Critical dynamics within the real-time fRG approach

The Schwinger-Keldysh functional renormalization group (fRG) developed in [1] is employed to investigate critical dynamics related to a second-order phase transition. The effective action of model A is expanded to the order of $O({\mathrm{\partial }}^2)$ in the derivative expansion for the $O(N)$ symmetry. By solving the fixed-point equations of effective potential and wave function, we obtain static and dynamic critical exponents for different values of the spatial dimension $d$ and the field component number $N$. It is found that one has $z\ge 2$ in the whole range of $2\le d\le 4$ for the case of $N=1$, while in the case of $N=4$ the dynamic critical exponent turns to $z<2$ when the dimension approach towards $d=2$.