Pulsars, a special type of neutron star, often move at very high speed ($\sim 450 \mathrm{km/s}$), referred to as the pulsar kick phenomenon. However, why this high speed occurs is unclear. In this study, starting from realistic cooling model and equation of state, we analyzed the scattering between the neutrinos emitted during the cooling stage of a proto-neutron star and the anomalous axial current induced by the chiral separation effect in the star, and calculated the resulting time-dependent pulsar kick velocity.The result is: for isotropic axial current, the velocity is zero; however, with a 10\% anisotropic component of the axial current and the magnetic field strength $4\times 10^{11} \mathrm{T}$, the velocity is $223 \textrm{km/s}$. This value explains the well-known difference between the observed pulsar kick velocity ($\sim 450 \textrm{km/s}$) and the hydrodynamic simulation of supernova explosions ($\sim 200 \textrm{km/s}$). Hence, for the understanding of neutron star evolution during its early age, quantum anomaly is a crucial mechanism.