| The bulk-boundary correspondence is an important principle for studying topological matter.In general,the topological boundary states of open-boundary systems can be described by the topological invariants in Bloch Hamiltonian.However,in some non-Hermitian systems,there is the non-Hermitian skin effect,the eigenstate of open boundary systems are local states on the boundary rather than extended states.This phenomenon is called nonHermitian skin effect.The non-Hermitian skin effect breaks the conventional bulk–boundary correspondenc,and the corresponding non-Hermitian topological boundary states cannot be described by the topological invariants in Bloch Hamiltonian.In order to restore the bulk–boundary correspondence in non-Hermitian systems,people extended the concept of Brillouin zone,a complex-valued wave vector is introduced to construct the generalized Brillouin zone.It is found that the non-trivial boundary states of open-boundary systems can be described faithfully by the topological invariants defined on the generalized Brillouin zone.However,it is very difficult to use generalized Brillouin region to find the topological invariants of analytic form for partial non-Hermitian systems.Therefore,we try to construct and study two special cases of the non-Hermitian Su–Schrieffer–Heeger(SSH)model,in order to find a more convenient method to solve the topological phase.At first,we study topological phases of a non-Hermitian coupled SSH ladder.The model originates from the brick-wall lattices in the two-row limit.The Hamiltonian can be brought into block off-diagonal form and the winding number can be defined with the determine of the block off-diagonal matrix,analytical form of the topological phase transition points are obtained by winding number.We find the determine of the off-diagonal matrix has nothing to do with the interleg hopping of the ladder.So the topological phases of the model are the same as those of the chains.Further numerical simulations verify the analysis.And then,we construct a partner of a non-Hermitian system by getting rid of the nonHermitian skin effect.Through adjusting the imbalance hopping,we find that the existence of zero-energy boundary states still dictate the bulk topological invariants based on the bandtheory framework.Two non-Hermitian SSH models are used to illuminate the ideas.Specially,we obtain the winding numbers in analytical form without the introduction of the generalized Brillouin zone.The work gives an alternative method to calculate the topological invariants of non-Hermitian systems. |
