The Study On The Fundamental Properties Of Non-hermitian Systems

Beyond the Landau theory of phase transition with the spontaneous symmetry breaking,the theory of topological phase and topological phase transition of matter have exploited a new field of condensed matter physics.After decades of development,the concepts and classifications of topological insulators and topological superconductors have been gradually completed.With the introduction of crystalline symmetries,the theory of higher-order topological insulators was established.Since there are a lot of open systems in real physical systems,the Hermitian Hamiltonian is no longer valid,thus,the effective non-Hermitian Hamiltonian provides a simple and intuitive path to study the real systems.If the periodic-boundary energy spectrum of a one-dimensional non-Hermitian system has point gap,there exist a large number of bulk eigenstates localized at the boundary of the system under open boundary condition.This is unique property to non-Hermitian systems and called the non-Hermitian skin effect.In order to depict the non-Hermitian skin effect analytically,the Bloch band theory is generalized to the non-Bloch band theory,and the Brillouin zone is correspondingly extended to the generalized Brillouin zone.In non-Hermitian systems,due to the existence of complex energy spectrum,band degeneracy may be accompanied by the occurrence of exceptional points,which is also impossible in Hermitian systems.Due to these novel properties of non-Hermitian systems,the study of the properties of non-Hermitian systems has become one of the frontier fields of recent international condensed matter physics.In this research field,we systemically and analytically study the open-boundary energy bands,eigenstates,and general properties of degeneracy and exceptional points in nonHermitian systems,and obtain some novel results.This thesis is organized as follows:In chapter 1,we introduce the background of topological insulators and higherorder topological insulators.Based on integer quantum Hall effect,anomalous quantum Hall effect,and quantum spin Hall effect,the concepts and classifications of topological insulators and topological superconductors are gradually established.After introducing crystalline symmetries,higher-order topological insulators with richer properties have emerged.In chapter 2,we discuss our studies on general theory of one-dimensional(1D)non-Hermitian systems with open boundary.In 1D non-Hermitian systems,the nonHermitian skin effect appears due to the existence of topologically nontrivial point gap of periodic-boundary spectra.We develop a systematically general theory under open boundary condition,which gives the general formalism of the eigenenergies and eigenstates of 1D non-Hermitian systems.We claim that the open-boundary energy spectra of 1D non-Hermitian systems is constituted by continuous energy bands(CEBs)and isolated energy bands(IEBs).The CEBs deduce the non-Bloch band theory,and the general forms of localized bulk eigenstates of non-Hermitian skin effect;the IEBs correspond to topologically protected edge states in most situations.We also clarify the important difference between the topological invariant of topological origin of nonHermitian skin effect and that of bulk-boundary correspondence.The topological invariant characterizing non-Hermitian skin effect is the winding number of periodicboundary spectrum,while the topological invariant protecting the topological edge state is obtained through the eigenstates of non-Bloch Hamiltonian.In chapter 3,we discuss our studies on band degeneracy and exceptional points of non-Hermitian systems.After analyzing the fundamental properties of Hermitian gapless points and non-Hermitian exceptional points in momentum space,we utilize the general theory of non-Hermitian systems with open boundary constructed by ourselves,to study the band degeneracy of IEBs and CEBs.When the dimension of null space of boundary matrix is less than the number of degenerate bands at the degenerate point,the exceptional point of open-boundary spectra emerges.We analytically study the band degeneracy and exceptional points of the generalized non-Hermitian SuSchrieffer-Heeger model,and solve the eigenstates of degenerate zero energy.When the generalized Brillouin zone and non-Bloch band theory are invalid,we discover that there exist novel degenerate points in 1D non-Hermitian systems,where the number of degenerate bands is scaled with the number of lattice sites.We call these degenerate points the infernal points,and we generalize the infernal points of 1D systems to the infernal knots of four-dimensional systems.In chapter 4,we discuss our studies on second-order non-Hermitian systems.We construct the nested tight-binding formalism,to analytically study the localized modes of second-order non-Hermitian systems.Based on the localized non-Hermitian skin effect and topological edge states of 1D non-Hermitian systems,we analyze the existence of the second-order skin-skin(SS),skin-topological(ST),and topologicaltopological(TT)corner-localized modes rigorously in two-dimensional(2D)secondorder non-Hermitian systems.We apply our nested tight-binding formalism to the 2D non-Hermitian single-,two-,and four-band model.We analytically find that there exist SS corner-localized modes in non-Hermitian single-band model,ST corner-localized modes in non-Hermitian two-band model,and SS,ST,and TT corner-localized modes in non-Hermitian four-band model.Moreover,We obtain the analytic forms of these corner-localized modes and the topological invariants protecting them.In chapter 5,we give the conclusion and outlook of this thesis.