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.