Majorana And Fractional Fermions In One Dimensional Systems

In systems where low energy properties can be described by Dirac-like Hamilto-nian, i.e., the one with linear dispersion, Majorana fermions and fractional fermions may appear at the boundaries between topological nontrivial and trivial areas. This can be understood by considering the scalar field coupled with Dirac spinor. It turns out that the scalar field induces a mass gap and also forms a kink near the boundary that separates two topologically distinct areas. The field with a kink structure cannot be deformed smoothly into a uniform one, and thus is topological nontrivial. Majo-rana and fractional fermions emerge when such a nontrivial background appears. In the first chapter, we introduce several well-known models, either realistic or not, and demonstrate that such is the case for both Majorana fermions and fractional fermions. Also, this particular chapter provides some background information for the chapters following.In the second chapter, we consider a one dimensional Rashba system in presence of a spatially varying magnetic field and show that Majorana fermions as well as frac-tional fermions can be induced therein. While the emergence of Majorana fermions re-lies on superconductivity, that of fractional fermions occurring in Jackiw-Rebbi model doesn’t. We find that a spatially varying field induces multiple gaps in the energy spec-trum. Whenever chemical potential resides in one of these gaps, Majorana fermions may appear if superconductivity paring potential is switched on. Fractional fermions, however, appear when two of such gaps overlap, in which case a band gap could be formed. With further studies, we demonstrate that the specific shape of external field is irrelevant in producing these exotic zero mode bound states. But the period of the field has to be an integer multiple of Rashba spin orbit coupling length in order for the system to have fractional fermions at the boundaries.In the third chapter, we consider a single chain lattice composing of Majorana fermions on each site, and show that it exhibits a supersymmetric quantum critical point corresponding to a tricritical Ising (TCI) phase transition, which separates a critical phase transition in the Ising universality class from a first-order one. We verify our predictions with numerical density-matrix-renormalization-group computations. Our main result is that the TCI phase transition occurs when the interaction among Majorana fermions is very strong comparing to bilinear hopping terms. However, we show that this can be achieved by tuning the chemical potential only.In the fourth chapter, we introduce a two-ladder lattice model with interacting Ma-jorana fermions that could be realized on the surfaces of a topological insulator film. We study this model by a combination of analytical and numerical techniques and find a phase diagram that features both gapless and gapped phases as well as interesting phase transitions including a quantum critical point in the tricritical Ising universality class. The latter occurs at an intermediate coupling strength at a meeting point of a first-order transition line and an Ising critical line and is known to be described by a su-perconformal field theory. The advantage of this specific model is that the occurring of TCI phase transition doesn’t rely on very strong interactions. In the end we discuss the experimental feasibility of constructing the model and tuning parameters to the vicinity of the TCI point where signatures of the elusive supersymmetry can be observed.We summarize our main findings in the last chapter and briefly discuss how our study can be further extended.