The Majorana Fermions In One-dimensional Chains

Majorana fermion is a peculiar particle that its antiparticle is itself with electrical neutrality.Because of its good resistance to environmental interference,Majorana fermions are considered to be the best candidate for realizing quantum topology calculation,and they have attracted more and more attention in the field of condensed matter physics.The physicist Kitaev has proposed that Majorana fermions can be found in some superconductors under specific conditions,and he constructed a Kitaev model in which the Majorana fermions exist.Through one-dimensional Kitaev model,one-dimensional SSH model and their variant types'symmetries,topological class and topological properties,we give the phase diagram,and then we can find the parameter areas of the Majorana fermions.Here we divide our paper into five aspects:the first part is the introduction,we mainly introduce the topological materials,BCS theory,topological superconductor and topological classes.In the second part,we talk about symmetries,topological class and topological properties in one-dimensional SSH model and the one-dimensional Kitaev model in which carries the phase?=0??=k??and??k?,and we also talk about the zero mode in the one-dimensional Kitaev model.Then we give an account of the detailed introduction about the Ising model.In the third part,we build the new SSH model,talk about corresponding symmetries,topological class and topological properties under two conditions?=0 and??0.The forth part is the description of the one-dimensional Kitaev model with alternating potentials,and we focus on the Majorana zero modes.In the fifth part,we talk about the one-dimensional Kitaev model that carries the phase?^-i?2n+1??,?^-in?and cosn?.In the end,we have a brief summary for the above models.