Majorana Zero Modes in Solid State Systems

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Majorana zero modes are zero-energy excitations that are their own anti-particles, and obey non-Abelian statistics which could be harnessed for topological quantum computation. There are many theoretical proposals to realize them in solid state systems, but experimental realizations are confronted by a number of non-idealities. In this thesis, we theoretically investigate such complications, thereby suggesting improvement and directions that could be pursued. We first develop a theoretical framework to analyze the effect of ensemble-averaged disorder on the Majorana zero modes, generalizing the Eilenberger theory to handle 1D systems while retaining short-distance fluctuations. We then consider disordered topological insulator-based heterostructures, showing that extra subgap states are potentially induced, obscuring the density-of-states signature of the Majorana zero mode. We also analyze in depth the experimentally observed soft gap feature, suggesting that a cleaner interface in the semiconductor-based proposal can harden the gap.

In view of some of the limitations of the proposals based on semiconductors or topological insulators, we look into a new class of systems in which a ferromagnetic atomic chain is put on the surface of a bulk spin-orbit-coupled superconductor. This system is analyzed in two limits, corresponding to weak or strong inter-atomic hopping on the chain. In each of these cases, the topological criteria are obtained. We also find that in the limit of strong chain-superconductor coupling, the length scales of the effective Hamiltonian of the chain are significantly suppressed, potentially explaining some of the recent observations in experiments.