Chiral symmetry and axial U(1) symmetry in finite temperature QCD with domain-wall fermion

We study the restoration of the spontaneously broken chiral symmetry and the anomalously broken axial U(1) symmetry in finite temperature QCD at zero chemical potential. We use 2 flavors lattice QCD with optimal domain-wall fermion on the ${16}^3\times 6$ lattice, with the extent $N_s=16$ in the fifth dimension, in the temperature range $T=130-230$ MeV. To examine the restoration of the chiral symmetry and the axial $U(1)$ symmetry, we use diluted $Z_2$ noises to calculate the chiral condensate, and the chiral susceptibilities in the scalar and pseudoscalar meson channels, for flavor singlet and non-singlet respectively. From the degeneracy of the chiral susceptibilities around $T_c$, it suggests that the axial $U(1)$ symmetry is restored in the chirally symmetric phase. Moreover, we examine the spectral density $\rho ({\lambda }_c)$ of the 4D effective Dirac operator with exact chiral symmetry, which is obtained by computing zero modes plus (180+180) conjugate pairs of low-lying modes for each gauge configuration. The suppression of low modes in the spectral density provides a consistency check of the restoration of axial $U(1)$ symmetry in the chirally symmetric phase.