  We study the axial U(1) symmetry at finite temperature in two-flavor lattice
QCD. Employing the Mobius domain-wall fermions, we generate gauge
configurations slightly above the critical temperature Tc with different
lattice sizes L = 2-4 fm. Our action allows frequent topology tunneling while
keeping good chiral symmetry close enough to that of overlap fermions. This
allows us to recover full chiral symmetry by an overlap/domain-wall
reweighting. Above the phase transition, a strong suppression of the low-lying
modes is observed in both of overlap and domain-wall Dirac spectra. We,
however, find a sizable violation of the Ginsparg-Wilson relation in the Mobius
domain-wall Dirac eigenmodes, which dominates the signals of the axial U(1)
symmetry breaking near the chiral limit. We also find that the use of overlap
fermion only in the valence sector is dangerous since it suffers from the
artifacts due to partial quenching. Reweighting the Mobius domain-wall fermion
determinant to that of the overlap fermion, we observe the axial U(1) breaking
to vanish in the chiral limit, which is stable against the changes of the
lattice volume and lattice spacing.
