Critical Behavior and Dimension Crossover of Pion Superfluidity

We investigate the critical behavior of pion superfluidity in the frame of functional renormalization group. By solving the flow equation in the SU(2) linear sigma model with pion superfluidity at finite temperature and isospin density, and making comparison with the fixed point analysis of the flow diagram in a general $O(N)$ model at zero temperature and density but with continuous dimension, the pion superfluidity is a second order phase transition subject to a $O(2)$ universality class with a dimension crossover from $d_{\text{eff}}=4$ to $d_{\text{eff}}=3$. The usually used large $N$ expansion gives a temperature independent critical exponent $\beta$ and agrees with the renormalization group only at zero temperature.