The Study On The Properties Of Non-hermitian And Topological Systems

Last century,Landau phase transition theory has achieved great success in describing the phase transition.However,the quantum Hall effect shows that some phase transition cannot be described by local order parameter.In further research,people found that these systems beyond the Landau paradigm can be described by the concept and theory about topology in mathematics.Thus,these systems are called as topological systems.Studying on topological systems becomes an important field of condensed matter physics.Generally,the phase of topological system is described by the topological invariants which depend on the global properties of the system.Systems with non-trivial topological invariants usually have isolated boundary states,this property of topological system is called bulk-boundary correspondence.For topological systems,topological invariants can always be protected by symmetries.Thus,symmetry and boundary state are two important topics in the field of topological systems.In traditional condensed matter physics,people always concern isolated systems,whose Hamiltonian are Hermitian operators.However,there are many open systems in the research.These open systems can be described by non-Hermitian effective Hamiltonian.In recent research,people find that the spectrum of the non-Hermitian system under periodic boundary condition can be totally different from the spectrum under open boundary condition,and the non-Hermitian system can have enormous states localized at the boundary.These properties have no counterpart in Hermitian systems.Because of these novel properties,non-Hermitian system becomes an important field of condensed matter physics.This thesis systematically introduces my research results about topological insulator and non-Hermitian system.This thesis is organized as follows:In chapter 1,starting with the quantum Hall effect,we review the development history about the research of the topological insulator.We introduce the topological state and the bulk-boundary correspondence of the topological system,topological field theory,the topological classification of topological system based on the internal symmetry and the research about the high-order topological insulator.In chapter 2,we introduce the fundamental properties of non-Hermitian system.Through non-Hermitian Su-Schrieffer-Heeger model,we introduce the non-Hermitian skin effect,generalized Brillouin zone and non-Bloch band theory.Based on these theories,the bulk-boundary correspondence of 1-dimensional non-Hermitian system is established.Then we introduce the internal symmetries of non-Hermitian system and the topological classification based on them.In the last,we introduce the exceptional point of the non-Hermitian system.In chapter 3,we introduce and discuss my studies on the origination of nonHermiticity of the system.Under long-wave limit,we can transform the Schrodinger equation of the 1-dimensional generalized non-Hermitian Hatano-Nelson model to an equation about a continuous system.Then we compare this continuous equation with the Schrodinger equation of the free particle system in a 2-dimensional manifold,and we can obtain the metric of the 2-dimensional manifold analytically and the correspondence relation of the wave function of the two systems.After that,we find this correspondence relation can be generalized to the situation that does not satisfy the long-wave limit.More generally,we find a d-dimensional non-Hermitian system with single band can always correspond to a free particle system in a 2d-dimensional manifold and the metric of the manifold can be obtained analytically.Since The free particle system is isolated,there does not exist gain and loss,the non-Hermiticity of the system can originate from the curved space.In chapter 4,we introduce and discuss my studies on the topological properties of space-time crystal systems:we obtain the topological classification of the gapped and gap-preserving rational space-time crystal systems.We find that the Hamiltonian of a rational space-time crystal system can be transformed to a Hamiltonian of a traditional Floquet system.Different from the previous methods,we use an ansatz to solve the Floquet equation analytically and get the effective Hamiltonian whose spectrum is the quasispectrum of the system.According to the quasi spectrum,there are 3 types of Hermitian rational space-time crystal systems:gapped,half gapped and gapless.Similarly,there are 3 types of non-Hermitian rational space-time crystal systems:gap preserving,gap breaking and gapless.By analyzing the internal symmetry of the effective Hamiltonian,we find that the Hermitian gapped space-time crystal systems can be classified by the tenfold Altland-Zirnbauer class,and the non-Hermitian gap preserving spacetime crystal systems under periodic boundary condition can be classified by the 38-fold topological classification of non-Hermitian system.Chapter 5 is a brief summary and outlook of this thesis.