super-merging points, one can see in Fig. 1.9a regular Dirac point and the two triplets of satellites associated to it.

Expanding the Hamiltonian in small momenta q around K reveals that the super-merging has a cubic dispersion,

feff(K+ q) ∝ zz2. (1.59)

Note that although thedispersion is cubic, thepower counting gives a topological chargeof one. is indicates that the Berry phase gained by an electron is only π. is is dueto the fact that the super-merging is due to two sets of satellites with different chiral-ity. en the sum rule yields accordingly the chirality of the band touching at K :1− 3 + 3 = 1.

N7 Graphene and beyond

In N7 gaphene the position of the additional band touchings is given by a cubic equa-tion. It is therefore harder to analyze the solution that follows. ough it easier to visu-alize it from the super-merging point, which can be determined from the condition that

32

1. QAH 2D C

−π −π/2 0 π/2 π−π

−π/2

0

π/2

π

Fig. 1.9: e zero lines of h1 (in green) and h2 (in red) in N4 graphene. A small pertur-bation (+0.001) of t4 at the merging point t3 = 2/5, t4 = 1/5 creates 6 Dirac pointsaround the stable Dirac points K. In the inset there is a zoom around K. e Diracpoints are represented by full circles, •; there is a central K Dirac point in black, and 2sets of satellite Dirac points, in blue and red.

x = −1/2 is four time degenerate solution to the equation (1.50). e super-merging isunique at

t3 =7

12, t4 =

1

4, t7 =

1

12, (1.60)

such that a perturbation will produce a maximum of 9 Dirac points around a regularDirac point. e effective Hamiltonian shows now a quartic dispersion near K

feff(K+ q) ∝ z3z. (1.61)

Again, although there is a high-order dispersion, the low topological charge is explainedby the alternating chirality for the 3 sets of satelliteDirac points near the super-merging.

ere are no longer unique super-merging points for graphene Nm, with m > 7.is is due to the fact that already band touchings inN8 graphene have an equation that

remains cubic in x. erefore there is no single unique choice of parameters for whichthe solution x = −1/2 has multiplicity four. erefore, even if a more distant-neighborhopping will add one order to the equation, the order of the equation will never matchagain the number of coefficients.

Factoring out the trivial solution at x = −1/2, the equation for Dirac points in N8graphene reads

8(t7+ t8)x3+4(t4− t8)x

2+2(t3−4t7−3t8)x+1− t3−2t4+2t7+3t8 = 0 (1.62)

33

1.4 L C H

hopping physical distance chemical distancet2

√3 2

t5 3 4t6 2

√3 4

Table 1.6: First hopping integrals between equivalent sites.

ere are still super-merging points possible, but they are no longer unique. Note thatit is possible to have partial merging points where there are solutions with high multi-plicity. Regardless, as the order of the polynomial grows and the space of parametersbecomes larger, we conjecture that it will always be possible to nd solutions that are allreal and smaller than one in absolute value. at will lead to a multiplication of tripletsof satellite Dirac points.

1.4.2 Phases of the Haldane modele originalHaldanemodel is built on the hexagonal lattice forNNgraphene by adding

N2 hopping t2, such that when hopping is performed clockwise in the unit cell an elec-tron gains a phase ϕ. It is enough to consider the mass term h3σ3 and throw away theidentity term, which just breaks the particle-hole symmetry and shi s (topologically)trivially the bands.

e mass term, h3σ3, which breaks time reversal and inversion symmetry, reads

h3 =M − 2t2 sinϕ[sin(k · a2)− sin(k · a1) + sin(k · (a1 − a2))]. (1.63)

When hopping between distant sites is allowed, the generalized mass term reads

h3 =M −∑n

2t(n) sin(nϕ)[sin(nk · a2)− sin(nk · a1)

+ sin(nk · (a1 − a2))], (1.64)

where n is an integer that indicates that hopping takes place between AA or BB sitessituated at a distance of n

√3a. Only the rst two terms in this expression are consid-

ered in the following; they correspond to a maximal two-unit-cell hopping. e termcontaining the hopping integral t5 multiplies the identity Pauli matrix and is neglectedin the following. Interesting for the topology of the problems are hoppings along thelinks where the electrons gain the phase ϕ. e rst two components of the mass termwhich have this property are the t2 and t6 terms.

Gapping 2nDirac points can yieldChern number phases n. In the following subsec-tions, cases where different mass term gaps the system are studied for different underly-ing graphene models.

34

1. QAH 2D C

−π −π/2 0 π/2 π

φ

−6

−4

−2

0

2

4

6

M/t

2 −1 1

Fig. 1.10: Phase diagram for the Haldane Hamiltonian as a function of the on-site en-ergyM , divided by the hopping integral t2, and the ux ϕ. e topologically nontrivialinsulating phases are color-identi ed and have the topological index denoted inside therespective regions. e topologically insulating regions, C = 0, are white.

Haldane t2 model

NN graphene with a hopping t2 constitutes the original Haldane model. e phase di-agram is obtained by observing that h3 changes sign between the Dirac points (K(′)) ofgraphene. erefore the Hamiltonian exhibits three topological phases; a trivial insu-lating phase and two |C| = ±1QAH phases. Eq. (1.30) yields in this case

C =1

2(sgnM+ − sgnM−), (1.65)

where M± = M ∓ 3√3t2 sinϕ is the mass term at K′, and K respectively. e phase

diagram is represented in Fig. 1.10. e lines M± = 0 represent topological transitionlines where the bulk gap closes at least at one of the K and K′ points.

Larger Chern phases become possible when the underlying model is N3 graphene.e mass term has the same sign for a regular Dirac point and its satellites, and opposite

sign at the time reversed points. erefore, when the satellites exist, the gapped phaseswill be indexed by Chern number, |C| = 2.

Momentum κ(′) locates any satellite point of K(′) and, manifestly, the expression forχ(κ(′)) holds in the range of existence of satellite points.

Let us de ne the mass at the regular Dirac pointsM± = M(∓ ( 4π

3√3, 0)

). e mass

at the satellite Dirac points is denoted bym+(−), if it is associated to the regular Diracpoint K′ (K). en it follows from Eq. (1.30) and Tab. 1.4 that

C =1

2

[(sgnM− − sgnM+)− 3(sgnm− − sgnm+)

], (1.66)

35

1.4 L C H

−π −π/2 0 π/2 π

φ

−6

−4

−2

0

2

4

6

M/t

2

2 −2

−1

−1

1

1

Fig. 1.11: Phase diagram for theN3HaldaneHamiltonian. e hopping parameters aret2 = 1/3 and t3 = 0.35 in units of t1.

where the mass of the Dirac points reads

M± =M ∓ 3√3t2 sinϕ,

m± =M ∓ 2t2t3(1 + t3)

√1−

(1− t32t3

)2

sinϕ. (1.67)

Eq. (1.66) yields thephase diagram for the systemwhenall eightDiracpoints are present.When there are no satellite Dirac points (t3 ∈ (−1, 1/3)), the topology of the system isin fact identical to the original system t3 = 0 and therefore it has the phase diagram rep-resented in Fig. 1.10. When t3 is varied to go outside the region (−1, 1/3), two phasesof higher Chern number develop around theM = 0 line. A typical phase diagram forthe case where satellite Dirac points exist is represented in Fig. 1.11. For example, fromEqs. (1.67), it follows that atM = 0 a regular Dirac point and its satellites will have thesame mass. erefore the Chern number reduces to C = sgnM+ − sgnM−. is yieldstopological phases indexed by±2. By increasing |M |, one crosses a transition line wheretheHaldanemass of all satellite points in the system becomes identical, while it remainsdifferent for the regular Dirac points. is transition is given by

m± = 0. (1.68)

is region extends up to the the last topological transition line given by M± = 0. Inthis region the Chern number reduces again to the original case (t3 = 0) with C =1/2(sgnM−− sgnM+). WhenM is increased even further, all Dirac points are gappedidentically and therefore this is the topologically trivial region.

Note that the C = ±1 phases completely vanish at the merging point t3 = 1/2, andthe phase C = ±2 would have maximal area delimited byM = ±3

√3t3 sinϕ. en, at

36

1. QAH 2D C

kx

−3−2−1 0 1 2 3ky−3−2−1 0 1 2 3

E

−4

−2

0

2

4

kx

−3 −2 −1 0 1 2 3ky−3−2−1 0 1 2 3

E

−4−3−2−101234

Fig. 1.12: Energy dispersion for the Haldane t2 model on N3 graphene at a topologicaltransition between two QAH phases (at ϕ = π/2 and t2 = 1/3). (a) Topological tran-sition between C = −2 and C = 0 phases at the merging point between the regular K′

and its three satellites κ′ in N3 graphene. e energy dispersion in N3 Haldane showsa quadratic band touching at K′ forM =

√3, t3 = 0.5. (b) Topological transition be-

tween C = 1 and C = −2 phases. ree Dirac cones form at the satellite points of K′

for t3 = 0.35 andM ≈ 0.95, indicating a change of the Chern number by 3 units at thetopological transition.

the topological transition from |C| = 2 phase to the trivial insulator, there is a quadraticband touching that is represented in Fig. 1.12(a).

e phase diagram in N3 Haldane model (see Fig. 1.11) has the nice feature that itaccommodates lines of transition where Chern number changes by 3 units. is is real-ized by the formation of three Dirac points at the topological transition. ese bandtouchings come from the vanishing of the Haldane mass at the three satellite Diracpoints previously found in N3 graphene. For example, let us take parameters t1 = 1,t2 = 1/3 and t2 = 0.35 from the phase diagram in Fig. 1.11. en xing ϕ = π/2, thereare two transition points between C = −2 and C = 1 phases nearK orK′. In particular,near K′, the Dirac points form at the satellites wherem+ = 0. e energy dispersion atthe topological transition is illustrated in Fig. 1.12(b).

Similarly, one can take as underlying model the N4 graphene model which containsthe t3 hopping. In Fig. 1.13 are represent the QAH phases that can appear for a partic-ular choice of parameters.

Haldane t6 model

e existence of 2n Dirac points for a submodel containing only two sigma matricesallows in principle to build topological insulators with Chern phases C = n. e N3graphenemodel with eight Dirac points can present a large Chern number, C = ±4. Toactualize all possible topological phases it is sufficient to add a t6 mass term. It has theeffect to produce oscillations in the phase-dependent Haldane mass, such that the termchanges sign between a regular graphene Dirac point and its satellites in N3 graphene.As expected, all phases are attainable under this modi cation of the Hamiltonian.

37

1.4 L C H

−π −π/2 0 π/2 π

φ

−2.0

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

2.0

M 1−1

11

1−1−1

−1

4

4

−4

−4

Fig. 1.13: Phase diagram showing the existence of 2 sets of satellite Dirac points. eparameters are t1 = 1, t2 = 1/3 t3 = 0.59 and t4 = 0.4.

e mass term becomes

h3 =M − 2t2 sinϕ)[sin(k · a2)− sin(k · a1) + sin(k · (a1 − a2))]− 2t6 sin(2ϕ)[sin(2k · a2)− sin(2k · a1) + sin(2k · (a1 − a2))]. (1.69)

e new phase diagram is computed by considering themass term (1.69) at the eightN3 graphene Dirac points. en the topological transition lines are given by the zeroesof the newmass terms,M′

±,m′±, expressed as a function of the previousmass terms from

Eq. (1.67)

M′± = M± ± 3

√3t6 sin 2ϕ

m′± = m± ∓ 2t6 sin 2ϕ(2 sin 2κ− sin 4κ), (1.70)

where κ = arccos[(t3 − 1)/(2t3)] holds in the domain of existence for satellite Diracpoints in N3 graphene.

e dependence of themass term on sin 2ϕ allows largeChern number phases, |C| =±4. Because the mass can now change sign not only between a regular Dirac point andits time-reveredpartner, but also between the regular one and its satellites (see Fig. 1.15).

erefore, when systemparameters are varied,N6Haldanemodel can present all Chernphases between −4 and 4. A phase diagram that illustrates this point is represented inFig. 1.14(a).

38

1. QAH 2D C

−π −π/2 0 π/2 π

φ

−3

−2

−1

0

1

2

3

M 2 −2

−1

−1

1

1

3

3

−3

−3−44

1

1

−1

−1

(a)

ϕ

−π −π/2 0 π/2 π

φ

−4

−3

−2

−1

0

1

2

3

4

M 5−5

2−2

2−2

3−3

3−3

1−1

−22

−22

1−1

1−1

−110000

0

0

(b)

ϕ

Fig. 1.14: (a)All QAH phases possible for N3 graphene become available in Haldane t6model. Phase diagram for the choice t1 = 1, t2 = 1/3, t3 = 0.35, and t4 = 0.26. (b)Haldane model from N4 graphene with a t6 mass term. Hopping integrals are t1 = 1,t2 = 1/3, t3 = 0.43, t4 = 0.3, t6 = 0.35 and t7 = 1/3.

−π −π/2 0 π/2 π

−π

−π/2

0

π/2

π

K−K

Fig. 1.15: ADirac point that is represented by • () has chirality+ (−). e colored linesrepresent lines of zeros for h1 (green), h2 (red), and themass term h3 (blue). e regularDirac points placed at

(± 4π

3√3, 0)are gapped by a Haldane mass that has opposite sign.

Also the mass term changes sign between the regular Dirac point and its satellites. Forparameters t1 = 1, t2 = 1/3, t3 = 0.35, t6 = 0.26, M = 0 and ϕ = π/8 the phase isC = −4.

Let us consider shortly the case ofN4 andN7 graphenewith t6 mass term. ere aretwo new free parameters t4 and t7. e parameters space becomes quite large to describeanalytically the dynamics of theDirac points and to track at the same time the sign of the

39

1.5 D

mass at the Dirac points. e general thesis remains however correct: larger and largerQAH phases become possible. In the case of N4 graphene there is a maximum of sixDirac points near aK point; for N7 graphene there are nine possible Dirac points. isindicates that with a proper mass term one can have the largest Chern phases |C| = 7(in N4 graphene) or |C| = 10 (in N7 graphene). In Fig. 1.14(b), it is represented a casewith large C = 5 phases.

1.5 Discussione polynomial whose zeroes determines the position of the Dirac fermions becomes

quickly of too high degree for analytical prehension. Nevertheless, what transpired hereis that truly the topological phases with high Chern number can be indeed inducedthrough addition of distant-neighbor hopping. As a side result, a model for graphenewas studied and was shown that adding distant-neighbor hoppings increases the num-ber of Dirac points. ey move under a variation of parameters and there are uniquesuper-merging points in parameter space where all additional Dirac points merge withthe K and K′ from the regular graphene. ey form points with a high-order energydispersion. However, the absolute value of the topological charge associated to thesuper-merging never grows larger than two because the satellite points always come intriplets with opposite charges. Unfortunately the study is mostly academic and serves asa proof of principle. e degree of control required to realize the super-mergings and thepresence of extreme distant-neighbor hopping remain rather problematic. However itwould be possible to nd multi-layer materials which are be mapped to two-band mod-els with effective-long range hopping. ese systems could use the methods developedin this chapter to analytically chart their topological phase diagram [37].

40

Chapter 2Edge states in aChern two insulator

e present chapter focuses on the toy model from Sec. 1.3. It was shown previouslythat this model supports topological phases with Chern number |C| = 2. e phasediagram of the system was determined by considering only bulk properties. When thesystem is realized on a nite geometry, there will be edges where the system connectswith a trivially gapped phase, here the vacuum. As a consequence there will be edgestates equal in number to the Chern index [15].

e question posed here is what quantum number differentiates the two edge statesthat can appear in the model. Here it is shown through different numerical and analyt-ical techniques that it is a valley number that demarcates the edge (Secs. 2.1 and 2.2).Moreover, the model is extended by adding spin degrees of freedom, such that it is con-verted in a time-reversal invariant Z2 insulator. e edge states from the |C| = 2 phaseare explicitly gapped by one-particle TRI perturbations (Sec. 2.3). is con rms thatevenChernnumbers produce trivialZ2 phases [52, 57, 58]. In the nal Sec. 2.4, there arean in nite number of interfaces engineered between two alternating topological phasesof the sameHamiltonian. It is possible to deduce the topological phase of overall modelfrom knowledge of the Chern number of the constituent stripes.

e momentum space Hamiltonian of the model 1.41 reads

H = 2t1 cos(kx)σ1+2t1 cos(ky)σ2+[m+2t2 cos(kx+ky)+2t3(sin(kx)+ sin(ky))]σ3(2.1)

with a topological invariant given by

C = sgn(−m− 2t2) +1

2[sgn(m− 2t2 + 4t3) + sgn(m− 2t2 − 4t3)]. (2.2)

eHamiltonian (2.1) is implemented on a cylinder and, subsequently, edge states format the bottom and top sides of the cylinder where the lattice terminates abruptly. efollowing sections investigate the edge state wave functions and their respective energydispersion.

41

2.1 N

2.1 Numerical experiments

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kx

(c)

−2 −1 0 1 2t3

−4

−2

0

2

4

m −2

0

0

−1 −1

1 1

a b c

(d)

Fig. 2.1: Zig-zag edge energy dispersionwhen thebulkHamiltonian is indifferentChernphases. e simulation is done for the system on a cylinder with height of 40 sites andcircumference of 180 sites. e number of edge states is 2 × |C| because there are twoedges. Energy dispersions for (a) t3 = 0.4 (C = −2), (b) t3 = 0.85 (topological phasetransition at closing bulk gap), (c) t3 = 1.6 (C = −1). e other parameters are t1 = 1,t2 = 1,m = −1.4. (d) e representation of chosen points on the phase diagram.

e nite geometry chosen for the numerical study is a cylinder. It is constructed outof a patch of the lattice having the shape of a parallelogram with Bravais vectors a1 anda2 as edges. Subsequently the edges parallel to a2 are glued together to obtain the thenal cylindrical shape. Because translational invariance is maintained in the direction

parallel to a1, kx remains a goodquantumnumber. erefore, one can xkx and considerthe resulting one-dimensional problem. Let us write the one-particle solutions of thecorresponding stationary Schrödinger equation as

|ψ(kx, j)⟩ =∑kx

ψj(kx)c†kx,j

|0⟩, (2.3)

42

2. E C

where j denotes the layers of sites in a2 direction, and ψj is a spinor due to the fact thatthere are two orbitals in the problem.

en for a given quasi-momentum kx, the Schrödinger equation reads

Γ1ψj + Γ2ψj+1 + Γ†2ψj−1 = Eψj, (2.4)

where

Γ1 = mσ3 + 2t1 cos(kx)σ1 + 2t3 sin(kx)σ3,Γ2 = t1σ2 + (t2e

ikx − it3)σ3. (2.5)

e cylinder has the edges at j = 1 and j = Ly. ere are hard wall boundary con-ditions, to the effect that the amplitudes ψ0, ψLy+1 must vanish. e energy dispersionas a function of kx is obtained by numerically solving Eq. (2.4) for the given boundaryconditions and for different choices of the parameters (see Fig. 2.1). In our numericalexperiments the cylinder circumference is Lx = 180 sites and with height Ly = 40 sites.All the energies are measured in units of t1.

e bulk-edge correspondence is illustrated by sampling several regions of the phasediagram in Fig. 1.5. In non-trivial topological regions, edge states appear around theends of the cylinder. e number of edge states at a given end equals the absolute valueof the Chern number. For example, three sets of parameter values along the constantm = −1.4 line are taken such that the transition between the C = −1 and C = −2phases is explored (see Fig. 2.1(d)).

While the bulk remains insulating there are states crossing the gap. ese are theedge states and their total number is 2× |C| since the cylinder has two edges. Note thatthe edge states at zero energy, cross the gap at kx = ±π/2. At any topological transitionthe bulk closes at least in one of the special points kx = ±π/2. A transition changingthe Chern number by two requires that the gap closes at both points, while for a changeof one, only one Dirac cone forms.

2.2 Analytical solutionA greater insight into the model is gained by solving Eq. (2.4) analytically. In this way,the gapless states are clearly identi ed as edge states and their penetration length into thebulk is determined. e edge state dispersion law can be found either by directly solv-ing the Schrödinger equation or indirectly by studying the bulk Hamiltonian througha method described in Ref. [59]. Both approaches are explored in the general setting ofthe model and are exempli ed for a particular choice of parameters corresponding tothe phase C = −2.

2.2.1 Edge states from bulk HamiltonianAs it was elegantly proved in Ref. [59], the condition of existence and edge state dis-persion can, under certain provisions, be found from a simple analysis of bulk Bloch

43

2.2 A

Hamiltonians. e method developed by the authors applies when an in nite ribbonor a cylinder is cut out of the in nite 2D system. e direction of the cut must follow aBravais lattice vector. In this case the momentum parallel to the cut k∥ is conserved andthe system splits into a set of 1D Hamiltonians describing the motion of the electronbetween the layers of sites parallel to the cut. e nal prerequisite to apply themethodis that there are only nearest-neighbor layer hopping terms. Eq. (2.4) shows that it isindeed the case in the present model with k∥ = kx.

e key information about the edge states can be revealed by studying the curvestraced by h as a function of k⊥ with xed k∥. For the case with only nearest-neighborlayer hopping allowed, these curves are planar (actually they are ellipses). erefore, hcan be decomposed in two parts, h⊥ perpendicular to the ellipse plane and h∥ the in-plane component. Each component yields some important piece of information aboutthe edge states. Namely, the edge state with a given k∥ exists if and only if the ellipsetracedbyh∥ encloses theprojectionof theoriginonto theplaneof the ellipse. e energyof the state is equal to±|h⊥|.

In the present case k∥ = kx and k⊥ = ky. is yields

h∥ = (0, 2t1 cos(ky), 2t2 cos(kx) cos(ky)+2(t3−t2 sin(kx)) sin(ky)+m+2t3 sin(kx)).(2.6)

For a xed kx, the equation (2.6) describes an ellipse parametrized by ky ∈ [0, 2π). econdition that the ellipse encloses the origin reads

|m+ 2t3 sin(kx)| < 2|t3 − t2 sin(kx)|. (2.7)

is equation determines the range in kx where edge states exist. e energy of the stateis±2t1 cos(kx).

Although the edge dispersion is determined, it still remains to resolve the wave-function for the edges states. Also the existence condition for |C| = 2 phase must allowfour edge states, and therefore it is le to nd the extension in kx for each edge solutionseparately. ese limitations of the above method demand a more applied study of theedge states.

2.2.2 Edge states from the Schrödinger equatione equation characterizing the wave function amplitudes for the edge states have the

form of a recurrence relation. en one can nd a solution using the method of thegenerating functions. Here the solution follows closely the method used in Ref. [60].It involves determining the energy dispersion by studying the poles of the generatingfunction associated to the edge states. Subsequently, this is followed by constructingthe eigenvectors of the Schrödinger equation yielding localized wave functions at twoedges of the cylinder. It is shown that the solutions always cross at speci c points in theBZ and, subsequently, can have an associated “valley” quantum number.

44

2. E C

Let us write the Schrödinger equation for the edge states, complete with the bound-ary conditions. For the moment, assume that there is only one edge and the other oneis pushed to in nity. at is enough to nd the energy dispersion and “spinor” wavefunction for the edge states.

e Schroödinger equation from Eq. (2.4) with explicit boundary condition at theedge, j = 1, reads

0 = (Γ1 − E)ψj + Γ2ψj+1 + Γ†2ψj−1, j > 1,

0 = (Γ1 − E)ψ1 + Γ2ψ2, (2.8)

where

Γ1 = mσ3 + 2t1 cos(kx)σ1 + 2t3 sin(kx)σ3,Γ2 = t1σ2 + (t2e

ikx − it3)σ3. (2.9)

Multiply the jth equation by zj , where z is a complex number. Summing the equa-tions it follows that

∞∑j=1

zj−1[Γ†2ψj + (Γ1 − E)ψj+1 + Γ2ψj+2] = 0. (2.10)

Let us introduce the generating function

G(z) =∞∑j=1

zj−1ψj. (2.11)

en using the boundary condition in Eq. (2.10), it follows that the Schrödingerequation reads

[z2Γ2 + z(Γ1 − E) + Γ2]G(z) = Γ†2ψ1. (2.12)

Edge states exist only if its poles have all a modulus greater than one [60]. If all polesof the generating function have modulus less then one, then they correspond to wavefunction amplitudes that grow in the bulk. For an exponentially localized function atthe edge, all poles must be greater than one.

e generating function is given by

G(z) = [z2Γ2 + z(Γ1 − E) + Γ2]−1Γ†

2ψ1. (2.13)

Let us denote for convenience α = 2t1 cos(kx), β = m+ 2t3 sin(k), γ = t2eikx − it3

and Γ2ψ1 =(ϕ1

ϕ2

). e generating function reads

G(z) = N(z)/D(z),

N(z) =

(z2(γ∗ϕ1 − it1ϕ2) + z((E + β)ϕ1 + αϕ2) + γϕ1 − it1ϕ2

z2(−γ∗ϕ1 + it1ϕ2) + z((E − β)ϕ2 + αϕ1)− γϕ2 + it1ϕ1

).

45

2.2 A

e denominatorD(z) is a polynomial of order four in z. erefore it has four generallycomplex roots. It has the property that if z1 is a solution, then also 1/z∗1 is a solution. Letus assume that z1,2 are solutions with |z1,2| < 1. en a localized edge solution exists,if the roots z1,2 are simpli ed with the roots of the numerator. at is equivalent tosay that the two components of N(z) are linearly dependent and both proportional to(z − z1)(z − z2).

erefore the coefficient of z2 and z0 are proportional, yielding the energy indepen-dent relation between the components

(γ∗ − γ)(ϕ21 − ϕ2

2) = 0. (2.14)

e solution γ∗ = γ is unacceptable because the Viète relations require

|z1z2| = 1. (2.15)

is goes against the premise that both have modulus smaller than one. erefore itleaves to impose ϕ2

1 = ϕ22. Hence the edge states ψ1 can be chosen to be proportional to

eigenstates of σ1

|x±⟩ = 1√2

(1±1

). (2.16)

e energy of the edge state can be determined by returning in N(z) and applyingagain the linear dependence condition with the constraint that ϕ1 = ±ϕ2. is yieldsthe energy dispersion for edges localized at j = 1

E± = ±2t1 cos(kx), for |x±⟩. (2.17)

Let us create another edge at j = N . e system is nowa cylinderwith bottomat j =1 and top at j = N . ere are hardwall boundary conditions for thewave functions, andhence there are vanishing amplitudes ψ0 and ψN+1. e edge states localized at N willbe similarly eigenstates of σ1 with respective energies, E± = ±2t1 cos(kx). However,it remains to determine in the following the decay of the edge wave function in thebulk, and their extension inmomentum space. e latter is addressedwhen assessing theexistence conditions for the solutions. Because the number of edge solutions dependson the Chern phase, there are solutions that must vanish when model parameters arevaried.

To construct the edge wave function use the Ansatz

ψj = zj|x±⟩. (2.18)

Note that the equation in the bulk of the cylinder reads

[z2Γ2 + z(Γ1 − E±) + Γ†2]|x±⟩ =

(00

)(2.19)

46

2. E C

For the |x+⟩, and, respectively, |x−⟩ it follows that

z2(∓it1 + t2eikx − it3) + z[m+ 2t3 sin(kx)] + (∓it1 + t2e

−ikx + it3) = 0. (2.20)

Note that if z1 and z2 are solutions for the equation corresponding to eigenstates|x+⟩, then 1/z∗1 and 1/z∗2 are solutions for the equation corresponding to |x−⟩.

en at each edge there can be atmost two solutions one corresponding to |x+⟩ andone to |x−⟩. However to construct the wave function solution is necessary to apply theboundary condition. is is exempli ed in the following for a solution correspondingto |x+⟩ that is localized at the bottom edge, j = 1.

Because there are two eigenmodes z1,2 associated to a “spinor”, the wave functionreads

ψj = (c1zj1 + c2z

j)|x+⟩, (2.21)

where c1,2 are coefficients. e boundary condition, ψT0 = (0, 0), constrains the coeffi-

cients c+ := c2 = −c1. e coefficients c+,− correspond to |x±⟩ eigenstates. ey arexed by applying boundary conditions at the edges of the cylinder, but they are not of

concern here.us if there is an edge state solution, its form and energy are

ψ+b(t)j = c+(z

j1 − zj2)|x+⟩, E = 2t1 cos(kx),

ψ−b(t)j = c−(z

∗−j1 − z∗−j

2 )|x−⟩, E = −2t1 cos(kx), (2.22)

where the index b (t) denotes an edge state localized near the bottom (top) of the cylin-der. e eigenmodes are determined from the quadratic Eq. (2.20)

z1,2 =−b±

√b2 − 4ac

2a, (2.23)

with

a = −it1 + t2eikx − it3,

b = m+ 2t3 sin(kx),c = −it1 + t2e

−ikx + it3. (2.24)

It is manifest that the model can have zero, one or two solutions at an edge. Forconcreteness, let us examine the states localized at cylinder’s bottom (j = 1). If theeigenmodes |z1,2| = 1, then the solutions (2.22) describe extended bulk solutions. ereare no edge solutions, if for any kx,

|z1| < 1 < |z2|, or|z2| < 1 < |z1|. (2.25)

47

2.2 A

Fig. 2.2: Energy spectrum as a function of momentum kx on a cylindrical geometry(height 40 sites and circumference 180 sites). Two edge states located near j = 1 arerepresented in blue, and two at j = 40 in green. e position and chirality of the edgesis schematically represented in the inset.

In this case all solutions (2.22) diverge for j > 1. Manifestly, the same conclusions applyto the top edge.

Edge solutions can exist when both z1 and z2 are simultaneously smaller or greaterthan one. Note however that it is not possible to encounter a localized solutionψ+b andψ−b in the same range of kx. is is because if one is localized, then the other one hasdiverging eigenmodes z. is indicates that if ψ±b is a localized solution in a given kxrange, then ψ∓t is a localized top solution.

Hence there are a maximum of two wave function solutions for a given edge. ForC = 1, only one of the solutions ψ±b holds in the BZ and the other is diverging awayfrom the edge. For C = 2, both solutions ψ±b are possible in distinct ranges of kx.

Note that the energy dispersion always crosses the zero energy at kx = ±π/2. Ata given edge, if there are two solutions, then they are associated either to kx = π/2 orkx = −π/2. is answers the question: what quantum number distinguishes the edgestate solutions for |C| = 2? Any edge state can be indexed a “valley” quantum number±K corresponding to a zero crossing at±π/2.

e existence conditions and the respective valid edge wave functions are given bythe following inequalities:

|z1| < 1, |z2| < 1, ψ+bj and ψ−t

j ,

|z1| > 1, |z2| > 1, ψ−bj and ψ+t

j . (2.26)

Let us illustrate the above results for a special point t1 = 1, t2 = 1, t3 = 0,m = 0

48

2. E C

of the phase diagram in Fig. 1.5. is point corresponds to the “center” of the C = −2phase and is characterized by the largest gap and attest bands for the spectrum of thebulk states. One can expect two edge states at either endof the cylinder. e eigenmodesz are determined from Eq. (2.20)

z1,2 = ±i√

(−i+ eikx)(−i+ e−ikx)

−i+ eikx. (2.27)

e penetration length for the edge states is given by

ξ = −1/ ln(|z1,2|),

=4

ln[1 + sin(kx)]− ln[1− sin(kx)]. (2.28)

In this case the edge states have maximal extension and cover the entire BZ, becausethe existence condition (2.26) is always respected except at points kx = 0 and kx = π.At these particular points the edge states enter the bulk.

e above arguments allow to readily obtain the four edge state solutions in the C =−2 phase for t1 = t2 = 1 and m = t3 = 0. e wave functions and the respectiveenergies read

ψ−Kbj = c+(1− (−1)j)ρj1|x+⟩, E = 2 cos(kx), kx ∈ (−π, 0),ψKbj = c−(1− (−1)j)ρ∗−j

1 |x−⟩, E = −2 cos(kx) kx ∈ (0, π),

ψKtj = c+(1− (−1)j)ρj1|x+⟩, E = 2 cos(kx), kx ∈ (0, π),

ψ−Ktj = c−(1− (−1)j)ρ∗−j

1 |x−⟩, E = −2 cos(kx), kx ∈ (−π, 0). (2.29)

e indices t and b indicate whether the edge states live close to the top (j = Ly) or thebottom (j = 1) part of the cylinder. e index ±K designates edges states crossing thezero energy at±π/2 or, equivalently in this case, whether the solutions extended in theright, respectively le , half of the ([−π, π]) BZ. e coefficients c± are normalizationcoefficients which are not of interest here. e edge states’ energy dispersion (2.29) areplotted in Fig. 2.2 together with the numerical solution in order to show their perfectagreement.

2.3 Extension: Edges in a Z2 insulatore classA topological insulator thatwas constructed in the previous subsections can be

easily transformed into aZ2 topological insulator in the symplectic classAII by imposingtime-reversal symmetry (seeTab. 1). In the following, spin degrees of freedomare addedand aQSHmodel is created from two copies of theZ insulator. e object of the presentsubsection is to follow the fate of the two chiral edge sates obtained at the interface of|C| = 2 phases with the vacuum. Crucially, these states are not robust with respect toarbitrary one-particle, time-reversal invariant perturbations.

49

2.3 E: E Z2

emost straightforward route to create theZ2 insulator is the one taken inRef. [4].A TRI model is constructed from a Z insulator model and its time-reversal copy, withopposite Chern number. If there are no terms coupling the two models, then any edgestate predicted from one of the models will have a partner with opposite chirality fromthe other one. Let us add spin avor and ensure that the spin s3 Pauli matrix commuteswith the Hamiltonian. Each of the spinless Z model, now represents a polarized spinup, respectively down model, and each edge state is spin polarized.

Suppose that the spin up component is described by Eq. (1.41), while the spin downone represents its time reversed copy. is yields the following 4-band Hamiltonian:

H(k) = 2t1 cos(kx)σ1 + 2t1 cos(ky)σ2s3+ [m+ 2t2 cos(kx + ky)]σ3 + 2t3[sin(kx) + sin(ky)]σ3s3, (2.30)

where s Pauli matrices represent electronic spin, and σ, the orbital degrees of freedom.Because the spin Hamiltonian is created from two copies of the spinless Hamilto-

nian, with no spin mixing terms, the conditions for the energy gap are not changed.at means the previously found insulating phases remain insulating for the four-band

model.On a cylindrical geometry, there are edge states forming around the edges of the

cylinder. Because there are no spin mixing terms, the energy spectrum is trivially ob-tained by “doubling” the spectra already found for the spinless Hamiltonian. More pre-cisely it is obtained from the union of the spinless (now spin up)Hamiltonian spectrumand its re ection about kx = 0 underTRS. erefore the number of edge states will alsodouble such that each original edge state will get a Kramers partner.

Although every previously nontrivial Chern phase will show edge states in the spin-ful model, not all of them are robust. Indeed, the QSH insulator constructed from ofthe spinless model with |C| = 2 allows one-particle TRI perturbations that destroys theedge states [6, 61].

A small TRI perturbation gaps the edge states at the crossing point kx = ±π/2.Hence it is sufficient to analyze the system near the crossings to nd such a perturba-tion. e low-lying edge states are described by an effective Hamiltonian, obtained bylinearizing the solutions (2.29) near crossings kx = ±π/2. At a given edge, for the phasewith |C| = 2, this yields:

Heff(qx) = Ψ†KR↑vqxΨKR↑ +Ψ†

−KR↑vqxΨ−KR↑

−Ψ†−KL↓vqxΨ−KL↓ −Ψ†

KL↓vqxΨKL↓, (2.31)

where Fermi velocity reads v = 2t1. e indices of the fermionic creation and annihila-tion operatorsΨ† andΨ describe the valley (±K), the direction of motion (L orR) andthat of spin (↑ or ↓). Note also that the rst two terms inHeff describe the dynamics ofspin up electrons, and therefore correspond to the original 2-band Hamiltonian, whilethe spin down terms stem from of time reversal operator T; TΨKR↑T

−1 = Ψ−KL↓. elocking between the direction of the spin and that of motion means thatHeff describesa helical liquid [61].

50

2. E C

e edge states above arenot robust because one can create the followingone-particle,TRI, local perturbation that will gap the edge helical liquid in Eq. (2.31) (localmeaningthere is no inter-edge scattering)

Ψ†KR↑ΨKL↓ −Ψ†

−KR↑Ψ−KL↓ +H.c.. (2.32)

It is possible to buildmanyperturbations at the tight-binding level yielding the aboveform at low energy. It is noteworthy to observe that they all break the spin s3 symmetry.An example of TRS perturbation in the tight-binding formulation is

t4 sin(kx)σ3s1. (2.33)

For the phases with |C| = ±1 no one-particle, local, TRI perturbation can resultin backscattering of the edge states. e above example agrees with the statement thatmodels with an even number of Kramers pairs of edge states are Z2-trivial [62].

2.4 Extension: Stripes of a Z insulatorere are still ways in which the Chern number computed analytically for a two-band

model continues to be useful in multiband systems. Among the more simple ones is the“striped” topological insulator explored in the following.

Generally when two insulating phases with different Chern number are put in con-tact, there are edgemodes that form between the phases. Consider now that one createsstripes from the same systemwithwidth of the order of the length of the unit cell. ereare only two type of stripes, differentiated by parameter values (m, ti). A new 2D systemis constructed by alternating the two stripes, thus creating interfaces everywhere in thevolume.

e natural suppositionwould be that if the parameters are associatedwith differentChern phases then there are edge states forming everywhere in the model. is turnsout to be correct; the system becomesmetallic with transport everywhere in the bulk inthe direction of the edges. Let us study inmore detail how the phases can be determinedfrom an analysis of the phase diagram for each stripe Hamiltonian separately.

e model |C| = 2 insulator in Eq. (1.41) is considered again. To simplify the prob-lem, eliminate the |C| = 1 phase by forcing t3 = 0. en the Hamiltonian reads

H = 2t1 cos(kx)σ1+2t1 cos(ky)σ2+[m+2t2 cos(kx+ky)+2t3(sin(kx)+sin(ky))]σ3.(2.34)

e Chern number that indexes the insulating phases reads

C = sgn[−m− 2t2] + sgn[m− 2t2]. (2.35)

In order to create the stripes, an alternating value for t2 is introduced between twoadjacent unit cells. In Fig. 2.3 are represented two of the cells and the striped model

51

2.4 E: S Z

Fig. 2.3: e stripe model with alternating t2 and t′2 hopping term. e red contour in-dicates the original unit cell. A er the addition of t′2, the unit cell doubles its volumeto the blue contour. e two orbitals with different on-site energy ±m are not repre-sented; the Pauli matrices σ indicate the hopping integrals between orbitals along thethree directions in the model.

which is obtained by repeating to in nity the pattern. e change is equivalent to dou-bling the size of the unit cell, which now contains t2 and t′2 hopping terms. Hence inthe stripe model there are four orbitals in the unit cell. A new set of of Pauli matrices τare introduced to denote a “cell” degree of freedom.

en the new Hamiltonian in momentum space reads

H = 2t1 cos(kx2

)σ1τ1 + 2t1 cos(ky)σ2τ1

+

[m+ (t2 + t′2) cos

(kx2

+ ky

)]σ3τ0 + (t2 − t′2) cos

(kx2

+ ky

)σ3τ3.

(2.36)

erefore the energy dispersion reads

E = a2 + b2 + c2 ± 2√

(a2 + b2)c2, (2.37)

with

a = 2t1

√cos2(kx/2) + cos2(ky), b = m+ (t2 + t′2) cos(kx/2 + ky),

c = (t2 − t′2) cos2(kx/2 + ky)). (2.38)

Note that because the unit cell is doubled, the Brillouin zone shrinks by a factorof two. However, here kx was rescaled such it continues to run from π to −π. Dueto the folding of the BZ there are trivial solutions where the energy goes to zero, i.e.(kx, ky) ∈ (0,±π/2), (π,±π/2). ese points correspond to the edges of themetallicregion. ey correspond tom = ±2t2 andm = ±2t′2. But this lines in parameter spacecorrespond exactly to the topological transitions between the topological phases for the

52

2. E C

−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5t2(t′2)

−6

−4

−2

0

2

4

6

m −22

0

0(a)

.........................................

....................

-2.0

-1.5

-1.0

-0.5

0.0

ChernnuberC

-3 -2 -1 0 1 2 3

m

(b)

Fig. 2.4: (a) Superimposed phase diagram for each of the two stripe-Hamiltonian (onewith t2, the other with t′2) gives the complete phase diagram for the system. e Chernnumbers index the (non)trivial phases according to Eq. 2.35. In grey is represented themetallic phase, where Chern number is exchanged at every stripe edge in themodel. (b)Numerical Chern number calculation for the green line on the phase diagram (a), with(t2 = 0.5, t′2 = 1) andm varying in [−3, 3]. When the line crosses themetallic phase theChern number is not de ned; here this is seen as C taking fractional values between thewell-de ned C = 0 and C = −2 phases.

bulk Hamiltonian corresponding to the two stripes (see Eq. (2.35)). is solution in-dicates that, in order to determine the phase diagram of the model, it suffices to knowthe phase diagram for the two submodels.

When the two stripes are in the same connected topological phase, then no edgestate are expected to appear between them. erefore theoverall systemstays in the samegapped topological phase. Nevertheless, for disconnected phases described by the sameindex (for example, separated by a different phase), edge states might appear, renderingthe system metallic. In every case where a Chern number is exchanged at the interface,edge channels will form, rendering the systemmetallic. en the phase diagram can canbe seen by superposing the phase diagrams for the submodels (see Fig. 2.4(a))

e above result is rst tested by numerically computing the energy spectrum. iscon rms the presence ofmetallic phases in the exactwindowpredictedby the superposi-tion argument that brought the diagram in Fig. 2.4(a). A more discerning investigationinvolves computing the Chern number for different parameters of the Hamiltonian.

At half- lling, there are two occupied (indexed by n) and two unoccupied bands(indexed by n′). en the Chern number is determined numerically [17, 41]

C =1

2π

∫BZd2kΩxy(k), (2.39)

53

2.5 D

in terms of Berry curvature, Ωxy, for the occupied bands

Ωxy(k) = −2∑n occ.

∑n′ unocc.

Im[(vx)nn′(vy)n′n]

(En − En′)2, (2.40)

where (vx)nn′ is expectation value of the velocity operator between one occupied andone unoccupied band

(vx)nn′ = ⟨n|∇kxH|n′⟩. (2.41)

For example, t2 and t′2 are xed, andm is varied in Fig. 2.4(b) to yield the expectedvalues for the Chern number. Note that the Chern number is not well-de ned in themetallic region. is is re ected in spurious fractional values between well-de ned in-teger values (in the Fig. 2.4(b), C = 0 and C = −2).

In conclusion, the four-bandmodel phases can be entirely determined from the orig-inal two-band model. is is not a surprise as the striped topological insulators is ob-tained from doubling the original model. In the case of the Z2 insulator, the doublingof themodel kept all the gaps in the original model. However, in the striped topologicalinsulators, if a edge state forms between two stripes as a consequence of disconnectedtopological phases, then the entire 2D model becomes metallic.

2.5 Discussione analyticalmethods in studying the edge states in topological insulators need further

development. e encompassing generality in the description of bulk phases contraststhe short-sightedness of edge state investigations. For example, both methods used tocompute the edge state solution took advantage of the fact that the 1D Schrödingerequation that describes the system connects only near-neighbor sites in the directiontransverse to the edge. is is an artifact of the speci c way in which the edge was cut.An open question remains how to adequately describe generic edge states in topologicalinsulators. e decomposition effected in Chap. 1 of two-band models into gaplesssystemplus amass term gives hope that this could be used to describe such generic states.In perspective, it would take an extension of methods developed to study general edgestates in graphene [63, 64] by considering now the role of the mass term.

A nal word of caution is in order. Edge states can also exist between zones withidentical Chern number. For example, in topologically trivial gapped graphene, edgestates form at a domain wall due to a change in the on-site energy [65, 66]. e sameeffect was observed in simulations carried in Sec. 2.4, and led to the apparition ofmetal-lic phases when the stripes are created out of disconnected topological phases with thesame Chern number. is surprising effect needs further investigation to check the ro-bustness of the edge states. Moreover it adds a new level of difficulty in describing thephysics of the edge states, because it drastically amends the rule of thumb that equalsthe number of edge solutions to the variation of the Chern number across an interface.

54

Part II

Majorana fermions

55

Chapter 3Introduction

is introductory chapter contains a primer to Majorana fermions. First they are pre-sented as real solutions to the Dirac equation. Secondly they emerge as quasiparticlesin different condensed matter realizations. Among these, two one-dimensional systemsare particularly relevant in the context of this thesis and therefore are presented inmoredetail: the Kitaev model [67] and the spin-coupled semiconducting wire supportingMajorana fermions in proximity of an s-wave superconductor [24, 25]. Finally, thechapter also includes a presentation of the concept ofMajorana polarization as an orderparameter to describe a topological transition.

3.1 Majorana fermion primer

3.1.1 Dirac equation and theMajorana conditionA fermionic particle that is its own antiparticle is a Majorana fermion [68]. For a so-lution, Ψ, to the Dirac equation, it is possible to de ne the antiparticle solution, Ψc,obtained under the charge conjugation operation. When the two are equal, they corre-spond to a Majorana particle.

Let us detail the charge conjugation operation by following Ref. [69]. e three-dimensionalDirac equation for a spin-1/2 particle in the presence of an electromagneticeldAµ reads

(iΓµ(∂µ − ieAµ)−m)Ψ = 0. (3.1)

e Greek indices run over the spatial (1, 2, 3) and temporal (0) components, while theRoman indices run only over the spatial ones. e Γs are the 4 × 4 Dirac matrices andare conventionally chosen as

Γ0 = σ0 ⊗ τ3, Γj = iσj ⊗ τ2, (3.2)

57

3.1 M

whereσ and τ are the usual 2×2Paulimatrices. eΓmatrices obey theClifford algebra

Γµ,Γν = 2ηµν , (3.3)

with ηµν the Minkowski metric tensor, diag(1,−1,−1,−1).e charge conjugate solutionΨc will obey the same Dirac equation (3.1), but with

opposite charge −e. at allows one to de ne the charge conjugation operator C, thatyields the equation

C(iΓµ(∂µ − ieAµ)−m)Ψ = (iΓµ(∂µ + ieAµ)−m)CΨ,

= 0, (3.4)

where CΨ = Ψc. Note that the operator C is anti-unitary and therefore must containthe complex conjugation operatorK . It must also contain the only imaginary Γmatrixsuch that [C, iΓµ] = 0. Hence the charge conjugation operator can be represented up toa global phase by

C = Γ2K. (3.5)

Let us now consider the ful lled Majorana condition, that a particle is identical toits antiparticle

Ψc ≡ CΨ = Ψ. (3.6)

Crucially this condition is Lorentz invariant and it is therefore valid in any referenceframe [70]. Moreover, it is immediate from equation (3.1) that only a neutral spin-1/2fermion can obey the condition, e = 0. erefore both Ψ and Ψc are solutions to thesame Dirac equation

(iΓµ∂µ −m)Ψ(c) = 0. (3.7)

e original Dirac spinor Ψ is a complex four-component spinor. Hence it is de-scribed by eight free real parameters. Under the Majorana condition four are xed. Forexample, if the Dirac spinor is written asΨT = (ϕT , χT ), with ϕ and χ two-componentspinors, then the under Majorana condition

Ψ =

(ϕ

σ2ϕ∗

). (3.8)

Because ϕ is generally complex, aMajorana fermion has only four real components. Un-der a unitary transformation one can obtain purely real Majorana fermions, where eachof its four components is real. is happens in a Majorana representation, where all theΓ matrices, not only Γ2, are imaginary. Here the charge conjugation becomes complexconjugation and Majorana condition emerges as a strict reality condition

Ψ = Ψ∗. (3.9)

58

3. I

A possible choice for a Majorana basis is

Γ0 = σ2 ⊗ τ1, Γ1 = iσ3 ⊗ τ0, Γ2 = −iσ2 ⊗ τ2, Γ3 = −iσ1 ⊗ τ0, (3.10)

with Γs that continue to obey the Clifford algebra.Although Majorana particles are theoretically valid solutions to the Dirac equation,

they have never been found among the fundamental particle. To this moment the neu-trino remains an open possibility as a massive Majorana particle.

3.1.2 Particle-hole symmetryIn recent years, the idea of a Majorana fermion was resurrected in the context of con-densed matter where it could be realized as a quasiparticle in a superconductor [71].Intuitively it is natural to search in a superconductor because quasiparticle excitationscontain both particle and hole degrees of freedom. A hole is simply the “antiparticle”of an electron.

In the language of creation and annihilation operators, the charge conjugation op-eration discussed previously is simply the operation of taking the Hermitian conjugate.

at follows because creating an electron is identical to destroying a hole with oppositemomentum.

At mean eld level, the Hamiltonian for a superconductor in the second quanti ca-tion reads [13]

H =1

2C†HC, (3.11)

with C† = (c†, c) and c is a row (or column, depending on context) vector comprisingthe annihilation operators at all the lattice sites in a superconductor, eventually with aspin index (c↑ or c↓). e “ rst-quantized’’ Bogoliubov-de Gennes Hamiltonian reads

H =

(H0 ∆∆† −H∗

0

), (3.12)

where ∆ is the matrix containing the superconducting order parameters. e action ofthe charge conjugation transformation is to convert a particle into a hole and vice versa

CcC−1 = c†. (3.13)

e Hamiltonian exhibits the charge conjugation symmetry

[C, H] = 0. (3.14)

Together with the Hermiticity of H, it is sufficient to determine the consequences ofcharge conjugation symmetry on the BdG Hamiltonian,

CHC−1 = τ1Hτ1 = −H∗, (3.15)

59

3.1 M

where τ are the usual Pauli matrices acting in particle-hole space. is allows to de nea particle-hole symmetry represented by an anti-unitary operator C′ that anticommuteswith the Hamiltonian. Note that it is not a proper symmetry, due to the anticommuta-tion

C′,H = 0, with C′ = τ1K, (3.16)

andK the complex conjugation operator. e consequence of this property is that anyeigenstate of the BdG Hamiltonian has a particle-hole conjugate with opposite energy.

at is, the presence of the PHS immediately re ects itself as redundancy in the solu-tions to the BdG equations. For example, take the Schrödinger equation

HΨ = EΨ, ΨT = (uT , vT ), (3.17)

whereu (v) stands for the particle (hole) component of thewave function. e solutionsdescribe the quasiparticles excitations above the BCS ground state. en if (uT , vT ) isthe solution with energyE, then (v†, u†) is the particle-hole conjugate with energy−E.In contrast, a quasiparticle solution is aMajorana fermion if it is equal to its ownparticle-hole conjugate (anti-quasiparticle), at the same energy. is implies that a Majoranafermion can be realized only at zero energy.

A solution to the mean- eld equation can also be represented by a quasiparticle cre-ation operator

γ†E =∑j

uj,Ec†j + vj,Ecj, (3.18)

where j runs over all the sites (orbitals, and spins) in the superconductor. en theparticle-hole and Majorana conditions can also be concisely expressed as

γE =

γ†−E, PHS condition,γ†E, Majorana condition.

(3.19)

Trivially both conditions are satis ed at zero energy for uj,0 = v∗j,0.It is not always possible to impose the Majorana conditions for the zero modes in

superconductors. In a spin-singlet superconductor, an excitation has the structure [72]

γ†α =∑j

uj,αc†j,α + vj,−αcj,−α, (3.20)

where α is a spin index ↑ / ↓. Now j runs only over the site and orbital degrees offreedom. e Majorana condition reads γ†α = γα, and the quasiparticle and the anti-quasiparticle contain different electron and hole creation operators. Hence, no mat-ter the value of the coherence factors (ujα, vjα), it is not possible to create a Majoranafermion.

60

3. I

In conclusion, a spin-singlet superconductor does not allow naturally the forma-tion of Majorana fermions. Nevertheless, there are ways to create Majorana fermionsusing spin-singlet superconductivity in concert with different ingredients that will beenumerated a erwards. However up to now, the presence of superconductivity standsout as a necessary condition for nding Majorana fermions in condensed matter. emechanism involves always nding a quasiparticle that is its own antiparticle.

By inverting the relation (3.20), one can express the regular creation and annihila-tion operators in terms ofMajorana fermions. is decomposition is generally availableand resembles the representation of a complex number using two real numbers, its imag-inary and real parts. roughout this thesis the following representation of a creationoperator is used

c† =eiθ/2√

2(γ1 − iγ2), (3.21)

where the angle θ is an arbitrary degree of freedom in de ning the Majorana fermions.erefore the Majorana fermions obey the Clifford algebra

γA, γB = δA,B, (3.22)

where δA,B is the Kronecker symbol and A,B are Majorana indices, 1 or 2. Moreover,unlike complex fermions, Majoranas do not square to zero, but γ21,2 = 1

2.

Let us couple two Majorana fermions, γ1 and γ2. Working in the basis of the eigen-states of γ1 and γ2, the coupling 2iγ1γ2 will act as a Pauli σ2 matrix with eigenvalues±1 [73]. e occupation number of the fermionic operator created from theMajoranasthen reads

c†c =1

2+ iγ1γ2. (3.23)

erefore the occupation number c†c is either 0 or 1. In the limit of a very weak cou-pling ε between theMajorana fermions (for example they can be located far away), theycan still create a nonlocal fermionic state that is either full or empty. For vanishing cou-pling, ε→ 0, the ground state is effectively degenerate, and it costs ε to ll or empty thefermionic state. erefore if the system contains Nγ1 and Nγ2 Majoranas, the groundstate degeneracy will then be 2N . Systems with such a highly degenerate state were pro-posed inRef. [71], where theMajorana fermions are trapped in vortex cores of 2D p+ipsuperconductor. Moreover, interchanging the Majorana fermions through a continu-ous adiabatic process generates a different ground state wave function that depends onthe precise trajectory taken by the vortices. is dependence on the topology of pathstaken by the vortices indicates the non-Abelian character of the ground state.

One of the main interest in nding Majorana fermions lies in using the highly de-generate ground state to store information. en computations can be realized on it bytaking advantage of its non-Abelian character and the protection ofMajorana fermions.

e catalog of operations on the ground state form a representation of the braid group[74]. e goal of this research is to nally realize a topological quantum computer [73].

61

3.2 C

3.2 Condensedmatter realizations

3.2.1 Overviewe reader is referred to excellent reviews that provide a good panorama of the research

onMajorana fermions in condensedmatter systems [75, 76]. Here it suffices tomentiona few milestone in the evolution of the eld.

As seen before, in condensed matter physics Majorana fermions can be realized asquasiparticles in spinless or triplet-pairing superconductors. ey are possible as zeroenergy excitations in systems with particle-hole symmetry.

Naturally they were rst proposed in 2D p + ip superconductors [71], where theycould be realized as boundmodes in the vortices of the superconductor. A 1D variationwas soon a er treated by Kitaev [67], where Majorana states in a spinless tight-bindingmodel appear as bound states at the ends of the wire. Different proposals followed thesetwo paths: boundary states in p-wave superconductors or Caroli-de Gennes-Matriconstates [77], where theMajorana fermion appearwhen a amagnetic ux penetrates a typeII p-wave superconductor. Other materials in which Majorana fermions are thought toappear are 2D electron gases in the fractional Hall regime at 5/2 lling factor [78], andtwo-dimensional strontium ruthenate, Sr2RuO4 [79] (and [80] for experiments). eywere also proposed to form in p-super uids in ultracold atom gases [81, 82]

e problem with the 2D p-wave superconductors is that they are rare materials.Moreover, for the case ofMajorana states bound to vortices, there aremore excited stateswithin the bulk gap rendering fragile the protection of the zero modes [76].

A revolution of the eld came when it was realized that one can replace the p-wavesuperconductors with s-wave superconductors and one can have an effective p-wave su-perconducting pairing mediated through s-wave pairing. e rst example was pro-posed in Ref. [83]. ere Majorana end states appear in a 2D topological insulatordeposited on an ordinary s-wave superconductor. e chiral edge states in the topo-logical insulator can be coupled through the s-wave pairing such that inside the gapone can have a pair of Fermi points (or more precisely an odd number in half the BZ)and an effective p-wave pairing. Numerous proposals have followed which are usingtopological insulators and superconductors as a basic setup for creatingMajorana excita-tions [84, 85]. Some of the experiments following the theoretical proposals are reportedin Ref. [86].

Nevertheless, topological insulators remain exotic materials. More recently, propos-als have emerged where topological insulators are replaced by regular semiconductingwires with a strong-spin orbit coupling under a magnetic eld [24, 25, 87]. ey areone of the main subjects of the thesis and they will be described in more detail in thenext sections. Several articles expanded on this direction, in semiconducting quantumwells [88], or in multiband spin-orbit coupled wires [89, 90]. Experiments in thesedirections were pursued by several groups [91, 92]. ere are also variations to thesesystems which dispense with the spin-orbit coupling [93, 94].

Let us nally note the prediction of Majorana fermions in non-centrosymmetric su-

62

3. I

(a)

(b)

Fig. 3.1: Two topological phases of the Kitaev model. e red and blue denote twoMajorana fermions that compose a regular complex fermion, represented in black. (a)Trivial phase where the Majoranas are coupled inside each site. (b) Nontrivial phasewhere Majorana in adjacent sites hybridize to form a complex electron. Unpaired zeromodes are le at the end of the wire.

perconductors [95, 96] and or in ultracold atomic gases with s-wave superconductiv-ity [97].

For the detection of the Majorana modes the principal directions are: interferome-try [98, 99], measuring nonlocal tunneling [100, 101], fractional Josephson effect [67,102], the detection of zero bias conductance peak [103, 104], etc.

3.2.2 Kitaev modele present thesis is limited to the study of phenomena related to Majorana fermions

arising in one dimensional systems. e following Kitaev model inescapably arises asa paradigm for 1D systems supporting Majorana fermions. erefore it is worthy toprobe it in more detail.

e model was proposed in Ref. [67] and it is captured by the tight-binding Hamil-tonian

H =N∑j=1

(1/2− c†jcj

)µ+ (−tc†jcj+1 +∆cjcj+1 +H.c.), (3.24)

where j runs over all the sitesN of a nite lattice. It is a 1D toymodel describing spinlesselectrons that experience a superconducting pairing∆, treated at themean- eld level; µis the chemical potential and t is the hopping strength. It was argued that such systemcan present Majorana, zero energy states bound at the two ends of the wire. is workwas motivated by the search of a qubit that is protected with respect to perturbations.

e qubit is formed by the two entangled Majorana states. Nevertheless, the interest inthe following will be centered on the conditions and the description of the topologicalphase.

e Majorana fermions exist as quasiparticle excitations and that explains the needfor superconductivity. e system is spinless, yielding essentially a p-wave superconduc-tor, with the quasiparticles γ obeying the reality (Majorana) condition γ = γ†. More-over the system has a bulk gap that protects eventual zero energy modes.

63

3.2 C

rough a gauge transformation, the phase of the superconducting parameter can beabsorbed in the de nition of the creation and annihilation operators. For ∆ = |∆|eiφthe complex superconducting, the new operators are chosen such that

cjeiφ/2 → cj. (3.25)

en, without loss of generality,∆ is chosen here to be a real positive quantity.Any fermionic Hamiltonian supports a rewriting in terms of Majorana fermions:

γ1 =1√2(c† + c),

γ2 =i√2(c† − c). (3.26)

us any complex fermion is formally split into two Majorana fermions.en the Hamiltonian has the form

H = iN∑j=1

µγj2γj1 +

[(∆− t)γj1γ

j+12 + (∆ + t)γj2γ

j+11

]. (3.27)

e resulting tight-bindingHamiltonian can present bound states that consist in havinga zero mode Majorana fermion trapped at each edge. For example, a limit case followsfor µ = 0 and∆ = t, when the Hamiltonian reduces to

H = 2itN∑j=1

γj2γj+11 , (3.28)

andMajorana fermions are coupled only between neighboring sites. en two unpairedMajorana fermions remain at the ends of the chain, γ11 and γN2 (see Fig. 3.1(b)). eyare completely localized at the extremity sites and have zero energy as they are decou-pled from the Hamiltonian. However there will be a residual interaction between theend modes, rapidly decreasing with the length aN of the wireO(e−aN/ξ), where ξ is thesuperconducting coherence length and a = 1, the lattice constant. Note that in theabsence of the hopping t and the superconducting pairings ∆, the Hamiltonian cou-ples only Majoranas on the same site. is latter limit case represents a trivially gappedsystem, an atomic insulator.

e zeromodes, protected by the bulk gap and the PHS, subsist at zero energy whenparameters begin to vary from the values set above. ey will start to extend, but theywill remain Majorana fermions. e topological phase can be empirically tracked byknowingwhere the bulk gap closes in parameter space and if there are such exotic boundstates in between two bulk closings.

e topological phasewas characterized by aZ2 topological invariant denoted byMin Ref. [67]. If the wire supports Majorana fermions, then M = −1, and, if the systemis in a trivial gapped phase, thenM = 1. When the bulk gap closes,M is unde ned.

64

3. I

Any free fermionicHamiltonian can be written in the basis ofMajorana fermions as

H =i

2

∑m,n

Am,nγm1 γ

n2 , (3.29)

where A is a real, anti-symmetric matrix (due to the anticommutation of Majorana fer-mions). e way to characterize the topological properties of the system involves (anti)-periodizing the system in a ring form. en the topological invariant was shown [67]to be given by

M(H) = sgn(Pf[Ap]Pf[Aa]), (3.30)

whereAp,a represents the systemwithperiodic, respectively antiperiodic, boundary con-ditions and Pf denotes the Pfaffian of the matrix.

ere is an alternative way to characterize the topology of the system. Note that thesystem is time-reversal invariant in the class of chiral one-dimensional BDI systems (seeTab. 1). As such, the system equally supports a characterization by a winding numberw.

e system is made in nite such that there is translational invariance. Hence mo-mentum k remains a good quantum number. e Hamiltonian is Fourier transformedin momentum space, so that in the basis C†

k = (c†k, c−k) it reads

H =1

2

∑k

C†kHCk. (3.31)

e BdG Hamiltonian without the constant∑

j µ/2 takes the form

H =

(−µ− 2t cos k −2i∆ sin k2i∆ sin k µ+ 2t cos k

). (3.32)

e bulk energy dispersion reads

E± = ±√(µ+ 2t cos k)2 + 4|∆|2 sin2 k. (3.33)

erefore the bulk gap closes for k = 0 and µ = −2t, or k = π and µ = 2t.e Hamiltonian can be expanded in a basis of particle-hole Pauli matrices τ

H = h2τ2 + h3τ3, (3.34)

with h2 = 2∆ sin k and h3 = −µ − 2t cos k. Note that the system obeys the TRS,represented by the operator of complex conjugationK and the chiral symmetry, repre-sented by the operator τ1. Also note that the unit vector Hamiltonian, h, is a mapping

65

3.2 C

from the one dimensional torus T 1 to the circle S1. erefore the behavior of h can becharacterized by a winding number w (see also Sec. 6.1.2)

w = −1

2

∑k∈0,π

sgn[h3∂kh2],

=1

2(sgn[2t− µ] + sgn[2t+ µ]), (3.35)

where the sum was performed over the nodes of the dispersion (k = 0 and k = π). ewinding number is either zero, for |µ| > 2|t|, and there are no edge states inside the gap,or one, for |µ| < 2|t|, and it is a Majorana bound state inside the gap.

is allows to relate the winding number to theM topological index:

M =

−1, |µ| < 2|t|,1, |µ| > 2|t|,

(3.36)

where M = 1 denotes that the trivial strong-pairing phase and M = −1, the weak-pairing phase (the topological phase) [76].

Note that because the system is described by a Z invariant, it could sustain multi-ple Majorana modes at a given edge. e fact that there there are only two topolog-ical phases is entirely due to the fact that it involves only nearest-neighbor couplings.Chap. 6 treats an extension of the Kitaev model in the BDI class, with distant-neighborcouplings, which exempli es a case with winding number higher that one, |w| = 2 andtwo Majorana modes at a given edge.

A different evaluation of the topological invariantM is given by [24, 67]

M = (−1)ν(π)−ν(0), (3.37)

where ν(k) is the number of negative eigenvalues ofH at the k point. Here [0, π] is halfthe BZ and then ν(π) − ν(0) is the number (mod 2) of Fermi points in half-BZ. isde nition relies on the existence of the PHS symmetry that ensures that the other halfof the BZ has the same number of Fermi points.

erefore the conditions to have Majorana fermions in the system are, up to now:having a spinless or spin-triplet superconductor, a bulk gap, and an oddnumber of Fermipoints in half the BZ.

3.2.3 1D spin-coupled semiconductingwire in proximity to an s-wavesuperconductor

e Kitaev toy model needs to be implemented in a more realistic setting. Among dif-ferent proposals a special attention is given in the present thesis to one which uses a 1Dspin-coupled semiconducting wire in the proximity to an s-wave superconductor andunder the effect of amagnetic eld to realizeMajorana endmodes [24, 25]. Suchmodel

66

3. I

Fig. 3.2: Schematic setup for a 1D system supportingMajorana fermions. Semiconduct-ing wire (black) with strong spin-orbit coupling deposited on a s-wave superconductor(blue). A magnetic eldBz acts in z direction, perpendicular to the wire.

has received a lot of attention in the experimental community due to the use of the sim-plest ingredients. Instead unconventional superconductors, it needs the more availables-wave singlet superconductors. Moreover, by using semiconductors, instead of topo-logical insulators, it can take advantage of the tremendous experimental know-how inthe fabrication and manipulation of semiconducting wires.

e race to experimentally discoverMajorana fermionswas purportedly endedwhena zero-bias conductance signature was detected in a superconductor-normal system thatis based on the aforementioned model [91]. However, recent doubts have been raised,claiming that a robust zero-bias signature can be due to the presence of disorder [105,106] or Kondo resonances [107].

In the present section, the system and its phases will be presented more on a phe-nomenological level. e next chapter treats a direct extension of the model when aDresselhaus spin-orbit coupling is included. at will be the place for a more detailedanalytical and numerical treatment.

e system is described by the Hamiltonian (see also [108])

H =

∫dx

[ψ†

(p2

2m− µ+ αpσ2 +Bzσ3

)ψ + (∆ψ↑ψ↓ +H.c.)

], (3.38)

where σ are the Pauli spin matrices. is Hamiltonian models a 1D semiconductingwire extended in the x-direction at chemical potential µ (see Fig. 3.3). e wire experi-ences a proximity effect due to the presence of the s-wave superconductor. ese leadsto superconducting correlations inside the wire, which are expressed at mean- eld levelby the presence of the superconducting parameter ∆. ere is also Rashba spin-orbitcoupling α which tends to align the spins in the y-direction. Finally there is a magneticeldBz perpendicular to the spin-orbit eld. All these elements are necessary to repro-

duce at low energy the Kitaev model. First it is necessary to li the spin degeneracy toallow for the possibility of an odd number of Fermi points. Let us see how this happensby studying the bulk energy dispersion.

Due to the presence of the anomalous pairings ψ↑ψ↓ one can write the Hamiltonian

67

3.2 C

in a BdG form

H =1

2

∫Ψ†HΨdx, Ψ† = (ψ†

↑, ψ†↓, ψ↓,−ψ↑),

H =

(p2

2m− µ+ αpσ2

)τ3 +Bzσ3 −∆τ1, (3.39)

where τ are the Paulimatrices in particle-hole space. By squaring twice theHamiltonianH it follows that the bulk energy reads

E2 = ξ2p + α2p2 +B2z +∆2 ± 2(ξ2pα

2p2 + ξ2pB2z +∆2B2

z )1/2, (3.40)

with ξp = p2/2m − µ. Due to the particle-hole symmetry, any state at energy E has acounterpart at −E. Also any state with momentum p has a counterpart with the sameenergy and opposite momentum. Because of this redundancy, it suffices to analyze therst quadrant (E > 0 and p > 0) of the dispersion in Eq. (3.40). Without loss of

generality µ can be set to zero in the following.For vanishing magnetic eld and superconducting correlations, Bz = ∆ = 0, the

spin y states |y±⟩ are good eigenstates of the system. Hence the spin-orbit couplingyields two shi ed parabolas for the dispersion for the two spins (see Fig. 3.3(a)). How-ever the system still needs to be gapped. e superconducting gap is not sufficient togenerate Majorana fermions. is is because the system remains time-reversal invariantand therefore the gaps at p = 0 and p = pF (Fermi momentum) are identical. en forevery energy, there will be an even number of Fermi points for p ∈ [0,∞). erefore itis necessary to break the time reversal invariance by adding a magnetic eld Bz , whichgaps the system at p = 0 (see Fig. 3.3(b)).

Note that there are two gaps in the system. One at Fermi momentum, which is pro-portional to the induced gap∆(pF ) ∝ ∆, and a gap at p = 0,∆(0) = Bz−

√(µ2 +∆2).

Crucially the gap at p = 0 can change its sign. When the Zeeman energy dominates thesuperconducting gap, the gap at p = 0 and the gap at pF are of of opposite type. is isthe condition for the presence of Majorana fermions

B2z > µ2 +∆2. (3.41)

e above condition is obtained rigorously from an analysis of topological invariantsin the next chapter. It suffices to say that if |∆| < |Bz| there is a range of µ for whichthe system is in the topological phase. is quantity can be changed by gating and onewishes to have a window as large as possible in the chemical potential. However theZeeman eld cannot be increased ad lib as it tends to destroy theCooper pairs by polar-izing the electrons in z-direction. Moreover, the breaking of the time reversal symmetrymakes the system susceptible to disorder, which in turn can close the bulk gap [109]. Alarge SOC is needed to combat the effect of the magnetic eld by enforcing the anti-alignment of electron spin with opposite momentum. However, the downside is that alarge SOC suppresses the electron mobility [76, 110].

68

3. I

0.0

0.5

1.0

1.5

2.0

E

0 1 2

p

Bz = 0

∆ = 0

(a)

0.0

0.5

1.0

1.5

2.0

E

0 1 2p

Bz = 0.4∆ = 0

(b)

0.0

0.5

1.0

1.5

2.0

E

0 1 2p

Bz = 0.4

∆1 = 0.4

∆2 = 0.3

∆3 = 0.5

(c)

Fig. 3.3: A representation of the energy dispersion in the rst quadrant for differentvalues of magnetic eld Bz and induced superconducting gap ∆. e mass m and theRashba SOC α are set to 1, the chemical potential µ = 0. (a) Shi ed parabolas dueto the SOC. e energy eigenstates have a clear spin direction: blue ↑ and red ↓. (b)Nonvanishing magnetic eld opens a gap at momentum p = 0 allowing the possibilityof an odd number of Fermi points. (c) Topological transition at p = 0 between twogapped phases, ∆ = 0.3 nontrivial, ∆ = 0.5 trivial, through closing of the gap whenBz =

√∆2 + µ2, corresponding to∆ = 0.4.

Finally, it is necessary to clarify how can an s-wave superconducting pairing∆medi-ate p-superconductivity. For that one works in the window provided by the topologicalcondition in Eq. (3.41) [24]. Assuming that the superconducting parameter ∆ is verysmall in comparison toBz , one can diagonalize the Hamiltonian (3.38) to yield energy

E± = ξp ±√B2

z + α2p2, (3.42)

and the two eigenvectors are

ψ± =1

N±

(Bz ±

√B2

z + α2p2

iαp

), (3.43)

where N± are chosen to normalize the spinors. e Fermi energy is between the twobands. en the Hamiltonian with the superconducting pairing term can be projectedon the lower band, and then one has access to the physics at the Fermi momentum. eHamiltonian projected on the occupied band reads

H =

∫dpE−ψ−(p)ψ−(p) +

[∆−(p)ψ−(p)ψ−(−p) +H.c.

],

∆−(p) =iαp∆

2√B2

z + α2p2. (3.44)

e effective pairing is an odd function in themomentum p, pairing particles in the sameband with opposite momentum. us one has effective p-wave symmetry for spinlesselectrons induced through an s-wave superconducting pairing.

69

3.3 M

3.3 Majorana polarizationis nal introductory section advances the notionofMajorana polarization [111]. e

concept tries to answer the need for a local order parameter that can adequately describethe topological transition from a trivial gapped phase to a Majorana supporting gappedstate. It will be used in the next chapters as a means to identify zero modes in spinfultopological superconductors as Majorana states.

In short, the Majorana polarization amounts to having a measure of the degree ofanomalous triplet pairing in the system. As it was shown in the previous sections, thisunconventional pairing can be effectively produced even in semiconducting systems inproximity to a spin-singlet superconductor. Here superconductivity is treated only atthemean eld level. If at the Fermi energy the system is effectively a triplet superconduc-tor and there are zero modes protected by a bulk gap, then they are Majorana fermions.In this sense, Majorana polarization is an necessary (but not sufficient) condition forhaving Majorana fermions.

e Hamiltonian for a spinful 1D superconductor (or in proximity to a supercon-ductor) is given by

H =1

2

∫dxΨ†HΨ, Ψ†(x) = (ψ†

↑(x), ψ†↓(x), ψ↓(x),−ψ↑(x)), (3.45)

where ψ†α is a creation operator for an electron with spin α. e Bogolibov-de Gennes

equation at a particular point in a 1D system reads

HΦ(x) = EΦ(x), Φ† = (u↑, u↓, v↓, v↑)∗ (3.46)

with u, v amplitudes for the electron, respectively hole, components of the wave func-tion.

e BdGequation is amatrix equationwhere there are four degrees of freedom: spin↑, ↓, particle and hole. Generally there are four eigenvectors and eigenvalues solutionsto Eq. (3.46).

In this particular basis the PHS operator, which anticommutes with the Hamilto-nian, is represented by σ2τ2K . roughout σ are the Pauli matrices in the spin space andτ in the particle-hole space. Under the action of the particle-hole operator the wave-function transforms as

σ2τ2Φ∗(E) = Φ(−E). (3.47)

en the Majorana condition reads

σ2τ2Φ∗(E) = Φ(E). (3.48)

One can compute locally the overlap between any eigenvector and its particle hole con-jugate at the same energy to see how close the wave function respects theMajorana con-dition. However, the particle hole operator is de ned up to a phase which is not easily

70

3. I

determined in practical situations. Hence the focus is on the real part of the overlapΦTσ2τ2e

−iθΦ∗, which is generally a complex number. Let us consider this quantity as avector that can be decomposed on the “Majorana” x-axis (θ = 0) and y-axis (θ = π/2).

en to identify the Majorana fermions one can take the real and imaginary parts ofthe overlap as polarizations along the two axis. e Majorana x- and y-polarizations arede ned as

PMx(x) = Re[ΦT (x)σ2τ2Φ∗(x)],

PMy(x) = Im[ΦT (x)σ2τ2Φ∗(x)]. (3.49)

e Majorana polarization vector is then de ned as the absolute value of the Majoranapolarization vector

PM(x) = |(PMx , PMy)|. (3.50)

Here the normalization from a Majorana solution is conventionally chosen such thatthe overlap integrated over the region where the zero mode is extended reads∫

dxPM(x) =1

2, (3.51)

which echoes the particular squaring of a Majorana fermion, γ2 = 12.

Explicitly, in terms of the wave function components, the Majorana polarizationcomponents are

PMx = 2Re[u↓v∗↓ − u↑v∗↑],

PMy = 2Im[u↓v∗↓ − u↑v

∗↑]. (3.52)

Note that the de nitions involve pairing of electrons and holes with the same spin, in-dicating a spin-triplet pairing in the model. en a different way to think about theMajorana polarization is to see it as a measure of spin-triplet pairing in the model. Notethat it is possible to have non-zeroMajorana polarization, but aMajorana fermion existsonly when itsMajorana density integrates to 1/2 (see Eq. 3.51) In the 1D spinful systemthat will happen at zero energy.

It is well suited to compare the Majorana polarization with usual operators used toinvestigate the local structure of wave function. Let us neglect for the moment the holedegrees of freedom. en one can de ne familiar concepts as the local electronic densityof states operator ρ(x) and the local spin polarization operator S(x) at a given energyEas

ρ(x,E) =4∑

n=1

Ψ†n(x)

(σ0 ⊗

τ0 + τ32

)Ψn(x)δ(E − En), (3.53)

and

S(x,E) =4∑

n=1

Ψ†n(x)

(σ ⊗ τ0 + τ3

2

)Ψn(x)δ(E − En) (3.54)

71

3.3 M

where n indexes the eigenvalue and the respective eigenfunction. Note that S has a vec-tor structure with components in x, y and z-spin direction.

In analogy the Majorana polarization operators can be de ned from Eq. (3.52)

PMx(x,E) =∑n

Ψ†n(x)σ2 ⊗ τ2Ψn(x)δ(E − En), (3.55)

PMy(x,E) =∑n

Ψ†n(x)σ2 ⊗ τ1Ψn(x)δ(E − En). (3.56)

Note again that an electron (or hole) will always have zero Majorana polarization.Eq. (3.52) shows that the same is the case with wave functions with spin-singlet pairingof the type (u↑, 0, v↓, 0). is bringsmore support to the idea thatMajorana polarizationmeasures a degree of spin-triplet pairing in the system.

Majorana polarization and density are not physical quantities that one can measure,but they can be used to give a picture of Majorana excitations at zero energy. Two zeromodes with opposite Majorana polarizations provide a clear illustration to the fact thatthey are modes that can be “combined” to form a Majorana unpolarized electron (orhole). Two Majorana modes with the same polarization can coexist nearby, but tend tohybridize and form regular electronic states if their polarization is opposite. A rotationof Majorana polarization might be due to the variation of physical parameters as spin-obit coupling or the superconducting phase. ese cases can pose a problem as the angleof rotation can be locally dependent and difficult to interpret. en the absolute valueof the polarization vector, theMajorana density (3.50) would bemore suited to identifya wave function as Majorana.

72

Chapter 4Topological semiconducting wire with RashbaandDresselhaus spin-orbit coupling

is chapter reconsiders the 1D spin-orbit coupled semiconducting wire fromRef. [24,25] with an additional Dresselhaus spin-orbit interaction. e wire supports Majoranamodes that present a particular spin texture. e electronic degrees of freedom of theMajorana fermionshave a transverse polarization to themagnetic eld that is entirely de-termined by the direction of the spin-orbit coupling vector. ey are always opposite atthe two ends of the semiconductingwire. AlreadyMajoranamodes for twodimensionalspin-triplet topological superconductors has been shown to exhibit an Ising-like spindensity that may allow their detection via coupling to a magnetic impurity [112, 113].In the same vein it is suggested that the spin texture in the 1D superconductor might bedetected in a spin-polarized scanning tunneling microscopy experiment.

Furthermore, the Majorana polarization de ned in Sec. 3.3 is shown to be a goodorder parameter to describe the topological transition in the model.

4.1 Model HamiltonianLet us consider a semiconducting wire oriented along the x-direction, and in proximityto an s-wave superconductor. Due to bulk inversion asymmetry, semiconducting wirescan exhibit along with the Rashba SO interaction analyzed in Refs. [25, 87, 108], aDresselhaus SO interaction [114].

In the present case, Dresselhaus SOI to rst order inmomentum p in the x-directionalong the wire takes the form βpσ1. e coupling β can be of the same order of magni-tude with the Rashba SOC (∼ 0.1 eVÅ) [115]. eHamiltonian describing the systemreads

H =

∫dx

[ψ†

(p2

2m− µ+ αpσ2 + βpσ1 +Bzσ3

)ψ + (∆ψ↑ψ↓ +H.c.)

]. (4.1)

73

4.1 M H

e Pauli matrices σ represent spin. e identity matrices and the spin index for thefermionic eld are implied when absent. e chemical potential is denoted by µ, Bz isthe Zeeman eld,∆ is the induced superconducting pairing and α (β) characterizes themagnitude of the Rashba (Dresselhaus) SO interaction.

Let us consider for the moment a purely real superconducting parameter∆. Due tothe presence of superconducting correlations, the Hamiltonian can be equally casted ina Bogoliubov-de Gennes form

H =1

2

∫dxΨ†HΨ, Ψ† = (ψ†

↑, ψ†↓, ψ↓,−ψ↑),

H =

(p2

2m− µ+ αpσ2 + βpσ1

)τ3 +Bzσ3 −∆τ1. (4.2)

Pauli matrices τ act in the particle-hole space. e products of Pauli matrices that livein different spaces should be understood as a tensor products. e BdG representationis particularly useful as it allows diagonalization of the Hamiltonian in a basis of quasi-particle excitations.

e spin-orbit interaction tends to orient the spins in (x, y) plane with an in-planedirection nxy, while the magnetic eld remains perpendicular to the plane. Both Dres-selhaus and Rashba SOI tend to split the energy bands for states with opposite spins|nxy+⟩ and |nxy−⟩. For vanishingBz and∆, |nxy±⟩ are good eigenstates of the Hamil-tonian

H|nxy±⟩ = ξp ±√α2 + β2p|nxy±⟩, (4.3)

where ξp = p2

2m− µ.

e magnetic eld has the effect to open a gap at zeromomentum. When the Fermienergy is in this gap the system becomes effectively “spinless”. Finally, when supercon-ducting proximity effect is considered, and close to Fermi energy, the system can bemapped to the Kitaev model. e s-wave pairing ∆ has the effect of opening gaps atFermi momentum and to mediate p-wave pairing for the spinless model.

e presence of the Dresselhaus term only trivially modi es the spectrum for thetranslation invariant system[25]. e energyobtainedby squaring twice theBdGHamil-tonian reads

E2 = ξ2p + (α2 + β2)p2 +B2z +∆2 ± 2[ξ2p(α

2 + β2)p2 + ξ2pB2z +∆2B2

z ]1/2. (4.4)

A numerical analysis requires implementing the BdG Hamiltonian in Eq. (4.2) on alattice. roughout the section, the lattice constant a and ~ are set to 1. e quantitiescan be expressed in energy units of t = 1

m. e usual substitutions∫

dx→ L∑j

, ψ(x) → 1√Lcj, ∂xψ(x) →

cj+1 − cj−1

2√L

, (4.5)

74

4. T …

where L is the size of the system, allow to write the direct space lattice BdG Hamilto-nian:

H =∑j

C†j [(t− µ)τ3 +Bzσ3 −∆τ1]Cj −

1

2

[C†

j (t+ iασ2 + iβσ1)τ3Cj+1 +H.c.],

Cj = (c†j↑, c†j↓, cj↓,−cj↑). (4.6)

e lattice Hamiltonian reproduces the continuum Hamiltonian at low energy.

4.2 Topological invariant

e computation of the topological invariant can be carried out exactly in the case ofthe latticeHamiltonian. e calculation will show that the topological condition is notin uenced by the Dresselhaus SOC β.

Any one-particle fermionic Hamiltonian supports a representation in a Majoranabasis [67]. Let us consider on-site real quasiparticle excitationoperators (theMajoranas)

γ(j)1,α =

1√2(cj,α + c†j,α),

γ(j)2,α =

1

i√2(cj,α − c†j,α). (4.7)

e Majorana fermions obey the anti-commutation relation

γ(i)Aα, γ(j)Bβ = δijδαβδAB, (4.8)

where, (i, j), (α, β), and (A,B) are site, spin, and respectivelyMajorana indices. en inthe Majorana basis ΓT

j =(γ(j)1↑ , γ

(j)2↑ , γ

(j)1↓ , γ

(j)2↓)the lattice version of Hamiltonian (4.1)

is written

H =∑j

(t− µ) +i

2

∑i,j

ΓTi AijΓj (4.9)

whereAij is an anti-symmetric realmatrix. e constant termdoes not affect theHamil-tonian topology and can be neglected. e matrixAij encodes only on-site and nearest-neighbor hopping terms. erefore one can express theHamiltonian in a small numberof 4× 4 block matrices, A(i− j), which contain only spin and Majorana indices. Notethat due to anti-symmetry

A(i− j) = −A(j − i)T . (4.10)

75

4.3 M

en, without the constant term, the Hamiltonian reads

H =i

2

∑j

[ΓTj A(0)Γj + 2ΓT

j+1A(1)Γj],

A(0) = is2((t− µ) +Bzσ3) + i∆s1σ2,

A(1) = −1

2[s2(it+ ασ2)− iβσ1]. (4.11)

e Pauli matrices s are pseudo-spin matrices representing the Majorana (1, 2) degreeof freedom. Note that the matricesA(0) andA(1) are real.

IfH has a gap, then one can determine if zero-energy Majorana fermions live at theedge of the 1D system by computing a topological index. A. Kitaev has proved [67] thatthey can exist only when the Majorana numberM is negative

M(H) = sgn(Pf[A(0)])(Pf[A(π)]). (4.12)

Here A is the Fourier transform of Aij computed at two exceptional points 0 and π inthe BZ. In the present case, with only on-site and NN terms

A(0/π) = A(0)± [A(1)− AT (1)]. (4.13)

Note that both spin-orbit coupling terms are symmetric in Majorana and spin indices,and therefore they drop out from the topological index.

en the topological invariant reads

M(H) = sgn(µ2 −B2z + |∆|2)sgn((2t− µ)2 −B2

z + |∆|2). (4.14)

e conditions for the existence of the topological phase supportingMajorana fermionsare unaffected by the Dresselhaus SO interactions. As expected from the experimentalconsiderations, the bandwidth t is much larger than the other parameters of the sys-tem, (∆, Bz, α). Hence the second term is always positive, and thus a topological phasecontinues to exist for

B2z > ∆2 + µ2. (4.15)

It is interesting to note however that Majorana bound states can exist even in the ab-sence of the Rashba term, when only Dresselhaus SO interactions are present. Dressel-haus term has a similar effect as the Rashba term in removing the spin degeneracy of theenergy bands.

4.3 Majorana wave function solutionsTodirectly see the effect of theDresselhaus SOC it is opportune to study thewave func-tions for the Majorana fermions. It has been shown that Majorana bound states can

76

4. T …

Fig. 4.1: Majorana fermions, represented in red, form at the interface with the topologi-cally trivial states when the chemical potential creates a domain wall.

arise at the interface between trivial and topological regions of a one-dimensional wireby considering for example a position-dependent chemical potential [25]. Similarly, amodel wire is divided here in three regions with variable chemical potential. e chemi-cal potential can be changed through gating such that it takes the value µ1 in the centralregion [0, L] and µ0 outside

µ21 < B2

z −∆2, x ∈ [0, L],

µ20 > B2

z −∆2, x /∈ [0, L]. (4.16)

Hence a topological region is formed in the central region and Majorana fermions areexpected at the boundary with the outer, trivially gapped regions.

e in nite system exhibits gaps at p = 0 and a superconducting gap at the Fermimomentum pF = 0. e gap at zeromomentum is∆−

√B2

z − µ2, while the gap at pF isthe induced superconducting gap ∆. To solve the system analytically, one assumes thatchemical potentials µ1,2 are chosen such that the gap at p = 0 is much smaller than thesuperconducting gap. Equivalently, if the induced gap ∆ is close in magnitude to theZeeman energy, it is necessary only a small change in the chemical potential to switch aregion from a trivially gapped phase to a topologically nontrivial phase. Hence the lowenergy solutions can be obtained by linearizing the BdG Hamiltonian (4.2) in p.

H = (αpσ2+βpσ1)τ3+Bzσ3−∆τ1−∑

j∈0,1

µj[θ(x(2j−1)

)+θ

((x−L)(2j−1)

)]τ3,

(4.17)

where θ is the Heaviside step function.e Majorana wave function is determined by searching for zero energy solutions

bound to the ends of the topological region. If the length of the topological region isvery large, L ≫ 1, the localized Majorana states are found independently at the twoends. ey have the form of four component spinors, and the amplitude of the wavefunction must decay away from the interface. For example, the Ansatz for the localizedfunction at x = 0 is ψ0,1 ∝ e±k0,1x, with wave vectors k0,1 > 0 and 0, 1, denoting thele (topologically trivial), respectively, the right (topologically nontrivial) side of theinterface.

e allowed wave vectors for the complete system are obtained by solving for the

77

4.3 M

zero energy eigenvalues at each interface

k±j =∆±

√B2

z − µ2j√

α2 + β2, j ∈ 0, 1. (4.18)

e complete solutions are formed by matching the wave functions across each in-terface. Near the interfaces, x = 0, L the wave functions are given by

ψ(x ∼ 0) =

κu1(µ1)ek−1 x, x > 0,

κ2

[(1 + tanϕ1

tanϕ0)u1(µ0)e

k−0 x + (1− tanϕ1

tanϕ0)u2(µ0)e

k+0 x

], x < 0,

(4.19)

and

ψ(x ∼ L) =

κu3(µ1)e−k−1 (x−L), x < L,

κ2

[(1 + tanϕ1

tanϕ0)u3(µ0)e

−k−0 (x−L) + (1− tanϕ1

tanϕ0)u4(µ0)e

−k+0 (x−L)

], x > L.

(4.20)

e spin-orbit vector is

eiϑ =(α + iβ)√α2 + β2

(4.21)

and the angles ϕj are de ned as

eiϕj =1√2

(√1 + µj/Bz + i

√1− µj/Bz

). (4.22)

e Majorana eigenvectors are given by

u1(µj)T = (cosϕje

iϑ,− sinϕj, sinϕjeiϑ, cosϕj),

u2(µj)T = (cosϕje

iϑ, sinϕj,− sinϕjeiϑ, cosϕj),

u3(µj)T = −(cosϕje

iϑ, sinϕj, sinϕjeiϑ,− cosϕj), (4.23)

u4(µj)T = (− cosϕje

iϑ, sinϕj, sinϕjeiϑ, cosϕj).

Note that the obtained wave functions are indeed Majorana fermions respecting thereality condition through the phase choice (ϑ+π)/2 for the complex coefficient κ. emagnitude of κ is determined from the normalization conditions of the wave functionsand is of the order of (

√B2

z − µ21 −∆/

√α2 + β2)1/2.

ewave functions allowone to compute the spinpolarizationof theMajoranawavefunction and read directly the in uence of the Dresselhaus SOC. e spin polarization

78

4. T …

is recorded here only for the electronic degrees of freedom in the Majorana wave func-tion. It is computed by taking the zero energy expectation value

s(x) = ψ†(x)

(σ ⊗ τ3 + τ0

2

)ψ(x). (4.24)

For theMajoranawave functions fromEqs. (4.19) and (4.20) exactly at the interface,x = 0, L, it follows that

s(0) = |κ|2

2

(− sin(2ϕ1) cosϑ, sin(2ϕ1) sinϑ, cos(2ϕ1)

),

s(L) = |κ|2

2

(sin(2ϕ1) cosϑ,− sin(2ϕ1) sinϑ, cos(2ϕ1)

). (4.25)

e above results show that the wave functions have the same spin polarization in z-direction, which is due to the action of the Zeeman eldBz . However, they have equalin magnitude, but opposite transverse spin polarizations. In fact, the direction of thespin polarization at both interfaces is given entirely by the relative weight of the Rashbaand Dresselhaus SOC

s2s1

= −βα. (4.26)

e Majorana polarization vectors for the Majorana wave functions are also readilyavailable, PM = (PMx , PMy)

PM(0) = −PM(L) = −|κ|2(cosϑ, sinϑ cos(2ϕ1)). (4.27)

ey are also opposite for the two end Majorana fermions. is arguably allows us tocall the twomodes as “different”Majoranas. When the twomodes are brought together,they form a regular fermion with zero Majorana polarization.

For xed parameters µ, ∆ and Bz , PMx is proportional to s1, while PMy is propor-tional to s2. us, when only Rashba/Dresselhaus SOC is present, the total transversespin polarization is proportional to the Majorana polarization, with a proportionalityconstant which depends on the chemical potential potential and the applied Zeemaneld. When both components of the SOC are present, the Majorana polarization and

the transverse spin polarization vectors are no longer collinear.Eq. (4.26) indicates that the local spin density for the Majorana electronic degrees

of freedom should rotate in transverse direction under the in uence of the Dresselhausterm. It is important to stress that there is no transverse polarization of the system. isstatement remains true if one considers the entire Majorana function or if one puts to-gether an electron from its “fractionalized” components at the two ends.

79

4.4 N

4.4 Numerical studye numerical study supports the analytical study undertaken in the previous section.e model BdG Hamiltonian is given in Eq. (4.6), and is implemented in a 100-site

system.As mentioned before, the spin-orbit couplings for the physical system can be ex-

pected to be of order 0.1 eV Å (~ = 1) or α ∼ β ∼ 104 m/s. Hence the spin-orbit cou-pling energy,mα2, is of the order 1 K. e Zeeman energy can be of the order∼ 100K,while the superconducting proximity effect can create gaps of order∆ ∼ 1− 10K. ehopping strength t = ~2

ma2is of the order of the bandwidth ∼ 104 K [76, 108, 115]. In

the numerical simulations, the lattice constant a and the reduced Planck constant ~ aredimensionless and equal to one, a = ~ = 1. All the physical quantities are measured inunits of the hopping strength t = 1. Due to nite size effects on the 100-site system, itis hard to visualize the physics of the Majorana system for small Bz and ∆ with respectto the hopping strength t. erefore,Bz,∆, α are arti cially enhanced in the following,while maintaining them smaller than t. e reference values for the rest of the simula-tions are ∆ = 0.3, Bz = 0.4, α = 0.2 and µ = 0, and any deviation from these values isexplicitly noted.

Exact diagonalization of BdG Hamiltonian in Eq. (4.6) provides the local densityof states, and the local spin-polarized density of states along the x, y, and z directions.For example the local (site n) electronic i-spin polarization density at a given energy Eis de ned here as

sn(E) =4N∑j=1

Ψ†(j)n

(σ ⊗ τ0 + τz

2

)Ψ(j)

n δ(E − Ej), (4.28)

where N is the number of sites in the system, Ej is the jth eigenvalue of H and Ψ(j)n is

the site n component of the jth eigenvector,Ψ†n(j)

= (u(j)n↑ , u

(j)n↓ , v

(j)n↓ , v

(j)n↑ ). Similarly, it is

possible to compute the local Majorana polarizations as a lattice version of Eqs. (3.52)

PMx,n =4N∑j=1

2Re[u(j)n↓v∗(j)n↓ − u

(j)n↑v

∗(j)n↑ ]δ(E − Ej). (4.29)

e Dirac delta functions are implemented as Gaussians of width∼ 10−4~vF/a.e x and z components of the spin polarization, as well as the Majorana polariza-

tion, in a system without Dresselhaus SOC, are represented in Fig. 4.2. e analyticalsolution correctly predicts that there is no y-spin polarization, while the zero boundstates have opposite electronic x-spin polarization at the two ends. Due to the Zeemanmagnetic eld, both end modes are identically z-spin polarized. e zero-energy Majo-rana wavefunctions are extended over a small number of edge sites, and exhibit stronglydamped spatial oscillations.

80

4. T …z-axis Spin Polarization

0 20 40 60 80 100

Site

-0.2

-0.1

0

0.1

0.2

Energy

-0.14

-0.06

0

0.06

(a)

x-axis Spin Polarization

0 20 40 60 80 100

Site

-0.2

-0.1

0

0.1

0.2

Energy

-0.2

-0.1

0

0.1

0.2

(b)

Majorana Polarization PMxMajorana Polarization PMx

0 20 40 60 80 100

Site

-0.2

-0.1

0

0.1

0.2

Energy

-0.5

-0.25

0

0.25

0.5

0 20 40 60 80 100

Site

-0.2

-0.1

0

0.1

0.2

Energy

-0.5

-0.25

0

0.25

0.5

(c)

Fig. 4.2: e spin polarization along (a) the z and (b) x directions, and (c) the Majorana po-larization PMx , as a function of energy and position, ∆ = 0.3, Bz = 0.4, α = 0.2, β = 0 andµ = 0.

Note that the analytical result in Eq. (4.25) predicts that the z-polarization vanishesfor µ1 = 0,

cos(2ϕ1) = 0. (4.30)

is is not the case seen in the numerical simulations, where there is a appreciable z-spin polarization of Majorana modes (see Fig. 4.2(a)). However the analytic result wasobtained by neglecting the kinetic term in the Hamiltonian from Eq. 4.1. To leadingorder, the quadratic term contributes with an effective µ

⟨p2⟩ ≈ O((∆−Bz)2), (4.31)

which creates a negative effective potential in qualitative accord with the numerical re-sults. Moreover, this effective chemical potential is responsible for the spatial (quicklydamped) oscillations of the spin polarization observed numerically. Although these os-cillations are not captured by the continuum limit calculations, for any site i the ratios2,i/s1,i depends only on the spin-orbit couplings in agreement with Eq. (4.25).

Fig. 4.3 exempli es the case where only Dresselhaus SOC is present. e local den-sity of states reveals that zeromodes bound at the ends of thewire continue to be present(see Fig. 4.3(a)). Majorana and spin polarization support the analytical ndings. emodes are entirely polarized on the y-direction, i.e. orthogonal to the case α = 0 andβ = 0. For example, a plot of the Majorana polarization PMy identi es the zero modesas Majorana states in Fig. 4.3(b).

e numerical results for theMajorana polarization presented in Fig. 4.2 also followcloselyEq. (4.27). e values of theMajoranapolarization are always opposite at the twoends of the wire. When the Dresselhaus term is non-vanishing, the modes gain a PMy

component. Without Dresselhaus SOC, PMx is proportional in this case to the x-spinpolarization. However, in general there is a crucial difference fromthe spin-polarization.

e Majorana polarization vector rotates in the transverse direction from site to site.When both the Rashba and Dresselhaus SOC components are present, the Majoranapolarization inEq. (4.27)depends on the cos(2ϕ1) and, subsequently, the ratioPMy/PMx

can vary on the end sites over which the Majorana mode is extended, in contrast to thespin case. e precession of Majorana polarization makes it more favorable to register

81

4.4 N Local Density of States

0 20 40 60 80 100

Site

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Energy

0

0.05

0.1

0.15

0.2

0.25(a)

Majorana Polarization PMyMajorana Polarization PMy

0 20 40 60 80 100

Site

-0.2

-0.1

0

0.1

0.2

Energy

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0 20 40 60 80 100

Site

-0.2

-0.1

0

0.1

0.2

Energy

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

(b)

Fig. 4.3: (a) Local density of states and (b) Majorana polarization PMy at∆ = 0.3,Bz =0.4, µ = 0, α = 0, and β = 0.2. Majorana bound states are present at the ends of a wirecontaining only Dresselhaus SOC.

0

0.03

0.06

0.09

0.12

0.2 0.25 0.3 0.35 0.4

Energy

Zeeman field Vz

0.1

0.2

0.3

0.4

0.5

0.25 0.3 0.35

Bz

(a)

Majorana Polarization PMx

0 20 40 60 80 100

Site

0.2

0.25

0.3

0.35

0.4ZeemanfieldVz

-0.5

-0.25

0

0.25

0.5B

z(b)

Fig. 4.4: (a) Lowest-energy eigenvalues and the half-wire integral of the Majorana polarizationPMx (inset) as a function of Bz . (b) Majorana polarization PMx of the lowest-energy state as afunction of position andBz . Parameters: ∆ = 0.3, µ = 0, β = 0, and α = 0.2

the Majorana polarization density (3.50) and prove that it is 0.5 for a zero energy modebound at the wire extremities.

0

0.01

0.02

0.03

0.04

0.2 0.23 0.26 0.29 0.32 0.35

Energy

Chemical potential µ

0.1

0.2

0.3

0.4

0.5

0.22 0.27 0.32

(a)

Majorana Polarization PMy

0 20 40 60 80 100

Site

0.2

0.25

0.3

0.35

Chem

icalpotentialµ

-0.4

-0.2

0

0.2

0.4

(b)

Fig. 4.5: (a) e lowest-energy eigenvalues and the half-wire Majorana polarization PMx

integral (inset) are plotted as a function of µ. In the second panel the Majorana polar-ization PMx of the lowest-energy state is plotted as a function of position and µ. eparameters considered are∆ = 0.3,Bz = 0.4, α = 0.2, and β = 0.

In the following, the claim that Majorana polarization is a good order parameter to

82

4. T …

characterize the topological transition receives more numerical support. is is doneby varying each of the parameters (∆, Bz, µ) to drive the system in a trivial phase. InFig. 4.4(a),Bz is varied and it is shown that the systembecomes trivially gapped (noMa-jorana bound states) for Bz ≤ ∆. e inset describes the dependence of the half-wireintegral of theMajorana polarization for one of the lowest-energy states as a function ofBz (an integral of 0.5 is equivalent to a “full” Majorana state). e Majorana polariza-tion decreases smoothly to zero below the critical value of Bz . e same phenomenoncan be observed in Fig. 4.4(b), where the spatial distribution of the Majorana polariza-tion is plotted as a function of Bz . e transition becomes sharper for an increasingsystem size. e same qualitative features are obtained when ∆ and µ are varied acrossthe topological transition. For example, the variation of the chemical potential is repre-sented in Fig. 4.5.

Discussione present chapter has shown that the Majorana polarization (and density) is a good

local order parameter to identify the topological transition at B2z = ∆2 + µ2. Further-

more, it was shown that there is a spin texture in the transverse plane to the magneticeld and it depends on the relative strength of Dresselhaus and Rashba spin-orbit cou-

pling. e electronic degrees of freedom of theMajorana wave function at the two endsof the wire are polarized in opposite directions on the transverse plane. is could be inprinciple detected through a contact to an impurity [112, 113]. However it is necessaryto detail such claim. How is it possible to have access only to the electronic part of thewave function?

Note that the system remains spin unpolarized at zero energy, and that could be re-alized only if there is a compensation for the spin polarizations at the two extremities ofthe wire. en the explanation for the registered spin-texture must invoke a conserva-tion of the polarization in the transverse direction. However, it remains in perspectiveto understand in more detail the physical reasons for this particular spin-texture.

83

Chapter 5ExtendedMajorana states in Josephsonjunctions

e one-dimensional topological superconductor supporting Majorana fermions wasshown in the previous chapters to admit a pertinent implementation in a heterostruc-ture constructed from a spin-orbit coupled semiconducting wire in proximity to an s-wave superconductor. Detection schemes of Majorana fermions o en require buildingon this basic structure. In particular, for transport measurements, the heterostructurecan be connected to a normal metal region, thus forming a “superconductor”-normalmetal (SN) junction. ey were recently investigated experimentally in Ref. [91] anda zero-bias conductance peak thought to be associated with Majorana fermions [103,104, 116–118] was detected through tunneling spectroscopy. A different system can beconstructed by coupling two topological superconductors through a normalmetal suchthat Majorana fermions form in the normal region. e presence of Majorana fermionsin these “superconductor”-normal metal-“superconductor” (SNS) junctions gives riseto a fractional Josephson current with a 4π periodicity [67, 102, 119–121].

In the present chapter, several models for one-dimensional SN and SNS junctionsare investigated numerically. e interest lies in following the behavior of the Majo-rana states in these new geometries. e essential property acquired by the Majoranafermions due to the coupling to a normal metal is that they can become extended states(see Sec. 5.1 and Refs. [76, 118, 122, 123]) In the case of the SNS junction, this has theparticular effect that the normal region supports two extendedMajoranas (see Sec. 5.3).

is happens only at a phase difference π between the two superconductors; otherwisethe Majorana states hybridize and form Andreev bound states at higher energies. etwo extendedMajorana fermions are recognized by reading a total integratedMajoranapolarization of one over the normal region. Finally, in Sec. 5.4.2, the formation ofMajo-rana fermions is studied in linear and ring geometries under a uniform superconductingphase gradient.[? ] e ring geometry is of particular interest as the twisting of the phaseallows the formation zero energy bound states in a normal region, similar to the regular

85

5.1 S-

(a) (b)

Fig. 5.1: Schematic representations of (a) superconductor-normal and (b) supercon-ductor-normal-superconductor junctions. ey are created by having the spin-orbitcoupled semiconducting wire added on top of an s-wave superconductor (S). A per-pendicular magnetic eldBz acts on the wire.

SNS junction.

5.1 Superconductor-normal junctionsLet us consider rst the SN junction that is schematically represented in Fig. 5.1(a). esystem consists of a spin-orbit coupled semiconducting wire in proximity to an s-wavesuperconductor. However, the wire sits only partially on the superconductor. ere-fore it is assumed that the superconducting proximity effect does not affect the entirewire. is is modeled by having a constant induced gap ∆ only onN sites. A variationconsidered subsequently is that of a decaying induced gap, which can be associated to apenetration length of Cooper pairs in the normal region.

e Hamiltonian of the system is an amendment to the model in Eq. (4.6) withchemical potential µ, induced gap∆, magnetic eldBz and only the Rashba spin-orbitcoupling α

H =L∑

j=1

C†j [(t− µ)τ3 +Bzσ3 −∆τ1θ(N − j)]Cj −

1

2

[C†

j (t+ iασ2)τ3Cj+1 +H.c.],

Cj = (c†j↑, c†j↓, cj↓,−cj↑), (5.1)

where L is the total number of sites in the system andN , the sites with induced gap. Innumerical simulations, the system sizeL is 100 sites andN is 80 sites. Hence the normalregion extends over the last 20 sites. Unless explicitly speci ed, the model parametersare chosen the same as in Sec. 4.4, Bz = 0.4, ∆ = 0.3, α = 0.2 and µ = 0, thusplacing the system under the topological condition (3.41). e lattice constant a andthe reduced Planck constant ~ are set to one; all parameters are expressed in units wherethe hopping strength is one, t = 1. In contrast to Sec. 4.4, the delta-function enteringthe de nition of LDOS and Majorana polarization are implemented as Gaussians ofwidth∼ 10−5~vF/a.

e condition to have aMajorana fermions remains the same as the one inEq. (3.41)

B2z > ∆2 + µ2. (5.2)

e crucial difference is that the right Majorana fermion from the topological super-conductor extend uniformly over all sites of the normal region. e integrated Ma-

86

5. E M …

0 0.1 0.2 0.3 0.4

µ

-0.1

-0.05

0

0.05

0.1

Energy

0

0.1

0.2

0.3

0.4

0.5

(a)

0.2 0.3 0.4 0.5 0.6

Bz

-0.1

-0.05

0

0.05

0.1

Energy

0

0.1

0.2

0.3

0.4

0.5(b)

Fig. 5.2: Resperesentation of the Majorana density as a function of (a) the chemical po-tential µ and (b) the magnetic eld Bz . e topological transition where the Majoranafermion states disappear at µc ≃ 0.26 andBz = 0.3.

jorana density (3.50) (for lattice version (4.29)) over the normal region is representedin Fig. 5.2. e fact that it yields the 1/2 indicates that there is exactly one Majoranafermion extended over the normal region. Furthermore, the topological condition isveri ed by varying the parameters of the system. Fig. 5.2(a) exempli es the variation ofthe chemical potential µ, while Fig. 5.2(b), the variation of the magnetic eld Bz . Inboth cases the topological transition takes place at the values predicted fromEq. (3.41);when the other parameters are xed, the critical chemical potential is µc ≃ 0.26 and thecritical Zeeman eld is (Bz)c = 0.3.

70 75 80 85 90 95 100

Site

0

0.2

0.4

0.6

0.8

1

tSN

0

0.05

0.1

0.15

0.2

Fig. 5.3: Representation of the zero-energy LDOS dependence on the hopping strengthtSN between the superconducting and normal region. A localized Majorana fermionextends in the normal region when increasing coupling to the bulk value t = 1.

e extendednature of theMajorana fermions canbe visualized by varying smoothlythe coupling tSN between the normal and superconducting region. When the couplingis very weak, the superconducting region is effectively connected on the right side toa trivial insulator. Hence Majorana fermions remain localized at the end of the super-conducting region. However, increasing the coupling tSN leads to extended zero energymodes. is situation is illustrated in Fig. 5.3 by recording the zero-energy local densityof states in the normal region as a function of tSN .

87

5.2 F J

0 20 40 60 80 100

Site

-0.2

-0.1

0

0.1

0.2

Energy

-0.2

-0.1

0

0.05

0 20 40 60 80 100

Site

-0.2

-0.1

0

0.1

0.2

Energy

-0.2

-0.1

0

0.05

(a)

0 20 40 60 80 100

Site

-0.2

-0.1

0

0.1

0.2

Energy

-0.25

-0.1

0

0.1

0 20 40 60 80 100

Site

-0.2

-0.1

0

0.1

0.2

Energy

-0.25

-0.1

0

0.1

(b)

0 20 40 60 80 100

Site

-0.2

-0.1

0

0.1

0.2

Energy

-0.25

-0.15

0

0.15

0 20 40 60 80 100

Site

-0.2

-0.1

0

0.1

0.2

Energy

-0.25

-0.15

0

0.15

(c)

Fig. 5.4: Low-energy Majorana x-polarization illustrates the increasing localization ofrightMajorana fermions with the growth of the superconducting region by ramping upthe penetration length ξp. (a) ξp = 50, (b) ξp = 100, (c) ξp = 200.

Let us nally consider a smooth decay of the superconducting gap into the normalregion. For a small penetration length ξp of Cooper pairs into the normal region, thesystem supports, as before, extended Majorana fermions. When the penetration lengthbecomes large, the topological superconducting region is prolonged into the normalregion. erefore the Majorana fermions become more and more localized with thediminishing normal region. In Fig. 5.4 is presented a limit case of the above situation.

e wire is connected to a p-wave superconductor only at its le edge and Cooper pairsenter into thewire by proximity effect. In the same basis as before, theHamiltonian thatmodels the wire reads

H =L∑

j=1

C†j [(t−µ)τ3+Bzσ3−∆e−j/ξp ]Cj −

1

2

[C†

j (t+ iασ2)τ3Cj+1+H.c.]. (5.3)

Fig. 5.4 illustrates caseswith an increasing penetration length. Note that the penetrationlength can be seen as an estimate of the size of the topological superconducting regionin the wire. As the penetration length grows, the normal region diminishes and theMajorana fermions become more and more localized on the right side. Note again thatMajorana fermions at the two ends of the wire have opposite Majorana x-polarization,but the extension of the right Majorana does not weigh on the fact that the integralMajorana polarization in the normal region yields 1/2. Hence the “normal region” hostsa single Majorana fermion of variable extension.

e following sections aremostly centered on the exploration ofMajorana fermionsin SNS junctions. e presence ofMajoranas in the normal region is crucially displayedin the fractional Josephson effect. e essential features of this particular phenomenaare presented next.

5.2 Fractional Josephson effectA Josephson junction is created by bringing into proximity two superconductors (seeFig. 5.5). Between them can either be a weak metallic link, or an insulating barrier.However, Cooper pairs can tunnel between the two superconductors. is gives rise

88

5. E M …

Fig. 5.5: Schematic picture of a Josephson junction. Without loss of generality, one canconsider that the superconducting parameter has equal absolute value in the two super-conductors. e le superconductor is conventionally considered real, while there is asuperconducting phase ϕ in the right wire. In the middle there is an insulating barrieror a metallic region.

to the Josephson current, that depends on the phase difference between the supercon-ductors. Here the focus is on the DC Josephson effect [124], in which there is no volt-age difference across the junction and the phase between superconductors is consideredxed. is results in a direct current dependent on the phase difference ϕ

IJ ∝ sin(ϕ). (5.4)

e remarkable feature of theonedimensional systems supportingMajorana fermionsis the presence of a actional DC Josephson effect. e tunneling between the super-conducting islands takes place by fusingMajorana fermions that are close on the le andright side of the junction [67, 102]. is leads to tunneling events involving single elec-trons instead of Cooper pairs. e resulting current-phase relation has the distinctive4π-periodicity

IJ ∝ sin(ϕ/2). (5.5)

To understand this relation, consider a junction created using the Kitaev model [76,121]. e junction is modeled by the Hamiltonian

H = HL +HR +HT , (5.6)

where HL represents the Kitaev model (3.24) describing the le wire, with real super-conducting order parameter, and, respectively,HR, the Kitaev model for the right wire,with a generally complex superconducting parameter, having a phase ϕ,∆ → ∆eiϕ (seeFig. 5.5). A short junction will have the tunneling HamiltonianHT given by

HT = −Γc†RcL +H.c., (5.7)

where Γ is the coupling between the superconducting islands. e operators c†R/L cor-respond to the fermion creation operators on the right and, respectively, on the le sideof the junction.

Let us suppose that le and rightwires are in a topological phase, |µ| < 2|t|. en forwires placed far away from each other, there are zero energy Majorana fermions pinnedat the extremities of the two wires. When a junction is formed by bringing together the

89

5.2 F J

two wires, the two end Majorana fermions can couple. Furthermore, at low energy, thephysics is entirely determined by the two end Majoranas. is is because the wires areassumed long enough that the in uence from opposite end Majorana fermions remainsnegligible. Moreover, all the other states are energetically separated from the Majoranafermions by the bulk superconducting gap.

To understand the physics of the Josephson junction it is sufficient to treat a limitcase, when the Majorana fermions are completely localized at one extremity site (seeSec. 3.2.2) for µ = 0 and∆ = t. en at low energy one uses Eqs. (3.26) to obtain

cL → i√2γ2,

cR → 1√2γ1e

−iϕ/2. (5.8)

Note that in the above low-energy substitution, themissingMajorana fermions from thedecomposition of a complex fermion hybridize with the Majorana fermions from adja-cent sites to create bulk states. erefore the low-energy Hamiltonian of the junctionreads

Heff = −iΓ cos(ϕ/2)γ1γ2,

= −Γ

2cos(ϕ/2)(2c†c− 1). (5.9)

e second equality is obtained by using Eq. (3.23); the effectiveHamiltonianwas writ-ten as a function of the occupation number c†c of the electronic state due to the fusionof the Majorana fermions γ1 and γ2.

As explained in Ref. [76], one notes that the occupation number is trivially a con-served quantity because it commutes withHeff. erefore an occupied state has energyE = −Γ/2 at ϕ = 0 and goes to E = Γ/2 at ϕ = 2π. e phase ϕ needs to increaseby another 2π for a state to come back at the original energy. is is the 4π-periodicityof the energy spectrum characteristic to the fractional Josephson effect. As announced,the 4π-periodicity re ects itself also in the Josephson current

IJ ∝ ∂⟨Heff⟩∂ϕ

=Γ

2sin

(ϕ

2

). (5.10)

Note that at ϕ = π/2 there are two degenerate zero energy states: the Majoranafermions. At this angle there is a pair of Majorana fermions trapped at the junction.Away from ϕ = π/2, the energy states are li ed from zero and become Andreev boundstates inside the superconducting gap. e rest of the chapter is devoted to various in-carnations of the fractional Josephson effect in long junctions.

90

5. E M …

γ1 γ2 γ1 γ2∆ ∆

γ1 ∆ γ2 −γ2 ∆eiπ γ1

Fig. 5.6: A different view of fractional Josephson effect. An SNS junction is createdby connecting two topological superconducting regions with Majorana fermions at ex-tremities. e right region can have a complex phase. When the wires are connectedthrough a metallic section, two central Majorana fermions extend inside and hybridize.However two central Majorana fermions are identical at ϕ = π and do not hybridize.

5.3 Superconductor-normal-superconductor junction

e SNS junction considered here is schematically presented in Fig. 5.1(b). e wire isplaced on two s-wave superconducting islands. Due to the proximity effect, these twoouter regions of the wire are mapped to topological superconductors. Conventionally,the le region has a real superconducting order parameter, while the right region hasa complex superconducting order parameter with a phase ϕ. e central region of thewire does not experience the superconducting proximity effect and forms the normalregion of the junction.

emain purpose of the chapter is to illustrate the Josephson effect in this geometry.Taken separately, each superconducting region supports Majorana fermions at its ends.When they are connected through the normal region, the two end Majorana fermionscan become extended in the normal region. is phenomenon takes place at a phasedifference ϕ = π between the superconducting islands. For different values, the twocentralMajorana fermions formextendedAndreev states at energies lower than the bulksuperconducting gap.

Here the Majorana polarization is used to illustrate the fact that Majorana fermionsat the junction are of the same type forϕ = π. is suggests a differentway to look at theJosephson effect. A rotation of the phase π in a superconducting region manifests itselfas a change γ1 → −γ2 and γ2 → γ1. erefore, at ϕ = π, the two Majorana fermionsat the junction are of same type and do not couple (see Fig. 5.6). For any other value in[0, 2π] there is a hybridization into Andreev bound states. An analysis of the Majoranapolarization captures this result, because a rotation with ϕ produces a rotation of thepolarization vector (PMx , PMy). At ϕ = 0 the vector has only a PMx component (seeChap. 4). A change of phase by π rotates the polarization vector by π. Hence at ϕ = πthere are two zero modes with the same Majorana polarization in the junction, and,subsequently, the normal region will read a Majorana density of one.

In the numerical study, a L = 100 site system is considered. e model parame-ters are chosen as in the previous section. e induced superconducting parameter∆ ismodeled as aHeaviside step function. If a smooth decay is considered instead, then onlythe extension of Majorana fermions in the normal region is affected. e Hamiltonian

91

5.3 S--

0 π/2 π 3π/2 2π

φ

-0.1

-0.05

0

0.05

0.1

Energy

0

0.25

0.5

0.75

1(a)

0 0.1 0.2 0.3 0.4

µ

-0.1

-0.05

0

0.05

0.1

Energy

0

0.25

0.5

0.75

1

1.25

1.5(b)

Fig. 5.7: (a) Majorana polarization density view of the fractional Josephson effect. Atangle ϕ = π there are two Majorana fermions forming in the junction. (b) Density ofstates in the normal region atϕ = π. RobustMajoranamodes are present in the junctionwith the variation of µ. e critical chemical potential where the Majorana modes aredestroyed is predicted by the topological condition (3.41, µc ≃ 0.26.

that models the system in Fig. 5.1(b) reads

H =L∑

j=1

C†j [(t− µ)τ3 +Bzσ3]Cj −

1

2

[C†

j (t+ iασ2)τ3Cj+1 +H.c.]

−N1∑j=1

C†j∆Cj −

L∑j=N2

C†j∆e

iϕCj, (5.11)

where the normal region extends betweenN1 andN2. During the simulations the nor-mal region has 20 sites,N1 = 40 andN2 = 60.

e physics of the fractional Josephson effect is veri ed by having awire that respectsthe topological condition,B2

z > ∆2+µ2. In Fig. 5.7(a) is represented theMajorana po-larization density integrated over the normal region as a function of the superconduct-ing phase. is illustrates two essential properties: there are two Majorana fermions inthe junction (the density reaches the value one) and they exist only when the phase isequal to π. Additionally, the Majorana fermions that form at the junction are robustwhen changing the parameters of themodel. is means that the topological conditionfor the existence of Majorana fermions remains valid. For example, in Fig. 5.7(b), thechemical potential µ is varied. Consequently, the Majorana fermions in the junctionsurvive until reaching the critical chemical potential determined from the topologicalcondition (3.41), µc ≃ 0.26.

Finally, the Majorana polarization picture of the fractional Josephson effect is de-tailed in Fig. 5.8. For a zero phase difference between the superconducting islands, thereare noMajorana fermions forming in the junction. Majorana fermions remain at the ex-tremities of the wire because the system sits in a topological phase (3.41). e would-beMajorana states in the normal region hybridize and appear as Andreev bound states at

92

5. E M …

0 20 40 60 80 100

Site

-0.1

0

0.1

Energy

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0 20 40 60 80 100

Site

-0.1

0

0.1

Energy

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

(a)

0 20 40 60 80 100

Site

-0.1

0

0.1

Energy

-0.3

-0.2

-0.1

0

0.1

0 20 40 60 80 100

Site

-0.1

0

0.1

Energy

-0.3

-0.2

-0.1

0

0.1

(b)

-0.5

0.0

0.5

Majoranapolarizations

0 π/2 π 3π/2 2π

φ

(c)

Fig. 5.8: Majorana x-polarization, PMx , at (a) ϕ = 0 and at (b) ϕ = π. e Majoranapolarization rotates with the phase. At ϕ = π there are two extendedMajorana fermionforming in the normal region. (c) Majorana polarization PMx (red) and PMx (blue) forthe Majorana fermion trapped at the right extremity.

energies lower than the superconducting bulk gap (see Fig. 5.7(a)). When the phasevaries from ϕ = 0, the Majorana fermion at the right end of the wire responds by ro-tating its polarization vector. e Andreev states get gradually closer to zero energy asthe phase approaches ϕ = π and additionally change their polarization. At ϕ = π theAndreev states fuse to create extended Majorana fermions with the same polarization(see Fig. 5.7(b)). Hence an integrated Majorana density of 1 (i.e. 2 Majorana fermions)is recorded in the normal region. In Fig. 5.7(c) is represented the rotation of the polar-ization vector for the right endMajorana fermion. It was checked in Ref. [122] that theAndreev bound states, when approaching zero energy, gradually gain a Majorana polar-ization to compensate the change in polarization of the right end zero-bound energystate.

5.4 Ring with a uniform phase gradientis section develops the previous numerical study of the SNS junction to ring geome-

tries as the one schematically illustrated inFig. 5.9(a). e spin-coupled semiconductingwire is fashioned into a ring and placed at on an s-wave superconductor. One sectionof the wire does not touch the superconductor and, considering that it does not experi-ence the superconducting proximity effect, forms the “normal” region of the ring. erest of the ring realizes a topological superconductor. Even if the thewire parameters arechosen such that it falls under the topological condition (3.41), there will be no Majo-rana fermions at the interface between the normal and superconducting region. is isbecause the would-be Majorana fermions forming at the two ends of the superconduct-ing region communicate through the normal region and are li ed from zero energy.Nevertheless Majorana fermions could still form if the phase of the superconductor isallowed to vary along the wire.

In Ref. [125], it was shown that supercurrents in the bulk of the superconductorcould be used in principle to manipulate the Majorana fermions. More precisely, it hasbeen shown that a constant spatial gradient in the phase of the superconducting param-eter can drive the system from a topological phase supporting Majorana fermions to a

93

5.4 R

(a)φ

x

0

π(b)

Fig. 5.9: (a) Ring geometry where a uniform phase gradient varies the phase of the in-duced superconducting parameter from 0 to ϕ in the superconducting region S. enormal region (N) in the wire is represented by a dashed line. (b) Variation of the su-perconducting phase in a topological SCwire: the phaseϕ is set to zero in the le region,and a constant gradient is considered to be induced in the central region by bulk super-currents (in red) such as that the phase ϕ reaches a value of π in the le region of thewire.

trivial phasewithout zero-energy boundmodes. is is the necessary ingredient to formMajorana fermions in the ring geometry.

Before tackling the ring geometry, it is instructive to investigate rst the linear geom-etry in order to understand the action of the uniform superconducting phase gradient.

is study was carried out in Ref. [125] and it is illustrated here in the context of a one-dimensional tight-binding model. e linear geometry is identical to the one treated inSec. 3.2.3 with the essential adjustment that the superconducting phase is allow to twistalong the wire via a uniform phase gradient. One can also fashion a tripartite systemwhere the gradient acts only on the central region of the wire. A typical spatial depen-dence of the phase is presented in Fig. 5.9(b); there are two outer regions of constantphase, with a central region uniform experiencing a phase twist. e main interest liesin ndingwhether the uniformphase variationmay give rise to similar physics to the oneobserved in the SNS junction. e question is if through the action of the phase gra-dient alone it is possible to bring the central region in a “normal phase”. In the secondsubsection, the ring geometrywith auniformphase gradient inFig. 5.9(b) is investigatedin order to identify the Majorana fermions in the normal region.

5.4.1 Constant phase gradient in a wireConsider the system present in the previous sections with the same model parameters.

e system is composed of three parts with a central region that experiences the uniformphase gradient∇ϕ and is modeled by the Hamiltonian

H =L∑

j=1

C†j [(t−µ)τ3+Bzσ3−∆e−iϕj ]Cj −

1

2

[C†

j (t+ iασ2)τ3Cj+1+H.c.]. (5.12)

94

5. E M …

Note that the phase of the superconducting parameter, ϕj , is site-dependent. In thecentral region of the wire (between sitesN1 andN2) there is a phase gradient∇ϕwhichacts by increasing the phase, ϕj = ϕj−1 +∇ϕ.

Before launching into a numerical analysis of the tight-binding Hamiltonian inEq. (5.12), it is worthy to consider the effect of the constant phase gradient on the topo-logical invariant. It is not entirely surprising that the gradient of the superconductingpa-rameter can make the system switch between a topological trivial and nontrivial phase.For a uniform gradient, this can be readily understood in the limit of an in nite wire. Asshown in Ref. [125], the phase of the pairing term can be gauged away, with the effect ofadding a gradient-dependent correction to the canonical momentum and of renormal-izing the hopping parameter and the spin-orbit coupling. For an in nite tight-bindingwire, the condition to have a topological phase, computed using the method presentedin Sec. 4.2, yields[

µ− t

8(∇ϕ)2

]2−α2(∇ϕ)2

4−V 2

z +|∆|2×[

µ−2t− t

8(∇ϕ)2

]2−α2(∇ϕ)2

4−V 2

z +|∆|2

< 0.

(5.13)

When the bandwidth t is larger than the other parameters of the system, the secondterm of the product is always positive. us, for a zero chemical potential µ, the criticalphase gradient is the exact lattice analogue of the continuum expression determined inRef. [125]:

(∇ϕ)c = 2√2

(α

t

)2

+

[V 2z −∆2

t2+

(α

t

)4]1/21/2

. (5.14)

A phase gradient has a Cooper pair-breaking effect and can close the superconductinggap at the Fermi momentum. is leads to a second critical value for the phase gradi-ent (∇ϕ)gl above which the bulk gap closes and the system enters a gapless regime (seeFig. 5.10). Its exact value is determined by numerically studying the closing of the gapfor an in nite system that experiences a uniformphase gradient∇ϕ. e study is carriedhere on a momentum space Bogoliubov-de Gennes Hamiltonian

H =1

2

∑k

C†kHCk, Ck = (c†k↑, c

†k↓, c−k↓, c−k↑),

H = (t− µ)τ3 +Bzσ3 − |∆|τ1 (5.15)− [t cos(k) + α sin(k)σ2] cos(∇ϕ/2)τ3 − [t sin(k)− α cos(k)σ2] sin(∇ϕ/2).

For the system parameters (Bz = 0.4,∆ = 0.3, α = 0.2, µ = 0), the gradient for whichthe system enters the gapless phase is (∇ϕ)gl ≃ 0.27, while the topological conditiondetermines a critical phase gradient (∇ϕ)c ≃ 1.57.

Let us denote the central region which experiences the uniform phase gradient asGR. e wire starts in a topological phase at zero gradient and hence it supports twoMajorana fermions at its extremities. e question to be tested iswhether newMajorana

95

5.4 R

0 0.5 1

0.5

1

01 1

2

B/∆

TP

GLP

NTP

u∇φ

2∆

NTPTPNTP

GLP

µ/∆

u∇φ

2∆

Fig. 5.10: Phase diagram of the spin-orbit coupled semiconducting wire in proximity toa superconductor, under a phase gradient∇ϕ (taken fromRef. [125]). B is themagneticeld (Bz),∆ the induced superconductingparameter,µ the chemical potential andu the

spin-orbit coupling constant (α). e system presents gapped topological phases (TP)which supportMajorana fermions and trivial phases withoutMajorana fermions. ereare also gapless phases (GLP) (manifestly devoid ofMajorana fermions). Central to thepresent study is the fact that starting from a topological phase (TP) at zero gradient itis impossible to enter a trivially gapped state through the action of the gradient alone;the gradient pushes the system into a gapless (nontopologial) phase.

fermions can form at the interface of the GR with the outer regions, or extended in theGR as in an SNS junction. Note that the above considerations allow us to eliminatethe rst possibility. Because (∇ϕ)c > (∇ϕ)gl, the system enters rst a gapless regime,so there is no boundary to a trivially gapped phase at the edges of the GR. Inspectingthe phase diagram in Fig. 5.10 indicates that this result is general in nature. Alwaysan increasing phase gradient pushes the system from a topologically gapped phase to agapless regime and therefore no localized Majorana bound states are expected.

If the transition takes place to a gapless phase, then it still remains open the possi-bility that Majorana fermions could form as extended states in GR. However the gra-dient has a particle-hole breaking effect[76, 125] and, moreover, losing the protectionof the bulk gap leads to the destruction of Majorana fermions even for phase gradientvalues above (∇ϕ)c. In Fig. 5.11(a) the phase gradient over the 20 central sites is π/20,smaller than the (∇ϕ)gl value and the GR remains topologically gapped. ere is anunique topological phase in the system and hence there are only two Majorana stateswhich form at the wire ends. Because of the relative phase ϕ = π between the two outerregions, the Majorana fermions have identical Majorana polarizations. Increasing thegradient has the effect to push the system in a gapless regime and to drastically diminishthe gap to the rst excited states in the GR, however no Majorana fermions form in theGR.

96

5. E M …

0 20 40 60 80 100

Site

-0.2

-0.1

0

0.1

0.2

Energy

-0.25

-0.1

0

0.05

0 20 40 60 80 100

Site

-0.2

-0.1

0

0.1

0.2

Energy

-0.25

-0.1

0

0.05

(a)

0 20 40 60 80 100

Site

-0.2

-0.1

0

0.1

0.2

Energy

-0.25

-0.1

0

0.1

0 20 40 60 80 100

Site

-0.2

-0.1

0

0.1

0.2

Energy

-0.25

-0.1

0

0.1

(b)

Fig. 5.11: (a) e phase of the superconducting parameter is twisted by π over 20 centralsites. e central regions remains topologically nontrivial and Majorana fermions withthe same polarization form only at the ends of the wire. (b) Phase gain of 11π over 11sites. Two Majorana fermions form in the GR having an opposite polarization withrespect to the end modes.

Nevertheless, there is one special case in which Majorana fermions can form in theGR. is situation arises when the GR is constructed as a series of Josephson junctionswith aphase increase of (2n+1)π between twoneighboring sites, withnbeing an integer.

e GR must consist of an odd number of sites, to ensure a relative phase difference(2n′+1)π between the le and right regions. en extendedMajorana fermions form inthe GR. is situation is exempli ed in Fig. 5.11(b): two extended Majorana fermionsform for a phase difference of ϕ = 11π over an 11-site GR. Integrating the Majoranapolarization over the central region yields a total value of one, showing that only twoMajorana states form in this region. is limit case is the only one where Majoranafermions are formed as extended states in the GR. Naturally it is difficult to expect thatsuch strong gradient and such particular conditions can be found in a physical system.

It is noteworty to point the general fact that the system is invariant under a changeof 2π in the phase gradient. Hence for an N -site GR, there is a 2πN periodicity in thetotal relative phase ϕ between the le and right ends of the wire. In the special casewhen the gradient is over an odd number of sites with Majorana fermions forming at∇ϕ = (2n+ 1)π, the periodicity in the total phase is 4πN (see in Fig. 5.12(a)).

5.4.2 Ring with a uniform phase gradient

Let us investigate the presence of Majorana fermions in the ring geometry fromFig. 5.9(b). e phase of the superconducting phase is twisted with a phase gradient∇ϕ. e question is under what conditions do Majorana fermions form in the normalregion.

97

5.4 R

0.0

0.02

0.04

0.06

0.08

0.1

Energy

0 11 22 33

Total phase φ/π

(a)

0.0

0.02

0.04

0.06

0.08

Energy

0 π/2 π 3π/2 2π

Relative phase φ

(b)

Fig. 5.12: (a) Evolution of the rst two positive eigenvalues with the phase differenceϕ for a 11-site GR. e constant zero-energy mode corresponds to an end Majoranastate. e energy of the othermode evolves with ϕ and reaches zero only when there is a(2n+ 1)π phase difference between two neighboring sites (this happens here for n = 0and n = 1, corresponding to ϕ = 11π and respectively ϕ = 33π). (b) Evolution of thelowest-energy modes with the phase difference ϕ. Note the shi δϕ from the expectedformation of Majorana fermions at π.

e tight-binding Hamiltonian describing the system reads

H =∑j∈

C†

j [(t− µ)τ3 +Bzσ3]Cj −1

2

[C†

j (t+ iασ2)τ3Cj+1 +H.c.]

−N2∑

j=N1

C†j∆e

−iϕjτ1Cj. (5.16)

e rst sum runs over the entire ring (), while the second sum runs over the supercon-ducting region between the sites N1 and N2. In the superconducting region the phasegrows under a uniform gradient ϕj = ϕj−1 +∇ϕ.

While no Majorana states are formed at ϕ = 0, for peculiar values of the phase dif-ference accumulated over the superconducting region, and for phase gradients that arenot too large, Majorana fermions can form in the normal region. is can be seen inFig. 5.12(b), where are plotted the low-energy eigenvalues as a function of the totalphase difference. In principle, when the accumulated phase is π, the would-be Majo-rana states are of opposite type and could exist as extendedmodes in the normal region.Nevertheless, note that in simulations theMajorana fermions form at a phase differenceslightly larger thanπ (see Fig. 5.12(b)). Such deviation fromπ can be attributed to nitesize effects and to the communication of the zero modes through the superconductingregion.[67] In numerical simulation,Majorana fermions form at π+δϕ and the shi δϕdecreases with system size. e Majorana polarization is plotted at this particular valuein Fig. 5.13(a). Integration of the polarization shows that the two zero energy modeshave the same polarization, which is adding up to a value of one.

is 2(π+δϕ)periodicity in the formationofMajorana fermions is preserved for gra-

98

5. E M …

0 20 40 60 80 100

Site

-0.2

-0.1

0

0.1

0.2

Energy

-0.04

0

0.03

0 20 40 60 80 100

Site

-0.2

-0.1

0

0.1

0.2

Energy

-0.04

0

0.03(a)

0.0

0.02

0.04

0.06

0.08

Energy

0 10 20 30 40 50 60 70 80

GR width (sites)

(b)

Fig. 5.13: (a) Majorana polarization as a function of energy and position for a ring witha total phase difference of π + δϕ. e system is periodic with site rst and last siteequivalent. Extended Majorana modes form over the entire normal region. (b) Firsttwo positive eigenvalues for a decreasing GR length. e total phase difference is keptat ϕ = π, while the number of sites is varied. Close to the critical value of the gradient(∇ϕ)gl = 0.27 ≃ 11.6, the system passes into a gapless regime. Due to nite-size effectsthe numerical analysis recovers this transition at a smaller value than the expected 11 or12 sites.

dients smaller than the critical gapless transition gradient (∇ϕ)gl = 0.27, correspond-ing to a total phase difference of ϕ ≃ 21.6π. As described above, larger gradients arepredicted to drive the GR to a gapless regime. However, numerical simulations indi-cate that while larger gradient values do seem indeed to take the system into the gaplessphase, Majorana fermions may still form for peculiar gradient values; at this point wedo not understand the origin of this phenomenon.

Another way to illustrate the evolution of the systemwith the value of the phase gra-dient is to x the total phase gain ϕ = π, and to study the behavior of the low-energymodes with the number of sites in the GR (a larger number of sites is equivalent to asmaller phase gradient). As it has been illustrated in Fig. 5.12(b), due to nite size ef-fects, one has anminigap for thewould-beMajorana states; the energy gap for thismodebecomes smaller and smaller with increasing the number of sites. When the number ofsites is reduced, the constraint of constant total phase ϕ drives the system to larger andlarger gradients. us it is possible to attain the critical value of the phase gradient thatsignals the passing of the system into the gapless phase. For an in nite system this is(∇ϕ)gl ≃ 0.27, corresponding to π/11.6. us a phase transition is expected when GRreaches the size of 11 − 12 sites. Indeed a crossing of the bands and li ing of the low-energy modes happens for a size of the GR of about 9 sites (see Fig. 5.13(b)).

5.5 Discussione present chapter has studied the behavior ofMajoranamodes in several SN and SNS

junction constructed from semiconducting spin-coupled wires in proximity to an s-

99

5.5 D

wave superconductor. roughout, it was shown that coupling Majorana modes to anormal metal can lead to extended Majorana modes in the latter. e extension of themodes is controlled by the penetration length ξp of superconductivity in the normal re-gion. Increasing the penetration length leads to a reduction of the normal region. Inthe limit of a penetration length larger than the normal region size, the system becomeseffectively a topological superconductor with localized Majorana fermions.

In the context of SNS junction, it was shownhowMajorana polarization can be usedto illustrate the physics of the fractional Josephson effect. e superconducting phaseleads to a rotation of Majorana polarization. In this manner, the extended Majoranafermions forming at a phase difference ϕ = π have the same polarization and do nothybridize to form Andreev bound states.

Finally, cases with a uniform gradient in the induced superconducting phase weretreated in linear and ring geometries. In the linear case, a topological superconductor ina nontrivial phase had a central region subjected to a uniform gradient. However it wasimpossible to induce Majorana fermions at the interfaces with the outer regions. ecentral region could not be turned into a trivially gapped phase due to the fact that thesystem always enters rst into a gapless regime. Hence the possibility to have localizedMajorana fermions at the interface is excluded. Moreover, the gradient has a Cooperpair-breaking effect and without the protection of the bulk gap there are no extendedMajorana fermions in the “normal” region. Extended modes were shown to exist onlyin the case where the total phase gain is an odd multiple of π and, simultaneously, thegradient changes by π from site to site. en the system behaves as a series of shortJosephson junction each at phase difference ϕ = π.

In the case of the ring geometry carrying a “normal” region, it was shown that theuniform gradient can twist the phase in the topological superconducting region suchthat its two ends have a difference of π. en the systemmaps to a regular SNS junctionand extended Majorana modes form in the normal section. Finite size effects producea shi δϕ from the ideal value ϕ = π. It remains in the future to investigate the pres-ence ofMajorana fermions at higher gradient values above the transition into the gaplessregime; these are not predicted by the theory and do not seem to arise with the period-icity 2(π + δϕ).

100

Chapter 6MultipleMajorana fermions in a two-bandmodel

e spin-coupled semiconducting wire studied in the previous chapters supports Z2

topological phases, where Majorana modes appear due to proximity effect to an s-wavesuperconductor. e indexZ2 means theMajorana fermions are essentially solitary zerobound modes living at the interface of the wire with the vacuum. ey are li ed fromzero energy through coupling with another Majorana fermion and they form a regu-lar complex fermion. Is it possible to have multiple Majorana living in proximity toeach other in a 1D wire? is question has recently received a clear positive answer. InRef. [126] it was shown that in the class of chiral topological superconductors BDI, onecan in principle have a system described by aZ topological invariant (see Tab. 1). ere-fore multiple Majorana fermions could be accommodated at the ends of a wire. epossibility was made more concrete when a simple two-band tight-binding supercon-ducting model for spinless fermions in the BDI class was shown to hold two Majoranamodes [127].

In the present chapter the model proposed in Ref. [127] is treated as an ideal modelon par with the Kitaev model. It is actually an extension of Kitaev model with the cru-cial modi cation that there are next-nearest-neighbor hoppings and superconductingparings in the model. rough simple arguments it is shown here that this is generallya sufficient condition to allow for multiple Majorana fermions localized at an edge.

Furthermore, here are investigated the speci c signature due to presence of mul-tiple Majorana fermions. For once the presence of several Majorana fermions at oneedge of the superconducting wire opens several Andreev transport channels in SN junc-tions and therefore the conductance can reach the value 2e2/h × Q withQ ∈ Z [128].However, the focus here is on the question whether the fractional Josephson effect, forJosephson junction connecting wires which support multiple Majoranas, survives andthe anomalous 4π-periodicity of the phase/current dependence is maintained.[129]

Finally, the possibility to create newMajoranamodes through the addition of a uni-form superconducting phase gradient in the central region of the 1D wire is explored.

101

6.1 T - BDI

e gradient can locally push the system in a non-trivial Z2 phase, while the rest ofthe system remains a topologically trivial Z2 (but nontrivial Z). en new Majoranafermions form at the interfaces with the uniform gradient region, while multiple Majo-rana modes can subsist at the end of the wire.

Before analyzing the properties of H, let us provide some general symmetry argu-ments which explain why a general 1D Hamiltonian can sustain phases with more thanone Majorana end states.

6.1 Topological properties of a two-bandBDI topologicalsuperconductor

6.1.1 Symmetry constraintsA1Dsuperconducting system canhavemultipleMajorana bound states at its endswhenthe system exhibits particle-hole symmetry and, crucially is also time-reversal invariant(TRI) [126, 127]. For the two-band Bogoliubov-de Gennes (BdG) Hamiltonian pre-sented here, this can be seen from the following simple argument. A general two-bandBdGHamiltonianH obeys PHSby construction, and can bewritten in the particle holebasis as

H = h · τ , (6.1)

where τ s are the Pauli matrices in the particle-hole space. Note that under PHS symme-try the components of the vector Hamiltonian h observe

h1(k) = −h1(−k),h2(k) = −h2(−k),h3(k) = h3(−k). (6.2)

e time reversal operator for spinless fermions is just the complex conjugationoperator.Hence, if the system is TRI, the components of h obey the following constraints:

h1(k) = h1(−k),h2(k) = −h2(−k),h3(k) = h3(−k). (6.3)

Particle-hole and time-reversal symmetries impose the chiral symmetry represented bythe operator τ1 which anti-commutes with the Hamiltonian

H, τ1 = 0. (6.4)

If H obeys all these symmetries, then it follows that h1 must vanish. Hence H has onlytwo remaining components and, therefore, hde nes amapping from theBrillouinZone

102

6. M M …

(BZ) to the Bloch “circle”

h : T 1 → S1. (6.5)

Hence the mapping is characterized by a winding number w which is an integer [126].erefore a two-band BdG Hamiltonian belongs to the topological BDI class charac-

terized by a Z topological invariant [11–13].

6.1.2 Winding number of a circuit and role of distant site couplingse formof thewindingnumberwill give an insight tohowcanone increase thenumber

of Majorana fermions by adding distant-site coupling terms in the Hamiltonian. Let usclarify the computation of the winding number of a closed curve around the origin inthe 2D plane; the curve is a mapping f : T 1 → R2\0. By components the curve readsf(t) = (x(t), y(t)). Let t parametrize this curve with t ∈ [0, 2π). en the windingnumber w of the curve around the origin in R reads

w =1

2π

∫ 2π

0

dtxy − xy

r2, r2 = x2 + y2. (6.6)

e winding number can be computed using the Brouwer degree of a curve [45].In the case that the kernel of function x(t) has a nite number of points t, the windingnumber reduces to

w = −1

2

∑t∈kerx

sgn[x(t)y(t)

]. (6.7)

An equivalent formula for the winding number can be produced where now one uses asum over the zeros of y(t).

In the case of a general TRI, BdGHamiltonian presented in the previous subsection,the winding number is

w = −1

2

∑k∈kerh2

sgn[∂kh2h3]. (6.8)

Note from the symmetry constraints that the kernel ofh2 contains at least the special BZpoints 0 and π. To create more Majorana bound states at one end, the winding numbermust satisfy |w| > 1. is implies that kerh2 must contain other points than 0 and π.

is can happen by enlarging the unit cell through the addition of hopping and super-conducting pair terms coupling distant sites. en new nodes in the energy dispersiondevelop for k ∈ [0, π] and can lead to higher winding number. Hence the presence ofhigher-order hopping or pairing terms is a sufficient condition to have multiple Majo-rana bound states at the ends of a 1D wire.

103

6.2 M H

-2

-1

0

1

2

λ2

0 1 2 3 4

λ1

0

2

2

1

Fig. 6.1: Phase diagram for the Hamiltonian in Eq. (6.9). e insulating phases are sep-arated by black lines. Each insulating phase is characterized by a winding number w inred. e number of Majorana fermions bound at one end is given by |w|.

6.2 Model Hamiltonian and phase diagram

e model proposed in Ref. [127] is a two-band tight-binding model for spinless elec-trons. e Hamiltonian reads

H =∑i

[− (1− 2c†ici)µ− λ1(c

†ici+1 + c†ic

†i+1 +H.c.)

− λ2(c†i−1ci+1 + c†i+1c

†i−1 +H.c)

], (6.9)

where λ1 corresponds both to the nearest-neighbor (NN) hopping amplitude and tothe nearest-neighbor superconducting gap while λ2 denotes the next-nearest-neighbor(NNN) hopping amplitude and next-nearest-neighbor superconducting gap. In whatfollows, λ1 is assumed positive. e chemical potential µ is set to one in the following.When λ2 = 0, the Hamiltonian in Eq. (6.9) corresponds to the Kitaev model [67].

e phase diagram of H has been established in Ref. [127]. Here we recover thisphase diagram in a different manner, by using Eq. (6.8) to unambiguously characterizeeach topological phase in the (λ1, λ2) plane. is phase diagram is drawn for complete-ness in Fig. 6.1.

e phase diagram associated with H is characterized by phases with w = 0, 1, 2.Indeed, for λ2 > 1 + λ1 or λ2 < −1 and λ2 < 1 − λ1, H can sustain a phase with twoMajorana zero modes localized at each wire end [127].

104

6. M M …

6.3 Transport in SN junctionsBefore analyzing the physics of Josephson junctions made with wires supporting severalMajorana fermions, let us focus on an SN junction for a superconductor described bythe HamiltonianH in Eq. (6.9). For a junction between a wire with one Majorana endstate and a normalmetal, it has been predicted that the differential conductance exhibitsa zero-bias peak of height 2e2/h [103, 104]. A similar question for a junction betweena topological superconducting wire characterized by a topological index w > 1 and anormal metal has been recently addressed in Ref. [128]. e authors have shown thatfor such junctions the conductance G can reach a value of G = |w| × 2e2/h. is pre-diction is checked here by considering a junction between a wire described by Eq. (6.9)supporting 4Majorana fermions, 2 at each of its extremities, and a normal wire.

e low-energy properties of such a system are determined by the coupling of thetwo end Majorana fermions with the normal metal. is coupling can be captured bya 2 × 2 hybridization matrix Γ. In order to compute the low-bias transport proper-ties of this junction, one can directly use the S-matrix formalism developed by Flens-berg [104], and noting that the two wave-functions for the Majorana fermions are or-thogonal [127], such that there is no inter-Majorana coupling term.

e resulting expression for current can be written as [104]

I =e

h

∫dωM(ω)[f(−ω + eV )− f(ω − eV )], (6.10)

whereM(ω) = Tr[GR(ω)ΓGA(ω)Γ(ω)], and GR(ω) = 2[ω1 + 2iΓ]−1 denotes the re-tarded Green’s function. e differential conductance becomes

dI

dV= −e

2

h

∫dωM(ω)

[df(−ω + eV )

dω− df(ω − eV )

dω

]. (6.11)

which at T = 0 reduces todI

dV=

2e2

hM(eV ). (6.12)

Taking an explicit trace over the transmission matrix, it follows that

dI

dV=

8e2

h

8 det(Γ)2 + (eV )2Tr(Γ2)

[(eV )2 − 4 det(Γ)]2 + [2eVTr(Γ)]2 , (6.13)

which at zero bias becomesdI

dV=

4e2

h. (6.14)

e zero-bias value of the differential conductance is thus the double of that ex-pected for a junction with a single Majorana fermion at the interface. is is consistentwith each interface Majorana contributing a 2e2/h to the total conductance.

105

6.4 J

6.4 Josephson junctionsIn this section, it is analyzed how the Jospehson effect is affected by the presence of sev-eral Majorana zero modes. In particular, only short Josephson junctions are consideredbetween two wires which can sustain several Majorana end states. e model Hamilto-nian reads

H = HL +HR +HT . (6.15)

e Hamiltonian for the le wire, HL, is described by the HamiltonianH in Eq. (6.9)characterized by parameters (λL1 , λL2 ). e right wire is characterized by HR, which isobtained fromHL by changing (c†ic

†j → c†ic

†je

iϕ) in Eq. (6.9). Note that the same variablephaseϕ is attached to theNNandNNNpairing terms. e secondwire is characterizedby the parameters (λR1 , λR2 ). e tunneling Hamiltonian can be modeled as

HT = −(λL1 c†NcN+1 + λL2 c

†N−1cN+1 +H.c.), (6.16)

where the junction is made between site N (last of HL) and N + 1, ( rst of HR). Byconvention the hopping/pairing of the tunneling Hamiltonian are taken identical tothe ones in the le wire.

Each wire is labeled by a topological indexwα = 0, 1, 2with α = L,R. e absolutevalue of the winding number indicates the number of Majorana fermions at one end ineachwire, taken separately. It is useful to think about the junction as formed by bringingadiabatically the wires together. e low-energy physics of the junction reduces to ananalysis of the coupling of the several Majorana modes across the junction, as presentedin Fig. 6.2. In the gure are illustrated the various wL − wR junctions that are beingtreated in the following: the 1−1, 1−2 and 2−2 junctions. e numerical simulationsare made on a 100-site system with the junction at site 50.

Before analyzing the Josephson effect in junctionsmade fromwires supportingmul-tiple Majorana fermions, one may ask if a complex superconducting order parameter ina topological superconducting wire can have an effect on its phase diagram. However,whileHR(ϕ) is generically complex, an uniform phase ϕ can be gauged away, yielding areal Hamiltonian and the same phase diagram as the one depicted in Fig. 6.1.

6.4.1 e 1− 1 Josephson junctionLet us consider rst a Josephson junction between two wires with a topological indexwα = 1. is type of junction has been extensively studied and the model is checkedhere that it is consistent with the known physics. e model parameters are chosen as(λα1 , λ

α2 ) = (1, 1). Our numerical results for the dependence of the energy levels of this

junction with the phase difference recover a 4π periodicity consistent with the anoma-lous Josephson effect (see Fig. 6.3(a)).

106

6. M M …

γ1 γ2

γ1 γ2

γ2 γ3

γ3

γ4γ1φ0

Fig. 6.2: To form Josephson junctions, wires characterized by winding numbers 1 − 1,1 − 2 and 2 − 2 are brought into contact. Without loss of generality, the le -handside superconductor has real order parameters, while on the right-hand side they havea superconducting phase ϕ. e low-energy Hamiltonian is assumed to contain onlyphase-dependent coupling terms between the Majorana fermions.

0.0

0.2

0.4

0.6

0.8

1.0

Energy

0 π/2 π 3π/2 2π

φ

E(φ) = a|φ− π|

a = 0.363

(a)

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Energy

80 90 100 110 120

Energy Eigenvalue Index

φ = 0

φ = π/2φ = π

(b)

Fig. 6.3: (a) e dependence of the lowest-energy eigenvalues on the superconductingphase difference between the twowires. (b) e eigenvalue spectrum for three differentvalues of ϕ = 0, π/2, π. e system exhibits four zero-energy modes at ϕ = π. Modelparameters are λ1 = 1 and λ2 = 1

In Fig. 6.3(b) it is plotted the eigenvalue spectrum of the junction for ϕ = 0, π/2, π.Only for ϕ = π one recovers four zero-energy eigenvalues which corresponds to fourMajorana fermions: one at each extremity and two at the junction.

e 4π-periodicity can be understood from a simple effective low energy Hamilto-nian following Kitaev [67]. e overlap between the wave functions of the two Majo-rana fermions at the extremitieswith theMajorana fermions at the interface is neglected.

e simplest low-energy effectiveHamiltonian reads [24, 67, 102, 119–121, 130, 131].

H1−1eff = it12 cos(ϕ/2)γ1γ2, (6.17)

where γ1 is a Majorana fermion at the right end of the rst wire, and γ2, a Majoranafermion localized at the le end of the second wire. Here t12 is the effective tunnelingamplitude (see Fig. 6.2). One can check that the cosine behavior reproduceswell the lowenergy spectrum. It is worth emphasizing that iγ1γ2 is trivially a conserved quantity ofH1−1

eff .

107

6.4 J

0.0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Energy

0 π/2 π 3π/2 2π

φ

d+ = 0.473, d− = 0.135

|11〉

|10〉 |01〉

|00〉

(a)

-1.0

-0.5

0.0

0.5

1.0

Energy

80 90 100 110 120

Energy Eigenvalue Index

φ = 0

φ = π/2φ = π

(b)

Fig. 6.4: (a) e dependence of the lowest-energy eigenvalues on the SCphase differencebetween the two wires. e energy branches are labeled using the occupation numberrepresentation for the two-fermion states formed from fusing fourMajoranas across thejunction. At low energy the two energy branches are linearly tted: E2−2

+ ± E2−2− =

(d+ ± d−)|ϕ − π|/2 and the value off coefficients d± is displayed. (b) e eigenvaluespectrum for three different values of ϕ = 0, π/2, π. e system exhibits eight zero-energy modes at ϕ = π.

6.4.2 e 2− 2 Josephson junctionLet us now consider a junction between two superconducting wires characterized bywinding numbers wα = 2.

Analysis of the spectrum

e eigenvalue spectrum for a 2 − 2 junction is computed numerically. e result isshown inFig. 6.4(b). e rst important thing tonote is that the anomalous4π-periodicitystill holds. e only difference is that there are now four Majorana fermions forming atthe junction when the phase difference is ϕ = π. At each extremity of the system thereare also two Majorana fermions which subsist for any value of ϕ, making the groundstate eight-fold degenerate at ϕ = π (see Fig. 6.4(b)).

A low-energy Hamiltonian capable to describe the numerical results presented inFig. 6.4 must involve four Majorana fermions. Let us denote by γ1, γ2 the Majoranafermions on the le side of the junction and by γ3, γ4 the Majorana fermions on theright side of the junction. A general Hamiltonian involving the four Majoranas can bewritten as

H2−2eff = iγ1(t13γ3 + t14γ4) cos

ϕ

2+ iγ2(t23γ3 + t24γ4) cos

ϕ

2+ it34γ3γ4 sinϕ+ it12γ1γ2 sinϕ. (6.18)

e phase dependence ofH2−2eff is xed by enforcing a 2π-periodicity together with

the constraint that for ϕ = 0 there is no direct coupling between γ1 and γ2, nor betweenγ3 and γ4. e cos(ϕ/2) is required by gauge invariance. Here are formally included

108

6. M M …

some direct-tunneling terms between the Majorana fermions on the same side of thejunctions (t12 and t34), which may be non-zero when ϕ = 0, π (this explains the sin(ϕ)).However, one can argue that such termsmust be less signi cant since they are the resultof higher-orderhoppingprocesses between the two superconductors. In fact these termsturn out to be negligible for the description of the low-energy spectrum.

Contrary to the 1−1 junction, where therewas an obvious quantity commutingwiththe Hamiltonian for all values of ϕ, for the 2− 2 junction this is not obvious for ϕ = π.In the later case, there are four Majorana fermions fusing to form two regular fermionsc±. is can be seen from the spectral decomposition of the effective Hamiltonian

H2−2eff = E2−2

+ (2c†+c+ − 1) + E2−2− (2c†−c− − 1), (6.19)

where

E2−2± = d± cos(ϕ/2), (6.20)

with d± = 12√2

√b±

√b2 − 4a2 with a = t14t23 − t13t24 and b = t213 + t214 + t223 + t224.

Note that even if the f22 term inEq. (6.18)wasneglected, the formof the c± fermionsas a function of the original Majorana fermions remains complicated. For example, un-der the reasonable assumption that the coupling strength between Majorana fermionssituated at the same distance across the junction is identical, t13 = t14, if follows that

c± =1√

2 + 2g2±[γ4 + iγ1 + g±(γ3 + iγ2)], with

g± =1

2t13

[t23 − t14 ±

√4t213 + (t14 − t23)2

]. (6.21)

However theHamiltonian and the occupation numbers of the two fermionsn± = c†±c±trivially commute among each other. e occupation numbers for the two fermions areconserved and one can use Eq. (6.19) to label the energy branches in the occupationnumber representation of the two-particle states, |n+n−⟩ (see Fig. 6.4). e value ofE2−2

±can be found numerically from a linear t at low energy near ϕ = π in Fig. 6.4. Mostimportantly, the 4π periodicity of the DC Josephson effect is maintained.

Analysis of the Majorana polarization

Auseful tool to analyze the behavior ofMajorana fermions is theMajorana polarization,a local topological order parameter introduced in Ref. [111] and related to the degreeof anomalous pairing in a 1D topological wire.

e existence of a Majorana fermion is recorded as a zero-energy Majorana polar-ization density of 0.5. A Majorana fermion can have a x- and y-Majorana polarization.When the Hamiltonian is real at ϕ = 0 and ϕ = π, the y-Majorana polarization PMy

is 0. When the superconducting parameter acquires a phase, the polarization along they-direction is generally nonzero.

109

6.4 J

-2.0

-1.5

-1.0

-0.5

0.0

0.5

Polarizations

0 π/2 π 3π/2 2π

φ

Fig. 6.5: Zero-energyMajorana polarization integrated over the 40 sites around the junc-tion. Four Majorana states form at the junction for ϕ = π yielding a x-Majorana polar-ization of 2, and compensating the Majorana polarization of the four end bound state.

Here, similar to Ref. [122], the endMajorana fermions respond to the variation of ϕby rotating their Majorana polarization. is is due to the superconducting parameterbecoming complex. However at ϕ = π the superconducting parameter becomes realagain and the zero-energy end states have the sameMajorana polarization. For this par-ticular value of the phase, four newMajorana fermions form at the junction, two in eachwire, their x-Majorana polarization compensating theMajorana polarization of the endmodes (see Fig. 6.5).

6.4.3 e 1− 2 Josephson junctionIn the following, the focus is on the remaining 1 − 2 junction. is is a particularlyinteresting problem since two distinct topological sectors are brought into contact via aJosephson junction.

Analysis of the spectrum

e physics of this junction is expected to be dominated by three interacting Majoranafermions localized at the interface. One can show that for a winding number differenceof one between the right side and the le side of the junction, one Majorana mode isbound at the interface for any choice of ϕ. Moreover, at ϕ = π the system is six-folddegenerate with three zero energy Majorana in the junction region (see Fig. 6.6(b)).

Let γ1 denote theMajorana fermion on the le side of the junction, and by γ2, γ3 theMajorana fermions on the right side. e gauge invariance of the Hamiltonian suggeststhat the phase-dependent couplings between γ1 and γ2,3 are proportional to cos(ϕ/2)to compensate the sign change. A low-energy Hamiltonian describing the Majoranacoupling in the 1− 2 system can thus be written as

H1−2eff = iγ1(t12γ2 + t13γ3) cos

ϕ

2+ it23γ2γ3 sinϕ. (6.22)

110

6. M M …

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Energy

0 π/2 π 3π/2 2π

φ

E(φ) = a|φ− π|

a = 0.304

(a)

-1.0

-0.5

0.0

0.5

1.0

Energy

80 90 100 110 120

Energy Eigenvalue Index

φ = 0

φ = π/2φ = π

(b)

Fig. 6.6: (a) e dependence of the lowest-energy eigenvalues on the superconductingphase difference between the two wires. e dotted line indicate the t of the lowest-energy numerical eigenvalue with the analytical form in E+ from Eq. (6.23). e tparameter is a =

√t212 + t213/4. (b) e eigenvalue spectrum for three different values

of ϕ = 0, π/2, π. e system exhibits six zero-energy modes at ϕ = π.

is Hamiltonian has three zero-energy eigenstates at ϕ = π. For a phase difference ofϕ = π, the effective Hamiltonian (6.22) has one zero eigenvalue (required by the anti-symmetry of the 3× 3matrix) and two non-zero eigenvalues. e constant zero-energystate, which is a Majorana edge state bound at the interface between two topologicallynonequivalent regions, is the result of the difference of one unity between the topolog-ical indices of the two regions.

Similar to the 2− 2 junction, if the last term of Eq. (6.22) is neglected, the form forthe two eigenvalues becomes

E± = ±2a cos(ϕ/2). (6.23)

e hopping dependent parameter a = 14

√t212 + t213 can be determined from a low-

energy t of the numerical dispersion presented in Fig. 6.6(a). Note that taking intoaccount also the term t23 improves the quality of the t, especially in the vicinity ofthe superconducting gap. Note also that, while the t is accurate up to energies closeto the superconducting gap, the effective HamiltonianH1−2

eff is exclusively a low-energyeffectiveHamiltonian and should not be expected to recover the full dependence of theenergy eigenvalues on the superconducting phase difference.

Analysis of the Majorana polarization

In the le -side wire, described by a winding number w = 1, the superconducting pa-rameter is chosen to be real and the single Majorana fermion at the le end is alwaysfully x-polarized. For the right-hand side wire, at the right end there are two Majo-rana fermions that respond to the twisting of the phase ϕ by gaining a y-polarization; inFig. 6.7(a) it is presented the behavior of the x- and y-polarizations of these modes.

Let us now analyze what happens at the junction between the two wires. At ϕ = πit is expected to have 3 Majorana fermions. For any other ϕ there is always at least one

111

6.5 W

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-0.5

0.0

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Polarizations

0 π/2 π 3π/2 2π

φ

(a)

0.5

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1.5

LDOS

0 π/2 π 3π/2 2π

φ

(b)

-1.5

-1.0

-0.5

0.0

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0 π/2 π 3π/2 2π

φ

(c)

Fig. 6.7: (a) e Majorana polarization of the right-end Majorana fermions integratedover the last 30 sites. Note the oscillatory behavior of the x- (red) and y- (blue) Ma-joran polarization components with a total constant Majorana density of 2 × 0.5, cor-responding to two rotating Majorana modes. (b) Zero-energy density of states inte-grated over the 40 sites around the junction point. (c) Zero-energy Majorana polar-ization integrated over the 40 sites around the junction. e x−polarization (red) andthe y−polarization (blue). ree Majorana fermions having the same polarization aresupported at the junction for ϕ = π.

Majorana fermion stuck at the interface between the two topologically nonequivalentregions. is can be seen by plotting the zero-energy density of states, as well as thezero-energy Majorana polarization, integrated over the 40 sites around the junction. InFig. 6.7(b) it is plotted the integrated zero-energy density of states. Note that for ϕ = πthe density of states is constant and equal to 0.5 corresponding to a single Majoranamode bound at the junction. e sharp jump between 0.5 and 1.5 at ϕ = π describesthe contribution of the two extra Majorana modes which reach zero energy at ϕ = π(these two extra zero-energy states appearing at ϕ = π can also be seen in the spectrumdescribed in Fig. 6.6(b)). is is also con rmed by a plot of the zero-energy Majoranapolarization, integrated over the 40 sites around the junction, in Fig. 6.7(c); the jumpin the x-Majorana polarization at ϕ = π can be understood as coming from the twoMajorana modes that develop at zero energy. us, at ϕ = π there are three Majoranafermions at the two extremities of the wire (one at the le end and two at the rightend), fully x-polarized in the positive direction. ese fermions are compensated bythree Majorana fermions which form in the junction which are fully x-polarized in theopposite direction.

6.5 Wireswithan inhomogeneous superconductingphaseAs it was shown in Ref. [127], the system exhibits two Majorana modes at each end,provided that the time-reversal and chiral symmetries are not broken, namely as long asthe system remains in the BDI class [12, 13]. Breaking time-reversal symmetry leads tothe removal of the protection for two Majorana fermions since the system now belongsto the D symmetry class (see Tab. 1). en the system can return to the more typicalstate with at most one Majorana fermion at each end.

Here it is explored a different way to break TRS, i.e. by adding a constant phase

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φ

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0

π(a)

Majorana Polarization PMxMajorana Polarization PMx

0 20 40 60 80 100

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Majorana Polarization Density PMMajorana Polarization Density PM

0 20 40 60 80 100

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(c)

Fig. 6.8: Aconstant phase gradient over 40 sites twists the superconducting phase from0to π. (a) Schematic picture of the setup. Supercurrent in the bulk change the supercon-ductor phase ϕ in the (black) wire from 0 to π. (b) e constant phase gradient removesfour MBS in the middle region, leaving two MBS at each end, both x-polarized. (c)Majorana modes form at the ends of a 40 site central region driven in a nontrivial Z2

phase by a phase gradient∇ϕ = 2.

gradient. Let us take a wire characterized by the topological index w = 2. Instead ofan abrupt change in the superconducting phase, consider a long junction comprising aregion in which the phase can vary smoothly with a constant phase gradient. On thele side of this long junction, the superconducting phase is assumed to be constant andequal to 0 while on the right side it is supposed to be constant and equal to ϕ = π.

e phase of the central region is supposed to vary smoothly from 0 to π; such a uni-form phase gradient can be induced for example by the presence of supercurrents inthe bulk of the superconductor. is situation is schematically depicted in Fig. 6.8(a).

e phase gradient can be used to manipulate the creation or destruction of Majoranafermions [125].

Due to the phase gradient, the BdG Hamiltonian gains a TRS-breaking h1(k) oddcomponent:

h1(k) = λ1 sin(k) sin(∇ϕ/2) + λ2 sin(2k) sin(∇ϕ)h2(k) = −λ1 sin(k) sin(∇ϕ/2)− λ2 sin(2k) cos(∇ϕ) (6.24)h3(k) = 2− 2λ1 cos(k) cos(∇ϕ/2)− 2λ2 cos(2k) cos(∇ϕ),

where the gradient ∇ϕ is the change of phase over one site. Note that a phase gradi-ent also creates a non-vanishing h0 component that multiplies the identity Pauli matrix.Although it is neglected in the following, it indicates the breaking of PHS and the ten-dency of the gradient to destroy Cooper pairs. is poses a conceptual problem: howare the Majorana fermions protected in case of a broken PHS? In fact numerical simu-lations show spurious zero modes that appear when the phase gradient is present.

Let us neglect for the moment the breaking PHS, as h0 ≪ h3 when the gradient issmall. e system is no longer characterized by a winding number w but instead by theKitaev Z2 invariant [67]. Analyzing the topological invariant reveals that the system isin a topologically nontrivial phase if the condition

|1− λ2 cos(∇ϕ)| < |λ1 cos(∇ϕ/2)| (6.25)

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6.5 W

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-0.1

0.0

0.1

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0.3

Energy

0 π/2 π 3π/2 2π

Phase gradient ∇φ

Fig. 6.9: Illustration of the rst negative and the rst two positive eigenvalues in a 100-site superconducting wire subjected in its entirety to a phase gradient that varies from0 to 2π. Uniform disorder with a magnitude of ∼ 10% of the hopping strength λ1removes spurious zero modes leaving zero Majorana modes in the windows predictedby the topological condition (6.25) (marked with dashed lines). λ1 = 1 and λ2 = −1.5.

is satis ed. Note that at vanishing phase gradient, one recovers the phase diagram inFig. 6.1, with the only change that w = 2 and w = 0 are both trivial Z2 phases.

erefore adding a small phase gradient removes the Majorana fermions at the endsof the central region. is is due to the fact that locally both sides of the region aretopologicallyZ trivial. However, at both ends of thewire, in the regionswithout a phasegradient, twoMajoranamodes continue to be present because they are locally protectedby the time reversal symmetry.

In contrast, one can drive the central region to a topologically Z2 nontrivial state,by increasing the phase gradient. is happens when the inequality (6.25) is veri ed.

en the central region is connected to twoZ2 trivial regions and oneMajorana fermionforms at each interface. e pair ofMajorana fermions persist at the ends of thewire (seeFig. 6.8(c)).

Due to the twist of thephase byπ, the twoMajoranas at the ends are bothx-Majoranapolarized in the same direction.

It is noteworthy to realize that the validity of the topological invariant can be ques-tioned in the case where PHS is broken. However one can see that the topological in-variant continues to hold true. In numerical simulations, a 100-site system, originally inthe w = 2 topological phase, is subjected in its entirety to a uniform superconductingphase gradient wire. In Fig. 6.9 one can see that when the gradient is added the systembecomes a trivial Z2 topological insulator, marked by the loss of the four-fold degener-acy near ∇ϕ = 0. Spurious modes near zero energy are removed by disorder, and two

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6. M M …

robust Majorana modes develop in the window predicted by the topological invariant.At this moment, we lack an explanation of the origin of the spurious zero modes, or forthe perplexing effectiveness of the Z2 topological invariant in the absence of the PHSsymmetry. ese problems deserve further research.

6.6 Discussione study of the Josephson effect betweenwires supportingmultipleMajorana fermions

has revealed that the 4π anomalous periodicity is maintained in junctions characterizedby a sharp change of the superconducting phase. Moreover, a smooth phase gradientcan be used to destroy the Z phase and to create Majorana fermions at the interface be-tween trivial and nontrivial Z2 phases. Also a connection was made between the workon topological insulators with largeChern number andBDI class topological supercon-ductors. For the two-band models studied, it was shown that a sufficient condition tocreate multiple zero-energy bound states at the edges of system is to add coupling termsbetween distant sites. Let us nally remark that the Z Majorana modes that appearin these system should be fragile when considered in realistic systems as the spin-orbitcoupled semiconducting wire due to the necessity of time-reversal symmetry. e real-ization of Majorana fermions was shown to depend crucially on the presence of a time-reversal breaking magnetic elds that will naturally act to destroy the high Z phases.

115

Conclusion and Perspectives

Topological insulators and superconductors form a class of materials with a bulk gap inthe energy spectrum, inhabited by exotic edge states. A topological invariant character-izes the bulk properties of these systems and it is subsequently re ected in the numberof the edge states.

In the rst part of the thesis, the investigationwas centered around two-dimensionaltight-binding topological insulators indexed by a Chern number. ey are quantumanomalous Hall insulators that support an arbitrary number of edge states. However,large Chern number or, equivalently, multiple edge states are usually created by fashion-ing multiband systems. It was shown here that it is possible to vary the Chern numberin simple two-band models by adding distant-neighbor hopping terms.

e keyof the entire study lies inunderstanding that the topological invariant admitsa discrete formulation. is allows to see an insulator as a “sum” of a gapless systemwithDirac points plus a mass term. Crudely, the topological invariant reduces to a nitesum over the Dirac points weighed by the sign of the mass term. erefore a necessarycondition to increase the number of topological phases is to multiply the nodes in theenergy dispersion. is can be realized by having hopping terms between distant sites.

e consequences are explored in two models: a new arti cial model with ve Chernphases and also by adding distant-neighbor hopping in theHaldanemodel. emethodallows one to obtain complete topological phase diagrams for the models.

ere are two directions in which to extend the present study. e rst consists ingeneralizing to multiband systems. Already two four-band model extensions are dis-cussed: a Z2 insulator built from two Chern insulators and a “striped” Z topological in-sulator that can have largemetallic phases. e second direction is to search for possiblephysical realizations of the model. e existence of extreme distant-neighbor hoppingterms remains rather problematic; however multiband systemsmight effectively map tosuch models [37].

While the bulk admits generalmethods to easily discriminate between the topologi-cal phases, the edge-state investigations get quicklymired in details associated to the par-ticular geometry of the edges. Moreover, the fact that there could be several edge statesat an interface adds a new layer of difficulty to analytically nding the wave-function so-

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lutions. Chap. 2 contains a detailed analysis of a situation inwhich there are atmost twoedge states at an interface. In perspective, one needs to generalize to edges of differentgeometries. Most importantly, one has to clarify the nature of edge states which formbetween topologically gapped phases with the same Chern number. Conceivably, thedecomposition of a topological insulator in a gapless model plus a mass term could beused to extend previous studies of edge states in gapless systems [63, 64] to topologicalinsulators.

e second part of the thesis revolves around the subject ofMajorana fermions real-ized as quasiparticles in one-dimensional topological superconductors. ere is a deepanalogy to the previous study of topological insulators. e Majorana fermions are the“edge states” of topological superconductor and can be predicted from a knowledge ofsome bulk properties. ey are predicted to emerge in a variety of condensed mattersystems (see Sec. 3.2.1). However, Chaps. 4 and 5 are focused on a particular theoreti-cal proposal distinguished by the simplicity of its ingredients. e Majorana fermionsare predicted to appear as zero-energy excitations in a 1D spin-orbit coupled semicon-ducting wire under the effect of a magnetic eld and in proximity to an s-wave super-conductor [24, 25]. In Chap. 4, a Dresselhaus spin-orbit coupling term was added andit was shown that there is a spin texture to the electronic degrees of freedom that formtheMajorana quasiparticle. ey are spin-polarized in opposite direction in a transverseplane to the magnetic eld. e exact direction is determined by the relative weights ofRashba andDresselhaus spin-orbit couplings. is information could be used to detectMajorana fermions through the coupling to magnetic impurities. Nevertheless, it stillremains to explicit this idea in the form of a complete experimental proposal.

Chap. 5 continued the study to superconductor-normal metal and long supercon-ductor-normal metal-superconductor (SNS) junctions built from the semiconductingwire. eMajorana fermions can form as extended states into the normal part of the sys-tem. It was shown that the extension of themodes can depend on the coupling betweenthe normal and superconducting regions and also that leaking of Cooper pairs into thenormal metal can further localize the Majorana modes. In the SNS junction the Majo-rana fermions form only at phase difference π between the two superconductors, oth-erwise would-be Majorana fermions hybridize to form extended Andreev bound statesin the normal region. is phenomenon is a salient feature of the fractional Josephsoneffect. An order parameter, the Majorana polarization, introduced in Sec. 3.3 is used toinvestigate the extended zero-energy states in the junction. Finally, the study turned toa ring divided in two regions: a normal metal region and a topological superconductingone with a superconducting phase gradient. e system can be mapped in this case toan SNS junction and againMajorana fermions arise in the normal regionwhen the totalphase twist is close to π. It is crucial for future studies to understand in more detail theaction of the gradient. e phase gradient has a Cooper pair-breaking effect which canlead to gapless phases in the superconductor. ese are large-gradient phases devoid ofMajorana fermions. However there are peculiar values where Majorana fermions stillform in the normal region. ese events are not predicted by the theory and pose an

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open question for future studies.e nal Chap. 6 treats a particular model where the presence of time-reversal and

chiral symmetries allows the formation of multipleMajorana fermions at the same edgeof a wire [127]. It is shown that distant-neighbor pairing terms play the same role asin the 2D Z topological insulators studied before: they facilitates the increase of thetopological invariant and the multiplication of edge states. Moreover, the presence ofseveral Majorana fermions allows to construct SNS junctions between wires in differ-ent topologically nontrivial phases. e fractional Josephson effect still characterizesthe system and it maintains its signature 4π-periodicity. Furthermore, the multiple Ma-joranamodes can be destroyed by the addition of a phase gradient, which can transformthe Z insulator into a Z2 insulator with at most one Majorana fermion at an edge. epresence of a phase gradient poses some serious problems; because gradient tends tobreak the Cooper pairs, the presence of protected zero modes at high gradient valuesremains to be properly explained.

In perspective, this analysis needs to be carried over tomore realistic spinful systemsto assess the viability of phases with multiple Majorana. For instance, the spin-coupledsemiconducting wire studied before does not seem to be a good candidate, because themagnetic eld breaks the TRS necessary to have the Z insulator. A natural candidateto realize topological Z superconductors would be a heterostructure involving time-reversal invariant QSH insulators instead [83].

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