Non-hermitian Band Theory And Bulk-boundary Correspondence

Intriguing issues in non-Hermitian systems include complex eigen-energy,EP points,the breakdown of usual bulk-edge correspondence etc.In this thesis,after the brief introduction about the properties of non-Hermitian systems,the non-Hermitian topological effect of non-reciprocal lattice are studied.The non-Hermitian topological effects of non-reciprocal lattice would be discussed from the following aspects:the topological invariants?PBC?,the breakdown of bulk-edge correspondence?OBC?and the interplay between skin effect and localization.The chiral symmetry ensures the topological invariant of non-Hermitian systems can take half integers.And for understand this half-integer invariant,we provide a concise geometrical interpretation of the bulk topological invariants in terms of two independent winding numbers.Based on that,we study topological properties of one-dimensional non-Hermitian systems without chiral symmetry and give phase diagrams characterized by topological invariants?and v_tot,associated with complex energy vor-ticity and summation of Berry phases of complex bands,respectively.In the absence of chiral symmetry,we find that the phase diagram determined by?is different from v_tot.While the transition between phases with different?is closely related to the band-touching point,the transition between different v_totis irrelevant to the band-touching condition.We then generalize the fidelity approach to study the phase transition in the non-Hermitian system and find that transition between phases with different v_totcan be well characterized by an abrupt change of fidelity and fidelity susceptibility around the transition point.In order to understand breakdown of usual bulk-edge correspondence,we inves-tigate the topological phase diagrams and the zero-mode edge states of a generalized non-reciprocal Su-Schrieffer-Heeger model,based on some analytical results.For the system under the open boundary condition,we construct analytically the wavefunctions of zero-mode edge states by properly considering a hidden symmetry of the system and the normalization condition with the use of biorthogonal eigenvectors.Our analytical results directly give the phase boundary for the existence of zero-mode edge states and unveil clearly the evolution behavior of edge states.In comparison with results via exact diagonalization of finite-size systems,we find our analytical results agree with the numerical results very well.In addition,here we study the interplay of skin effect and the Anderson localization in a non-reciprocal quasiperiodic lattice,dubbed non-reciprocal Aubry-Andrémodel,and a rescaled transition point is exactly proved.The non-reciprocity can induce not only the NHSE,but also the asymmetry in localized states with two Lyapunov exponents for both sides.Meanwhile,this transition is also topological,characterized by a winding number associated with the complex eigenenergies under periodic boundary conditions?PBCs?,establishing a bulk–bulk correspondence.This non-reciprocity can be realized by an elaborately designed electronic circuit with only linear passive RLC devices instead of elusive non-reciprocal ones,where the transport of a continuous wave undergoes a transition between insulating and am-plifying.This initiative scheme can be immediately applied in experiments to other non-reciprocal models,and will definitely inspires the study of interplay of NHSEs and more other quantum/topological phenomena.