  Non-interacting spinless electrons in one-dimensional quasicrystals,
described by the Aubry-Andr\'{e}-Harper (AAH) Hamiltonian with nearest
neighbour hopping, undergoes metal to insulator transition (MIT) at a critical
strength of the quasi-periodic potential. This transition is related to the
self-duality of the AAH Hamiltonian. Interestingly, at the critical point,
which is also known as the self-dual point, all the single particle wave
functions are multifractal or non-ergodic in nature, while they are ergodic and
delocalized (localized) below (above) the critical point. In this work, we have
studied the one dimensional quasi-periodic AAH Hamiltonian in the presence of
spin-orbit (SO) coupling of Rashba type, which introduces an additional spin
conserving complex hopping and a spin-flip hopping. We have found that,
although the self-dual nature of AAH Hamiltonian remains unaltered, the
self-dual point gets affected significantly. Moreover, the effect of the
complex and spin-flip hoppings are identical in nature. We have extended the
idea of Kohn's localization tensor calculations for quasi-particles and
detected the critical point very accurately. These calculations are followed by
detailed multifractal analysis along with the computation of inverse
participation ratio and von Neumann entropy, which clearly demonstrate that the
quasi-particle eigenstates are indeed multifractal and non-ergodic at the
critical point. Finally, we mapped out the phase diagram in the parameter space
of quasi-periodic potential and SO coupling strength.
