Critical behavior and dimension crossover of pion superfluidity

Abstract

We investigate the critical behavior of pion superfluidity in the framework of the functional renormalization group (FRG). By solving the flow equations in the SU(2) linear sigma model at finite temperature and isospin density, and making comparison with the fixed point analysis of a general $O(N)$ system with continuous dimension, we find that the pion superfluidity is a second order phase transition subject to an $O(2)$ universality class with a dimension crossover from ${d}_{c}=4$ to ${d}_{c}=3$. This phenomenon provides a concrete example of dimension reduction in thermal field theory. The large-$N$ expansion gives a temperature independent critical exponent $\ensuremath{\beta}$ and agrees with the FRG result only at zero temperature.