Introduction to Majorana Zero Modes in a Kitaev Chain

TL;DR

This paper introduces Majorana zero modes in a Kitaev chain, explaining their significance for topological quantum computing and providing an accessible overview suitable for undergraduates.

Contribution

It offers an educational introduction to MZMs, covering key concepts and demonstrating their emergence in a simple Kitaev wire model.

Findings

MZMs enable non-abelian braiding for quantum computation

Conditions for MZMs to appear in a 1D chain are discussed

Spectrum analysis shows how MZMs arise in the Kitaev model

Abstract

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 intended 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.