Exceptional Topological Phenomena Of Non-hermitian Systems

In recent years,the investigations of non-Hermitian systems have shed light on their rich and diverse physical properties.These systems are not mere extensions of Hermitian systems but possess unique characteristics that distinguish them from their Hermitian counterparts.The discovery of the non-Hermitian skin effect has demonstrated significant differences in the eigen-energies between open and periodic boundary conditions,rendering the conventional bulk-boundary correspondence,which relies on topological invariants defined by the Bloch Hamiltonian,inadequate for characterizing the topological properties of non-Hermitian systems under open boundary conditions.To overcome this limitation,the Brillouin zone has been generalized to the generalized Brillouin zone,enabling the definition of non-Bloch topological invariants that accurately depict the existence of topological edge states in non-Hermitian systems.This approach has led to the establishment of non-Bloch bulk-boundary correspondence,which provides a faithful characterization of the topological properties of non-Hermitian systems.This academic paper aims to present exceptional topological phenomena in non-Hermitian systems within the framework of non-Bloch band theory.We begin by introducing important foundational concepts in non-Hermitian systems,such as point/line gaps,PT symmetry,and exceptional points（EPs）.Employing the non-Hermitian Su-Schrieffer-Heeger model as a prototypical example,the differences in energy spectra under varying boundary conditions,the non-Hermitian skin effect,and the generalized Brillouin zone are distinctive properties in non-Hermitian systems,which provide a fundamental explanation for the breakdown of the conventional bulk-boundary correspondence and explicate how to establish a non-Bloch bulk-boundary correspondence.By merging the concept of knots with the band structure,non-Bloch band structures can form topological knots by introducing a one-dimensional non-Hermitian tight-binding model,such as the Hopf link,induced by the non-Hermitian skin effect under open boundary conditions.The topological knot braided by energy bands is topologically robust against any perturbations without gap closing.At exceptional point,the phase transition occurs between topologically non-trivial knot and topologically trivial knot as the energy bands overlap.We define non-Bloch topological invariants and vorticity on the generalized Brillouin zone to accurately characterize the topological knot.The stability of the topological knots in the models we investigated could potentially facilitate future experimental demonstrations of this result.We establish a non-Bloch band theory for one-dimensional non-Hermitian topological superconductors.The universal physical properties of non-Hermitian topological superconductors are revealed based on the theory.According to the particle-hole symmetry,reciprocal particle and hole loops consist of the generalized Brillouin zone.The critical point of phase transition,where the energy gap closes,appears when the particle and hole loops intersect at Bloch points.If the non-Hermitian system has non-Hermitian skin effect,the corresponding eigenstates of particle and hole will localize at opposite ends of an open chain,it should be the Z₂ skin effect.The non-Bloch band theory is applied to two examples,non-Hermitian p-wave and s-wave topological superconductors.In terms of Majorana Pfaffian,a Z₂ non-Bloch topological invariant is defined to establish the non-Hermitian bulk-boundary correspondence for the non-Hermitian topological superconductors.For a two-dimensional non-Hermitian topological semimetal system,there are novel physical properties under open boundary conditions.Topological semimetal could not only transform to second-order topological insulator and normal insulator,but also have exceptional characteristics with the evolution of the gap closing point.In addition,the non-Hermitian skin effect in topological semimetal is robust,whose eigenstates are stable while against disorder.Modifying the geometric boundary,the eigenstates remain localized at the boundary.