The Study Of The Topological Phase Transition In Quasi-one Dimension Topological Superconductors

Quantum computer is the goal that people want to achieve in recent years.For the realization of quantum computer hardware,fault tolerance and resist decoherence is an urgent problem.And the Majorana zero mode in the condensed matter physics is likely to become the key to solve this problem.This is because the Majorana zero mode is a kind of non-Abelian quasi-particle which has topological protection.On the one hand,this kind of non-Abelian quasi-particle has strong anti-interference ability;on the other hand,it can change the quantum state by exchanging each other and then encode information.Therefore,the use of Majorana zero mode can achieve fault-tolerant topological quantum computer.With the development of topological superconductors,Majorana zero mode can be found in the end state of one dimensional or quasi-one dimensional topological superconductors.In this thesis,we study the topological phase transition of quasi-one dimensional topological superconductor induced by dimerizat.ion and periodical potential.Firstly,we analyze the symmetry which the Hamitonian satisfy when t.he superconducing pairing phase between the interchain and intrachain is ?=0 and ?=?/2.Then the Hamitonian can be classified into BDI class or D class according to the symmetry.The topological index in BDI class is Z index;we can use winding number W to calculate the topological phase diagrams.And the topological index in B class is Z2 index,we can use majorana number M to calculate the topological phase diagrams.Further,we calculate the energy spectrum corresponding to the topological phase diagrams.By analyzing the energy spectrum,we can verify that the topological phase diagrams accurately reflect the emergence of Majorana zero modes in the system.Finally,we use the recursive Green' s Function method to calculate the Andreev reflction at the NS junction.Then the differential conductance at zero bias can be obtained by the Andreev reflction?By comparing the topological phase diagrams and the differential conductance under the same parameters,we find that the phase diagrams and the differential conductance diagrams are the same.And it is the strong evidence that the process of topological transformation in the phase diagrams is the change process of Majorana zero modes.The main structure of this paper is as follows:In the first chapter,we introduce the origin of Majorana fermions and its development in the condense matter physics.Then we discuss the Majorana zero mode in spin polarizd p wave superconductor,and clarify the relation of topological superconductors and the Majorana zero mode.In this chapter,we also introduce the Andreev reflcetion on the contact surface of normal metal and superconductor.In the experiment,we can detect the exist of Majorana zero mode at the end of topological superconductor by using the Andreev reflection.In theory,to further prove the existence of Majorana zero mode,the recursive Green' s function method can usually be used to calculate the Andreev reflection.In the second chapter,we mainly discuss the topological phase transition of quasi-one dimentional topological superconductor induceded by dimerization.In the BDI class,we calculate the topological phase diagrams in parameter space(?,?)and(?,?).In the D class,we calculate the topological phase diagrams in parameter space(?,?).Further,we calculate the energy spetrums corresponding to topological phase diagrams by the diagonalization of Hamiltonian.Then compare with the topological phase diagrams and energy spectrums.In the third chapter,we mainly discuss the topological phase transition of quasi-one dimentional topological superconductor induced by periodical potential.In the BDI class,we calculate the topological phase diagrams in parameter space(u,ty)and(u,|?y|).In the D class,we calculate the topological phase diagrams in parameter space the same as in the BDI class.By comparing with the topological phase diagrams in the BDI class and the D class,We find t.hat the time reversal symmet.ry can play a key role for the system in which exist.multiple Majorana zero modes.The same as chapter 2,we also calculate the energy spetrums corresponding to topological phase diagrams by the diagonalization of Hamiltonian.And compare with the topological phase diagrams and energy spectrums.In the fourth chapter,we caculate the differential conductance of quasi-one dimensional topological superconductor induced by the dimerization and periodical potential respectively.And regardless of whether the Hamiltonian belongs to the BDI class or the D class,the differential conductance diagrams can coincide with the topological phase diagrams.At the end of this paper,we also make a brief summary of the work done.