Evidence of effective axial U(1) symmetry restoration at high temperature QCD

We study the axial $U(1)$ symmetry at a finite temperature in two-flavor lattice QCD. Employing the M\"obius domain-wall fermions, we generate gauge configurations slightly above the critical temperature ${T}_{c}$ with different lattice sizes $L=2--4\text{ }\text{ }\mathrm{fm}$. 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 overlap and domain-wall Dirac spectra. We, however, find a sizable violation of the Ginsparg-Wilson relation in the M\"obius 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 the overlap fermion only in the valence sector is dangerous since it suffers from the artifacts due to partial quenching. Reweighting the M\"obius 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.