Non-Hermitian systems, going beyond conventional Hermitian systems, have brought in intriguing concepts such as exceptional points and complex spectral topology as well as exotic phenomena such as non-Hermitian skin effects. However, previous studies on non-Hermitian systems predominantly focused on the properties of eigenstates, with rather limited discussions on non-Hermitian dynamic phenomena. Motivated by going beyond the parity-time symmetry in non-Hermitian physics, here we theoretically study a one-dimensional non-Hermitian lattice with the glide-time symmetry. We reveal that the glide-time symmetry leads to unique spectral and topological properties and enables rich dynamic phenomena such as dynamic non-Hermitian skin effects with diverse behaviors. We classify different dynamic phases and show that the model enables remarkable tuning of the dynamics in the bulk as well as at the edge boundaries. These include the directional wave propagation and amplification in the bulk, as well as the wave trapping and the dynamic patterns at the edges. These properties do not appear in conventional non-Hermitian models such as the Hatano-Nelson model and nonreciprocal Su-Schrieffer-Heeger model. With these findings, our work serves as a steppingstone for future research on non-Hermitian topology and dynamics enriched by the lattice symmetry.

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• Open quantum systems
• Topological phases of matter
• Non-Hermitian systems