Non-Hermitian dynamic strings and anomalous topological degeneracy on a non-Hermitian toric-code model with parity-time symmetry

In this paper, with the help of Hermitian/non-Hermitian dynamic strings, a theory of non-Hermitian topological orders is developed based on a non-Hermitian Wen-plaquette model. The effective models for bosonic topological excitations (e particles and m particles) are Hermitian tight-binding lattice models; the effective model for fermionic topological excitations (f particles) becomes a non-Hermitian tight-binding lattice model. In addition, the effective pseudospin model for topologically degenerate ground states is derived by calculating the expectation values of Hermitian/non-Hermitian topologically closed dynamic strings. For the topologically degenerate ground states of the non-Hermitian Wen-plaquette model on even-by-odd, odd-by-even, and odd-by-odd lattices, anomalous topological degeneracy occurs, i. e. , the number of topologically protected ground states may be reduced from 2 to 1. Now, the effective pseudospin 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.