Majorana States in Ferromagnetic Shiba Chains

Master’s Thesis Theoretical Physics

Majorana states in ferromagnetic Shiba chains

Kim P¨oyh¨onen 2015

Supervisor: Dr. Teemu Ojanen Examiners: Dr. Teemu Ojanen Prof. Kai Nordlund

HELSINKI UNIVERSITY DEPARTMENT OF PHYSICS

P.O. Box 64 (Gustaf H¨allstr¨omin katu 2) 00014 University of Helsinki HELSINGIN YLIOPISTO – HELSINGFORS UNIVERSITET – UNIVERSITY OF HELSINKI Tiedekunta/Osasto – Fakultet/Sektion – Faculty/Section Laitos – Institution – Department Faculty of Science Department of Physics Tekijä – Författare – Author Kim Pöyhönen Työn nimi – Arbetets titel – Title Majorana states in ferromagnetic Shiba chains Oppiaine – Läroämne – Subject Theoretical Physics Työn laji – Arbetets art – Level Aika – Datum – Month and year Sivumäärä – Sidoantal – Number of pages M. Sc. Thesis August 2011 64 Tiivistelmä – Referat – Abstract 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.

Avainsanat – Nyckelord – Keywords Topological superconductivity, Rashba effect, Topological phases, Majorana, Braiding Säilytyspaikka – Förvaringställe – Where deposited

Muita tietoja – Övriga uppgifter – Additional information

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 Möbiusstrip. 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 Möbius 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 Möbiusstrip 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 properties 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, particle 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 topological 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 experiments, 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 4 CHAPTER 1. INTRODUCTION thesis is as follows. The first chapter is the introduction. It contains a short introduction 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 integer 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 superconductors 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 quantum 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 systems. 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 properties. 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 HT T −1 = H, TT † = 1 T T = ±T PHS : PHT P−1 = −H, PP† = 1 PT = ±P (1.1)

−1 † 2 SLS : CSHCS = −H,CSCS = 1 CS = 1 6 CHAPTER 1. INTRODUCTION

Class TRS PHS SLS d = 1 d = 2 A (unitary) 0 0 0 - Z AI (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 Z C 0 –1 0 - Z DIII –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 2 in this case is the integer in the conductance, σH = ne /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 8 CHAPTER 1. INTRODUCTION 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 documented 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, several 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 introduce 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 |0i and |1i, analogously to a classical cubit. However, the actual state of a qubit can be a linear complex superposition of these:

|ψi = a0|0i + a1|1i (1.2)

2 2 where |a0| + |a1| = 1. For this superposition, as typical in quantum mechanics, the 2 probability of measuring a |0i is |a0| . 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 1 |ψABi = √ (|0Ai|0Bi + |1Ai|1Bi). (1.3) 2

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

N 1 Y X |ψi = ai1,i2,...,iN |i1, i2, ..., iN i (1.4) n=1 in=0 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 N qubits, |{i}i = |i1i ⊗ |i2i ⊗ ... ⊗ |iN i. The resultant Hilbert space is 2 -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. 10 CHAPTER 1. INTRODUCTION

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 ! ! 1 0 |0i = , |1i = (1.5) 0 1

- 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 |0i ↔ |1i: !" ! !# ! ! 0 1 1 0 0 1 a + b = a + b (1.6) 1 0 0 1 1 0

This is known as the σx-gate, and corresponds to the classical NOT-gate. Another important gate is the Hadamard gate ! 1 1 1 H = √ (1.7) 2 1 −1 which transforms a qubit from a set state to a superposition, for example H|0i = √ (|0i + |1i)/ 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   1 0 0 0   0 1 0 0 CNOT =   . (1.8) 0 0 0 1   0 0 1 0

The CNOT gate ﬂips the state of the qubit B assuming qubit A is in the state |1i, otherwise it does nothing. In general, n-qubit controlled gates are of the form ! 1 0 CG = 2×2 2×2 (1.9) 02×2 U 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- 2 ators with the property Pi = 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 {|ψii} is an ortonormal basis in the Hilbert space, the density matrix of some arbitrary system in that space can be written

X ρ = pihψi||ψii. (1.10) i

The density matrix above implies that the state |ψii occurs with the classical proba- ˆ bility pi. Then the expectation value of an operator O with respect to the system in question is ˆ X ˆ hOi = pihψi|O|ψii. (1.11) i Measuring the value of the operator Oˆ will give the value appropriate to the state 2 |ψii with the probability pi - not |pi| . In the simple case of a single-qubit system, the density matrix can be written

ρ = a|0ih0| + b|1ih1| (1.12) where a + b = 1. Unlike a superposition, the qubit really is in a deﬁnite state, either h0| or h1|, 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. 12 CHAPTER 1. INTRODUCTION

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

1 |ψABi = √ (|0A1Bi − |1A0Bi) (1.13) 2

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 = |ψABihψAB|,

1 X 1 ρ = hi |ρ |i i = (|0 ih0 | + |1 ih1 |) (1.14) A B AB B 2 A A A A i=0 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 e−βH ρ = , (1.15) tr(e−βH ) 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:

Pˆ2|12i = |12i → Pˆ|12i = ±|21i (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 Pˆ2 = 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] 14 CHAPTER 1. INTRODUCTION

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 nontrivial 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 Schrödingerequation of the system is 2 − ~ ∇2 + V (x, y) + V (z) Ψ(r) = EΨ(r) (1.17) 2m z 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, 1.3. TOPOLOGY AND BRAIDING 15 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 1 this concept we consider the simple example of a spin- 2 particle in a uniform magnetic ﬁeld. The Hamiltonian for this system is

H = −σ · B = −Bσ · nˆ, (1.19) where ˆn = (sin θ cos φ, sin θ sin φ, 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 H = −Bσ and 0 z ! cos θ e−iφ sin θ U = 2 2 . (1.21) iφ θ θ e sin 2 cos 2

The eigenstates of H0 are just | ↑i and | ↓i, so we denote the eigenstates of H by U(θ, φ)| ↑i ≡ | ↑ (θ, φ)i and U(θ, φ)| ↓i ≡ | ↓ (θ, φ)i. We now study the properties of this system under time evolution. Assume the particle is initially in the eigenstate | ↑ (θ0, φ0)i. If the magnetic field is allowed to change orientation adiabatically, the system will at all times be proportional to the † instantaneous eigenstate | ↑ (θt, φt)i 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.

−i R T E dt iγ T |ψ(T )i = e 0 ↑ e ↑ | ↑i. (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 Schrödingerequation, giving I γ↑(C) = i h↑ (θt, φt)|∇R(t)| ↑ (θt, φt)idR(t) (1.24) C 16 CHAPTER 1. INTRODUCTION 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 I eiϕ = eiEnT exp( A · dr) (1.25) where we have introduced the Berry connection ∂ A = hn|U †({λ}) U({λ})|ni. (1.26) µ ∂λµ 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 | ↑i and | ↓i are degenerate. In general, nontrivial 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 )i = 0.. However, due to the degeneracy it is now possible that hψ(0)|ψ(T )i = 0. Consider the time evolution operator resulting from adiabatically tracing the loop C,

h −i R T U(t)H U †(t)dti U(0,T ) = T e 0 0 , (1.28) where T denotes time ordering. By discretising time and demanding adiabatic evolution, |ψ(t)i = 0 for all t, one can show that [49] I |ψ(T )i = U(0,T )|ψ(0)i = P exp( A · dλ) |ψ(0)i (1.29) C 1.3. TOPOLOGY AND BRAIDING 17 where P denotes path ordering and the connection A is

∂U(λ) (A )αβ = hψα|U(λ)† |ψβi. (1.30) µ ∂λµ The time-evolution operator of the loop in path ordering formalism is called the holonomy ΓA, so that

|ψ(C)i = ΓA(C)|ψ(0)i. (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. 18 CHAPTER 1. INTRODUCTION 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, although the term Majorana fermion is also often seen used interchangeably. However, MBS do not adhere to Fermi-Dirac statistics; rather, they are nonabelian anyons, supporting 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

X † † † † † ∗ H = t(ânaˆn+1 +a ˆn+1aˆn) − µaˆnaˆn + ∆ânaˆn+1 + ∆ aˆn+1aˆn) (2.1) n wherea ˆ 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 1 X H = Ψ† H Ψ (2.2) 2 k k k k T † where Ψk ≡ aˆk aˆ−k and

Hk = (−2t cos(ka) + µ)σz + 2∆ sin(ka)σy (2.3) from which it is easy to ﬁnd the dispersion relation, q 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 eigenstate 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 =a ˆn +a ˆn γ2n = −iaˆn + iaˆn, (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

i X H = −µγ γ + (∆ + t)γ γ + (∆ − t)γ γ (2.6) 2 2n−1 2n 2n 2n+1 2n−1 2n+2 n 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 L L X 1 X H = −µ aˆ† aˆ = γ γ (2.7) n n 2i 2n−1 2n n=1 n=1 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

L−1 X † † † † X H = t(ânaˆn+1 +a ˆn+1aˆn +a ˆnaˆn+1 +a ˆn+1aˆn) = i tγ2nγ2n+1 (2.8) n n=1 We notice an important difference from the former case: now the Majorana operators 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 22 CHAPTER 2. TOPOLOGICAL CHAIN WITH SPIN-ORBIT COUPLING still, however, be combined into a delocalized fermionic operator: 1 aˆ = (γ + iγ ) (2.9) F 2 2L 1 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

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 1 1 |t| < 2 |µ| the chain is in the trivial phase and the system is gapped. At |t| = 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 ˆn +a ˆn † and γ2n = i(ˆan − aˆn). From this it is evident that

† † γ2n−1 = γ2n−1 and γ2n = 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 24 CHAPTER 2. TOPOLOGICAL CHAIN WITH SPIN-ORBIT COUPLING follow FD statistics:

{γi, γj} = 2δij (2.11)

2 It follows that, in particular, γ2n−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: ˆ ˆ† γ1 → B12γ1B 12 (2.12) ˆ ˆ† γ2 → B12γ2B12 ˆ where B12 is the braiding operator. Since we have changed the places of two identical ˆ ˆ† ˆ ˆ† particles we must have B12γ1B12 = αγ2 and γ2 → B12γ2B12 = 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 ˆ 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 ˆ ˆ† B12γ1B = −γ2 12 (2.14) ˆ ˆ† B12γ2B12 = γ1 √ ˆ 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 ˆ ˆ ˆ ˆ Bi,i+1Bj,j+1 = Bj,j+1Bi,i+1 |i − j| ≥ 2 ˆ ˆ ˆ ˆ ˆ ˆ Bi,i+1Bi+1,i+2Bi,i+1 = Bi+1,i+2Bi,i+1Bi+1,i+2 1 ≤ i ≤ N − 2 (2.15) ˆ ˆ−1 ˆ−1 ˆ Bi,i+1Bi,i+1 = Bi,i+1Bi,i+1 = e, ˆ ˆ−1 where Bi,i+1 corresponds to braiding the Majorana states i and i + 1, Bi,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 supports 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 superconducting qubits [58, 59]. 26 CHAPTER 2. TOPOLOGICAL CHAIN WITH SPIN-ORBIT COUPLING

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 systems 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 ferromagnetic 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 behaviour 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 2.2. NANOCHAIN WITH RASHBA SPIN-ORBIT COUPLING 27 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,

k2 X H = − µ + α (k σ − k σ ) τ + ∆τ − J (S · σ )δ(r − r ). (2.16) 2m R y x x y z x i i i The ﬁrst two terms in the Hamiltonian are properties of the bulk SC, whereas the sum 2 is over the impurity atoms. Here k /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. † † T The density is expressed in Nambu space, so that Ψ = (ψk↑, ψk↓, ψ−k↓, −ψ−k↑) , 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 = σzˆez for all i, where

êz 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 superconductor 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 ∝ iêx. With these restrictions we arrive at the equation E + τx X 1 − α√ (Sˆ · σ) Ψ(xi) = − JE(xij)(Sˆ · σ)Ψ(xj) (2.17) 2 2 ∆ − E i6=j where

1 − + − + JE(x) = JS (I (x) + I (x))τz + (I (x) − I (x))τzσy 2 1 1 2 2 (2.18) 1 − + − + + 2 JS(E + ∆τx) (I3 (x) + I3 (x)) + (I4 (x) − I4 (x))σy . 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. 28 CHAPTER 2. TOPOLOGICAL CHAIN WITH SPIN-ORBIT COUPLING above equation, we have introduced the functions

ν −1 I1 (x) = mNνIm J0((kF,ν + iξE )|x|) + iH0((kF,ν + iξE)|x|) ν −1 I2 (x) = −i mNνsgn(x)Re iJ1((kF,ν + iξE)|x|) + H−1((kF,ν + iξE )|x|)

ν mNν −1 I (x) = −√ Re J0((kF,ν + iξE)|x|) + iH0((kF,ν + iξ )|x|) 3 ∆2 − E2 E ν mNν −1 I (x) = −i√ sgn(x)Im iJ1((kF,ν + iξE)|x|) + H−1((kF,ν + iξ )|x|) , 4 ∆2 − E2 E (2.19) √ 2 2 where ξE = vF / ∆ − E is the energy-dependent coherence length, and

ς N± = 1 ∓ √ 1 + ς2 m α = (N + N )JS = 1 mJS 4 + − 2 (2.20) p 2 kF,± = kF 1 + ς ∓ ς

ς = mαR/kF

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 −1 linearize Eq. (2.17) in E and kF . 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, ! ∆(1 − α)δ + f(x ) g(x ) Heff = ij ij ij (2.21) ij ∗ g(xji) −∆(1 − α)δij − f(xij) where

1 2 + − f(x) = 2 JS∆ lim(I3 (x) + I3 (x)) E→0 (2.22) i + − g(x) = JS∆ lim(I2 (x) − I2 (x)) 2 E→0

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

6 Recall that we normalized the distance between lattice sites; the approximation made is kF a 1 2.2. NANOCHAIN WITH RASHBA SPIN-ORBIT COUPLING 29 approximation and as such not necessarily valid, especially when the slow, asymptot- − 1 ically x 2 , 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 previously 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 is   aλ2 − λ bλ cλ2 −λd    −bλ λ − a −λd c    Ψ = 0 (2.23)  −cλ2 −λd aλ2 + λ −bλ    −λd −c bλ −λ − a

Here we have introduced the parameter ∆ + E λ = √ . (2.24) ∆2 − E2 as well as the submatrices √ ∆2 − E2 a = αδ − α I−(x ) + I+(x ) ij ij 2m 3 ij 3 ij i − + bij = −α I2 (xij) − I2 (xij) 2√m (2.25) i ∆2 − E2 c = −α I−(x ) − I+(x ) ij 2m 4 ij 4 ij 1 d = α I−(x ) + I+(x ) . ij 2m 1 ij 1 ij 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 30 CHAPTER 2. TOPOLOGICAL CHAIN WITH SPIN-ORBIT COUPLING

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 7 these parameters, the coherence length for zero-energy states 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

X ik(i−j) ak = aije (2.26) j6=0 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:

2 4 2 2 2 2 2 2 2 2 4 2 λ (a λ − a (2b λ − 2c λ − 2d λ + λ + 1) − 8akbkckdkλ k k k k k (2.27) 2 2 2 2 2 4 2 2 2 2 4 2 +ck(2bkλ − 2dkλ − λ − 1) + λ (bk + dk − 1) + ckλ ) = 0. 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,

2 2 2 2 2 2 (ak + bk + ck + dk − 1) (akbk + ckdk) R = 2 2 − 4 2 2 + 2, (2.28) ak + ck ak + ck and using Eq. (2.24) this is easily expressed as s 2 2 2 2 2 2 (ak + bk + ck + dk − 1) − 4(akbk + ckdk) E = ±∆ 2 2 2 2 2 2 2 2 . (2.29) (ak + bk + ck + dk − 1) − 4(akbk + ckdk) + 4(ak + ck)

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 2.2. NANOCHAIN WITH RASHBA SPIN-ORBIT COUPLING 31 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 = q . (2.30) + − 2 + − 2 ∆(I3 (k) + I3 (k)) + 1 + (I1 (k) + I1 (k)) k=0,π

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 Z π 1 −1 N = dk tr CH ∂kH (2.31) 4πi −π 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,

X X ˆ H = Ei|Ei(k)ihEi(k)| = EiPi(k). (2.32) i i 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

Z π 1 h ˆ ˆ i N = dk tr CP+(k)∂kP+(k) , (2.33) πi −π ˆ where P+ is the projector to the positive-energy eigenstate. 32 CHAPTER 2. TOPOLOGICAL CHAIN WITH SPIN-ORBIT COUPLING

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, experimental 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 number, 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

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.

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. 34 CHAPTER 2. TOPOLOGICAL CHAIN WITH SPIN-ORBIT COUPLING logical choice of qubits is letting one state correspond to |0, 0i and the other to |1, 1i 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, 0i, |1, 1i). 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, 0i ↔ |1, 1i, 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 topological 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 analytically 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

Figure 3.1: Topological phases for the four-band model. a) Diagram plotting the ratio 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 system 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 3.1. NANOCHAIN WITH RASHBA SPIN-ORBIT COUPLING 37

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 38 CHAPTER 3. RESULTS

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 parameters 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 density 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 3.1. NANOCHAIN WITH RASHBA SPIN-ORBIT COUPLING 39

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∆.

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. 40 CHAPTER 3. RESULTS

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 −a −b amplitude is significantly slower, following a decay law of |ψ| ∝ x ln(x/x0) . 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 additionally 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 42 CHAPTER 3. RESULTS 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 (3.1) Set B : N = 641, n = 180, kF = 20, αR = 0.01, αT = 1, αN = 2, ξ = 30

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, 44 CHAPTER 3. RESULTS

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 46 CHAPTER 3. RESULTS

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 materials, 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 superconductor, 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. Bibliography

[1] A. Das, Y. Ronen, Y. Most, Y. Oreg, M. Heiblum and H. Shtrikman, Nature Physics 8, 887–895 (2012).

[2] V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M. Bakkers and L. P. Kouwenhoven, Science 336, 1003–1007 (2012).

[3] S. Nadj-Perge, I. K. Drozdov, J. Li, H. Chen, S. Jeon, J. Seo, A. H. MacDonald, B. A. Bernevig and A. Yazdani, Science 346, 602 (2014).

[4] Jin-Peng Xu et al, Phys. Rev. Lett. 114, 017001 (2015).

[5] E. J. H. Lee, X. Jiang, R. Aguado, G. Katsaros, C. M. Lieber, and S. De Franceschi, Phys. Rev. Lett. 109, 186802 (2012).

[6] H. O. H. Churchill, V. Fatemi, K. Grove-Rasmussen, M. T. Deng, P. Caroﬀ, H. Q. Xu, C. M. Marcus, Phys. Rev. B 87, 241401 (2013).

[7] D. I. Pikulin, J. P. Dahlhaus, M. Wimmer, H. Schomerus and C. W. J. Beenakker, New J. Phys. 14, 125011 (2012).

[8] J. Liu, A. C. Potter, K. T. Law, and P. A. Lee, Phys. Rev. Lett. 109, 267002 (2012).

[9] C.-H. Lin, J. D. Sau, and S. Das Sarma, Phys. Rev. B 86, 224511 (2012).

[10] G. Kells, D. Meidan, and P. W. Brouwer Phys. Rev. B 86, 100503 (2012).

[11] K. Pöyhönen,A. Westströmand T. Ojanen, in preparation.

[12] Jiannis K. Pachos, 2012: Introduction to Topological Quantum Computation. Cambridge University Press.

[13] T. Ando, Y. Matsumoto and Y. Uemura, J. Phys. Soc. Jpn. 39, pp. 279-288 (1975).

49 50 BIBLIOGRAPHY

[14] K. v. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 (1980)

[15] R. B. Laughlin, Phys. Rev. B 23, 5632 (1981).

[16] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett., 49, 405 (1982).

[17] K. v. Klitzing, Phil. Trans. R. Soc. A 363, 2203 (2005).

[18] D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev. Lett. 48, 1559 (1982).

[19] R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983).

[20] D. Arovas, J. R. Schrieﬀer, and F. Wilczek Phys. Rev. Lett. 53, 722 (1984).

[21] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005).

[22] A. Y. Kitaev, Annals Phys. 303, 2-30 (2003)

[23] J. Alicea, Y. Oreg, G. Refael, F. von Oppen, and M. P. A. Fisher, Nature Physics 7, 412-417 (2011).

[24] E. Dennis, A. Y. Kitaev, A. Landahl and J. Preskill, J. Math. Phys. 43, 4452-4505 (2002).

[25] J. Li, T. Neupert, B. A. Bernevig and A. Yazdani, arXiv:1404.4058

[26] X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, Phys. Rev. B 83, 205101 (2011).

[27] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys.82, 3045 (2010).

[28] X. G. Wen, Advances in Physics 44, 405 (1995).

[29] A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, Phys. Rev. B 78, 195125 (2008)

[30] M. R. Zirnbauer, J. Math. Phys. 37, 4986 (1996).

[31] A. Altland and M. R. Zirnbauer, Phys. Rev. B 55, 1142 (1997).

[32] A. Y. Kitaev, AIP Conf. Proc. 1134 22 (2009)

[33] M. K¨onig,S. Wiedmann, C. Br¨une,A. Roth, H. Buhmann, L. W. Molenkamp, X.-L. Qi, S.-C. Zhang, Science 318, 766 (2007). BIBLIOGRAPHY 51

[34] Roth A., C. Br¨une,H. Buhmann, L. W. Molenkamp, J. Maciejko, X.-L. Qi, and S.-C. Zhang, Science 325, 294 (2009).

[35] D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava and M. Z. Hasan, Nature 452, 970-974 (2008)

[36] J. K. Pachos and S. H. Simon, New J. Phys. 16, 065003 (2014).

[37] A. R. Mellnik et al., Nature 511, 449 (2014).

[38] P. W. Shor, SIAM J. Sci. Statist. Comput. 26, 1484, (1997).

[39] L. K. Grover, arXiv:quant-ph/0109116

[40] W. Roga, M. Fannes and K. Zyczkowski, Phys. Rev. Lett. 105, 040505 (2010).

[41] R. Raussendorf, D.E. Browne and H.J. Briegel, Phys. Rev. A 68, 022312 (2003).

[42] D.P. DiVincenzo, Science 270, 255 (1995)

[43] A.W. Cross, D.P. DiVincenzo, and B.M. Terhal, Quant. Inf. Comp. 9, 541 (2009).

[44] E. Knill, R. Laﬂamme, and W.H. Zurek, Proc. Roy. Soc. Lond. A 454, 365 (1998).

[45] A.Y. Kitaev, Russ. Math. Surv. 52, 1191 (1997). (Ref. Pachos)

[46] J. M. Leinaas and J. Myrheim, Il Nuovo Cimento B 37, 1 (1977).

[47] M. A. Cazalilla, R. Citro, T. Giamarchi, E. Orignac, and M. Rigol, Rev. Mod. Phys. 83, 1405 (2011).

[48] M. V. Berry, Proc. Roy. Soc., Ser. A 392, 45 (1984)

[49] F. Wilczek and A. Zee, Phys. Rev. Lett. 52, 2111 (1984).

[50] E. Majorana, Nuovo Cimento 5, 171 (1937) (Ref. Wilczek, 2009).

[51] F. Wilczek, Nature Physics 5, 614 (2009).

[52] A. Y. Kitaev, Phys. Usp. 44, 131 (2001).

[53] C.W.J. Beenakker, Annu. Rev. Con. Mat. Phys. 4, 113 (2013)

[54] M. Lejinse, K. Flensberg, arXiv:1206.1736v2

[55] D. A. Ivanov, Phys. Rev. Lett. 86, 268 (2001). 52 BIBLIOGRAPHY

[56] F. Hassler, A. R. Akhmerov, C.-Y. Hou, and C. W. J. Beenakker. New Journal of Physics, 12:125002, (2010).

[57] L. Jiang, C. L. Kane, and J. Preskill. Phys. Rev. Lett. 106, 130504, (2011).

[58] D. Rainis and D. Loss, Phys. Rev. B 85, 174533, (2012).

[59] J. M¨annikand J. E. Lukens. Phys. Rev. Lett. 92, 057004 (2004).

[60] E. Dumitrescu, B. Roberts, S. Tewari, J. Sau and S. Das Sarma, Phys. Rev. B 91, 094505 (2015)

[61] Y. Peng, F. Pientka, L. I. Glazman and F. von Oppen, Phys. Rev. Lett. 114, 106801 (2015).

[62] P. M. R. Brydon, S. Das Sarma, H.-Y. Hui and J. Sau, Phys. Rev. B 91, 064505 (2015)

[63] A. Heimes, D. Mendler and P. Kotetes, New J. Phys. 17 023051 (2015)

[64] J. R¨ontynen and T. Ojanen, Phys. Rev. Lett. 114, 236803 (2015).

[65] F. Pientka, L. Glazman and F. von Oppen, Phys. Rev. B 88, 155420 (2013).

[66] L. Yu, Acta Phys. Sin. 21, 75 (1965).

[67] H. Shiba, Prog. Theor. Phys. 40, 435 (1968).

[68] A. I. Rusinov, JETP Lett. 9, 85 (1969).

[69] A. Westström,K. Pöyhönenand T. Ojanen, Phys. Rev. B 91, 064502 (2015)

[70] Y. A. Bychkov and E. I. Rashba, JETP Lett. 39 78–81 (1984).

[71] C. Kittel, 1986: Introduction to Solid State Physics, 6th edition. New York: John Wiley & Sons, Inc.

[72] M. F. Crommie, C. P. Lutz, and D. M. Eigler, Science 262, 218 (1993).

[73] N. Nilius, T. M. Wallis, and W. Ho, Science 297, 1853 (2002).

[74] F. Pientka, L. I. Glazman, and F. von Oppen, Phys. Rev. B 89, 180505(R) (2014)

[75] J. R¨ontynen and T. Ojanen, Phys. Rev. B 90, 180503 (2014). 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, k2 X H = − µ + α (k σ − k σ ) τ + ∆τ − J Sσ δ(r − r ) (A.1) 2m R y x x y z x z i i

† † T in Nambu space, Ψ = (ψk↑, ψk↓, ψ−k↓, −ψ−k↑) . 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 X [E − (ξk + αR(kyσx − kxσy)) τz − ∆τx] Ψ(r) = −J Sσzδ(r − ri)Ψ(ri) (A.2) i

2 where we have introduced ξk = k /(2m) − µ. Going over to momentum space gives

X −ik·ri [E − (ξk + αR(kyσx − kxσy)) τz − ∆τx] Ψ(k) = −J Sσzδ(r−ri)e Ψ(ri). (A.3) i Inverting the Hamiltonian and transforming back to real space yields the equation X Ψ(r) = − JE(r − ri)σzΨ(ri) (A.4) i where JE is the integral Z dk J (r) = JS eik·r [E − (ξ + α (k σ − k σ )) τ − ∆τ ]−1 (A.5) E (2π)2 k R y x x y z x

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

X (1 − JE(0)σz)Ψ(ri) = − JE(ri − rj)σzΨ(rj). (A.6) j6=i

We move on to treat the integrals in JE. In order to facilitate this we ﬁrst diagonalize the integrand matrix in spin space,

Z " ! #−1 dk ik·r ξ+ 0 JE(r) = JS 2 e E − τz − ∆τx (A.7) (2π) 0 ξ− where ξ± ≡ ξk ± αRk. Multiplying with the conjugate of the inverse matrix and transforming back to the Nambu basis yields the expression JS Z dk J (r) = eik·r(M + M ) (A.8) E 2 (2π)2 + − where E + ξ±τz + ∆ ky kx M± = 2 2 2 1 ± σx ∓ σy . E − ξ± − ∆ k k 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

Z 2π Z ∞ ik±(ξ)r cos(θ) ± m ξe I1 (r) = 2 N± dθ 2 2 2 dξ (2π) 0 −∞ E − ξ − ∆ m Z 2π Z ∞ ξeik±(ξ)r cos(θ)+iθ I±(x) = N dθ dξ 2 (2π)2 ± E2 − ξ2 − ∆2 0 −∞ (A.9) Z 2π Z ∞ ik±(ξ)r cos(θ) ± m e I3 (x) = 2 N± dθ 2 2 2 dξ (2π) 0 −∞ E − ξ − ∆ Z 2π Z ∞ ik±(ξ)r cos(θ)+iθ ± m e I4 (x) = 2 N± dθ dξ 2 2 2 dξ (2π) 0 −∞ E − ξ − ∆ √ √ where N = 1 ∓ √ ς and k = k ( 1 + ς2 ∓ ς) + ξ/(v 1 + ς2). The expressions ± 1+ς2 ± F F 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 I3 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 E + τx X 1 − α√ (Sˆ · σ) Ψ(xi) = − JE(xij)(Sˆ · σ)Ψ(xj) (A.10) 2 2 ∆ − E i6=j with the function JE(x) being deﬁned as

1 − + − + JE(x) = JS (I (x) + I (x))τz + (I (x) − I (x))τzσy 2 1 1 2 2 (A.11) 1 − + − + + 2 JS(E + ∆τx) (I3 (x) + I3 (x)) + (I4 (x) − I4 (x))σy . where xij = xi − xj, and

± −1 I1 (x) = mN±Im J0((kF,± + iξE )|x|) + iH0((kF,± + iξE)|x|) ± −1 I2 (x) = −i mN±sgn(x)Re iJ1((kF,± + iξE)|x|) + H−1((kF,± + iξE )|x|)

± mN± −1 I (x) = −√ Re J0((kF,± + iξE)|x|) + iH0((kF,± + iξ )|x|) 3 ∆2 − E2 E ± mN± −1 I (x) = −i√ sgn(x)Im iJ1((kF,± + iξE)|x|) + H−1((kF,± + iξ )|x|) . 4 ∆2 − E2 E (A.12)

In the above, we have also introduced the energy-dependent coherence length ξ = √ E 2 2 vF / ∆ − E . 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

N S = + J ((k + iξ−1)|x|) + iH ((k + iξ )|x|) 2 0 F,+ E 0 F,+ E N− −1 + J0((kF,− + iξE )|x|) + iH0((kF,− + iξE)|x|) 2 (A.13) N A = + sgn(x) iJ ((k + iξ )|x|) + H ((k + iξ−1)|x|) 2 1 F,+ E −1 F,+ E N − − sgn(x) iJ ((k + iξ )|x|) + H ((k + iξ−1)|x|) . 2 1 F,− E −1 F,− E The real and imaginary parts of S and A are easily seen to be 1 ImS = (I− + I+) 2m 1 1 i − + ReA = − (I2 − I2 ) √2m 2 2 (A.14) ∆ − E − + ReS = − (I3 + I3 ) √ 2m i ∆2 − E2 ImA = − (I− − I+). 2m 4 4 56 APPENDIX A. SHIBA CHAIN WITH RASHBA SOC

Using these expressions, Eq. (2.18) can now be written in the form E + ∆τx E + ∆τx J = α τzImS − √ ReS + iτzσyReA + i√ σyImA (A.15) ∆2 − E2 ∆2 − E2

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 |+ ↑i = |+iτx ⊗ | ↑iσz , the transformed basis is

T Ψj = (h+ ↑ |Ψjih− ↓ |Ψjih+ ↓ |Ψjih− ↑ |Ψji) .

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

a = α(ReS(x ) + δ ) b = αReA(x ) ij ij ij ij ij (A.16) cij = αImA(xij) dij = αImS(xij) using which the problem can be stated in terms of the nonlinear eigenvalue equation   aλ2 − λ bλ cλ2 −λd    −bλ λ − a −λd c    Ψ = 0 (A.17)  −cλ2 −λd aλ2 + λ −bλ    −λd −c bλ −λ − a for the eigenvalues ∆ + E λ = √ . (A.18) ∆2 − E2

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 X ik(i−j) ak = aije , (A.19) j6=0 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

2 4 2 2 2 2 2 2 2 2 4 2 λ (a λ − a (2b λ − 2c λ − 2d λ + λ + 1) − 8akbkckdkλ k k k k k (A.20) 2 2 2 2 2 4 2 2 2 2 4 2 +ck(2bkλ − 2dkλ − λ − 1) + λ (bk + dk − 1) + ckλ ) = 0. 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

2 2 2 2 2 2 (ak + bk + ck + dk − 1) (akbk + ckdk) R = 2 2 − 4 2 2 + 2, (A.21) ak + ck ak + ck from which it is easy to obtain the transcendental equation for the energy by substi- tution, s 2 2 2 2 2 2 (ak + bk + ck + dk − 1) − 4(akbk + ckdk) E = ±∆ 2 2 2 2 2 2 2 2 . (A.22) (ak + bk + ck + dk − 1) − 4(akbk + ckdk) + 4(ak + ck)

In the limit ξE → ∞ this directly gives the energy dependence on k. 58 APPENDIX A. SHIBA CHAIN WITH RASHBA SOC 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)  −cλ2 −λd aλ2 + λ −bλ    −λd −c bλ −λ − a by linearizing the equations using the approximation kF >> 1 >> E, as long as the determinant of the block-oﬀ-diagonal is (nearly) zero, ! cλ2 −λd det = 0. (B.2) −λd c It is perhaps most convenient to consider the system in reciprocal space. As seen −1/2 in Ref. [62], the terms in this 2 × 2 matrix are asymptotically proportional to kF .

59 60 APPENDIX B. COMPARISON TO THE TWO-BAND MODEL

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.

−1/2 In general, then, the two models converge as kF 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 important. 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 topological 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 convergence. 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 topological invariant N in terms of the positive-energy eigenstate |E+(k)i. Our starting point is the integral formula for the topological invariant1, Z π 1 −1 N = dk tr CH ∂kH . (C.1) 4πi −π 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

X X ˆ ˆ ˆ H = Ei|EiihEi| = EiPi = E+P+ + E−P−. (C.2) i i 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 ˆ ˆ 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 Z π 1 h ˆ ˆ ˆ ˆ i N = dk tr C(P+(k) − P−(k))∂k(P+(k) − P−(k)) . (C.3) 4πi −π

Writing out the product, using the relation C|E+(k)i = |E−(k)i 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 Z π/a 1 h ˆ ˆ ˆ ˆ i N = dk tr CP+∂kP+ − P+C∂kP+ . (C.4) 2πi −π/a Integration by parts ﬁnally yields the desired result,

Z π/a 1 h ˆ ˆ i N = dk tr CP+∂kP+ . (C.5) πi −π/a

To find the projection operator it is sufficient to find the eigenvector corresponding to positive energy. This is done by solving

 2 2    aλ − λ bλ cλ −λd x1      −bλ λ − a −λd c  x2     = 0 (C.6)  −cλ2 −λd aλ2 + λ −bλ  x     3 −λd −c bλ −λ − a x4 for the vector components. Solving the components as a function of x1, we obtain (up to a normalization factor)

d2λ2 − (ad − bc)2 x = −λ c2 − d2 + 1 λ2 − (a2 + c2) λ(ab + cd + bλ)(bc − ad + dλ) x3 = (ac − bd)λ − c + 2 2 2 λ − (a + c ) (C.7) λ(ab + cd + bλ) λ(ad − bc + dλ) x = x + x 2 λ2 − (a2 + c2) 1 λ2 − (a2 + c2) 3 1 x = [bλx + (a − λ)x + dλx ] . 4 c 1 2 3 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.