Ring Frustration Effect And Topological Extend-kink Phase

In condensed matter physics,the research on novel quantum states and their related phase transition problems has been a hot topic for a long time,and these researches have potential applications in quantum information and quantum computation.The properties of quantum states include topological and thermodynamic properties.Topological properties can be judged by the time inversion symmetry,particle-hole symmetry and chiral symmetry of the system in which the quantum state is located.Through the existence of these three different symmetry,we can know which category the topological properties of Quantum States belong to.There are ten different classifications for free Fermi subsystem,namely periodic table of topological insulators and superconductors.The topological properties of quantum states can also be identified by topological indexes,such as the winding number,the chern number and the Majorana number,and so on.Besides topological index,we can also mark the topological properties of quantum states by entanglement entropy,entanglement spectrum and degeneracy.The thermodynamic properties can be characterized by the state density,specific heat,critical behavior and correlation function.The quantum Hall effect found in the experiment is the embodiment of topological quantum state in the related materials.It makes people see the dawn of discovering and manufacturing topological materials.Researchers are also actively looking for topological superconductors which have been used in theory for a long time,because they can realize the strange Majorana zero states,which have potential applications in the field of topological quantum computing.In addition,in theory,we can achieve the purpose of preparing the quantum states by adjusting the boundary conditions of Hamiltonian.Based on the above background,we study three one-dimensional systems with ring frustration.The ring frustration is realized by special periodic boundary conditions,which can produce novel topological extended-kink states(phases).The research on the properties of these systems is the focus of this paper.The research we have carried out is as follows:1.study on the transverse field Ising modelIn quantum statistical physics,transverse field Ising model has the advantages of simple,effective and widely used.When we study the anti-ferromagnetic transverse field Ising model,we limit that the total number of lattice size is odd and it is in the periodic boundary condition.We find that the system can realize a novel gapless topological extended-kink phase.By analyzing the energy spectrum and calculating the number of kinks and the correlation function,we can get the important characteristics of the ring frustration and the topologicalextended kink phase.We found that the kink bound state can also be realized by introducing defects.2.the study of fermion chains with interactionKitaev chain is a one-dimensional system without spin fermions.Its ground state is the Majorana zero mode.Recently,people are more and more interested in the Majorana zero model presented in the interaction model.However,it is suggested that the interaction term may lead to the transformation of onedimensional topological phase into non-topological phase,which does not satisfy the classification of non-interaction fermion model.Therefore,there are many unknown physical phenomena in the interaction Kitaev chain.Our work is to study the influence of special periodic boundary conditions on the fermion chains with interaction.We design a scheme to solve the model by using the Quaternary Jordan-Wigner mapping between fermion chain and spin chain.It is found that the fermion chains with interaction will be affected by the ring resistance and frustration effect,and a novel gapless topological extended-kink phase will be formed.By analyzing the energy spectrum and calculating the correlation function and entanglement entropy,we get the important characteristics of the ring frustration and the topological extended-kink.We found that fermion kink bound state was realized by introducing defects.3.Cluster-Ising modelCluster-Ising model has been widely concerned once it is put forward.Under the open boundary conditions,the ground state of the model is highly degenerate and is symmetry protected topology state.The model has interesting statistical and entanglement properties and topological properties.Cluster-Ising model has a wide application prospect in quantum computer.Our work is to add special periodic boundary conditions to Cluster-Ising model,and study the influence of special conditions on the original topological state protected by symmetry or symmetry breaking state,By analyzing the energy spectrum and calculating the string correlation function,we find that when the periodic boundary conditions,the total number of lattice size and the length of cluster ising interaction are odd,the model will be affected by the ring frustrantion effect and form a novel topological kink phase.We find that the topological extended-kink states are different from the usual symmetry protected topological states by calculating the entanglement spectrum,and they can be distinguished by the characteristics of the entanglement spectrum.