TY - BOOK

T1 - Aspects of Majorana Bound States in One-Dimensional Systems with and without Time-Reversal Symmetry

AU - Wölms, Konrad Udo Hannes

PY - 2015

Y1 - 2015

N2 - In recent years there has been a lot of interest in topological phases of matter. Unlikeconventional phases of matter, topological phases are not distinguished by symmetries,but by so-called topological invariants which have more subtle physical implications. Itcomes therefore as no surprise that for a long time only a few topological phases werestudied and those that were, were not studied in the full topological context, which isonly known now. One of the topological phases that has been know for a very longtime is the quantum Hall eect. The quantum Hall eect is a topological phase intwo-dimensions without any symmetries. Even though the bulk of a quantum Hall systemis insulating, it exhibits gapless edge modes. It is therefore dierent from other insulatingtwo-dimensional materials. It was soon realized after the discovery of the quantum Halleect, that there is a quantized invariant (topological invariant) associated with thequantum Hall eect [48], but only much later such invariants were found and studied inother systems.By now other topological systems are also being studied from an experimental andtheoretical point of view [7]. There exist topological phases in any number of dimensions.One of the topological phases that received a lot of attention in recent years, is theone-dimensional topological superconducting phase, without time-reversal symmetry [5].Similar to the quantum Hall eect, this phase exhibits edge excitations, which are zerodimensionalfor one-dimensional systems. For this particular phase the edge excitationsare called Majorana bound states and they are interesting in themselves. There has beena lot of eort in detecting Majorana bound states in the lab. One reason is that theseexcitations provide evidence that a system is indeed in a topological phase. It is thereforerequired to have unambiguous experimental evidence for the presence Majorana boundstates, which in turn requires a good theoretical understanding of the physics associatedwith Majorana bound states. In particular for the most common experimental methodsthat are used to study them, the signature of Majorana bound states in the measurementstill has to be understood better. And example would be the frequently performedtunnel probe measurement on Majorana bound states [26, 40, 41]. A second reason whyMajorana bound states are interesting is their potential application to a certain quantumcomputation scheme. This scheme, called topological quantum computation, relies on thebraiding of so-called non-abelian anyons in order to perform computations [18]. Majoranabound states are the simplest example of such non-abelian anyons. No other non-abeliananyons have been realized experimentally yet, which puts further focus on the study ofMajorana bound states. Additionally to probing Majorana bound states, their use intopological quantum computation also requires them to be manipulated. This also posesan interesting problem for both experimentalists and theorists [25, 27].We can summarize the challenges presented so far as being related to generatinga topological phase, probing Majorana bound states and manipulating them. Thesechallenges are actually important beyond the intensively studied topological phase ofone-dimensional superconducting systems without time-reversal symmetry. In particularthey are very important for the closely related phase of one-dimensional topologicalsuperconductors with time-reversal symmetry. This phase also exhibits Majorana boundstates, and we will study some of its aspects in this thesis. We will discuss some issuesrelated to obtaining this topological phase and how electron-electron interactions mayhelp in achieving this. We will also discuss issues related to using this phase and its edgestates for topological quantum computation, by calculating the result of an exchange oftwo such edge states. Finally we will return to the broken time-reversal-symmetry caseand discuss aspects related to tunnel probing Majorana bound states.

AB - In recent years there has been a lot of interest in topological phases of matter. Unlikeconventional phases of matter, topological phases are not distinguished by symmetries,but by so-called topological invariants which have more subtle physical implications. Itcomes therefore as no surprise that for a long time only a few topological phases werestudied and those that were, were not studied in the full topological context, which isonly known now. One of the topological phases that has been know for a very longtime is the quantum Hall eect. The quantum Hall eect is a topological phase intwo-dimensions without any symmetries. Even though the bulk of a quantum Hall systemis insulating, it exhibits gapless edge modes. It is therefore dierent from other insulatingtwo-dimensional materials. It was soon realized after the discovery of the quantum Halleect, that there is a quantized invariant (topological invariant) associated with thequantum Hall eect [48], but only much later such invariants were found and studied inother systems.By now other topological systems are also being studied from an experimental andtheoretical point of view [7]. There exist topological phases in any number of dimensions.One of the topological phases that received a lot of attention in recent years, is theone-dimensional topological superconducting phase, without time-reversal symmetry [5].Similar to the quantum Hall eect, this phase exhibits edge excitations, which are zerodimensionalfor one-dimensional systems. For this particular phase the edge excitationsare called Majorana bound states and they are interesting in themselves. There has beena lot of eort in detecting Majorana bound states in the lab. One reason is that theseexcitations provide evidence that a system is indeed in a topological phase. It is thereforerequired to have unambiguous experimental evidence for the presence Majorana boundstates, which in turn requires a good theoretical understanding of the physics associatedwith Majorana bound states. In particular for the most common experimental methodsthat are used to study them, the signature of Majorana bound states in the measurementstill has to be understood better. And example would be the frequently performedtunnel probe measurement on Majorana bound states [26, 40, 41]. A second reason whyMajorana bound states are interesting is their potential application to a certain quantumcomputation scheme. This scheme, called topological quantum computation, relies on thebraiding of so-called non-abelian anyons in order to perform computations [18]. Majoranabound states are the simplest example of such non-abelian anyons. No other non-abeliananyons have been realized experimentally yet, which puts further focus on the study ofMajorana bound states. Additionally to probing Majorana bound states, their use intopological quantum computation also requires them to be manipulated. This also posesan interesting problem for both experimentalists and theorists [25, 27].We can summarize the challenges presented so far as being related to generatinga topological phase, probing Majorana bound states and manipulating them. Thesechallenges are actually important beyond the intensively studied topological phase ofone-dimensional superconducting systems without time-reversal symmetry. In particularthey are very important for the closely related phase of one-dimensional topologicalsuperconductors with time-reversal symmetry. This phase also exhibits Majorana boundstates, and we will study some of its aspects in this thesis. We will discuss some issuesrelated to obtaining this topological phase and how electron-electron interactions mayhelp in achieving this. We will also discuss issues related to using this phase and its edgestates for topological quantum computation, by calculating the result of an exchange oftwo such edge states. Finally we will return to the broken time-reversal-symmetry caseand discuss aspects related to tunnel probing Majorana bound states.

UR - https://soeg.kb.dk/permalink/45KBDK_KGL/fbp0ps/alma99122943881505763

M3 - Ph.D. thesis

BT - Aspects of Majorana Bound States in One-Dimensional Systems with and without Time-Reversal Symmetry

PB - The Niels Bohr Institute, Faculty of Science, University of Copenhagen

ER -