Introduction to Majorana Zero Modes in a Kitaev Chain

Majorana zero modes (MZMs) realize a representation of non-abelian braid groups that enable topological quantum computation, wherein the storage and manipulation of information occur in decoherence-free degrees of freedom. This paper is meant to build off existing texts in the topic [1, 2] to serve as an introduction to MZMs at the undergraduate level, providing an overview of many important concepts in condensed matter physics, such as the Bogoliubov-de Gennes formalism and topological phases. We provide an overview of the motivation behind finding MZMs and discuss how the nature of these modes can provide an elegant solution to various challenges in quantum computing. We then show how MZMs can appear in a toy model known as a Kitaev wire by solving its spectrum, and conclude with a discussion on the conditions under which they arise in a one-dimensional chain.