PT symmetry in the non-Hermitian Su-Schrieffer-Heeger model with complex boundary potentials

We study the parity-and time-reversal (PT) symmetric non-Hermitian Su-Schrieffer-Heeger (SSH) model with two conjugated imaginary potentials +/- i gamma at two end sites. The SSH model is known as one of the simplest two-band topological models which has topologically trivial and nontrivial phases. We find that the non-Hermitian terms can lead to different effects on the properties of the eigenvalues spectrum in topologically trivial and nontrivial phases. In the topologically trivial phase, the system undergos an abrupt transition from the unbroken PT-symmetry region to the spontaneously broken PT-symmetry region at a certain gamma c, and a second transition occurs at another transition point gamma(c)' when further increasing the strength of the imaginary potential gamma. But in the topologically nontrivial phase, the zero-mode edge states become unstable for arbitrary nonzero gamma and the PT symmetry of the system is spontaneously broken, which is characterized by the emergence of a pair of conjugated imaginary modes.