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 M¨obiusstrip. 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 M¨obius 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 M¨obiusstrip unless you cut the ribbon and twist it before reconnecting it - which would not be a continuous transformation. Thus, objects and spaces are divided into topological equivalence classes. During the past few decades the concept of topology has become increasingly ubiquitous in physics, vital to such diverse areas as general relativity, string theory, and stochastic processes. The focus in this thesis, however, will be on the application of topology to condensed-matter physics, where topological materials have been the focus of much research. Topological superconductors are a specific class of materials that display proper- ties different from those seen in the by-now familiar states of ordinary matter. In 1 2 CHAPTER 1. INTRODUCTION A PC xPu kxPP kPC6 pendent spectroscopyC wAB dIZdVversusVat 7P mK and BC The ZBPishighl wfromP to x9P mTin ~PkmTsteps;tracesareoffset the gap edgesC At ~PC6 ZB lowest6 trace at B = PBC Data are from device ~C slope equal to ~q meVZ ed gapPCq peaksC wBB Colorkscale plot of dIZdVversusV wAB are extracted from w w6 Pu Zu ~Pu ~u w66ZB D PC PCq qPPu PC~ kxPP k6PP P 6PP xPP V wµB Figure 1.1: Differential conductivity of an InSb nanowire on an s-wave superconductor with Rashba SOC as a function of gate voltage and magnetic field. Different lines correspond to magnetic fields from 0 to 490 mT in steps of 10 T, with offset added for clarity; the green arrows indicate the edges of the gap. Relevant parameters are T = 70mK, ∆ ≈ 250µeV. Figure taken from ref. [2] particular, they may support zero-energy modes known as Majorana bound states, quasiparticles with interesting properties such as zero spin, and, more excitingly, par- ticle statistics that do not conform to the traditional cases of Fermi-Dirac or Bose- Einstein. In particular, is is theorized that these states could appear in systems of one effective space dimension, facilitating construction of wire networks that could be used in quantum computation. With the exciting prospect of new physics ahead, the area has seen much progress within the past 15 years, but so far experimental progress has lagged far behind the theory. This thesis is motivated by recent experimental developments in the field of topo- logical superconductors. Within the past five years, several different groups have published results indicating that quasiparticles known as Majorana bound states may be present [1, 2, 3, 4]. In 2012, Mourik et al [2] conducted measurements on InSb nanowires contacted by a normal and a superconducting electrode. Through tun- neling spectroscopy, they detected a zero-bias peak (ZBP) for intermediate-strength magnetic fields1, as seen in Fig. 1.1. The peak was only present when the theoretical 1The measured differential conductance at a given voltage is proportional to the density of states at the corresponding energy E = eV . 3 Figure 1.2: Conductance map of a Fe atomic chain on top of a Pb superconductor. Relevant parameters are ∆s = 1:36meV, T = 1:45K. The grey scale bar is of length 10 A.˚ The conductance shows a zero-bias peak localized at the end of the chain, while higher-energy modes are delocalized. Figure taken from ref. [3] requirements for the presence of Majorana bound states were fulfilled, indicating a possible discovery of the Majorana quasiparticles. However, various alternative expla- nations for the observed ZBPs have been suggested, such as zero-bias Kondo peaks [5, 6] and disorder [7, 8], and the experiment is therefore not viewed as conclusive. In addition to the fact that their energy is zero, another property of Majorana bound states in one-dimensional nanowires is that they are expected to be localized at the ends of the wire, which should also be observable experimentally. In 2014, Nadj-Perge et al conducted a new experiment in which they measured the spatially resolved differential conductance in Fe-based atomic chains on a Pb superconductor. They also observed ZBPs, which, as seen in Fig. 1.2, are clearly located at the ends of the wires. However, as was the case for the the previously mentioned experiment, the topological nature of the zero-bias peaks are in doubt [9]; in particular, under certain conditions the zero-bias peaks may be located at the ends of the chain even in the absence of Majorana modes [10]. Nevertheless, while deemed inconclusive, these experiments indicate that progress is being made and a physical realization of a topological superconducting system may soon be feasible, inspiring the research done in this work. This thesis, largely based on Ref. [11], has two main aims.