We consider a generalization of the Su-Schrieffer-Heeger (SSH) model by including next-nearest-neighbour (NNN) interaction and balanced loss-gain (BLG), and subjecting the whole system to an external uniform magnetic field. We study the band structure, edge states, and persistent current in this extended SSH model under general boundary condition (GBC) of which the periodic, antiperiodic, and open boundary conditions appear as special cases. It is shown that the point band gap decreases with increasing strength of the NNN interaction and vanishes beyond a critical value for both topologically trivial and nontrivial phases. Further, the line gap exhibits closed-loop-like structures for nonvanishing NNN interaction under the periodic boundary condition (PBC). The Zak phase receives no contribution from the NNN interaction under the PBC. We show that the NNN interaction has no effect on the persistent current in the half-filled limit for the case of PBC, while for other fillings less than the half-filling, it enhances the magnitude of the current significantly. We numerically study the variation of the persistent current with respect to the system parameters under the GBC, for the case in which the Hamiltonian admits entirely real spectra, and show that its magnitude increases with the increasing strength of the BLG. We show that the model without the NNN interaction is exactly solvable for a class of GBC, of which PBC, antiperiodic boundary condition (APBC) and anti-Hermitian boundary condition (AHBC) arise as special cases. We obtain analytic expressions for the edge states in the case of open boundary condition (OBC) and AHBC for vanishing NNN interaction. We show numerically for OBC that edge states in the topologically trivial phase appear for nonvanishing NNN interaction only when the strength of the loss-gain term is greater than the modulus of the difference between the intercell and intracell hopping strengths. In the topologically nontrivial phase, the edge states under OBC exist only up to a critical value of the NNN strength and vanish beyond this critical value. The bulk-boundary correspondence (BBC) for unbroken $\mathcal{PT}$ phase is similar to Hermitian SSH model, while the non-Hermitian skin effect (NHSE) is observed for broken $\mathcal{PT}$ phase.

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• Edge localized mode
• Topological materials
• PT-symmetry
• Exact solutions for many-body systems
• Su-Schrieffer-Heeger model