The aim of this thesis is to study non-Abelian physics in condensed matter systems; in particular, the examples of Möbius band and Majorana fermions.

We start with a simple model of a two-level atomic chain and show with numerical results the existence of non-Abelian phenomenon in this model: a particle is initialized in one of the bounded states in the vicinity of a STM probe, and then the evolved state may change after a cyclic operation of STM around the center of the chain. We argue that the model is a good representation of the Möbius band.

Secondly, we study one type of the non-Abelian particles: Majorana fermions. There are two works about Majoranas in this thesis. One is about operations on MKP(Majorana Kramer's pair) based qubits. As we know, decoherence always exist...[ Read more ]

The aim of this thesis is to study non-Abelian physics in condensed matter systems; in particular, the examples of Möbius band and Majorana fermions.

We start with a simple model of a two-level atomic chain and show with numerical results the existence of non-Abelian phenomenon in this model: a particle is initialized in one of the bounded states in the vicinity of a STM probe, and then the evolved state may change after a cyclic operation of STM around the center of the chain. We argue that the model is a good representation of the Möbius band.

Secondly, we study one type of the non-Abelian particles: Majorana fermions. There are two works about Majoranas in this thesis. One is about operations on MKP(Majorana Kramer's pair) based qubits. As we know, decoherence always exists for any system in presence of time dependent potential, such as D- or DIII-class topological superconducting system(TSC). By using the perturbation theory, one can get that the decoherence in both cases has a similar first order term. However, an extra second order term exists in DIII-class TSC which induces the so-called 'local rotation' among MKP. The local rotation only occurs if the correlation function of the perturbation fulfill some special condition, therefore we may argue that MKP based qubits are still a good choice for quantum computation, which may have other advantages. Since the braiding process is a time dependent operation, an experimental protocol is derived for ideal braiding operations without local rotations, which will lead to a quantized phase for each Majorana state after Braiding twice. This work may be of great importance as a first study on the non-Abelian braiding operation in symmetry-protected topological systems. Besides, we also show a controllable and adiabatic operation to rotate the MKP, serving as a method to change the information in MKP based-qubits.

The other work about Majroanas is an experimental proposal of MFB(Majroana flat band) by using multi-ferromagnetic layers on the top of s_± wave superconductor, in the absence of spin-orbital coupling. We studied the band structure to show the reasons of existing MFB in this system and give a numerical calculation based on a heterostructural model. Potential benefits of this proposal are pointed out.