Non-Hermitian Dynamic Strings and Anomalous Topological Degeneracy on non-Hermitian Toric-code Model with Parity-time Symmetry

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.