  We study localization and many-body localization transition in one
dimensional systems in the presence of deterministic quasi-periodic potential.
We use single-particle excitations obtained through single-particle Green's
function in real space to characterize the localization to delocalization
transition. A single parameter scaling analysis of the ratio of the typical to
average value of the local density of states (LDOS) of single particle
excitations shows that the critical exponent with which the correlation length
$\xi$ diverges at the transition point $\xi \sim |h-h_c|^{-\nu}$, coming from
the localized side, satisfies the inequality $\nu \ge 1$ for the
non-interacting Aubry-Andre (AA) model. For the interacting system with AA
potential, we study single particle excitations produced in highly excited
many-body eigenstates across the MBL transition and found that the critical
exponent obtained from finite-size scaling of the ratio of the typical to
average value of the LDOS satisfies $\nu \ge 1$ here as well. This analysis of
local density of states shows that the localization and MBL transition in
systems with quasi-periodic potential belong to a different universality class
than the localization and MBL transition in systems with random disorder where
$\nu \ge 2$. In complete contrast to this, finite-size scaling of the level
spacing ratio is known to support the same universality class for MBL
transitions in systems with quasiperiodic as well as random disorder
potentials. For the interacting systems with quasiperiodic potentials, though
finite-size scaling of the level spacing ratio shows a transition at
$h_c^{lsr}$ which is close to the transition point obtained from LDOS within
numerical precision, the critical exponent obtained from finite-size scaling of
level spacing ratio is $\nu \sim 0.54$ in close similarity to the MBL systems
with random disorder.
