Majorana Fermionic Representations Of Two Integrable Models And Their Studies

This thesis is mainly on the study of two integrable models.The analytical method we use are all based on the Majorana representation.The first one the Kitaev chain,in which it would holds two absolutely localized unpaired Majorana fermions at both ends.While there would only has fermions in the trivial superconducting state.Two Kitaev chains would coupled end-to-end and the formation of quantum entanglement can be studied,and this process can be described by mutual information.This kind of coupling is equivalent to induce a local quench in the Kitaev chain.Another one is the BCS-Hubbard model.This spin fermionic model can be map to a spinless fermionic model by two times of refermion.This in turn reduces a complex many-body problem to a free-fermion model.We study the quantum phase transition by adjusting the the magnetic field and Hubbard interaction.Although the origins of quantum entanglement are difficult to understand,it remains the foundation of modern physics,especially in quantum information.This is because quantum entanglement is the basis of quantum communication,quantum computing,etc.,in which the transmission and processing of information are all on the base of quantum mechanic laws.Furthermore,quantum entanglement is also spread from quantum information to quantum optics,condensed physics,and high energy physics.Especially in condensed matter physics.There are many complex relationships among the subsystems in the system,and many novel phases are induced.They include the phases induced by the collective phenomena such as topological order,quantum spin liquid,high-temperature superconductor,and fractional quantum Hall effect.These strange phases are always related to the long-range quantum entanglement and are beyond the framework of Landau phase transition theory.Assume there have two subsystems which do not overlap in the space,and do not entangled originally.If some modulations are introduced suddenly to induce the entanglement between the two subsystems,how will the entanglement established.The thesis focus on the establishment of quantum entanglement among subsystems.In Chap.1,the background of Majorana fermions is introduced.Also,the applications of Majorana fermion representation in two integrable models.In Chap.2 is on the building of quantum entanglement between the two edge modes in the quenched Kiatev chain by the time-depend variational principle.We find that the entanglement is built by propagation in the trivial superconducting state.This propagation depends on the moving of the pair of coherent quasi-particles.In contrast,in the nontrivial superconducting state,The establishment of entanglement depends on the macroscopic delocalization of entanglement,which can be interpreted as topological excitation in Kitaev chains.We also find that this entanglement is synchronized the entanglement between the sites of the local quench.These two,together imply that the entanglement in the nontrivial superconducting is not because of the propagation.We conjecture that this can be used to estimate is the system short-range entanglement or long-range entanglement.In Chap.3,the static edge mode density distribution after local quench is presented by exactly diagnoalization.Lastly,a strongly correlated model with an external field is introduced.This model is on the base of the traditional Hubbard model and added with an equal spin pairing term and a magnetic field term.This model can be exactly solved along the ’symmetric lines’.By splitting the fermion into two Majorana fermions,the corresponding local conserved qualities can be obtained.By refermionization,the model can be map to a bilinear spinless model.We find that a disordered phase arises in this model,and it is termed as glass-like phase.That is because there would have residua entropy by ergodic calculation in a small size.This indicates that this phase is highly degenerate in its ground state.Further studies on this phase would rely on power computation resources.