  In this paper, with the help of Hermitian/non-Hermitian dynamic strings, the
theory of non-Hermitian topological order is developed based on a non-Hermitian
Wen-plaquette model. The effective models for bosonic topological excitations
(e-particle and m-particle) are Hermitian tight-binding lattice model; the
effective model for fermionic topological excitation (f-particle) becomes a
non-Hermitian tight-binding lattice model. In addition, the effective
pseudo-spin model for topologically degenerate ground states is derived by
calculating the expectation values of Hermitian/non-Hermitian topological
closed dynamic strings. For the topologically degenerate ground states of
non-Hermitian Wen-plaquette model on an even-by-odd, odd-by-even and odd-by-odd
lattice, anomalous topological degeneracy occurs, i.e., the number of the
topologically protected ground states may be reduced from 2 to 1. Now, the
effective pseudo-spin model turns into the typical PT-symmetric non-Hermitian
Hamiltonian with spontaneous PT-symmetry breaking. At exceptional points, the
topologically degenerate ground states merge with each other and the
topological degeneracy turns into non-Hermitian degeneracy. In the end, the
application of the non-Hermitian Z2 topological order and its possible physics
realization are discussed.
