Multifarious Topological Quantum Phases And Characterizations Of Topology In Two-dimensional Chiral Topological Superconductors

Topological superconductor is one of the hot issues in condensed-matter field in recent years due to the Majorana zero modes on the boundary.These zero modes obey non-Abelian statistics and can be used in the topological quantum computations.In the thesis,we study the multifarious topological quantum phase transitions and the topol-ogy in two-dimensional chiral topological superconductor realized on checkerboard lattice model.The relationship of topological characterizations among Chern numbers,edge states and Majorana zero modes is in particularly concerned.The thesis is orga-nized as follows:In the first chapter,we briefly introduce the background of Majorana fermions proposed by E.Majorana and the possible realization in the superconductors.The the-oretical and experimental proposals on topological superconductors and Majorana zero modes are further introduced.We illustrate these proposals specifically in chapter two,including the most straightforward chiral p-wave superconductors and the coupling of s-wave superconductors with other materials.In chapter three,we study the two-dimensional chiral p-wave topological super-conductor of spinless fermions in the checkerboard-lattice Chern-insulator model.Mul-tifarious topological superconducting phases,including the high Chem number states up to ±3,are generated even with the nearest-neighbor px+ipy pairing in this sim-ple two-sublattice model.Higher Chern number = ±4 topological superconducting phase is further obtained when the second-nearest-neighbor p-wave pairing is included.We propose a feasible way to generate multifarious topological phase in the quasi-one-dimensional chains by introducing the harmonic potential,which controls the effective chemical potential.The Majorana zero modes localize on the phase boundaries and are well separated.In chapter four,we study the topology in two-dimensional chiral topological super-conductors.In general,the topological nature of the chiral topological superconductors can be characterized by the Chern numbers from the bulk states,the edge states,and the Majorana zero modes.However,the relationship among these characteristic quantities and whether they are equivalent remain unknown.Based on the topological supercon-ducting phases established in chapter three,we revealed the mismatch among the Chern number,edges states,and Majorana zero modes.The intrinsic reason is that some of the edge states in the superconducting phases inherited from the Chem-insulator phase are not protected by particle-hole symmetry.We further check our results in vortex states and Josephson junctions.In chapter five,we give a conclusion and outlook.