modified Haldane model, pseudo-Hermitian and anti-PT symmetries, topological phase transitions, long-rang hoppings, gain-loss effects, Physics, QC1-999

We investigate a non-Hermitian modified Haldane model on a honeycomb lattice incorporating third nearest neighbor hopping t3. The results indicate that the system satisfies the pseudo-Hermitian and anti-PT symmetries, ensuring the reality and orthogonality of the eigenstates. We present the phase diagrams in the m/t2-ϕ and m/t2-t3 planes to elucidate the topological phases of the system. For specific values of t3, the system reveals Chern insulating phases characterized by Chern numbers ±2 and ±1, alongside trivial insulating phases. Upon introducing gain and loss γ in the on-site energy, additional phases with Chern number ±1 emerge between the two Chern insulating phases. The edge states possess topological properties and their number corresponds to the value of the Chern number by calculating and analyzing the edge states of a semi-infinite honeycomb lattice. Due to symmetry breaking caused by the truncation of the real-space lattice, the edge states acquire a significant amount of imaginary energy, while most of the bulk state energies remain almost real. Our work enhances understanding of the influence of long-range hoppings and gain-loss effects on the topological phases of non-Hermitian modified Haldane models.