  We investigate the transition from the many-body localized phase to the
ergodic thermalized phase at an infinite temperature in an $XY$ spin chain with
$L$ spins, which experiences power-law decaying interactions in the form of
$V_{ij}\propto1/\left|i-j\right|^{\alpha}$ ($i,j=1,\cdots,L$) and a random
transverse field. By performing large-scale exact diagonalization for the chain
size up to $L=18$, we systematically analyze the energy gap statistics,
half-chain entanglement entropy, and uncertainty of the entanglement entropy of
the system at different interaction exponents $\alpha$. The finite-size
critical scaling allows us to determine the critical disorder strength $W_{c}$
and critical exponent $\nu$ at the many-body localization phase transition, as
a function of the interaction exponent $\alpha$ in the limit
$L\rightarrow\infty$. We find that both $W_{c}$ and $\nu$ diverge when $\alpha$
decreases to a critical power $\alpha_{c}\simeq1.16\pm0.17$, indicating the
absence of many-body localization for $\alpha<\alpha_{c}$. Our result is useful
to resolve the contradiction on the critical power found in two previous
studies, $\alpha_{c}=3/2$ from scaling argument in Phys. Rev. B \textbf{92},
104428 (2015) and $\alpha_{c}\approx1$ from quantum dynamics simulation in
Phys. Rev. A \textbf{99}, 033610 (2019).
