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Fate of zero modes in a finite Su-Schrieffer-Heeger Model with $\mathcal{PT}$ Symmetry

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.