Non-Hermitian dynamic strings and anomalous topological degeneracy on a non-Hermitian toric-code model with parity-time symmetry
Cui-Xian Guo, Xiaoran Wang, Can Wang, Su-Peng Kou
Abstract
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 $\mathcal{PT}$-symmetric non-Hermitian Hamiltonian with spontaneous $\mathcal{PT}$ 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 ${Z}_{2}$ topological order and its possible physics realization are discussed.