Non-Bloch Band Theory And Non-hermitian Skin Effect

Recently,with the development and maturity of topological band theory in Hermitian systems,the research interest has gradually been extended to non-Hermitian systems.The bulk-boundary correspondence,as a cornerstone of topological band theory,refers to that topological invariance from Bloch wave function can faithfully characterize the appearance of topological edge states under open boundary condition.However,such bulk-boundary correspondence breaks down in some non-Hermitian systems.Zhong Wang’s group at Tsinghua University pioneered the way of the generalized Brillouin zone.Specifically,The topological invariance defined by non-Bloch waves characterized by generalized Brillouin zone can precisely predict the emergence of edge states with open boundary,which recovers the bulk-edge correspondence in non-Hermitian systems.Meanwhile,when the generalized Brillouin zone and Brillouin zone are not equal,an extensive number of open-boundary eigenstates are localized on the boundary,a phenomenon known as non-Hermitian skin effect.The spectrum of a non-Hermitian Bloch Hamiltonian can be complex,thus the spectral topology can be defined,which is always trivial in Hermitian counterpart.Although the topological classification of non-Hermitian Bloch Hamiltonian has been established,the physical meaning of the bulk-edge correspondence is still unclear.In the first work,chapter 3 of this thesis,we exactly establish the correspondence between spectral winding number and non-Hermitian skin modes,that is,there are skin modes iff the spectral winding is nonzero,which is also a simple criterion for the appearance of skin effect.The validity of this correspondence depends on the analytical property of the generalized Brillouin zone,that is,the generalized Brillouin zone always encloses the same number of zeros and poles(where the order has been counted).We also show that both the nonzero winding and the presence of skin modes share the common physical origin that is the nonzero current functional under the periodic boundary condition.The numerical errors in solving the open boundary spectrum of some non-Hermitian systems are sensitive to the system size.Therefore,It is difficult to obtain asymptotic spectrum in thermodynamic limit using numerical method.Consequently,the numerically solved generalized Brillouin zone may be quite inaccurate.In the second work,chapter 4 of this thesis,we develop an analytical approach for solving generalized Brillouin zone and open boundary continuum bands,namely the auxiliary generalized Brillouin zone method.First,we relax the generalized Brillouin zone condition into the auxiliary generalized Brillouin zone condition that can be obtained analytically by using the resultant method mathematically.Then,we select a set of analytic arcs that satisfy specified ordering,and these arcs exactly constitute the generalized Brillouin zone.Finally,the open boundary continuum bands can be obtained from the generalized Brillouin zone.This analytical method avoids the issue of intrinsic spectral instability in some non-Hermitian systems,explaining the non-perturbative effect of continuum bands,and reveals sub-generalized Brillouin zones in multiband systems.In one dimension,the non-Bloch band theory based on the generalized Brillouin zone has been established.Especially,the one-dimensional skin effect has been experimentally observed,inspiring further studies on their higher-dimensional generalizations.In the third work,chapter 5 of this thesis,we establish the theorem of universal non-Hermitian skin effect,which reveals the universality of the skin effect in higher dimensions.Furthermore,we classify the universal skin effect into two categories by the current functional,namely,non-reciprocal skin effect and generalized reciprocal skin effect.We propose the corner-skin effect and the geometry-dependent-skin effect as representative phenomena of these two categories,respectively.Finally,a corollary of this theorem can be obtained,that is,all lattice systems with stable exceptional points must have universal skin effect.