Localization, PT symmetry breaking, and topological transitions in non-Hermitian quasicrystals

Abstract

The topological phase transition in a Hermitian system is associated with a change in the topological invariant that characterizes the band structure of the two distinct phases. Recently, the delocalization-localization transition in a quasicrystal described by the non-Hermitian $\mathcal{PT}$-symmetric extension of the Aubry-Andr\'e-Harper (AAH) Hamiltonian has been related to a topological phase transition. Interestingly, the $\mathcal{PT}$ symmetry also breaks down at the same point. In this article, we have shown that the delocalization-localization transition and the breaking of $\mathcal{PT}$ symmetry are not connected to a topological phase transition. We have studied the non-Hermitian $\mathcal{PT}$-symmetric AAH Hamiltonian in the presence of Rashba spin-orbit coupling. It has been demonstrated that except in some particular cases, the delocalization-localization and the topological transition points are not identical. In fact, it precedes the topological transition point whenever they do not coincide. However, the breaking of $\mathcal{PT}$ symmetry occurs simultaneously with the delocalization-localization transition. Moreover, we have discovered that the Lyapunov exponent, which is often used as a primary signature to characterize the delocalization-localization transition, does not always determine it correctly. The Lyapunov exponent rather identifies the topological transition point in quasicrystals.