  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.
