Majorana fermions in superconductors are the subgap quasiparticle excitations that are their own antiparticles.They occur on the boundary of so-called topological superconductors and are an equal... Show moreMajorana fermions in superconductors are the subgap quasiparticle excitations that are their own antiparticles.They occur on the boundary of so-called topological superconductors and are an equal weight quantum superposition of a positively charged electron and a negatively charged hole, that makes them charge-neutral but not an eigenstate of charge. Thus, the expectation value of charge vanishes, but the fluctuations of charge stay finite.Study of the latter ones would provide a purely electrical detection of Majorana states. In this thesis, we show that the electrical shot noise (the second moment of charge fluctuations) originated from the surface of a three-dimensional topological superconductor is related to the thermal conductance with a coefficient which consists of temperature and fundamental constants.The second part of the thesis is devoted to the strongly interacting fermionic zero modes (both Dirac and Majorana) described by the Sachdev-Ye-Kitaev model, which can be realized in disordered quantum dots in magnetic fields. The theoretical predictions of the spectroscopic properties of such quantum dots are presented in this thesis. Show less
Topological superconductors are a novel type of superconductors that carry Majorana particles at their boundary. These surface states are equal superpositions of electrons and holes, and hence are... Show moreTopological superconductors are a novel type of superconductors that carry Majorana particles at their boundary. These surface states are equal superpositions of electrons and holes, and hence are their own anti-particles. There has been a recent surge of theoretical and experimental effort to realize these special particles in the lab. While first observations support the theoretical predictions, fail-safe experimental evidence for Majoranas is still needed. Part of the challenge is that due to their vanishing charge they are not easily detected electrically. The topic of this thesis is the proposal and study of electrical signatures of Majoranas that are present in spite of their charge neutrality. By applying scattering and random matrix theory we first examine their generic properties. With the tool of numerical simulations we then put our predictions to test on realistic systems. Show less
This thesis deals with characterizing topological phases as well as the transitions between them, focusing on transport properties and the effects of disorder. In Chapters 2 and 3 we derived... Show moreThis thesis deals with characterizing topological phases as well as the transitions between them, focusing on transport properties and the effects of disorder. In Chapters 2 and 3 we derived scattering matrix expressions for the topological invariants of systems. This approach is oftentimes numerically easier to evaluate than Hamiltonian expressions. In Chapter 4 we predict novel transport features of the quantum Hall plateau transition, and efficiently estimate the associated critical exponent. In Chapter 5 we examine the universal properties of phase transitions in two-dimensional helical topological superconductors. We compute the critical exponents characterizing the divergence of the localization length, as well as the critical conductance. In Chapter 6, we model a one-dimensional topological superconductor in a bottom-up fashion, as an array of coupled quantum dots. We show how to tune this system deep within the non-trivial phase, with well localized Majorana bound states at its ends. In Chapter 7, we find a new class of disordered topological insulators protected not by an exact symmetry, but by an average symmetry of the disordered ensemble. This greatly increases the range of non-trivial phases, as every topological phase transition gives rise to infinitely many higher-dimensional topological phases Show less
Topological phases of matter are exceptional because they do not arise due to a symmetry breaking mechanism. Instead they are characterized by topological invariants -- integer numbers that are... Show moreTopological phases of matter are exceptional because they do not arise due to a symmetry breaking mechanism. Instead they are characterized by topological invariants -- integer numbers that are insensitive to small perturbations of the Hamiltonian. As a consequence they support conducting surface states that are protected against disorder and other imperfections. Furthermore, a variety of unusual transport properties arise due to the presence of topology. In this work the interplay between topology and sample imperfections is investigated with a focus on transport phenomena. Show less
