Topology and $\mathcal{PT}$ Symmetry in a Non-Hermitian Su-Schrieffer-Heeger Chain with Periodic Hopping Modulation

TL;DR

This paper investigates how periodic hopping modulation and non-Hermitian effects influence the topological phases and edge states of a Su-Schrieffer-Heeger chain, revealing complex eigenvalue behaviors and localization properties.

Contribution

It introduces a detailed analysis of PT symmetry breaking and topological phase transitions in a non-Hermitian SSH model with periodic hopping modulation, highlighting new eigenvalue and localization phenomena.

Findings

In-gap states can have real or imaginary eigenvalues depending on parameters.

Localization of edge states varies with hopping modulation and dissipation strength.

Bulk states localize at the maximally dimerized limit, influenced by dissipation.

Abstract

We study the effect of periodic but commensurate hopping modulation on a Su-Schrieffer-Heeger (SSH) chain with an additional onsite staggered imaginary potential. Such dissipative, non-Hermitian (NH) extension amply modifies the features of the topological trivial phase (TTP) and the topological nontrivial phase (TNP) of the SSH chain, more so with the periodic hopping distribution. Generally a weak potential can respect the parity-time (PT ) symmetry keeping the energy eigenvalues real, while a strong potential breaks PT conservation leading to imaginary end state and complex bulk state energies in the system. We find that this PT breaking with imaginary potential strength \gamma show interesting dependence on the hopping modulation \Delta for different hoping modulations. In-gap states, that appear also in the \gamma = 0 limit, take either purely real or purely imaginary eigenvalues depending on the strength of both \gamma and \Delta. The localization of end states (in-gap states) at the boundaries are investigated which show extended nature not only near topological transitions (further away from |\Delta/t| = 1) but also near the unmodulated limit of \Delta = 0. Moreover, localization of the bulk states is observed at the maximally dimerized limit of |\Delta/t| = 1, which also have a {\gamma} dependence. Analyzing further the dissipation caused by the complex eigenvalues in this problem with different hopping periodicity can be essential in modulating the gain-loss contrast in optical systems or in designing various quantum information processing and storage devices.