  We present the first lattice calculation of the nucleon isovector helicity
parton distribution function (PDF) in the framework of large-momentum effective
theory (LaMET) that uses the hybrid scheme with self-renormalization. We use
ensembles generated by the MILC collaboration at lattice spacings
$a=\{0.1207,0.0888,0.0582\}$ fm, with $N_f=2+1+1$ flavors of highly improved
staggered quarks at sea pion mass of $M_{\pi}\approx 315$ MeV. We use
clover-improved action for our valence quarks with nucleon boost momentum
$P_z\approx 1.75$ GeV and high-statistics measurements for the LaMET matrix
elements. We perform an extrapolation to the continuum limit and improve the
handling of systematic errors using renormalization-group resummation (RGR) and
leading-renormalon resummation (LRR). Our final nucleon helicity PDF is
renormalized in the $\overline{\text{MS}}$ scheme at energy scale $\mu=2.0$
GeV. We compare our results with and without the two systematic improvements of
RGR and LRR at each lattice spacing as well as the continuum limit, and we see
that the application of RGR and LRR greatly reduces the systematic errors
across the whole $x$ range. Our continuum results with both RGR and LRR show a
small positive antiquark region for the nucleon helicity PDF as well as a
change of as much as a factor of two in the central values compared to results
with neither RGR or LRR. By contrast, the application of RGR and LRR only
changes the central values by about 5\% in the quark region. We compare our
lattice results with the global fits by the JAM, NNPDF and DSSV collaborations,
and we observe some tension between our results.
