Edge states and persistent current in a PT-symmetric extended Su-Schrieffer-Heeger model with generic boundary conditions

Abstract: 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, anti-periodic and open boundary conditions appear as special cases. It is shown that the point bandgap decreases with the increasing value of the strength of the NNN interaction and vanish beyond a critical value. Further, the line gap exhibits closed-loop like structures for non-vanishing 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. We show that the model without the NNN interaction is exactly solvable for a class of GBC of which PBC, anti-periodic 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 non-vanishing NNN interaction in the parametric regions where PT-symmetry is broken under PBC. In the topologically non-trivial phase, the edge states under OBC exists only up to a critical value of the NNN strength and vanishes beyond a critical value. The bulk-boundary correspondence(BBC) for unbroken PT-phase is similar to hermitian SSH model, while non-Hermitian skin effect(NHSE) is observed for broken PT-phase.