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Our work constructs a non-Hermitian system with long range hopping under periodic driving. The Hamiltonian has chiral symmetry, which implies that the topological invariant can be defined. Based on the non-Bloch band theory and the Floquet method, relevant operators and topological number can be defined, providing quantitative approaches for studying topological properties. For example, by calculating the non-Bloch time-evolution factor, the Floquet operator, etc., it is found that the topological invariant is determined by the change of the phase of $U^{+}_{\epsilon=0,\pi}(\beta)$ as it moves along the generalized Brillouin zone, corresponding to the emergence of quasi-energy zero mode and $\pi$ mode.
Results show that the topological structure of the static system can be affected by periodic driving significantly. The topological phase boundary of the zero mode can be changed. When there is no periodic driving, there is no $\pi$ mode in the energy spectrum. After the introduction of periodic driving, a gap appears at the quasi-energy $\epsilon=\pi$, inducing a non-trivial $\pi$-mode phase and enriching the topological phase diagram. Further, the next nearest neighbor hopping has a unique effect in this system. It can induce large topological numbers. However, different from the static system, large topological numbers only appear in specific parameter intervals under periodic driving. As the strength of the next nearest neighbor hopping increases, the large topological number phase disappears instead, reflecting the non-monotonic regulation characteristics of the Floquet system. In addition, the introduction of the phase of the next nearest neighbor hopping can change the topological phase boundary, providing new ideas for experimentally regulating topological states.
This research is of significance in the field of topological phase transitions in non-Hermitian systems. Theoretically, it reveals the synergistic effect of long-range hopping and periodic driving, and improves the theoretical framework for the cross-research of long-range and dynamic regulation in non-Hermitian systems. From an application perspective, it provides theoretical support for experimentally realizing the controllable modulation of topological states, which is helpful to promote the development of fields such as low energy consumption electronic devices and topological quantum computing.-
Keywords:
- non-Hermitian /
- Floquet /
- long-range hopping /
- topological phase transition
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