Study of Majorana Fermions in topological superconductors and vortex states through numerically efficient algorithms

Abstract

Recent developments in the study of Majorana fermions through braid theory have shown that there exists a set of interchanges that allow for the realization of true quantum computation. Alongside these developments there have been studies of topological superconductivity which show the existence of states that exhibit non-Abelian exchange statistics. Motivated by these developments we study the differences between Abelian and non-Abelian topological phase in the vortex state through the Bogoliubov de-Gennes (BdG) formalism.

Due to our interests in low-energy states we first implement computationally efficient algorithms for calculating the mean fields and computing eigenpairs in an arbitrary energy window. We have shown that these algorithms adequately reproduce results obtained from a variety of other techniques and show that these algorithms retain spatial inhomogeneity information. Our results show topological superconductivity and vortex states can coexist; providing a means to realize zero-energy bound states, the number of which corresponds to the topological phase. With the use of our methods we present results contrasting the differences between Abelian and non-Abelian topological phase.

Our calculations show that an increase in Zeeman field affects numerous parameters within topological superconductors. It causes the order parameter to become more sensitive to temperature variations in addition to a reduced rate of recovery to the bulk value from a vortex core. The increased field suppresses spin-up local density of states (LDOS) in close proximity to the vortex core for low-energy states. Further, it narrows the spectral gap at the lattice centre. Both energy spectrum and LDOS calculations confirm that trivial topological phase have no zero-energy bound states, Abelian phases have an even number, while non-Abelian phases have an odd number.