Why does bulk boundary correspondence fail in some non-hermitian topological models

The bulk-boundary correspondence is crucial to topological insulators. It associates the existence of boundary states (with zero energy and possessing chiral or helical properties) with the topological numbers defined in bulk. In recent years, topology has been extended to non-hermitian systems, opening a new research area called non-hermitian topological insulator. In this paper, however, we will illustrate that the bulk-boundary correspondence does not hold in these new models. This is because a prerequisite condition: 'the boundaries cannot alter most of the bulk states, so as to the topological numbers defined on them' does not hold any longer. This cuts out the correspondence between the topological numbers and the boundary states. We will illustrate that, as approaching the open boundary condition by eliminating the strength of the hopping between the two ends of a chain, a new series of exceptional points must be passed through and the topological structure of the spectrum in the complex plane has been changed. This makes the spectrum topology different for the chains with and without boundaries. We also discuss that such exotic behavior does not emerge when the open boundary is replaced by a domain-wall. So the index theorem can be applied to the systems with domain-walls but cannot be further used to those with open boundaries. (paper)