Correlated Dirac eigenvalues around the transition temperature on $N_τ=8$ lattices
We investigate the criticality of chiral phase transition manifested in the first and second order derivatives of Dirac eigenvalue spectrum with respect to light quark mass in (2+1)-flavor lattice QCD. Simulations are performed at temperatures from about 137 MeV to 176 MeV on $N_τ=8$ lattices using the highly improved staggered quarks and the tree-level improved Symanzik gauge action. The strange quark mass is fixed to its physical value $m_s^{\text{phy}}$ and the light quark mass is set to $m_s^{\text{phy}}/40$ which corresponds to a Goldstone pion mass $m_π=110$ MeV. We find that in contrast to the case at $T\simeq 205$ MeV $m_l^{-1} \partial ρ(λ, m_l)/\partial m_l$ is no longer equal to $\partial ^2ρ(λ, m_l)/\partial m_l^2$ and $\partial ^2ρ(λ, m_l)/\partial m_l^2$ even becomes negative at certain low temperatures. This means that as temperature getting closer to $T_c$ $ρ(λ, m_l)$ is no longer proportional to $m_l^2$ and thus dilute instanton gas approximation is not valid for these temperatures. We demonstrate the temperature dependence can be factored out in $\partial ρ(λ, m_l)/ \partial m_l$ and $\partial^2 ρ(λ, m_l)/ \partial m_l^2$ at $T \in [137, 153]$ MeV, and then we propose a feasible method to estimate the power $c$ given $ρ\propto m_l^{c}$.