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

TL;DR

This paper investigates how $\\mathcal{PT}$-symmetric potentials influence the fate of zero modes in finite SSH models, revealing conditions under which zero modes can be recovered or broken, with implications for topological non-Hermitian systems.

Contribution

It clarifies the role of edge mode energies in the recovery of zero modes in finite $\\mathcal{PT}$-symmetric SSH models, connecting infinite system properties to finite system behavior.

Findings

Edge mode energies in infinite systems determine zero mode recovery in finite systems.

Zero mode recovery fails if edge mode energies are non-zero in the thermodynamic limit.

Results are applicable across various experimental platforms for topological non-Hermitian systems.

Abstract

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.