Symmetries and Topological Phase Transitions Modified Haldane Models with Long-Range Hoppings and Gain-Loss Effects

Xiao-Cui Chen ¹ Xiao-Feng Nie ¹ Ye-Wei-Yi Li ¹ Wen-Xue Cui ^1,2*

• ¹ Yanbian University, Yanji, China
• ² Fudan University, Shanghai, Shanghai Municipality, China

We investigate a non-Hermitian modified Haldane model on a honeycomb lattice incorporating third nearest neighbor hopping t 3 . We demonstrate that the system satisfies the pseudo-Hermitian and anti-PT symmetries, ensuring the reality and orthogonality of the eigenstates. To elucidate our findings on the topological phases, we present the phase diagrams in the m/t 2 -φ and m/t 2 -t 3 planes. For specific values of t 3 , the system reveals Chern insulating phases characterized by Chern numbers ±2 and ±1, alongside trivial insulating phases. Notably, upon introducing gain and loss γ in the on-site energy, we observe the emergence of additional phases with Chern number ±1 situated between the two Chern insulating phases. Furthermore, we explore the characteristics of a semi-infinite honeycomb lattice ribbon geometry and illustrate the edge state properties of the system. This work enhances our understanding of the influence of long-range hopping on the topological phases of non-Hermitian modified Haldane models.