  We investigate the topology and localization of one-dimensional Hermitian and
non-Hermitian Su-Schrieffer-Heeger chains with quasiperiodic hopping
modulations. In the Hermitian case, phase diagrams are obtained by numerically
and analytically calculating various topological and localization characters.
We show the presence of topological extended, intermediate, and localized
phases due to the coexistence of independent topological and localization phase
transitions driven by the quasiperiodic disorder. Unlike the gapless and
localized TAI phase in one-dimensional random disordered systems, we uncover
three types of quasiperiodic-disorder-induced gapped topological Anderson
insulators (TAIs) with extended, intermediate (with mobility edges), and
localized bulk states in this chiral chain. Moreover, we study the
non-Hermitian effects on the TAIs by considering two kinds of non-Hermiticities
from the non-conjugate complex hopping phase and asymmetric hopping strength,
respectively. We demonstrate that three types of TAIs preserve under the
non-Hermitian perturbations with some unique localization and topological
properties, such as the non-Hermitian real-complex and localization transitions
and their topological nature. Our work demonstrates that the disorder-induced
TAIs in Hermitian and non-Hermitian quasiperiodic systems are not tied to
Anderson transitions and have various localization properties.
