  Due to the boundary coupling in a finite system, the zero modes of a standard
Su-Schrieffer-Heeger (SSH) model may deviate from exact-zero energy. A recent
experiment has shown that by increasing the system size or altering gain or
loss strength of the SSH model with parity-time ($\mathcal{PT}$) symmetry, the
real parts of the energies of the edge modes can be recovered to exact-zero
value [Song \emph{et al.} Phys. Rev. Lett. \textbf{123}, 165701 (2019)]. To
clarify the effects of $\mathcal{PT}$-symmetric potentials on the recovery of
the nontrivial zero modes, we study the SSH model with $\mathcal{PT}$-symmetric
potentials of different forms in both infinite and finite systems. Our results
indicate that the energies of the edge modes in the infinite size case decide
whether or not the success of the recovery of the zero modes by tuning the
strength of $\mathcal{PT}$-symmetric potential in a finite system. If the
energies of the edge modes amount to zero in the thermodynamic limit under an
open boundary condition (OBC), the recovery of the zero modes will break down
by increasing the gain or loss strength for a finite system. Our results can be
easily examined in different experimental platforms and inspire more insightful
understanding on nontrivial edge modes in topologically non-Hermitian systems.

Quantum Physics;Mesoscale and Nanoscale Physics