Quantum Phase Transitions In One-dimensional P-wave Superconductors

The most remarkable feature of the Majorana fermion is that its antiparticle is itself and due to the non-Abelian statistics of the Majorana fermion,it becomes the one of the powerful candidates of the fault-tolerant topological quantum computing.One has never stopped looking for the Majorana fermion since it was born.Recently,one finds that the zero-energy quasiparticle excitations in the topological superconductor have the similar properties as the Majorana fermion,which provides the theoretical basis for the study of the Majorana fermion.We mainly study the Majorana zero modes in the one-dimensional p-wave superconducting quantum wire with the modulated chemical potentials.We first introduce the related concepts of the topological insulators and the topological superconductors and discuss the phenomenon of the Anderson localization and the properties of the one-dimensional Aubry-André?AA?model.Due to the self-duality property,the well-known AA model presents a metal to insulator transion without mobility edges.Some generalized AA models which have the mobility edges in compactly analytic forms are found.By adding the p-wave superconductivity,those models show rich topological quantum phases and complex mobility edges.In this thesis,we consider a one-dimensional p-wave superconducting quantum wire with the specific modulated chemical potentials which can be solved by the Bogoliubov-de Gennes method.When the parameter b ?0 and ? is a rational number,the system undergoes a transition from topologically nontrivial phase to topologically trivial phase which is accompanied by the disappearance of the Majorana fermions and the changing of the Z₂ topological invariant of the bulk system.We find the phase transition strongly depends on the strength of potential V and the phase shift ?.For some certain special parameters ? and ?,the critical strength V of the phase transition is infinity.For the incommensurate case,i.e.???,the phase diagram is identified by analyzing the low-energy spectrum,the amplitudes of the lowest excitation states,the Z₂ topological invariant and the inverse participation ratio?IPR?which characterizes the localization of the wave functions.Three phases emerge in such case for ? =0,topologically nontrivial superconductor,topologically trivial superconductor and topologically trivial Anderson insulator.For a topologically nontrivial superconductor,it displays zero-energy Majorana fermions with a Z₂ topological invariant.By calculating the IPR,we find the lowest excitation states of the topologically trivial superconductor and topologically trivial Anderson insulator show different scaling features.For a topologically trivial superconductor,the IPR of the lowest excitation state tends to zero with the increase of the size,while it keeps a finite value for different sizes in the trivial Anderson localization phase.