Localization, $\mathcal{PT}$-Symmetry Breaking and Topological Transitions in non-Hermitian Quasicrystals

TL;DR

This paper investigates the relationship between localization, $ ext{PT}$-symmetry breaking, and topological transitions in non-Hermitian quasicrystals, concluding that delocalization-localization and $ ext{PT}$-symmetry breaking are not generally topological phase transitions.

Contribution

It provides an analytical expression for the topological transition point in non-Hermitian $ ext{PT}$-symmetric Aubry-André-Harper models with Rashba Spin-Orbit coupling and clarifies their non-equivalence to localization transitions.

Findings

Delocalization-localization transition occurs earlier than topological transition.

Topological transition points generally do not coincide with $ ext{PT}$-symmetry breaking points.

Analytical expression for the topological transition point derived.

Abstract

According to the topological band theory of a Hermitian system, the different electronic phases are classified in terms of topological invariants, wherein the transition between the two phases characterized by a different topological invariant is the primary signature of a topological phase transition. Recently, it has been argued that the delocalization-localization transition in a quasicrystal, described by the non-Hermitian $\mathcal{PT}$-symmetric extension of the Aubry-Andr\'{e}-Harper (AAH) Hamiltonian can also be identified as a topological phase transition. Interestingly, the $\mathcal{PT}$-symmetry also breaks down at the same critical point. However, in this article, we have shown that the delocalization-localization transition and the $\mathcal{PT}$-symmetry breaking are not connected to a topological phase transition. To demonstrate this, we have studied the non-Hermitian $\mathcal{PT}$-symmetric AAH Hamiltonian in the presence of Rashba Spin-Orbit (RSO) coupling. We have obtained an analytical expression of the topological transition point and compared it with the numerically obtained critical points. We have found that, except in some special cases, the critical point and the topological transition point are not the same. In fact, the delocalization-localization transition takes place earlier than the topological transition whenever they do not coincide.