Non-hermitian Band Theory And Bulk-boundary Correspondence

The bulk-boundary correspondence has played a fundamental role in the development of topological band theory.In general,the topological nontrivial boundary states can be characterized by the topological invariants defined by the Bloch Hamiltonian.For example,the chiral edge states can be faithfully predicted by the Chern number,which is defined as the integral of Berry curvature over the entire two-dimensional Brillouin zone.Recently,the celebrated bulk-boundary correspondence is challenged in some nonHermitian systems.The open boundary and periodic boundary spectra can be totally different,and the corresponding topological nontrivial edges protected by the band topology can never be characterized by the Bloch Hamiltonian.Further studies revealed that the open boundary eigenstates of these systems are not extended states,but localized states at the boundary.This phenomena is dubbed non-Hermitian skin effect.In order to recover the bulk-boundary correspondence,the concept of Brillouin zone,which is defined in the periodic boundary systems,is generalized to the open boundary systems,namely,the concept of generalized Brillouin zone.In the Hermitian systems,the generalized Brillouin zone coincides with the Brillouin zone.However,in the nonHermitian systems,the generalized Brillouin zone can be totally different with the Brillouin zone.Based on the concept of generalized Brillouin zone,the nontrivial boundary states protected by the band topology can be successfully characterized by generalized Brillouin zone Hamiltonian.The first theme of this thesis is the non-Hermitian bulkboundary correspondence and the non-Hermitian skin effect.We first show that the following concepts or phenomena are equivalent:(i)the distinct between periodic and open boundary spectra;(ii)the emergence of non-Hermitian skin effect;(iii)the discrepancy between generalized Brillouin zone and Brillouin zone.After that,we show the above three equivalent phenomena can be predicted by the spectra(or energy)topology of the Bloch Hamiltonian.We give a geometrical meaning of the generalized Brillouin zone and the non-Hermitian band structure,and propose an analytic method to calculate the generalized Brillouin zone,namely the auxiliary generalized Brillouin zone method.Furthermore,we clarify the relation between non-Hermitian skin effect and symmetries,and propose how to realize the non-Hermitian skin effect with the experimental tunable non-Hermitian terms,namely,the on-site dissipations.Finally,we study the linear and non-linear self-energy effect in the Floqeut proximity superconductivity.When the generalized Brillouin zone coincides with the Brillouin zone,the open boundary Hamiltonian can be approximated by the Bloch Hamiltonian.The second theme of this thesis is to study the topological properties of the non-Hermitian band degeneracies in these systems,including the degeneracy points and degeneracy lines.Among them,there exist two types of novel band degeneracies which are called exceptional points and exceptional lines.At these points or lines,the non-Hermitian Hamiltonian is non diagonalizable.We first give a correct definition of these non-Hermitian band degeneracies,and then,prove that the stable exceptional points must come in pairs,which is call the no-go theorem of exceptional points.In three dimension,we show that the stable degeneracies are exceptional lines.These exceptional lines can be linked or knotted together.We first use the Jones polynomial to classify them,and then,prove the Jones polynomial of the perturbation generated phases around the critical phase is finally determined by the local evolutions around every touching points of the exceptional lines.