Majorana Fermions in Synthetic Quasi One-Dimensional Systems: Quantum Computer Driven Simulation Tools

Abstract

Majorana fermions promise potential applications in quantum computing, superconductivity, and related fields. In this thesis, an analysis of A. Y. Kitaev's “Kitaev Chain”, a quasi-one-dimensional quantum wire in contact with a p-wave superconductor, designed as a model exhibiting unpaired Majoranas, is performed. Described by tunneling of spinless fermions between quantum dots, and formation of Cooper pairs on neighboring dots, Kitaev's chain Hamiltonian serves as a basis for emergent Majorana Zero Modes (zero energy Majorana fermions localized at either end of the chain) and artificial gauges (phases) to appear. By exact diagonalization, energy spectra and wavefunctions of a chain of spinless fermions on discrete quantum dots described by Kitaev's Hamiltonian are generated. By transforming the system into a basis of Majorana fermions and "bond fermions", where Majoranas on neighboring dots are paired, emergent Majorana Zero Modes (MZMs) are found at the ends of the chain. These emergent MZMs are paired in a non-local, zero energy bond fermion, which is found to allow degenerate energy states of the system to occur. Joining the ends of the chain by allowing tunneling and pairing of fermions on end sites, a ring topology is considered, where an "artificial gauge" emerges. This artificial gauge, or phase, causes a phase change on tunneling and Cooper pairing Hamiltonian matrix elements as a result of operator ordering within the Hamiltonian's ring terms. These required operator orderings are derived by comparison of energy spectra of the Kitaev ring in the fermion and bond fermion bases. Matching of calculated energy spectra in the Majorana and fermionic bases is used to confirm the presence of the artificial gauge, where this phase is found to be necessary in order to maintain a consistent energy spectra across the transformation between bases. This analysis is performed in order to understand the concept of Majorana Zero Modes and the emergence of Majorana fermions in 1D chains. By doing so, it is determined what Majorana fermions are, where they come from, and why Majorana Zero Modes are considered to be zero energy. These results contribute to the understanding of Kitaev chains and rings, as well as serve as a starting point for discussions regarding physical implications of the artificial gauge's effect, fermion statistics, and the emergence of Majorana Zero Modes in quasi-one-dimensional systems.