department of physics - ltl.tkk.filtl.tkk.fi/~teemuo/kp_pro_gradu.pdf · ticle statistics that do...

Master’s Thesis

Theoretical Physics

Majorana states in ferromagnetic Shiba chains

Kim Poyhonen

2015

Supervisor: Dr. Teemu Ojanen

Examiners: Dr. Teemu Ojanen

Prof. Kai Nordlund

HELSINKI UNIVERSITY

DEPARTMENT OF PHYSICS

P.O. Box 64 (Gustaf Hallstromin katu 2)

00014 University of Helsinki

HELSINGIN YLIOPISTO – HELSINGFORS UNIVERSITET – UNIVERSITY OF HELSINKI Tiedekunta/Osasto – Fakultet/Sektion – Faculty/Section Laitos – Institution – Department

Tekijä – Författare – Author

Työn nimi – Arbetets titel – Title Oppiaine – Läroämne – Subject

Työn laji – Arbetets art – Level Aika – Datum – Month and year

Sivumäärä – Sidoantal – Number of pages

Tiivistelmä – Referat – Abstract

Avainsanat – Nyckelord – Keywords

Säilytyspaikka – Förvaringställe – Where deposited

Muita tietoja – Övriga uppgifter – Additional information

August 2011

Topological superconductivity, Rashba effect, Topological phases, Majorana, Braiding

Faculty of Science

Topological superconductors, combining the principles of topology and condensed-matter physics, are a new field which has seen much progress in the past two decades. In particular, they are theorized to support Majorana bound states, a type of quasiparticle with several interesting properties – most notably, they exhibit nonabelian exchange statistics, which has applications in fault tolerant quantum computing. During the past few years, several groups have observed effects in topological superconductors indicating that an experimental confirmation of their existence may be imminent. Recently experimental focus has been on ferromagnetic systems with spin-orbit coupling, serving as the motivation for our research. In this thesis, we study the topological properties of a system consisting of magnetic adatoms implanted on a two-dimensional superconducting substrate with Rashba spin-orbit coupling. Starting from the mean-field Bogoliubov-de Gennes Hamiltonian, we derive a nonlinear eigenvalue problem describing the system, generalizing previous results which considered a linearized version. In the reciprocal space, we obtain a transcendental equation for the energy of the system. Through numerical solution of these equations in the limit of long coherence length we obtain the topological phase diagram of the system. We further analyse the spatial decay of the Majorana wavefunctions as well as the dependence of their energy splitting on the length of the adatom chain. As an application, we study a prototype topological qubit constructed by intersecting two one-dimensional adatom chains to obtain a cross-shaped geometry that supports two pairs of Majorana bound states. The design allows for braiding of the individual quasiparticles, providing a possible platform for experimental verification of their nonabelian exchange statistics. Using numerical methods, we simulate moving the topological phase boundaries to enact a braid of two Majorana bound states and calculate the system energy for each step. We find that throughout the process the zero-energy modes are separated from the bulk states by a finite energy gap, as required for adiabatic braiding.

Theoretical Physics

64

Kim Pöyhönen

Department of Physics

Majorana states in ferromagnetic Shiba chains

M. Sc. Thesis

Contents

1 Introduction 1

1.1 Topological Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Quantum Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.1 Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.2 Quantum gates . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.3 Quantum decoherence . . . . . . . . . . . . . . . . . . . . . . . 11

1.3 Topology and Braiding . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3.1 Anyon statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3.2 The Berry phase . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Topological chain with spin-orbit coupling 19

2.1 Majorana bound states . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1.1 The spinless p-wave superconductor . . . . . . . . . . . . . . . . 20

2.1.2 Properties of Majorana operators . . . . . . . . . . . . . . . . . 23

2.2 Nanochain with Rashba spin-orbit coupling . . . . . . . . . . . . . . . . 25

2.3 Cross geometry for nonabelian braiding . . . . . . . . . . . . . . . . . . 32

3 Results 35

3.1 Nanochain with Rashba spin-orbit coupling . . . . . . . . . . . . . . . . 35

3.2 Wavefunction decay and energy splitting . . . . . . . . . . . . . . . . . 40

3.3 Adiabaticity of MBS braiding . . . . . . . . . . . . . . . . . . . . . . . 42

4 Conclusions 47

Appendices 53

A Shiba chain with Rashba SOC 53

B Comparison to the two-band model 59

1

C Derivation of the winding number 63

Chapter 1

Introduction

Assume one were to connect the ends of a ribbon of paper in order to create a loop.

By simply connecting the ends without twisting the ribbon, one obtains a ”trivial”

band, similar to a thin hollowed-out cylinder. However, by rotating the ribbon by

half a turn - π radians - before connecting the ends, one obtains what is known as

a Mobius strip. The crucial point here is that starting from the normal, non-twisted

ribbon, once the ends are connected, it is impossible to twist it to obtain the Mobius

strip - in fact, one needs to cut up the closed ribbon and reconnect it in a different

way to switch between the two. This is due to something which is known as topology.

Topology is the area of mathematics which concerns itself with those properties

of space that are unchanged by continuous transformations. These properties are

known as topological invariants. If two shapes or spaces differ in the value of one

or more topological invariants, they are regarded as topologically inequivalent; as a

consequence, it is not possible to continuously deform the one into the other. From a

topological point of view, then, a cube and a sphere are equivalent to each other, but

not to a torus, as the handle in the latter cannot be created in a continuous fashion.

Similarly, twisting a ”trivial” connected ribbon will never result in a Mobius strip

unless you cut the ribbon and twist it before reconnecting it - which would not be

a continuous transformation. Thus, objects and spaces are divided into topological

equivalence classes. During the past few decades the concept of topology has become

increasingly ubiquitous in physics, vital to such diverse areas as general relativity,

string theory, and stochastic processes. The focus in this thesis, however, will be on

the application of topology to condensed-matter physics, where topological materials

have been the focus of much research.

Topological superconductors are a specific class of materials that display proper-

ties different from those seen in the by-now familiar states of ordinary matter. In

1

2 CHAPTER 1. INTRODUCTION

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Figure 1.1: Differential conductivity of an InSb nanowire on an s-wave superconductor

with Rashba SOC as a function of gate voltage and magnetic field. Different lines

correspond to magnetic fields from 0 to 490 mT in steps of 10 T, with offset added

for clarity; the green arrows indicate the edges of the gap. Relevant parameters are

T = 70mK, ∆ ≈ 250µeV. Figure taken from ref. [2]

particular, they may support zero-energy modes known as Majorana bound states,

quasiparticles with interesting properties such as zero spin, and, more excitingly, par-

ticle statistics that do not conform to the traditional cases of Fermi-Dirac or Bose-

Einstein. In particular, is is theorized that these states could appear in systems of

one effective space dimension, facilitating construction of wire networks that could

be used in quantum computation. With the exciting prospect of new physics ahead,

the area has seen much progress within the past 15 years, but so far experimental

progress has lagged far behind the theory.

This thesis is motivated by recent experimental developments in the field of topo-

logical superconductors. Within the past five years, several different groups have

published results indicating that quasiparticles known as Majorana bound states may

be present [1, 2, 3, 4]. In 2012, Mourik et al [2] conducted measurements on InSb

nanowires contacted by a normal and a superconducting electrode. Through tun-

neling spectroscopy, they detected a zero-bias peak (ZBP) for intermediate-strength

magnetic fields1, as seen in Fig. 1.1. The peak was only present when the theoretical

1The measured differential conductance at a given voltage is proportional to the density of states

at the corresponding energy E = eV .

3

Figure 1.2: Conductance map of a Fe atomic chain on top of a Pb superconductor.

Relevant parameters are ∆s = 1.36meV, T = 1.45K. The grey scale bar is of length

10 A. The conductance shows a zero-bias peak localized at the end of the chain, while

higher-energy modes are delocalized. Figure taken from ref. [3]

requirements for the presence of Majorana bound states were fulfilled, indicating a

possible discovery of the Majorana quasiparticles. However, various alternative expla-

nations for the observed ZBPs have been suggested, such as zero-bias Kondo peaks

[5, 6] and disorder [7, 8], and the experiment is therefore not viewed as conclusive.

In addition to the fact that their energy is zero, another property of Majorana

bound states in one-dimensional nanowires is that they are expected to be localized

at the ends of the wire, which should also be observable experimentally. In 2014,

Nadj-Perge et al conducted a new experiment in which they measured the spatially

resolved differential conductance in Fe-based atomic chains on a Pb superconductor.

They also observed ZBPs, which, as seen in Fig. 1.2, are clearly located at the ends

of the wires. However, as was the case for the the previously mentioned experiment,

the topological nature of the zero-bias peaks are in doubt [9]; in particular, under

certain conditions the zero-bias peaks may be located at the ends of the chain even

in the absence of Majorana modes [10]. Nevertheless, while deemed inconclusive,

these experiments indicate that progress is being made and a physical realization of a

topological superconducting system may soon be feasible, inspiring the research done

in this work.

This thesis, largely based on Ref. [11], has two main aims. Firstly, to analyse the

properties of a system that is reasonably similar to the ones used in the above exper-

iments, and obtain analytical and numerical results describing this system. Secondly,

to consider the suitability of such a system for applications in topological quantum

computing, specifically for constructing a topological qubit. The structure of this


thesis is as follows. The first chapter is the introduction. It contains a short intro-

duction to topological superconductors, along with some basic concepts in particle

statistics and topological quantum computing. The treatment in the latter two are

largely based on Pachos, Introduction to topological quantum computation [12]. In the

second chapter, focus is placed on the concepts that will be treated in this work. First,

Majorana bound states are explained starting by the example of the one-dimensional

p-wave superconductor; following this, we move on to the system that will be the

main focus on this thesis, obtaining analytical results as well as introducing the basic

concept and intended structure of the topological qubit. In the third chapter, we

present numerical results both for the one-dimensional chain and for simulations of

the qubit. Finally, we state our conclusions and consider possible venues for future

research.

1.1 Topological Materials

The importance of topology to condensed-matter physics was first realized around 40

years ago. In the late 1970s, it was noticed that the conductance in certain Hall effect

setups displayed unusual plateaus [13], and in 1980, von Klitzing demonstrated [14]

that the conductance is exactly quantized, resulting in what is now known as the in-

teger quantum Hall effect (IQHE). In the IQHE, the transverse conductance σxy only

takes values that are integer multiples of e2/h, with e and h being, respectively, the

elementary charge and the Planck constant. What is interesting is that the bulk of the

material is still insulating despite the surface displaying this quantized conductance.

The mechanism behind this effect was first explained by Laughlin in 1981 [15]. Five

years later, Thouless, Kohmoto, Nightingale and den Nijs (TKNN) showed that the

IQHE could be explained by calculating a topological invariant, known as the Chern

number, of its first Brillouin zone [16]. This paper gave rise to the idea that there

are states of matter topologically distinct from the vacuum, and that this topological

distinction can have significant effects. Notably, topology ensures that these effects

are unusually robust to perturbations; in particular, the IQHE has been measured to a

precision of around 2 · 10−9 [17]. Around the same time, in 1983, the fractional Quan-

tum hall effect (FQHE) was discovered and explained [18, 19]. The FQHE showcased

the potential of topology in condensed-matter physics: the associated quasiparticles

carry fractional charge and do not obey either Fermi-Dirac or Bose-Einstein statistics

[20]. This opened up the possibility of developing practical applications based on

anyonic exchange statistics, and fuelled interest in the field of topological materials.

1.1. TOPOLOGICAL MATERIALS 5

The real expansion started in the 2000s, with such new theoretical discoveries as the

quantum spin Hall effect, three-dimensional topological insulators, and topological su-

perconductors being made. Although decades have passed since the discovery of the

IQHE, the field is still in the early stages of its development; topological insulators,

one of the most active areas of research, were discovered as late as 2005 [21]. Neverthe-

less, many potential applications have already been suggested or discovered. Perhaps

most significantly, topological systems could provide a venue for fault-tolerant quan-

tum computing [22], overcoming some of the issues faced by non-topological quantum

computers. Inspired by this possibility, countless proposals for topological quantum

bits have been presented, and the theoretical side is well documented [23, 24, 25].

Today, the field of topological materials is incredibly diverse, incorporating several

different kinds of topological invariants observed in both liquids and solid-state sys-

tems. In this thesis, we will focus exclusively on solid-state systems that are gapped

in the bulk, forgoing entirely the treatment of liquids and gapless systems. Hence,

in the remainder of this text, ”topological material” will be used to describe solids

with topological invariants that require a bulk energy gap. Consequently, when two

systems are described as topologically equivalent, this is taken to mean that their

Hamiltonians are equivalent up to a transformation which does not close the bulk

gap of the system. We stress that there are topological materials that do not fit into

this description, such as the gapless Weyl semimetals [26]. The presence of a single-

particle band gap in the type of topological materials treated in this thesis means that

electron interactions can generally be neglected [27], and hence topological materials

– as the term is used here – can be understood in terms of the band theory of solids,

in contrast to interacting systems such as the FQHE which are based on the concept

of intrinsic topological order [28].

Topological materials can be divided into several classes depending on their prop-

erties. The most common classification scheme is the one introduced by Schnyder

et al based on the Altland-Zirnbauer symmetry classes in random matrix theory

[29, 30, 31]. Materials are placed into ten different categories depending on three

discrete symmetries: time reversal symmetry (TRS), particle-hole symmetry (PHS)

and chiral or sublattice symmetry (SLS). These symmetries restrict the form of a

real-space Hamiltonian as follows:

TRS : T HTT −1 = H, T T † = 1 T T = ±T

PHS : PHTP−1 = −H, PP† = 1 PT = ±P

SLS : CSHC−1S = −H, CSC

†S = 1 C2

S = 1

(1.1)


Class TRS PHS SLS d = 1 d = 2

A (unitary) 0 0 0 - ZAI (orthogonal) +1 0 0 - -

AII (symplectic) –1 0 0 - Z2

AIII (chiral unitary) 0 0 1 Z -

BDI (chiral orthogonal) +1 +1 1 Z -

CII (chiral symplectic) –1 –1 1 Z -

-

D 0 +1 0 Z2 ZC 0 –1 0 - ZDIII –1 +1 1 Z2 Z2

CI +1 –1 1 - -

Table 1.1: Classification of topological insulators and superconductors in one and

two dimensions. In the symmetries column, a 0 indicates absence of the symmetry,

whereas a ±1 indicates presence of a symmetry with the symmetry operator squaring

to ±1. In the dimensions column, the symbol indicates the number of topologically

distinct phases; a dash indicates a trivial system, whereas Z2 allows two different

phases and Z any integer number.

Here, T and P are antiunitary operators of the form UK, where U is a unitary matrix

and K denotes complex conjugation; CS is a unitary matrix. The signs seen in front of

the transposes on the antiunitary operators are called the eigenvalues of the symmetry.

These three symmetries are not completely independent: a system with both TRS and

PHS, of either sign, must necessarily also have SLS. A system with only one of the two

antiunitary symmetries cannot support chiral symmetry, whereas it can be present

or absent in a system lacking both. All in all, this results in 10 symmetry classes.

Table 1.1 displays the symmetries and consequent topological classification of one- and

two-dimensional systems. As an example, the IQHE is a two-dimensional system in

class A, indicating a lack of all three symmetries. The Z-valued topological invariant

in this case is the integer in the conductance, σH = ne2/h where n can be any integer.

A one-dimensional system of the same symmetry class, on the other hand, would be

topologically trivial, showing that dimensionality is important to topology. While the

table included here only has two dimensions, the original classification by Schnyder

et al. covered dimensions 0 to 3; in 2009, Kitaev extended it to what is known as

1.1. TOPOLOGICAL MATERIALS 7

the Periodic table for topological insulators and superconductors [32], which shows

how the topology depends on the dimension d mod 8. In this thesis we will focus on

one-dimensional systems, and have hence omitted higher-dimensional systems from

the table.

For this type of topological materials, a topological equivalence class is defined as

the set of all materials which support a given subset of the above three symmetries. In

this prescription, it is understood that all materials within one topological class can

be transformed to each other without closing the bulk gap of the system - for example,

a conventional insulator can be transformed into the vacuum this way. Conversely,

it is not possible to transform a material of one class to that of another with a

transformation that does not close the bulk gap somewhere. The most significant

property of these topological effects is that they only depend on the existence of a bulk

gap and the discrete symmetries mentioned above. Because of this, perturbations that

do not close the gap or break these symmetries will have no effect on the properties

determined by the topology, namely the values of the topological invariants. This

is known as symmetry protected topological order. The protection also extends to

thermal fluctuations as long as T is low enough that the excitations past the bulk

gap are exponentially suppressed2; in this thesis, we will only treat systems at zero

temperature to avoid complications. It is important to be clear about the meaning of

the concept bulk gap, however: a topologically nontrivial system will generally support

zero-energy edge states that are mutually degenerate, but the first excited state will

be separated from this ground state by a finite, non-negligible energy. In contrast,

systems that extend to infinity in all directions or sport periodic boundary conditions

generally lack edges, and consequently for those the ground state lies at the gap rather

than at zero energy.

The effects of the topological classification, then, is most readily seen from what is

known as the bulk-boundary correspondence. As previously mentioned, the edges of

topological materials can often support zero-energy bound states. This is in general

something that occurs on the interface between two zones of different topological

invariants, and is a consequence of how we defined the appropriate transformations

in topological materials: in order to switch between two topological phases, the gap

must close. In topological insulators, for example, this leads to a surface which may

be conducting even while the bulk material is insulating. In one dimension, in turn,

the interface between two zones may support bound states at zero energy; these are

2The probability is proportional to e−βE , which is small as long as E >> β−1


not conductive in the traditional sense, unless the interface is mobile, but they may

affect the material - including its conductance - in any number of ways.

As in many branches of physics, however, the experimental side has not progressed

quite as far as the theory. However, progress is being made, and topological materials

are not limited to only theoretical ideas. Systems supporting IQHE and FQHE are

well known, and topological insulators are by now well-known with countless docu-

mented realisations [33, 34, 35], though conclusive experimental evidence of of anyonic

statistics has not yet been obtained [36]. It is clear that the unusual properties of

topological insulators and superconductors lend themselves well to a wide array of

practical applications, such as magnetic memory manipulation [37] and, potentially,

quantum computing.

1.2 Quantum Computing

Classical computers are notoriously inefficient at simulating quantum systems3. Orig-

inally quantum computers were conceived as a method of bypassing this problem by

including the quantum nature into the hardware of the computers. Since then, sev-

eral quantum algorithms have been devised that allow quantum computers to perform

more efficiently than the best classical algorithms we know of now. The most famous

of these are likely Shor’s algorithm for factorizing integers [38] and Grover’s algorithm

for searching databases [39]. While a classical computer can in principle, given infinite

time, calculate everything a quantum computer can4, it is evident that in many areas

a quantum computer has crucial advantages. In this section, following Ref. [12], we

will briefly cover some of the basic concept in the field. In section 1.2.1, we will talk

about qubits, the building blocks of a quantum computer. In section 1.2.2 we intro-

duce the concept of quantum gates, the operations used to encode qubits. Finally in

section 1.2.3 we focus on mixed states and decoherence, the latter of which serves as

one of the major motivation for the field of topological quantum computing.

1.2.1 Qubits

Quantum computers are in many ways similar to classical computers, in that they

are logic-based computational models based on the universal Turing machine. One of

3Fundamentally this is because the phase space of N qubits is 2N -dimensional whereas for N

classical bits it is 2N -dimensional.4Due to the Church-Turing conjecture

1.2. QUANTUM COMPUTING 9

the principal ways in which they differ is in the properties of the encoding elements.

Whereas classical computers make use of bits5, which can take on values of either 0

or 1, quantum computers are based on qubits, units of information that behave in a

fundamentally quantum-mechanical manner. Generally, the base states of a qubit are

denoted as |0〉 and |1〉, analogously to a classical cubit. However, the actual state of

a qubit can be a linear complex superposition of these:

|ψ〉 = a0|0〉+ a1|1〉 (1.2)

where |a0|2 + |a1|2 = 1. For this superposition, as typical in quantum mechanics, the

probability of measuring a |0〉 is |a0|2. This is very much unlike classical bits, which

are always in one definite state; it is worth noting that in principle qubits can also

be kept in classically definite states all the time, so a hypothetical error-free quantum

computer could perfectly simulate a classical computer.

Another key point in which quantum computers work differently from classical ones

is entanglement between qubits: several qubits can have states that are intrinsically

different from one another. The simplest case of this is two-qubit entanglement, for

example

|ψAB〉 =1√2

(|0A〉|0B〉+ |1A〉|1B〉). (1.3)

Here, if we measure qubit A, either outcome still has a 50% chance of occurring;

however, whatever the result, it is immediately certain that qubit B is in the same

state. Generally, a system with N qubits can be written

|ψ〉 =N∏n=1

1∑in=0

ai1,i2,...,iN |i1, i2, ..., iN〉 (1.4)

where, again, the total sum over the amplitudes must equal to 1. The entangled

state seen previously, for example, results when all a{i} are zero except for those

when all i are equal. The composite qubit is a tensor product of the individual

qubits, |{i}〉 = |i1〉 ⊗ |i2〉 ⊗ ...⊗ |iN〉. The resultant Hilbert space is 2N -dimensional,

in principle resulting in an encoding space exponentially larger than that used by

classical computers; in practice, access to this space is limited [40], but algorithms

utilizing the quantum nature of the qubits can in some cases nevertheless operate

faster than their classical counterparts.

5Or, in nonstandard architectures, trits etc.


1.2.2 Quantum gates

In order to utilize quantum computation it is not sufficient to only have qubits. The

ability to perform operations on the qubits and thus encode quantum information is

also required. Typically, the processing is done by quantum gates, unitary operations

that reversibly transform the state of one or more qubits. An n-qubit gate corresponds

in general to an element of U(2n) acting on the Hilbert space of n qubits. Often, the

qubits are described as vectors, for example

|0〉 =

(1

0

), |1〉 =

(0

1

)(1.5)

- note that this is just another way of acting on qubits. In this representation, a

one-qubit gate could for example be represented by the Pauli matrix σx, acting to

switch |0〉 ↔ |1〉: (0 1

1 0

)[a

(1

0

)+ b

(0

1

)]= a

(0

1

)+ b

(1

0

)(1.6)

This is known as the σx-gate, and corresponds to the classical NOT-gate. Another

important gate is the Hadamard gate

H =1√2

(1 1

1 −1

)(1.7)

which transforms a qubit from a set state to a superposition, for example H|0〉 =

(|0〉+ |1〉)/√

2.

Multiple-qubit gates allow for more interesting results, such as creating entangled

states out of previously independent ones. An important case are the controlled gates,

which perform a single-qubit operation on qubit B depending on the state of qubit A.

For example, the controlled-NOT (CNOT) gate is

CNOT =

1 0 0 0

0 1 0 0

0 0 0 1

0 0 1 0

. (1.8)

The CNOT gate flips the state of the qubit B assuming qubit A is in the state |1〉,otherwise it does nothing. In general, n-qubit controlled gates are of the form

CG =

(12×2 02×2

02×2 U

)(1.9)

1.2. QUANTUM COMPUTING 11

with U being a gate on n− 1 qubits and thus unitary. With these gates, it is possible

to create maximally entangled states starting from independent states. Generically a

quantum computer will make use of both single-qubit and two-qubit gates.

Occasionally it is also useful to have the ability to perform irreversible operations

on qubits. Among these operations are projectors, corresponding to projection oper-

ators with the property P 2i = Pi. Projectors acting on a state return its component

within some subspace of its Hilbert space; the other components are essentially lost

in a process. As an example, measuring the z spin of an electron will return ±1, and

any information regarding its original spin in other directions is lost. In quantum

computing, projectors can for example be used similarly for measuring qubits, for

example in one-way quantum computers [41].

1.2.3 Quantum decoherence

In many cases the exact state of a quantum system is not known. If so, the system

is in a mixed state, which has a certain classical probability of being in one of several

quantum states. The concept of a mixed state can be understood through the density

matrix ρ. If {|ψi〉} is an ortonormal basis in the Hilbert space, the density matrix of

some arbitrary system in that space can be written

ρ =∑i

pi〈ψi||ψi〉. (1.10)

The density matrix above implies that the state |ψi〉 occurs with the classical proba-

bility pi. Then the expectation value of an operator O with respect to the system in

question is

〈O〉 =∑i

pi〈ψi|O|ψi〉. (1.11)

Measuring the value of the operator O will give the value appropriate to the state

|ψi〉 with the probability pi - not |pi|2. In the simple case of a single-qubit system,

the density matrix can be written

ρ = a|0〉〈0|+ b|1〉〈1| (1.12)

where a+ b = 1. Unlike a superposition, the qubit really is in a definite state, either

〈0| or 〈1|, but which one it is is unknown, and the state is not useful for quantum

computation. In practice, mixed states often appear when some subset of a system

of qubits cannot be accessed, as in that cases the reduced density matrix is required.


For example, consider a two-qubit system in the pure (but entangled) state

|ψAB〉 =1√2

(|0A1B〉 − |1A0B〉) (1.13)

Assuming the qubit B is inaccessible, we can obtain the density matrix for A ρA by

tracing out the states of B from the composite density matrix ρAB = |ψAB〉〈ψAB|,

ρA =1∑i=0

〈iB|ρAB|iB〉 =1

2(|0A〉〈0A|+ |1A〉〈1A|) (1.14)

which is a maximally mixed state, in contrast to the original pure state. From the

point of view of quantum computing, the environment of a system can be considered an

inaccessible qubit, as it is generally impossible to know its state precisely. Interaction

with the environment will hence reduce the amount of information available about

a quantum system. If allowed to interact for a long time, the system of qubits will

achieve thermal equilibrium with its environment, resulting in a thermal state with

the density matrix

ρ =e−βH

tr(e−βH), (1.15)

often seen in quantum statistical mechanics.

This presents the main challenge in quantum computing. A system of qubits that

has entered a mixed state due to quantum decoherence cannot be reliably used in

algorithms due to the potential for errors. Typically decoherence will set in within

nanonseconds [42], so quantum computers will necessarily have to employ rigorous

error correction procedures to extend the available computation time. While error

correction algorithms for quantum computers do exist, in many cases the threshold

error rate these allow for is low [43, 44], motivating the need for an alternative. It is

therefore necessary to minimize the probability of decoherence on the hardware level,

which could, as originally suggested by Kitaev in 1997 [45], be accomplished using a

topological quantum computer. As the quasiparticle properties of topological materials

are robust to perturbations that do not close the energy gap, topological quantum

computers that function through braiding these quasiparticles gain an inherent fault-

tolerance [22].

1.3 Topology and Braiding

In this section, we will discuss the statistics of the quasiparticles that emerge in

topological materials. In subsection 1.3.1 we introduce the concept of anyons, while

1.3. TOPOLOGY AND BRAIDING 13

in subsection 1.3.2 we provide a short explanation of the Berry phase, which is useful

in understanding the statistics of anyons. We broadly follow the treatment in Ref [12]

throughout the section.

1.3.1 Anyon statistics

Dimensionality plays a crucial role in the topology of particle exchanges. In one

dimension, it is not meaningful to speak of exchange statistics 6, as two particles can

only be exchanged if they are allowed to pass through each other. In the familiar

case of three-dimensional systems, the available models are trivial. Indistinguishable

particles in three dimensions can have two different kinds of exchange statistics, owing

to the fact that bringing one particle around another and back to its starting position

should be equivalent to an identity operation:

P 2|12〉 = |12〉 → P |12〉 = ±|21〉 (1.16)

The upper sign corresponds to Bose-Einstein statistics, whereas the lower corresponds

to Fermi-Dirac statistics. The sign difference seen above is responsible for nearly all

of the important differences between bosons and fermions. However, while the two

types of particles are clearly different from each other, they are nonetheless abelian,

which is imposed by the restriction seen in eq. (1.16).

Two-dimensional systems, interestingly, allow for much more texture in particle

statistics [46]. In fact, the restriction P 2 = 1 no longer holds in two dimensions.

Topologically, the reason is that in three dimensions, any loop around a point can

be continuously deformed to a trivial loop. In two dimensions, the procedure is no

longer possible: in two dimensions, a loop which encloses a point is fundamentally

different from one which does not. This distinction can be seen in figure 1.3. With

the unit square requirement removed, exchanging two particles can change the system

much more fundamentally than just by a sign. In the case of abelian anyons, the

wavefunction can pick up an arbitrary complex U(1) phase. Non-abelian anyons

are, in turn, even more complicated, as exchanging two such particles transforms

the systems by a unitary matrix, and as such exchange operations generally do not

commute.

By this point it is relevant to consider whether anyonic systems ever occur in

nature. The main obstacle to realizing nonabelian statistics is the fact that nature

6Fermions and bosons still exist, but hard-core interacting bosons can be mapped onto noninter-

acting fermions [47]


Figure 1.3: Particle statistics as a result of the fundamental group of the (d−1)-sphere.

a) In three dimensions, any loop around a point can be continuously deformed to the

trivial loop by ’threading’ the particle through it. b) In two dimensions, the point

cannot be removed from inside the loop, enabling the possibility of assigning non-

trivial winding numbers to loops.

is inherently three-dimensional, and it is in principle impossible to manufacture a

genuinely two-dimensional structure. However, quantum-mechanical effects enable the

construction of a system which for quasi-particles is effectively 2D. Assume the system

is confined in the z direction, so that the time-independent Schrodinger equation of

the system is (− ~2

2m∇2 + V (x, y) + Vz(z)

)Ψ(r) = EΨ(r) (1.17)

where V (z) is a confining potential, for example similar to that of the particle-in-a-

box problem. In this case, the equation can (in principle) be solved by separation of

variables,

Ψ(r) = ψ(x, y)ψz(z) (1.18)

If the particle is strongly confined in the z direction, the separation between two

energy levels for the z wavefunction will typically be large - for the classical one-

dimensional particle-in-a-box problem, inversely proportional to the square of the

z width of the system. Consequently, the z-axis part of the wavefunction will be

restricted to the ground state as long as this energy gap is not bridged. Of course,

while the electrons are essentially confined to two dimensions, they are still three-

dimensional particles and there is no specific reason to believe that they should behave

as anyons. Quasiparticles made out of fermions confined thusly, however, can be

genuinely two-dimensional, resulting in possible venues for designing anyonic systems.

It is evident that these systems are not as robust as inherently two-dimensional ones,


as a strong enough perturbation may excite the electrons so that the system once

again becomes three-dimensional.

1.3.2 The Berry phase

One way of understanding anyon statistics is through the introduction of geometric

phases that cannot be removed through a simple gauge transformation. To illustrate

this concept we consider the simple example of a spin-12

particle in a uniform magnetic

field. The Hamiltonian for this system is

H = −σ ·B = −Bσ · n, (1.19)

where n = (sin θ cosφ, sin θ sinφ, cos θ) is the unit vector parallel to the orientation

of the magnetic field and σ = (σx, σy, σz is the Pauli spin vector. It is convenient to

write this in matrix form as

H = UH0U † (1.20)

where H0 = −Bσz and

U =

(cos θ

2e−iφ sin θ

2

eiφ sin θ2

cos θ2

). (1.21)

Since U is a unitary matrix, it is clear that if |n〉 is an eigenstate of H0, then U|n〉 is

an eigenstate of H:

UH0U † (U|n〉) = UH0|n〉 = EnU|n〉. (1.22)

The eigenstates of H0 are just | ↑〉 and | ↓〉, so we denote the eigenstates of H by

U(θ, φ)| ↑〉 ≡ | ↑ (θ, φ)〉 and U(θ, φ)| ↓〉 ≡ | ↓ (θ, φ)〉.We now study the properties of this system under time evolution. Assume the

particle is initially in the eigenstate | ↑ (θ0, φ0)〉. If the magnetic field is allowed to

change orientation adiabatically, the system will at all times be proportional to the

instantaneous eigenstate | ↑ (θt, φt)〉 of the new Hamiltonian H(t) = U(t)H0U †(t). If

the magnetic field finally returns to its original orientation, the final state must be

proportional to the initial, i.e.

|ψ(T )〉 = e−i∫ T0 E↑dteiγ↑T | ↑〉. (1.23)

The first proportionality factor is just the standard time evolution operator, and in

this case it simplifies to exp(E↑T ). The other factor contains a geometric phase, which

can be solved through insertion into the Schrodinger equation, giving

γ↑(C) = i

∮C

〈↑ (θt, φt)|∇R(t)| ↑ (θt, φt)〉dR(t) (1.24)


where C is the path taken by the angles during the time evolution, parametrised by

R(t). This is known as the Berry phase [48]. In a slightly more general notation, the

state has gained the phase

eiϕ = eiEnT exp(

∮A · dr) (1.25)

where we have introduced the Berry connection

Aµ = 〈n|U †({λ}) ∂

∂λµU({λ})|n〉. (1.26)

In the example used here, λ represents the angles of the magnetic field, compare

Eq. (1.24). In the example given here, the phase φ corresponds to two times the

solid angle spanned by the loop C with respect to the point B = 0. At that point in

the parameter space of B, the states | ↑〉 and | ↓〉 are degenerate. In general, non-

trivial Berry connections only arise when the parameter space of the path contains

points with ground-state degeneracy . While in the simple example given here, the

parameter space in question is that of the orientation of B, it can also be, for example,

the coordinate space of a set of quasiparticles [12].

The Berry connection discussed above can be extended to also explain nonabelian

statistics. Consider a D-dimensional Hamiltonian

H(λ(t)) = Uλ(t)H0U †λ(t) (1.27)

where λ denotes a set of adjustable parameters in the parametric space {λµ, µ =

1, ..., d} and Uλ(t) ∈ SU(D). Assume that the ground state of H0 is degenerate

at energy 0 and the lowest excited state has an energy ∆E > 0. Similarly to the

previously considered case, let the initial state of the system be an eigenstate of H0,

and let λ plot a loop in the parametric space so that the time evolution is slow

with respect to ∆E. In this case, the adiabatic theorem holds, and the system will

remain in the ground state, H|ψ(T )〉 = 0.. However, due to the degeneracy it is now

possible that 〈ψ(0)|ψ(T )〉 = 0. Consider the time evolution operator resulting from

adiabatically tracing the loop C,

U(0, T ) = T[e−i

∫ T0 U(t)H0U†(t)dt

], (1.28)

where T denotes time ordering. By discretising time and demanding adiabatic evolu-

tion, |ψ(t)〉 = 0 for all t, one can show that [49]

|ψ(T )〉 = U(0, T )|ψ(0)〉 = P

[exp(

∮C

A · dλ)

]|ψ(0)〉 (1.29)


where P denotes path ordering and the connection A is

(Aµ)αβ = 〈ψα|U(λ)†∂U(λ)

∂λµ|ψβ〉. (1.30)

The time-evolution operator of the loop in path ordering formalism is called the

holonomy ΓA, so that

|ψ(C)〉 = ΓA(C)|ψ(0)〉. (1.31)

This relation is important in topological quantum computing, as Eq. (1.31) could be

interpreted as a quantum gate operating on a qubit formed by the state vector.

Chapter 2

Topological chain with spin-orbit

coupling

2.1 Majorana bound states

Majorana fermions were originally introduced by the Italian physicist Ettore Majorana

as a self-adjoint solution to the Dirac equation [50]. The possibility of Majorana

fermions was originally of interest to particle physicists, as the self-adjoint nature of

the MFs would correspond to elementary particles that are their own antiparticles.

So far, they have not been detected as elementary particles, though it is theorized

that neutrinos may be Majorana fermions, as are the hypothetical supersymmetric

partners of bosons [51].

In condensed-matter physics, however, the picture is different. Superconductivity

imposes particle-hole symmetry, meaning for every energy eigenvalue E there exists

an eigenstate of opposite energy −E, which is obtained by complex conjugation of

the positive-energy state. There is then the possibility of Bogoliubov quasiparticles,

consisting of the superposition of an electron and a hole, at exactly 0 energy. These

quasiparticles are called Majorana bound states (MBS) or Majorana zero modes, al-

though the term Majorana fermion is also often seen used interchangeably. However,

MBS do not adhere to Fermi-Dirac statistics; rather, they are nonabelian anyons, sup-

porting nontrivial exchange statistics. Starting from a superconductor, it is possible

to obtain a Hamiltonian in therms of Majorana quasiparticles through use of a simple

operator transformation. In general, the transformation from fermionic to Majorana

representation is rather trivial and not in itself a sign of a topological phase: every

fermion can always be decomposed into two Majorana modes. Being (effectively) their

19

20 CHAPTER 2. TOPOLOGICAL CHAIN WITH SPIN-ORBIT COUPLING

own antiparticles, adjacent interacting MBS will cancel out to annihilate or create a

single fermion. However, it is possible to construct systems in which some mechanism

- spatial separation of the states, or some topological effects - prevent the MBS from

fusing, possibly allowing manipulation of these states.

2.1.1 The spinless p-wave superconductor

The archetypical example of a 1D system in which MBS appear is the spinless p-wave

superconductor, first proposed by Kitaev in 2000 [52]. It is also known as ”Kitaev’s

toy model”, as it is mainly constructed specifically to realize certain properties rather

than to describe an existing physical system. Here we will largely follow the reasoning

found in his seminal paper. The model in question is a spinless superconducting chain

with nearest-neighbour hopping, described by the Hamiltonian

H =∑n

t(a†nan+1 + a†n+1an)− µa†nan + ∆a†na†n+1 + ∆∗an+1an) (2.1)

where a is the fermionic annihilation operator, and t, µ and ∆ correspond to hopping

amplitude, chemical potential and the superconducting gap, respectively. The index

n runs over all sites in the lattice. We will henceforth choose ∆ to be real for clarity;

in general it is complex, but the relevant properties stay the same.

In case of an infinite system, we can express the operators in terms of their

reciprocal-space equivalents in order to obtain the Bogoliubov-de Gennes Hamilto-

nian

H =1

2

∑k

Ψ†kHkΨk (2.2)

where Ψk ≡(ak a†−k

)Tand

Hk = (−2t cos(ka) + µ)σz + 2∆ sin(ka)σy (2.3)

from which it is easy to find the dispersion relation,

E(k) = ±√

(2t cos(ka)− µ)2 + 4∆2 sin2(ka) (2.4)

where the presence of particle-hole symmetry is seen in the positive/negative eigen-

state pairs. However, it may be more interesting to assume a finite chain so that

n = 1, .., L. We can make the operator transformation

γ2n−1 = an + a†n γ2n = −ian + ia†n, (2.5)

2.1. MAJORANA BOUND STATES 21

Figure 2.1: The effect of the operator transformation into the Majorana basis. a) With

the first parameter set, Eq. (2.4), the Majorana operators on each site are paired up

in the Hamiltonian. b) As in Eq. (2.5), the Majorana operators with indices 1 and

2L do not appear in the Hamiltonian.

expressing each electron as a sum of two Majorana quasiparticles. Applying this

transformation, the Hamiltonian in Eq. (2.1) turns into

H =i

2

∑n

−µγ2n−1γ2n + (∆ + t)γ2nγ2n+1 + (∆− t)γ2n−1γ2n+2 (2.6)

The Hamiltonian is now expressed in terms of Majorana operators. To gain a better

understanding of the properties of the system we consider two special cases, which

will turn out to be topologically distinct. First, consider the case ∆ = t = 0. We then

have

H = −µL∑n=1

a†nan =1

2i

L∑n=1

γ2n−1γ2n (2.7)

This transformation is physically rather trivial; we have n fermions or, equivalently,

2n adjacent interacting Majorana modes. For contrast, let us now consider another

set of parameters: µ = 0,∆ = t. This results in

H =∑n

t(a†nan+1 + a†n+1an + a†na†n+1 + an+1an) = i

L−1∑n=1

tγ2nγ2n+1 (2.8)

We notice an important difference from the former case: now the Majorana opera-

tors γ1 and γ2L do not appear in the Hamiltonian. This separation is illustrated in

Fig. (2.1). While normally the way Majorana sites are paired up into fermionic sites

is arbitrary, in this case we see there is a significant difference: in the second case, the

Hamiltonian includes no interaction for the sites at the edges of the chain. They can


still, however, be combined into a delocalized fermionic operator:

aF =1

2(γ2L + iγ1) (2.9)

This fermion is not only highly delocalized, but we notice that it does not appear in

the Hamiltonian - its presence does not affect the energy of the system. The ground

state of the Kitaev model is hence degenerate, with two equal-energy states of different

fermion parity corresponding to the presence (odd)/absence (even) of this edge mode.

While the two different states of the system were introduced with the aid of a

very specific set of parameters, the system will necessarily be in one of the two states

for arbitrary values of these parameters, and most of the qualitative properties we

derived remain unchanged. The main difference is that as the system parameters are

moved away from these two special points, the MBS are no longer localized exactly on

the single sites at the edges of the chain, but rather extend into the bulk obeying an

exponential decay law [52]. The transition between the two states occurs at |µ| = 2|t|,with t < µ/2 corresponding to the topologically non-trivial phase with edge modes

present. We illustrate this in Fig. 2.2. We conclude that as the bulk gap closes1

at |µ| = 2|t|, the system undergoes a topological phase transition. At |µ| > 2|t|the system is in the trivial phase, displaying no unusual behaviour. The ground

state is delocalized with a finite energy. However, when |µ| < 2|t|, we find that

the ground state is degenerate at E = 0, and that the wavefunction of this state

is localized at the ends of the chain. Notably, the bulk states of the chain are still

gapped, consistent with the topological character of the system. While away from the

phase transition points, whether the parameters selected are topologically nontrivial

or not, the reciprocal-space Hamiltonian is gapped everywhere, as the zero-energy

states exclusively appear at edges which are not present in infinite or periodic systems.

We also noticed that the operators corresponding to the localized ground state are

most conveniently expressed in the so-called Majorana operator formalism. While

this simple model is ideal for introducing the concept, MBS are not unique to the

model treated here nor even to one-dimensional systems. They emerge generically as

quasiparticles in topological superconductors [53] as long as symmetry considerations

are satisfied (see table 1.1); notably, any T symmetry present must necessarily be

spinless, as Kramer’s degeneracy will otherwise cause the MBS to merge into a normal

fermion. The p-wave superconductor treated in this section is in symmetry class

D, giving it a Z2 topological invariant. This invariant can be calculated from the

1This can easily be seen from the energy equation; specifically, E(0) = 0 for these parameter

values

2.1. MAJORANA BOUND STATES 23

20 40 60 80 1000

0.5

0 1 2 3-8

-4

0

4

8

Figure 2.2: Topological properties of Kitaev’s toy model. a) Wavefunction of the

zero-energy modes (red) and the bulk state with the lowest positive energy (blue).

Parameters used are N = 100, t = ∆ = 1, µ = 1.5. The exponential decay of the MBS

wavefunctions are clearly visible here; if we set µ = 0 as in Eq. (2.8) the wavefunctions

would be exactly localized at the single sites on the edges. b) Evolution of the energy

states as a function of t. The red lines correspond to the states with lowest E. While

|t| < 12|µ| the chain is in the trivial phase and the system is gapped. At |t| = 1

2|µ|,

marked by the vertical blue line, the system undergoes a topological phase transition

(as noted in the main text), where the bulk gap closes. Further increasing |t| the

bulk gap opens again, leaving two degenerate states around zero energy; these are the

Majorana states of the systems.

Pfaffian of the Hamiltonian matrix [52], but in simpler terms, the two available states

correspond to the absence or presence of MBS.

2.1.2 Properties of Majorana operators

In the previous section we introduced the concept of Majorana operators and showed

how they appear in the case of the Kitaev p-wave model. We will now discuss their

properties in more detail, and thus explain the reasoning behind their importance in

the remainder of this thesis.

We recall the definition of the Majorana operators from Eq. (2.5), γ2n−1 = a†n + an

and γ2n = i(a†n − an). From this it is evident that

γ†2n−1 = γ2n−1 and γ†2n = γ2n, (2.10)

which explains the connection to the Majorana fermions of particle physics. However,

examining their commutation relations we notice that these quasiparticles do not


follow FD statistics:

{γi, γj} = 2δij (2.11)

It follows that, in particular, γ22n−1 = 1 and hence it is meaningless to speak of an

occupation number for Majorana quasiparticles. The occupation number for fermions,

in contrast, is physical, and is conserved modulo 2 due to superconductivity.

When MBS are exchanged, then, they do not simply acquire a minus sign as is

the case for normal Fermions. Rather, the effect of braiding depends on the choice

of MBS exchanged and the order in which the exchanges are done. However, these

exchange operations cannot change the fermion parity of the system, as the modulo

2 conservation still applies. As the physically observable quantity here is the fermion

number, braiding a single pair of MBS will simply result in a phase factor. However, if

we have several pairs of MBS, and exchange two MBS from different pairs, the result

is a unitary matrix [23, 54]. To illustrate this, let us first consider a system with two

MBS, denoted γ1 and γ2. Permuting them will transform the Majorana operators:

γ1 → B12γ1B†12

γ2 → B12γ2B†12

(2.12)

where B12 is the braiding operator. Since we have changed the places of two identical

particles we must have B12γ1B†12 = αγ2 and γ2 → B12γ2B

†12 = βγ1, where α, β ∈

{±} to preserve the appropriate normalization and MBS properties. The remaining

constraint is the fermion parity conservation. By requiring that the parity operator2

P = −iγ1γ2 [52] be preserved by the transformation, we find

−iγ1γ2 = −iαβγ2γ1 (2.13)

which immediately gives αβ = −1. Consequently, in a system with only two MBS

present, parity conservation means braiding results in a simple sign change on one of

them. The choice of MBS to change sign is arbitrary; note, however, that braiding

in the opposite direction should change the sign on both α and β. Choosing the sign

change to affect γ1, the transformation yields

B12γ1B†12 = −γ2

B12γ2B†12 = γ1

(2.14)

where B12 = (1 + γ1γ2)/√

2. With more than one MBS pair, it is possible to obtain

non-trivial braiding operations, as parity conservation can be satisfied through an even

2The parity operator is often denoted P; we have chosen to forgo this to avoid confusion with the

particle-hole conjugation operator.

2.2. NANOCHAIN WITH RASHBA SPIN-ORBIT COUPLING 25

number of fermion occupation changes [55]. These braids do generally not commute,

and the result depends on what MBS are chosen as bases for each fermionic state.3 The

behaviour under general braids can be deduces from the corresponding elements in

the braid group; particularly, braids made with MBS obey the Yang-Baxter equations

[12], which for a system with N MBS can be expressed

Bi,i+1Bj,j+1 = Bj,j+1Bi,i+1 |i− j| ≥ 2

Bi,i+1Bi+1,i+2Bi,i+1 = Bi+1,i+2Bi,i+1Bi+1,i+2 1 ≤ i ≤ N − 2

Bi,i+1B−1i,i+1 = B−1i,i+1Bi,i+1 = e,

(2.15)

where Bi,i+1 corresponds to braiding the Majorana states i and i+ 1, B−1i,i+1 is a braid

in the opposite direction, and e is the identity operator. Using these equations a braid

between any two MBS can be expressed as a combination of operations on adjacent

states.

The main disadvantage with applications based on braiding MBS is the fact that

the unitary matrices thus obtained do not span the complete set of quantum gates,

and as a consequence, universal quantum computing is inaccessible if one is limited to

MBS exchanges [54]. In general, MBS braiding operations result in Clifford gates such

as the σx mentioned in subsection 1.2.2. It is possible to expand the Hilbert space

and enable universal computing by including exchanges with other qubit systems

[56, 57]. Even omitting that option, however, MBS and their exchange operations

are nevertheless interesting as a possible base for a quantum memory. Due to the

delocalized nature of the MBS, the system is robust to perturbations unless they

either couple the ends of the chain - something which grows more unlikely as chain

length is increased - or change the fermion parity of the superconductor.4 Because

of this, MBS-based qubits are an active area of research, which in turn serves as the

main motivation for the research presented here.

2.2 Nanochain with Rashba spin-orbit coupling

During the past few years multiple models for realizing MBS in general and Majorana-

based qubits in particular have been studied. While Kitaev’s model conveniently sup-

ports MBS for a large region of parameter values and even rather short chains, realistic

3Any two MBS can be combined into a fermionic state. The fact that the choice of basis states

impacts the obtained results is, however, not unphysical; in practice, to measure the parity the MBS

need to be fused, which corresponds to this choice.4The latter of these is known as quasiparticle poisoning, and is a known problem of supercon-

ducting qubits [58, 59].


Figure 2.3: Schematic figure of the ferromagnetic Shiba chain. The system consists

of magnetic adatoms on a two-dimensional superconducting substrate. The arrows

represent the magnetic moment of the impurity atoms.

systems are generally not as optimal, sporting longer-range inter-site interactions as

well as potentially not having the suitable discrete symmetries to support topological

phases at all. Several more realistic and potentially topologically nontrivial systems

have been suggested, but despite this, no functioning topological qubits have yet been

constructed, and the evidence for the existence of MBS in a real system is so far not

conclusive. However, as mentioned in the introduction, recent experiments [2, 3] show

results that, while not conclusive, indicate that ferromagnetic spin-orbit-coupled sys-

tems are a promising candidate for detecting MBS. Similar setups have been studied

extensively in recent years [60, 61, 62, 63, 64]. In the remainder of this thesis, we

will focus on the system that was introduced by Brydon et al. [62], in turn partly

based on Ref. [65] which studied helical chains. The system in question is a ferromag-

netic chain of adatoms embedded on a two-dimensional s-wave superconductor with

Rashba spin-orbit coupling; a graphical representation can be seen in Fig. 2.3. We

will here utilize a mean-field approach with the assumption that the configuration of

the magnetic atoms is frozen in place, so that we can focus exclusively on the be-

haviour of the electrons. The atoms are treated as classical magnetic moments in a

ferromagnetic configuration, whose magnetic moments give rise to Yu-Shiba-Rusinov

subgap states [66, 67, 68]. The system is in many ways similar to the setup used in

recent experiments, Ref. [3], although the one treated here is both more ordered and

more dilute, which is useful as the physical properties of each individual chain are

easier to control. The goal of this section is to expand on the work done in [62] by

deriving the full 4×4 matrix equation describing the system similarly to the approach

used in [69], avoiding the need for the deep-impurity limit taken by Brydon et al. We

are interested in the topological properties of the system, particularly the presence

of MBS. The hybridization and decay of the Majorana states is also relevant, as a


possible topological qubit will be more robust the less the MBS are able to overlap.

Our starting point is the Hamiltonian density for a two-dimensional Rashba spin-

orbit coupled superconductor with embedded magnetic adatoms. Because of the fixed

impurity configuration, the Hamiltonian is for the electrons only,

H =

(k2

2m− µ+ αR(kyσx − kxσy)

)τz + ∆τx − J

∑i

(S · σi)δ(r− ri). (2.16)

The first two terms in the Hamiltonian are properties of the bulk SC, whereas the sum

is over the impurity atoms. Here k2/2m− µ is the kinetic energy of the electrons, αR

is the spin-orbit coupling and ∆ describes the pairing amplitude of Cooper pairs. The

vector r is the position of the electron, whereas ri describes the impurity positions.

The density is expressed in Nambu space, so that Ψ = (ψk↑, ψk↓, ψ†−k↓,−ψ

†−k↑)

T , where

ψkσ are the electronic field operators. The Pauli matrices in particle-hole (spin) space

are expressed by τ (σ); this allows us to remove Kronecker product notation with

no risk of confusion, so that, for example, σz ⊗ σz → τzσz. For simplicity we have

normalised the distance between impurity atoms (the chain lattice constant5) a ≡ 1.

Here, and in the remainder of the thesis, we have also set ~ = 1.

As mentioned previously, we will focus on the ferromagnetic phase, in which all

spins point in the z direction; this corresponds to setting σi = σzez for all i, where

ez is the unit vector in the z direction. The ferromagnetic Hamiltonian is in the BDI

symmetry class, so the system is significantly different from the p-wave superconduc-

tor presented earlier; in particular, the topological invariant N is Z-valued, allowing

phases with multiple MBS to emerge. To continue, we insert H into the BdG equation

HΨ = EΨ. To fulfil the single-axis symmetry requirements of the Rashba effect [70]

we will assume the superconducting system as a whole is two-dimensional, with the

magnetic atoms arranged in a one-dimensional chain so that ri ∝ iex. With these

restrictions we arrive at the equation(1− α E + τx√

∆2 − E2(S · σ)

)Ψ(xi) = −

∑i 6=j

JE(xij)(S · σ)Ψ(xj) (2.17)

where

JE(x) =12JS[(I−1 (x) + I+1 (x))τz + (I−2 (x)− I+2 (x))τzσy

]+1

2JS(E + ∆τx)

[(I−3 (x) + I+3 (x)) + (I−4 (x)− I+4 (x))σy

].

(2.18)

This result was previously obtained by Brydon et al. using Green’s functions [62]; a

derivation more closely following that used in [65] is included in appendix A. In the

5This is not necessarily the lattice constant for the superconductor.


above equation, we have introduced the functions

Iν1 (x) = mNνIm[J0((kF,ν + iξ−1E )|x|) + iH0((kF,ν + iξE)|x|)

]Iν2 (x) = −imNνsgn(x)Re

[iJ1((kF,ν + iξE)|x|) +H−1((kF,ν + iξ−1E )|x|)

]Iν3 (x) = − mNν√

∆2 − E2Re[J0((kF,ν + iξE)|x|) + iH0((kF,ν + iξ−1E )|x|)

]Iν4 (x) = −i mNν√

∆2 − E2sgn(x)Im

[iJ1((kF,ν + iξE)|x|) +H−1((kF,ν + iξ−1E )|x|)

],

(2.19)

where ξE = vF/√

∆2 − E2 is the energy-dependent coherence length, and

N± =

(1∓ ς√

1 + ς2

)α =

m

4(N+ +N−)JS = 1

2mJS

kF,± = kF

(√1 + ς2 ∓ ς

)ς = mαR/kF

(2.20)

The functions Jn and Hn are Bessel and Struve functions, respectively; the resultant

hopping terms are long-range, suppressed as |i − j|−1/2 over large distances. The

inter-site interaction, described here by JE(xij), is hence vastly different from the

simple nearest-neighbour interaction seen in Kitaev’s toy model. If the eigenstates

of interest are near the center of the gap, and the chain is dilute, it is possible to

linearize Eq. (2.17) in E and k−1F . This approach allows projection of the system

onto the single-impurity eigenstates [65], as the inter-site hopping decreases6. with

increasing kF . The result is the effective Bogoliubov-de Gennes Hamiltonian of the

two-band model,

Heffij =

(∆(1− α)δij + f(xij) g(xij)

g(xji)∗ −∆(1− α)δij − f(xij)

)(2.21)

where

f(x) = 12JS∆2 lim

E→0(I+3 (x) + I−3 (x))

g(x) = i2JS∆ lim

E→0(I+2 (x)− I−2 (x))

(2.22)

This effective Hamiltonian was originally obtained by Brydon et al. in Ref. [62]. How-

ever, while it can be used to obtain energies and eigenvalues of the system, it is an

6Recall that we normalized the distance between lattice sites; the approximation made is kFa� 1


approximation and as such not necessarily valid, especially when the slow, asymptot-

ically x−12 , convergence of the Bessel and Struve functions in the block off-diagonals

is taken into account. To ensure the reliability of the obtained results it is to find a

solution which relies on fewer approximations. We therefore diverge from the previ-

ously obtained results and aim to solve Eq. (2.17) away from the deep dilute limit.

To this end it is possible to obtain a nonlinear eigenvalue problem (NLEVP) which

is valid outside the deep dilute limit, similarly to the approach used in Ref. [69]. The

precise details of the derivation are somewhat involved and therefore left to Appendix

A; here we just state the resultant equation, which isaλ2 − λ bλ cλ2 −λd−bλ λ− a −λd c

−cλ2 −λd aλ2 + λ −bλ−λd −c bλ −λ− a

Ψ = 0 (2.23)

Here we have introduced the parameter

λ =∆ + E√∆2 − E2

. (2.24)

as well as the submatrices

aij = αδij − α√

∆2 − E2

2m

[I−3 (xij) + I+3 (xij)

]bij = −α i

2m

[I−2 (xij)− I+2 (xij)

]cij = −αi

√∆2 − E2

2m

[I−4 (xij)− I+4 (xij)

]dij = α

1

2m

[I−1 (xij) + I+1 (xij)

].

(2.25)

As mentioned previously, Eq. (2.23) is a non-linear eigenvalue problem for E and Ψ,

and difficult to solve even numerically. The main issue is that the energy dependence

enters the equation not only through λ, but also through the submatrices a, b, c, d as

they contain a non-trivial energy dependence hidden in ξE. Nevertheless, following

the procedure outlined in Ref. [69], it is possible to extract information about the

system using Eq. (2.23) in some limiting cases. In particular, in the limit ξE →∞ the energy dependence is limited to λ terms only, and the resultant polynomial

eigenvalue problem (PEVP) can be solved numerically. As the coherence length in

most superconductors tends to be large compared to the lattice spacing, the limit

is physically relevant rather than a mathematical curiosity. Consider for example a

Shiba chain embedded on a lead superconductor. Lead – used for the experiment in


Ref. [3] – has a superconducting gap of ≈ 2.7 meV, a lattice constant of ≈ 0.5 nm and

a Fermi velocity of ≈ 1.8 · 106 m/s (Kittel [71], p. 134, 23 and 328 respectively); with

these parameters, the coherence length for zero-energy states7 ξ0 = ~vF/∆ ≈ 440 nm.

Letting the lattice constant of the Shiba chain a = 4 nm (eight times that of the

underlying SC), we obtain ξ0/a ≈ 110, which can be increased further by using a less

dilute chain.

The topological properties of the one-dimensional chain are most conveniently

obtained by going over to the reciprocal space where they can be extracted from the

Bogoliubov-de Gennes Hamiltonian. As the submatrices in Eq. (2.23) are translation

invariant, it is in principle possible to Fourier transform them using

ak =∑j 6=0

aijeik(i−j) (2.26)

which reduces the problem from 4N × 4N to a 4 × 4 NLEVP. Transforming each

submatrix in this manner, we can find an equation for λ through requiring that the

determinant of the matrix be zero:

λ2(a4kλ2 − a2k(2b2kλ2 − 2c2kλ

2 − 2d2kλ2 + λ4 + 1)− 8akbkckdkλ

2

+c2k(2b2kλ

2 − 2d2kλ2 − λ4 − 1) + λ2(b2k + d2k − 1)2 + c4kλ

2) = 0.(2.27)

where ak, bk, ck, dk are the Fourier transforms of the corresponding submatrices in Eq.

(2.23). Obtaining these FTs is a nontrivial task, as the elements of the real-space

submatrices are sums of special functions and hence difficult to treat analytically. In

practice the FTs will be obtained through numerical treatment of the matrices, but

this is unimportant for the analytical treatment here. As λ 6= 0 for states within the

gap, by dividing by λ4 and introducing the new variable R ≡ λ2 + λ−2 we can solve

the equation explicitly,

R =(a2k + b2k + c2k + d2k − 1)2

a2k + c2k− 4

(akbk + ckdk)2

a2k + c2k+ 2, (2.28)

and using Eq. (2.24) this is easily expressed as

E = ±∆

√(a2k + b2k + c2k + d2k − 1)2 − 4(akbk + ckdk)2

(a2k + b2k + c2k + d2k − 1)2 − 4(akbk + ckdk)2 + 4(a2k + c2k). (2.29)

Thus we have found an equation giving the energy of the system as a function of

k. This equation is transcendental and lacks an analytical solution outside the limit

ξ → ∞, but can be solved through numerical methods; in that limit, the right-hand

7We have here restored ~ to obtain the correct units


side is no longer energy dependent and the equation hence gives the energy directly.

However, as changes in topological phases occur at E = 0, the phase boundaries for

any coherence length can be obtained by simply extracting them from this equation;

in this case the inconvenient energy dependence vanishes trivially. In particular, the

energy crossings at k ∈ {0, π} are easy to find, as the xij-antisymmetry of b and c

ensures their FTs vanish in these points; we obtain

α =2m√[

∆(I+3 (k) + I−3 (k)) + 1]2

+ (I+1 (k) + I−1 (k))2

∣∣∣∣∣k=0,π

. (2.30)

In a class-D topological superconductor - such as the Kitaev model treated previously -

the solution for these two k points would give a complete description of the topological

phase boundaries of the system. However, the chain here is in the topological class

BDI, and supports a Z-valued invariant which we will denote N . In general, the

gap closing between two zones corresponding to different values of N need not occur

at k = 0, π, and it is possible to have other closings as well; those require solving

Eq. (2.29) for E = 0 at arbitrary moments, which does not lend itself well to analytical

treatment. Nevertheless Eq. (2.30) is convenient in the case that knowledge of the

N = 1 phase is useful. However, this does not tell us anything about the other values

for the topological invariant, so in the general case another approach is needed. The

Z-valued topological invariant N corresponds to the winding number

N =1

4πi

∫ π

−πdk tr

[CH−1∂kH

](2.31)

where C = C† = C−1 is the particle-hole conjugation operator. As we do not have

an explicit H matrix for the chain, we will need to adopt a different approach here.

We can circumvent this by defining a topologically equivalent effective Hamiltonian

in terms of projection operators,

H =∑i

Ei|Ei(k)〉〈Ei(k)| =∑i

EiPi(k). (2.32)

Explicit expressions for the energy eigenstates can be obtained through solution of

the NLEVP. As detailed in Appendix C, this allows us to rewrite the expression for

the winding number in the more tractable form

N =1

πi

∫ π

−πdk tr

[CP+(k)∂kP+(k)

], (2.33)

where P+ is the projector to the positive-energy eigenstate.


The equations derived here – Eqs. (2.23), (2.29) and (2.33), in particular – consti-

tute the culmination of our analytical treatment of the model. In the next chapter,

we will present a numerical analysis of the chain based on the results derived in this

section.

2.3 Cross geometry for nonabelian braiding

One of the main appeals in topological superconductor research is the prospect of

utilizing the nonabelian statistics of MBS for quantum computing. However, experi-

mental confirmation of the statistical behaviour of MBS has not yet been presented.

In this section we introduce a possible construction for experimentally testing the

nonabelian statistics of MBS using a Rashba spin-orbit coupled Shiba chain in a cross

geometry. The same scheme could, once functional, also act as a topological qubit,

and hence this also acts as a test for the potential of the chain as a platform for a

topological quantum computer. For our prototype qubit, we need to take into account

at least the following considerations:

1. The base system should be physically realistic

2. The qubit should support non-trivial braiding operations

3. The construction of the qubit should be a feasible task

4. The braiding operation must be possible to perform adiabatically

The first of these we have already discussed in the previous section; the concept

of ferromagnetic atoms on a superconductor is reasonably realistic, and as such the

main question here is the validity area of the mean field approximation underlying

the Hamiltonian. This ties into point 3, as it mainly requires low temperature and

materials with suitable parameters. The second point, as noted previously, requires

the qubit to include at least two pairs of MBS, so that two MBS from different pairs

can be braided without looping around the others. In order to satisfy this requirement

we have chosen a cross-shaped geometry in which two different arms are initially in

the topological phase. The intended braiding operation is seen in figure 2.4.

As noted previously, the physically measurable quantity here is the Fermion num-

ber, particularly the parity induced by each MBS pair, measurable through the usual

number operator c†c, where now the operators correspond to the delocalized fermion

which is a composite of the Majorana states in the sense of Eq. (2.9). Hence the

2.3. CROSS GEOMETRY FOR NONABELIAN BRAIDING 33

1

2

1

11

2

2 2

Figure 2.4: Schematic picture of the qubit prototype used in this thesis. The yellow

circles represent MBS, and the orange areas are within topological parameters. The

system evolves as shown by the arrows, resulting in a braid of two MBS. The actual

operation will require significantly more than four steps; the specific pictures were

chosen to convey the braiding operation with maximal clarity.

1

2

Figure 2.5: Schematic picture of the braid enacted by the operation treated in this

section. The braiding of two MBS from different pairs, corresponding to the MBS

labeled 1 and 2 in Fig. 2.4, enacts an operation that changes the Fermion occupation

number of each pair from 0 to 1.


logical choice of qubits is letting one state correspond to |0, 0〉 and the other to |1, 1〉in the product space of the parities of both MBS pairs. The braiding operation con-

stitutes a NOT-gate on these states, expressed as a σx operator on the vector basis

(|0, 0〉, |1, 1〉). The basic concept can be understood as follows: as the MBS are ini-

tiated, the Fermion number n for both pairs are zero as the MBS appear from the

vacuum. The braiding process changes n for both pairs to 1, which does not violate

the parity conservation inherent to superconductors. A measurement on the system -

done, for example, by fusing the MBS - will reveal this change in n. If two operations

are performed, the system returns to the initial state and the MBS would again fuse

to the vacuum. Hence, the braiding operation here corresponds to a NOT gate, or

|0, 0〉 ↔ |1, 1〉, if the basis states of the qubit are defined as previously. A pictorial

representation of this is shown in Fig. 2.5.

A small-scale version of the system should be possible to construct with current

technology by manually depositing the magnetic atoms using an STM [72, 73]. The

specific shape was chosen with the intention of making this as simple as possible, with

only the ability to construct straight lines required. The fourth point is crucial and the

main challenge in this case. In order for the braiding operation to function at all, it is

imperative that the evolution be adiabatic, so that there are no transitions between

the MBS and the bulk states. If this requirement is fulfilled, any change to the MBS

will necessarily result in their final state being another MBS superposition, enabled

by the degeneracy. In order to avoid excitations, the energy of the MBS must at all

times be well separated from the gap edge. Further, any change to the parameters

of the Hamiltonian must be slow in the sense of the adiabatic theorem, though in

principle this can always be done by just slowing down the change of the system as

long as the gap is well defined. However, as in a finite system the states that would

otherwise be zero-energy modes will acquire a finite energy splitting, the evolution

must also be fast enough that these states can be mixed. We hence find the following

two minimum requirements for realising nonabelian statistics: A parameter must be

tweaked locally to expand and contract the topological zones gradually without closing

the energy gap or exciting the ground state; the gap energy must be large compared

to the energy splitting of the zero-energy modes to allow for a wider margin of error

as the parameter is changed. In addition, due to the long-range effective hopping

in this model, the MBS will tend to hybridize easily, necessitating longer inter-zone

distances as well as careful manipulation of parameters. We expect all these effects

to play into the difficulty of constructing a well-defined adiabatic evolution.

Chapter 3

Results

In this chapter we will present numerical results based on the work done in the previous

chapter. In section 3.1 we will first analyze the topological properties of the Rashba

spin-orbit-coupled Shiba chain. In section 3.2 we will focus on the properties of the

MBS wavefunctions. In section 3.3 we will present results related to the cross-shaped

qubit.

3.1 Nanochain with Rashba spin-orbit coupling

In order to extend on our previous analytical results, we have examined the topo-

logical properties of the chain by the use of equations (2.23) and (2.29). Our first

priority is to deduce the topological phases of the chain and to examine whether there

are parameter sets that allow a topological invariant N 6= 0. This is complicated

by the fact that the standard analytical methods of solving for topological invariants

require a Hamiltonian matrix, whereas our polynomial eigenvalue problem is formu-

lated in different terms. While in principle the solution for the energy, Eq. (2.29),

contains information about gap closings and would allow us to deduce the form of the

topological phase diagram, this would require explicitly Fourier transforming and an-

alytically manipulating the functions involved, which are combinations of Bessel and

Struve functions. It is therefore necessary to solve Eq. (2.23) numerically to obtain

information about the phase of the system. In order to compare this to our analytical

results we have Fourier transformed the system submatrices numerically by explicitly

taking the sum and cutting it off for small terms. However, the sums involved do

not converge conventionally at k = 0, so the numerical FT requires special care for

small values of k. Because of this, information obtained from the analytical results is

not always quantitatively exact, and in each case we make sure to compare with the

35

36 CHAPTER 3. RESULTS

1.4

1.2

1

0.8

0.64 5 6 7

1

0.5

0

1.4

1.2

1

0.8

0.64 5 6 7

Figure 3.1: Topological phases for the four-band model. a) Diagram plotting the ra-

tio of the ground state to the first excited state, obtained through numerical solution

of the NLEVP in each point. Parameters used are N = 100, ξ0 = ∞, ς = 0.01.

Resolution 120×120 points.b) Diagram of the winding number invariant of the sys-

tem calculated using Eq. (2.33). Blue, teal and yellow correspond to N = 0, 1, 2

correspondingly. Parameters used are ς = 0.01, with a coherence length of 500 for

convergence of the Fourier sums.

real-space results. The analytical equation are , however, still useful - for instance, the

N = 1 phase borders calculated by use of Eq. (2.30) are valid for arbitrary coherence

lengths ξ, requiring no approximations in that regard. Numerical results are not easy

to obtain for finite coherence lengths, as mentioned in the previous section; for any

ξ ∈ R+, the block matrices in Eq. (2.23) are transcendental functions of E resulting

in a complicated non-linear eigenvalue problem.

In Fig. 3.1 we present topological phase diagrams for the four-band version of the

model. To minimize errors, we have approached the problem both in real space and

in reciprocal space.

The left-hand figure, Fig. 3.1 a), is a map of the ratio between the ground state

and the first excited state in a finite system. In the dark blue sections of the figure,

this ratio is small, corresponding to the N = 1 phase - the energy of the MBS states

is negligible compared to the first excited state. The brighter area near the top of the

figure, where the ratio is 1, corresponds to the trivial insulating phase which lacks

zero-energy edge states. The most interesting behaviour is seen at the bottom of

the figure, where the ratio is high but oscillates. This area corresponds to the phase

N = 2, and the ratio compares the two MBS energies instead of MBS versus first

excited state. This behaviour would not be seen in a D-class superconductor like the

p-wave model treated earlier, and is a confirmation of the BDI classification of this


-1

0

1

-1

0

1

Figure 3.2: a) Energy spectrum in the N = 1 phase. Parameters used are kF =

17.3, ς = 0.01, α = 1, with 500 lattice points. The line colored red represents the zero-

energy modes; in total, there is one pair of MBS with energies around ±3 · 10−6∆.

The bulk gap energy in this phase is ≈ 0.1∆. b) Energy spectrum in the N = 0

phase. Parameters used are kF = 17.3, ς = 0.01, α = 2, with 500 lattice points. The

gap energy is approximately 0.33∆.

system.

In contrast, in Fig. 3.1 b) we have plotted the topological invariant N directly,

using Eq. (2.33) which is valid for infinite or periodic systems. The dark blue areas

correspond to the trivial phase N = 0, whereas the teal and yellow areas correspond

to N = 1 and N = 2 respectively, indicating that the system can support that

number of MBS pairs. Comparison between the two figures shows that the phases

obtained through comparing real-space energies are in agreement with those given by

the topological invariant. The wide area of transition from N = 0 to N = 1 appears

in both the finite- and infinite-chain phase diagrams; in this area, the gap size varies

rapidly, occasionally hitting values very low compared to ∆.

To ensure the topological nature of the system, we will more closely study certain

points in the phase diagram. We select the parameters N = 500, α = 0.01, kF = 17.3,

which according to Fig. 3.1 corresponds to the N = 1 phase. Solving for the energy,

we are primarly interested in the presence of zero-energy modes as well as a finite

energy gap. The energy levels are displayed in Fig. 3.2. For comparison we have also

added a similar figure for doubled α, corresponding to the trivial phase with no MBS.

From the figure it is evident that the zero-energy mode does appear while the

system is in the topological phase, and is absent in the trivial phase. In the topological

phase the gap is large compared to the energy splitting of the MBS, which is sufficient


0

0.6

0 250 5000

0.08

Figure 3.3: Wave function amplitudes in the N = 1 phase. Parameters used are

kF = 17.3, ς = 0.01, α = 1 with 500 lattice points. The upper subfigure corresponds

to the MBS wavefunction, while the lower corresponds to the first (positive) excited

state.

for the system to be topological as long as the temperature is low. As we are interested

primarly in MBS it is also of some interest to check whether these zero modes are

localized at the ends of the chain, and how fast they decay in the bulk. As seen in

Fig. 3.3, when in the N = 1 phase, the zero-energy modes are localized to the edges

of the chain, consistent with what is expected of MBS. The first state above the gap,

in contrast, is delocalized and bears similarity to the ground state of a particle in a

box; were we to plot the wavefunctions in the trivial phase, the MBS state would be

absent and the delocalized state at the gap would correspond to the lowest positive

energy.

Having established the presence of MBS in the system, we are interested in con-

firming that the two-MBS zones do support several zero-energy modes as expected

from the BDI classification of the system. For this purpose, we have selected the pa-

rameters kF = 17.74, ς = 0.01, α = 0.752, with 800 lattice points, which is within the

expected N = 2 zone, based on Fig. 3.1. Numerical solution of the NLEVP indicate

that two MBS pairs are indeed present, and the bulk states are gapped. The den-

sity of states is seen in Fig. 3.4. The reason we have increased the number of lattice

points compared to the N = 1 phase is that the MBS appear to hybridize easier for

these parameters, so that the N = 2 states require longer Shiba chains – notably,

the MBS energy in both cases is approximately equal even though the latter chain is

significantly longer. This is caused by the higher hybridization of the MBS. The wave

functions for the N = 2 case are shown in Fig. 3.5, supporting this conclusion. The

wavefunctions are visibly less localized than in the N = 1 case. As the wavefunction

overlap is relevant for MBS hybridization, this difference is of some interest, and we


-1

0

1

-0.1

0

0.1

Figure 3.4: Energy spectrum in the N = 2 phase. a) Spectrum for the entire system.

b) The same figure, but zoomed in to show the gap. Parameters used are kF = 17.74,

ς = 0.01, α = 0.752, with 800 lattice points. In total, there are four MBS with energy

of the order of 5 · 10−6∆. The energy of the first excited state is ≈ 0.026∆.

0

0.2

0

0.2

0 200 400 600 8000

0.06

Figure 3.5: Wave function amplitudes in the N = 2 phase. Parameters used are

kF = 17.74, ς = 0.01, α = 0.752, with 800 lattice points. The red curves corresponds

to the two zero-energy state (MBS) wavefunctions, whereas the blue curve corresponds

to the first (positive) excited state.


0

0 100 200 300 400 50010-5

10-4

10-3

10-2

10-1

10

0 200 400 600 80010-5

10-4

10-3

10-2

10-1

100

Figure 3.6: Decay properties of wavefunctions. The blue line corresponds to the

wavefunction of the system; the red line is a least-square-distance fit to the envelope.

a) N = 1 phase. System parameters are N = 500, kF = 20, ς = 0.01, α = 1.

The fitted function is f(n) ≈ 0.43e−0.16n + 6.84n−1 ln(n/0.0011)−2. b) N = 2 phase.

System parameters are N = 800, kF = 17.74, ς = 0.01, α = 0.73. The fitted function

is f(n) ≈ 0.31e−0.026n + 1.01n−0.79 ln(n/0.07)−2.

will examine this behaviour in the next section.

3.2 Wavefunction decay and energy splitting

For the purpose of experimental detection and manipulation of MBS, the scaling

relations of the MBS wavefunctions are of crucial importance. In finite systems, the

energy of the quasiparticles is not locked at zero1, but rather there is a finite energy

splitting caused by the overlap of the MBS wavefunctions. As smaller systems are

generally easier to construct, it is therefore of interest to know how the wavefunctions

decay over the chain. In this section we will present results regarding the localization

of the wavefunctions as well as the dependence of the energy splitting of zero-energy

states on chain length.

In Fig. 3.6 we present wavefunction profiles for the N = 1 and N = 2 phases. We

have numerically fitted lines to the wavefunction envelopes in order to study how the

amplitude behaves as a function of site index. For all parameter values studied the

wavefunction will at first decay as |ψ| ∝ e−kx, but at longer distances the change in

amplitude is significantly slower, following a decay law of |ψ| ∝ x−a ln(x/x0)−b. The

size of the region over which the exponential decay mode dominates varies with the

parameters. As seen in the figure, the decay of the N = 2 wavefunctions generally

1Except for very specific parameters, see subsection 2.1.1.

3.2. WAVEFUNCTION DECAY AND ENERGY SPLITTING 41

Figure 3.7: Evolution of the energy of the ground state and first excited state as

chain length varies from 10 to 200. a) N = 1 phase. The red curve corresponds

to the ground state (MBS) energy, the blue to the first excited state (Gap) energy.

Parameters used are kF = 20, ς = 0.01, α = 1. b) N = 2 phase. The red curves

correspond to the energy of the two MBS, the blue to the gap energy. Parameters

used are kF = 17.74, ς = 0.01, α = 0.73.

remains exponential over a longer region of the chain, but compared to the N = 1

phase the coefficient in the exponent is small resulting in overall slower decay. Note

that in Fig. 3.6 the N = 2 system consists of 800 sites rather than the 500 of the

N = 1 one – a similar figure for N = 500 would show only an exponential decay (never

entering the regime of modified polynomial decay), but the minimum amplitude of

the wavefunction would be higher. The long-range behaviour of the wavefunctions,

compared to the simple exponential localization in Kitaev’s model, is caused by the

effective long-range hopping seen in the Hamiltonian, which asymptotically converges

to zero as x−1/2. The exponential-logarithmic decay behaviour was previously found

by Pientka et al. [74] in the case of a helical Shiba chain. The helical chain addition-

ally has parameter values that support rapid exponential decay of the wavefunctions

throughout, attributed to the resonance between the Shiba oscillations and the helix;

this is obviously not applicable to the ferromagnetic chain. In experimental setups, as

a consequence, a ferromagnetic chains will likely need to be longer in order to keep the

splitting of the MBS as low as possible, which must be weighed against its advantages

in being easier to construct.

Having established the decay properties of the MBS wavefunctions, we turn to

consider the dependence of their energy splitting on the length of the chain. Fig 3.7

shows the MBS energies as well as the energy of the first excited state as a function

of chain length. From the figure it is evident that energy of the MBS is not strictly


decreasing as a function of chain length, but rather has additional oscillations on top

of the decreasing behaviour. This is a consequence of the oscillatory behaviour of the

wavefunctions seen previously in this section; depending on the length of the chain,

the edge MBS may interfere constructively or destructively, and hence may increase

or decrease the energy splitting. For longer chains this effect is minor compared to

the gap size, and as such should not present a serious problem to implementations of

the chain.

3.3 Adiabaticity of MBS braiding

Based on our results analysing the Rashba chain, we established certain parameter

areas in which the low-energy states of the four-band model derived in this thesis

agreed with the deep dilute limit model presented in Ref. [62] with respect to topology.

As the polynomial eigenvalue solver is resource intensive, the results presented here

were obtained using the simpler approximation. In addition to the fact that the

problem is linearized, another advantage with the two-band model is the ability to

solve it for a finite coherence length. This reduces the interaction between MBS and

thus the minimum system size which allows for smaller systems. As seen in Appendix

B, use of the deep-dilute approximation could potentially have significant effects on

the properties of the system, including the topological phase boundaries. In order to

avoid problems potentially caused by using the deep dilute approximation, we have

selected parameters for which two-band and four-band models are qualitatively in

agreement, with both models having the same topological phase and sporting gap

energies of the same order of magnitude. Additionally, the parameters were selected

to ensure that the topological phase would only support a single pair of MBS at each

edge, so that the time evolution corresponds to the appropriate braid.

One challenge is in finding a suitable parameter to manipulate in order to expand

and contract the non-trivial regions of the system. We elected to use the parameter α

for simplicity, as it acts locally on the magnetic sites; while the ease with which one

can modify any of these parameters experimentally is not clear, α can for example be

tweaked through letting a supercurrent flow through the system2.

Our first goal is to ascertain that the topological zones follow the parameters in

that the cross geometry does not affect the qualitative behaviour of the system. This

2See Ref. [75] for the case of a helical chain; the derivation for the ferromagnetic case can be

done similarly assuming αR and ∇φ are both small, as the supercurrent acts on the particle-hole

subspace.

3.3. ADIABATICITY OF MBS BRAIDING 43

Figure 3.8: Ground state wavefunctions of the numerical qubit for different locations

of the topological zone boundaries. Parameters used are N = 1001, kF = 20, αR =

0.03, αT = 1, αN = 1.5. The topological zone boundaries are either at the edges of

the chain or 180 steps from an edge.

is most easily seen by plotting the wave functions and energy spectra at different

stages of the time evolution and ensuring that the MBS character remains and that

the MBS are localized at the intended boundaries. To test this we have selected two

parameter sets:

Set A : N = 1001, n = 180, kF = 20, αR = 0.03, αT = 1, αN = 1.5, ξ = 30

Set B : N = 641, n = 180, kF = 20, αR = 0.01, αT = 1, αN = 2, ξ = 30(3.1)

Here N is the total amount of sites, n is the initial width of the topological zones, kF

is the Fermi wave vector, αR the SOC, and αT , αN correspond to the parameter α in

the normal and superconducting phases respectively. In both cases, the cross arms in

the x and y directions are of equal length, as there is no particular reason to make the

system asymmetric; we expect the qualitative behaviour to be mostly dependent on

the minimum distances between MBS during the evolution, so in practice the size of

asymmetric systems would likely be governed by the shorter arm. However, it may be

advantageous to have the crossing points of the arms offset from their center points,


Figure 3.9: The energies of the ground state and lowest excited state during the

braiding process. Parameters used are N = 1001, kF = 20, ς = 0.03, and α = 1 for

the topological state, α = 1.5 for the normal state. The schematic pictures above

the figure show the topological zone and MBS locations at the steps corresponding to

each discrete change of the gap energy.

as seen later for parameter set B.

To confirm that the MBS can be manipulated, in Fig. 3.8 we have plotted the

ground state wavefunction amplitudes for parameter set A for selected time steps. The

Majorana wavefunctions move with the time evolution as required for the braiding

operation (compare Fig. 2.4), remaining localized throughout; we draw the conclusion

that the topological zone boundaries follow the change in the parameter α as expected.

Our second goal is to make sure the braiding operation is possible to conduct

adiabatically. We therefore calculate the energy of both the MBS and the first excited

state for each step in the time evolution to ensure that there is always a well-defined

gap. While there are four MBS, we need only consider the one with the highest energy

to ensure the validity of the results. We will do this calculation separately for both of

the parameter sets listed earlier in this section. The time evolution of the system will

vary the parameters so that each step moves the edge of the topological zone with

3.3. ADIABATICITY OF MBS BRAIDING 45

one lattice site3.

The results for parameter set A can be seen in Fig. 3.9. At each point in the

time evolution, the ground state and first excited states are well separated. The gap

energy is seen to be strictly dependent on how many topological zones are crossing

the center at any given point, and largely unrelated to the length of these zones. For

this parameter set, it appears the magnitude of the gap is changed by a near-discrete

jump whenever a topological zone boundary (and corresponding MBS) crosses the

center, as seen in the figure. A closer look at the MBS energy reveals an oscillatory

behaviour similar to that seen in the previous section; the minimum distance between

two MBS is the leading cause of splitting here. Of some concern is the fact that the

maximum splitting is relatively large, as well as the irregular behaviour of the gap

energy seen when one of the MBS is near the center of the cross. While the gap

never closes in this case, in an experimental setup the sharp peaks connected with the

near-discontinuities in energy combined with temperature could conceivably lead to

problems, though more detailed knowledge of the exact process used is needed to make

any further conclusions. Importantly, the figure confirms our initial predictions that

the MBS splitting is primarily dependent on the minimum distance between MBS.

We conclude that the size of the system could be reduced as long as the minimum

MBS separation remains unchanged.

A similar plot for parameter set B is seen in Fig. 3.10. Based on the results for

parameter set A, we have chosen the new parameters to increase the gap energy and

simultaneously reduce system size. We have shortened the arms that are initially

topologically trivial (bottom and right arms in Fig. 2.4), so that distance from the

MBS to the crossing point is equal at the start and extremes of the time evolution.

While the system thus constructed is overall smaller, the gap energy is nevertheless

higher and the MBS splitting lower. The latter is easier to control in this case since the

minimum distance between MBS is not much lower than the maximum. The sharp

jumps at the crossing points are significantly less pronounced, and the transitions

smoother. While based on the simulation this set of parameters appears ideal for the

purpose of adiabatic braiding, it is important to notice that the parameter α = 2,

where the derivation of the two-band model requires the approximation α ≈ 1. Hence

while the numerical results using the two-band model indicate this set of parameters

is promising, the reliability of the model used is questionable. To estimate the error,

we have compared the energy spectra of the two-band and four-band models for these

3As a special case, when the second zone reaches the middle the edge moves two sites in one step

as the two zones are connected


100 200 300 400 5000

0.04

0.08

0.12

0.16Gap eneryMBS energy

Figure 3.10: The energies of the ground state and lowest excited state during the

braiding process. Parameters used are N = 641, kF = 20, ς = 0.01, and α = 1 for

the topological state, α = 2 for the normal state. The shorter arms of the cross are

70 sites long, equal to the distance from the initial MBS positions to the center of the

cross.

parameters, for a system consisting of a single chain. The numerical results indicate

that the topology of the two models agree for these parameters. For the trivial-phase

parameters used, the four-band model sports an energy gap which is approximately

75% of that of the two-band model; in the topological phase, the four-band gap is

in fact larger by a small margin, and the energy splitting of the MBS is lower in

the four-band model. While this comparison was done in the limit ξE → ∞, the

topological phases of the four-band model remain the same for a finite coherence

length, and the energy splitting of MBS generally decreases with decreasing ξ; it is

therefore reasonable to expect that the situation for finite coherence length would be

qualitatively similar.

Chapter 4

Conclusions

In this thesis we have derived the equations describing the Rashba spin-orbit-coupled

Shiba chain, obtaining analytical and numerical results describing the properties of the

system. Additionally, we have studied the possibility of using such a system as a basis

for testing the nonabelian statistics of Majorana quasiparticles. The original results

presented in this thesis, forming the basis of Ref. [11], are the analytical treatment of

the four-band ferromagnetic Shiba chain together with the numerical analysis found

in the Results chapter.

In the introductory part of the thesis, we presented some of the ideas behind

the concepts treated in this thesis. We started by introducing topological materi-

als, including the classification of symmetry-protected topological phases. We further

outlined the theory of quantum computing and nonabelian quasiparticles, which in

combination form the basis of topological quantum computing. Following this, we fo-

cused on one-dimensional topological superconductors and the Majorana bound states

that emerge in those. As an example, we considered Kitaev’s seminal p-wave super-

conductor, introducing many concepts crucial to understanding later sections. We also

presented some properties of the Majorana operators in topological superconductors.

Having presented the theoretical background we moved over to the research part

of the thesis, presenting the Rashba spin-orbit-coupled Shiba chain and obtaining

several new results. We found [11] that the system can be described by a 4N × 4N

nonlinear eigenvalue problem, which can be numerically solved in the limit of infinite

coherence length. We further found a transcendental equation for the energy of the

system in reciprocal space. As an application we introduced a prototype for a qubit

based on the chain, able to support a topological NOT-gate based on the braiding

properties of Majorana bound states introduced previously.

Finally, in the Results chapter, we presented numerical results based on the Shiba

47

48 CHAPTER 4. CONCLUSIONS

chain and the prototype qubit. In the limit of infinite coherence length, we obtained

the topological phase diagram of the system. The presence of topological phases

supporting multiple zero modes was confirmed. Additionally, we analysed the decay

behaviour of the MBS wavefunctions as well as the dependence of their energy splitting

on the length of the Shiba chain. We then moved on to consider the qubit prototype.

We confirmed that local manipulation of parameters allowed moving the topological

zone boundaries in the manner required for braiding while still keeping the Majorana

bound states separated from the bulk states by a well-defined energy gap.

We conclude that the results presented in this thesis indicate that the Rashba spin-

orbit-coupled ferromagnetic Shiba chain presents a promising candidate for verifying

the nonabelian statistics of Majorana bound states. Experimental confirmation is, of

course, the most obvious venue for future research, and the prototype qubit presented

here should be feasible to construct with current technology. Another approach would

be an extension of this work through numerical analysis of the qubit holonomy.

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Appendix A

Shiba chain with Rashba SOC

In this Appendix we will derive the relevant equations for the Shiba chain with Rashba

SOC in more detail. The results up to equation (A.10) were first obtained in Ref. [62],

although the approach used here more closely follows that in Ref. [65]. Our starting

point is the Hamiltonian density,

H =

(k2

2m− µ+ αR(kyσx − kxσy)

)τz + ∆τx − J

∑i

Sσzδ(r− ri) (A.1)

in Nambu space, Ψ = (ψk↑, ψk↓, ψ†−k↓,−ψ

†−k↑)

T .

As mentioned previously, we will focus on the ferromagnetic phase, in which all

spins point in the z direction. Consequently the Hamiltonian is in the BDI symmetry

class, as we have a time-reversal operator T = K, so the system is significantly differ-

ent from the p-wave superconductor presented earlier; in particular, the topological

invariant is Z-valued, allowing phases with multiple MBS to emerge. To continue, we

insert H into the BdG equation HΨ = EΨ, resulting in

[E − (ξk + αR(kyσx − kxσy)) τz −∆τx] Ψ(r) = −J∑i

Sσzδ(r− ri)Ψ(ri) (A.2)

where we have introduced ξk = k2/(2m)− µ. Going over to momentum space gives

[E − (ξk + αR(kyσx − kxσy)) τz −∆τx] Ψ(k) = −J∑i

Sσzδ(r−ri)e−ik·riΨ(ri). (A.3)

Inverting the Hamiltonian and transforming back to real space yields the equation

Ψ(r) = −∑i

JE(r− ri)σzΨ(ri) (A.4)

where JE is the integral

JE(r) = JS

∫dk

(2π)2eik·r [E − (ξk + αR(kyσx − kxσy)) τz −∆τx]

−1 (A.5)

53

54 APPENDIX A. SHIBA CHAIN WITH RASHBA SOC

– note that the underlying bulk SC is two-dimensional, for reasons discussed in the

main text. We are primarily interested in the wave function on the impurity sites, so

letting r = ri for some i, we find

(1− JE(0)σz)Ψ(ri) = −∑j 6=i

JE(ri − rj)σzΨ(rj). (A.6)

We move on to treat the integrals in JE. In order to facilitate this we first diagonalize

the integrand matrix in spin space,

JE(r) = JS

∫dk

(2π)2eik·r

[E −

(ξ+ 0

0 ξ−

)τz −∆τx

]−1(A.7)

where ξ± ≡ ξk ± αRk. Multiplying with the conjugate of the inverse matrix and

transforming back to the Nambu basis yields the expression

JE(r) =JS

2

∫dk

(2π)2eik·r(M+ +M−) (A.8)

where

M± =E + ξ±τz + ∆

E2 − ξ2± −∆2

(1± ky

kσx ∓

kxkσy

).

This leaves us with four types of integrals to calculate. By switching integration

variables from k to ξ± in the respective integrals, and assuming k is close to the Fermi

surface, it is sufficient to integrate

I±1 (r) =m

(2π)2N±

∫ 2π

0

dθ

∫ ∞−∞

ξeik±(ξ)r cos(θ)

E2 − ξ2 −∆2dξ

I±2 (x) =m

(2π)2N±

∫ 2π

0

dθ

∫ ∞−∞

ξeik±(ξ)r cos(θ)+iθ

E2 − ξ2 −∆2dξ

I±3 (x) =m

(2π)2N±

∫ 2π

0

dθ

∫ ∞−∞

eik±(ξ)r cos(θ)

E2 − ξ2 −∆2dξ

I±4 (x) =m

(2π)2N±

∫ 2π

0

dθ

∫ ∞−∞

dξeik±(ξ)r cos(θ)+iθ

E2 − ξ2 −∆2dξ

(A.9)

where N± = 1 ∓ ς√1+ς2

and k± = kF (√

1 + ς2 ∓ ς) + ξ/(vF√

1 + ς2). The expressions

utilize the normalized spin-orbit coupling ς = mαR/kF , which will be used henceforth.

In order to avoid divergences, we will assume E < ∆, which is a sensible limit for states

dependent on superconductivity. When r = 0, the integrals can be straightforwardly

calculated, and all but I±3 vanish due to having an odd integrand. Hence, the left-

hand side in Eq. (A.6) gives the single-impurity result also seen in Ref. [65], with no

dependence on the spin-orbit coupling. The other integrals give an answer expressible

55

with the aid of special functions, yielding (for a chain parallel to the x axis) the

equation(1− α E + τx√

∆2 − E2(S · σ)

)Ψ(xi) = −

∑i 6=j

JE(xij)(S · σ)Ψ(xj) (A.10)

with the function JE(x) being defined as

JE(x) =12JS[(I−1 (x) + I+1 (x))τz + (I−2 (x)− I+2 (x))τzσy

]+1

2JS(E + ∆τx)

[(I−3 (x) + I+3 (x)) + (I−4 (x)− I+4 (x))σy

].

(A.11)

where xij = xi − xj, and

I±1 (x) = mN±Im[J0((kF,± + iξ−1E )|x|) + iH0((kF,± + iξE)|x|)

]I±2 (x) = −imN±sgn(x)Re

[iJ1((kF,± + iξE)|x|) +H−1((kF,± + iξ−1E )|x|)

]I±3 (x) = − mN±√

∆2 − E2Re[J0((kF,± + iξE)|x|) + iH0((kF,± + iξ−1E )|x|)

]I±4 (x) = −i mN±√

∆2 − E2sgn(x)Im

[iJ1((kF,± + iξE)|x|) +H−1((kF,± + iξ−1E )|x|)

].

(A.12)

In the above, we have also introduced the energy-dependent coherence length ξE =

vF/√

∆2 − E2. The functions Jn and Hn are Bessel and Struve functions, respectively.

Diverging from the treatment in Brydon et al., instead of taking the deep-impurity

limit, we introduce the new expressions

S =N+

2

[J0((kF,+ + iξ−1E )|x|) + iH0((kF,+ + iξE)|x|)

]+N−2

[J0((kF,− + iξ−1E )|x|) + iH0((kF,− + iξE)|x|)

]A =

N+

2sgn(x)

[iJ1((kF,+ + iξE)|x|) +H−1((kF,+ + iξ−1E )|x|)

]−N−

2sgn(x)

[iJ1((kF,− + iξE)|x|) +H−1((kF,− + iξ−1E )|x|)

].

(A.13)

The real and imaginary parts of S and A are easily seen to be

ImS =1

2m(I−1 + I+1 )

ReA = − i

2m(I−2 − I+2 )

ReS = −√

∆2 − E2

2m(I−3 + I+3 )

ImA = −i√

∆2 − E2

2m(I−4 − I+4 ).

(A.14)


Using these expressions, Eq. (2.18) can now be written in the form

J = α

[τzImS −

E + ∆τx√∆2 − E2

ReS + iτzσyReA+ iE + ∆τx√∆2 − E2

σyImA

](A.15)

It is convenient at this point to switch over to the eigenstates of τx ⊗ σz, which is

the eigenbasis of the single-impurity problem. Defining |+ ↑〉 = |+〉τx ⊗ | ↑〉σz , the

transformed basis is

Ψj = (〈+ ↑ |Ψj〉〈− ↓ |Ψj〉〈+ ↓ |Ψj〉〈− ↑ |Ψj〉)T .

In this basis, Eq. (A.10) can be expressed as a matrix equation. The derivation is

straightforward using basic matrix algebra and the anticommutation properties of

Pauli matrices. The result is most conveniently expressed in terms of the matrices

aij = α(ReS(xij) + δij) bij = αReA(xij)

cij = αImA(xij) dij = αImS(xij)(A.16)

using which the problem can be stated in terms of the nonlinear eigenvalue equationaλ2 − λ bλ cλ2 −λd−bλ λ− a −λd c


Ψ = 0 (A.17)

for the eigenvalues

λ =∆ + E√∆2 − E2

. (A.18)

Recall that the matrices a, b, c, d depend on the energy through ξE. In the limit

vF >> ∆, that is, the long coherence length limit, this energy dependence vanishes,

and the equation simplifies to a polynomial eigenvalue problem for λ.

As the matrices above only depend on the difference xi − xj they are translation

invariant. It is then possible to Fourier transform the individual submatrices according

to

ak =∑j 6=0

aijeik(i−j), (A.19)

which reduces the NLEVP to 4 × 4 and allows further analytical treatment of the

system. As mentioned in the main text, in practice the FT has to be done numerically,

as the functions involved do not lend themselves to analytical treatment. Assuming

this has been done, we can explicitly take the determinant of the resultant matrix,

obtaining the equation

λ2(a4kλ2 − a2k(2b2kλ2 − 2c2kλ

2 − 2d2kλ2 + λ4 + 1)− 8akbkckdkλ

2

+c2k(2b2kλ

2 − 2d2kλ2 − λ4 − 1) + λ2(b2k + d2k − 1)2 + c4kλ

2) = 0.(A.20)

57

As λ 6= 0 for states within the gap, by dividing by λ4 and introducing the new variable

R ≡ λ2 + λ−2 we can simplify the equation to

R =(a2k + b2k + c2k + d2k − 1)2

a2k + c2k− 4

(akbk + ckdk)2

a2k + c2k+ 2, (A.21)

from which it is easy to obtain the transcendental equation for the energy by substi-

tution,

E = ±∆

√(a2k + b2k + c2k + d2k − 1)2 − 4(akbk + ckdk)2

(a2k + b2k + c2k + d2k − 1)2 − 4(akbk + ckdk)2 + 4(a2k + c2k). (A.22)

In the limit ξE →∞ this directly gives the energy dependence on k.

Appendix B

Comparison to the two-band model

One of the main original results in this thesis is a solution of the Shiba chain with

Rashba spin-orbit coupling that did not require taking the deep impurity limit to

obtain a two-band model. In this Appendix we will include a short discussion on

the impact of this approximation, which was also used in this thesis for the braiding

simulations.

The approximation to a two-band model is done starting from Eq. (2.17). In

short,the approximation is done by assuming α ≈ 1 and kF >> 1, so that the energy

of the single-impurity states is near the center of the gap and the coupling between

impurity sites is small. Then the equation can be linearized in both E and the

coupling between impurity sites, as seen in Ref. [65]. Due to the low inter-impurity

coupling, it is then assumed that the system remains approximately in the single-

impurity eigenstates, so that the projection onto τx ⊗ σz eigenstates only needs to

include these.

In practice, the Hamilton equation of the two-band model can be recovered from

Eq. (2.23), aλ2 − λ bλ cλ2 −λd−bλ λ− a −λd c


Ψ = 0, (B.1)

by linearizing the equations using the approximation kF >> 1 >> E, as long as the

determinant of the block-off-diagonal is (nearly) zero,

det

(cλ2 −λd−λd c

)= 0. (B.2)

It is perhaps most convenient to consider the system in reciprocal space. As seen

in Ref. [62], the terms in this 2 × 2 matrix are asymptotically proportional to k−1/2F .

59

60 APPENDIX B. COMPARISON TO THE TWO-BAND MODEL

-2.5

0

2.5

-0.1

0

0.1

Figure B.1: Comparison between the two-band and four-band DOS. The upper half of

the figure corresponds to the two-band model, the lower to the four-band reflected over

the x-axis for ease of comparison. Energy bin width 10−4. a) Full energy spectrum

used. Parameters used are, for both models, N = 300, kF = 20, ς = 0.01, and α = 1.

b) Energy spectrum comparison closeup for parameters N = 500, kF = 15.5, ς = 0.01,

and α = 1.15. The energies on the y axis have been limited to the interval [−0.1, 0.1]

to highlight the desired features. The two-band model is in the N = 2 phase whereas

the four-band model is trivial.

In general, then, the two models converge as k−1/2F for low enough energies E. This

convergence is much slower than the linear one treated in Ref. [69], and consequently

the distinction between the four-band and two-band models is significantly more im-

portant. To establish this, we will make some numerical comparisons between the

models. Fig. (B.1) shows a comparison between the energy spectra of the two models

for two different parameter sets. The most noticeable feature is the fact that the two-

band model supports energies higher than ∆; these are unreliable, as the integrals in

Eq. (A.9) have been calculated under the assumption that E < ∆ (for convergence).

For the purpose of distinguishing topological phases, however, it is the low-energy

states that are more important. For the parameters used in Fig. B.1 a), while the

two-band model has a lower gap energy, both models are in the topological phase

N = 1 and the spectra are reasonably similar for low enough energies. In general,

however, the topological phase transitions do not agree between the two models. In

Fig. B.1 b) we see that for certain parameter sets, even the lower energies can be

strikingly dissimilar; here, the two-band model supports two MBS pairs where the

four-band model is a topologically trivial insulator. In Fig. B.2, we present the topo-

logical phase diagram for the N = 1 phase of the two models, which we have obtained

by using Eq. (2.30) - note that the N = 2 phase is not included here. Though there is

61

Figure B.2: Topological phase diagrams of the two models. The red corresponds to

the N = 1 phase of the four-band model (compare Fig. 3.1), and the blue to the same

phase of the two-band model. Overlapping regions are purple. Parameters used are

ς = 0.01, ξ0 =∞

significant overlap, the difference between the two models is quite nevertheless clearly

visible, even when α = 1, and persists to high values of kF due to the slow conver-

gence. It is clear that the two-band model is not always reliable even qualitatively;

results obtained using it must be confirmed by comparison to the four-band model.

62 APPENDIX B. COMPARISON TO THE TWO-BAND MODEL

Appendix C

Derivation of the winding number

In this Appendix we will derive Eq. (2.33) from the main text, expressing the topo-

logical invariant N in terms of the positive-energy eigenstate |E+(k)〉.Our starting point is the integral formula for the topological invariant1,

N =1

4πi

∫ π

−πdk tr

[CH−1∂kH

]. (C.1)

where C = C† = C−1 is the chiral symmetry operator. To avoid the need for H we will

express it in terms of the projection operators

H =∑i

Ei|Ei〉〈Ei| =∑i

EiPi = E+P+ + E−P−. (C.2)

In principle, solution of the nonlinear eigenvalue problem yields two degenerate states

for each energy, one of which is unphysical. However, the two eigenstates are simply

related by a unitary rotation, meaning the resultant projectors are identical. We can

hence safely disregard these states as their effect on the topological invariant is trivial.

Further, because we are only interested in the topological properties of the system,

the precise details of the energy bands are irrelevant and we can flatten them with no

loss of generality to get H = E0(P+− P−), where E0 is an arbitrary positive number.

Upon insertion into the expression for N the energies cancel, giving

N =1

4πi

∫ π

−πdk tr

[C(P+(k)− P−(k))∂k(P+(k)− P−(k))

]. (C.3)

Writing out the product, using the relation C|E+(k)〉 = |E−(k)〉 as well as the cyclic

properties of the trace operation, we can write this in terms of just one projection

1The prefactor is in fact arbitrary here, and chosen to give an integer; conventionally the factor

is 1/8πi, but we have changed this in order to account for the number of states used as discussed in

this Appendix.

63

64 APPENDIX C. DERIVATION OF THE WINDING NUMBER

operator

N =1

2πi

∫ π/a

−π/adk tr

[CP+∂kP+ − P+C∂kP+

]. (C.4)

Integration by parts finally yields the desired result,

N =1

πi

∫ π/a

−π/adk tr

[CP+∂kP+

]. (C.5)

To find the projection operator it is sufficient to find the eigenvector corresponding

to positive energy. This is done by solvingaλ2 − λ bλ cλ2 −λd−bλ λ− a −λd c


x1

x2

x3

x4

= 0 (C.6)

for the vector components. Solving the components as a function of x1, we obtain (up

to a normalization factor)

x1 = −λ[c2 − d2 +

d2λ2 − (ad− bc)2

λ2 − (a2 + c2)

]x3 = (ac− bd)λ− c+

λ(ab+ cd+ bλ)(bc− ad+ dλ)

λ2 − (a2 + c2)

x2 =λ(ab+ cd+ bλ)

λ2 − (a2 + c2)x1 +

λ(ad− bc+ dλ)

λ2 − (a2 + c2)x3

x4 =1

c[bλx1 + (a− λ)x2 + dλx3] .

(C.7)

The eigenvector corresponding to positive energy can then be found by inserting the

appropriate solution of Eq. (2.29) into the expressions for the components.

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