  Bulk boundary correspondence is crucial to topological insulator as it
associates the boundary states (with zero energy, chiral or helical) to
topological numbers defined in bulk. The application of this correspondence
needs a prerequisite condition which is usually not mentioned explicitly: the
boundaries themselves cannot alter the bulk states, so as to the topological
numbers defined on them. In non-hermitian models with fractional winding
number, we prove that such precondition fails and the bulk boundary
correspondence is cut out. We show that, as eliminating the hopping between the
boundaries to simulate the evolution of a system from the periodic boundary
condition to the open boundary condition, exceptional points must be passed
through and the topological structure of the spectrum has been changed. This
makes the topological structures of a chain with open boundary totally
different from that without the boundary. We also argue 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 boundary.
