  Non-Hermiticity enriches the contents of topological classification of matter
including exceptional points, bulk-edge correspondence and skin effect. Gain
and loss can be described by imaginary diagonal elements in Hamiltonians and
the topological phase transition for a crosslinked chain in the presence of
such non-Hermiticity is investigated in this work. We obtain the phase diagram
in term of a winding number analytically. The boundaries of the phases coincide
with the surfaces of exceptional points in the parameter space. The
topologically original edge states locating mainly at the joints between
domains of different phases hold on even for the long chain. The non-Hermitian
topological feature can also be reflected by vortex structures in the vector
fields of complex eigenenergies and expected values of Pauli matrices or the
trajectories of these quantities. This model can be implemented in coupled
waveguides or photonic crystals. And the edge states are immune to various
kinds of disorders until the topological phase transition occurs. This work
benefits our insight into the influence of gain and loss on the topological
phase of matter.
