  We analyze the finite-size scaling of the average gap-ratio and the
entanglement entropy across the many-body localization (MBL) transition in one
dimensional Heisenberg spin-chain with quasi-periodic (QP) potential. By using
the recently introduced cost-function approach, we compare different scenarios
for the transition using exact diagonalization of systems up to 22 lattice
sites. Our findings suggest that the MBL transition in the QP Heisenberg chain
belongs to the class of Berezinskii-Kosterlitz-Thouless (BKT) transition, the
same as in the case of uniformly disordered systems as advocated in recent
studies. Moreover, we observe that the critical disorder strength shows a clear
sub-linear drift with the system-size as compared to the linear drift seen in
random disordered models, suggesting that the finite-size effects in the MBL
transition for the QP systems are less severe than that in the random
disordered scenario. Moreover, deep in the ergodic regime, we find an
unexpected double-peak structure of distribution of on-site magnetizations that
can be traced back to the strong correlations present in the QP potential.
