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.
