Chiral symmetry at finite temperature: Linear versus nonlinearσmodels

The linear $\mathrm{O}\mathrm{}\left(N\right)\mathrm{}$ $\sigma$ model undergoes a symmetry-restoring phase transition at finite temperature. We show that the nonlinear $\mathrm{O}\mathrm{}\left(N\right)\mathrm{}$ $\sigma$ model also undergoes a symmetry-restoring phase transition; the critical temperatures are the same when the linear model is treated in the mean field approximation and the nonlinear model is treated to leading plus subleading order in the $\frac{1}{N}$ expansion. We also carefully define and study the behavior of $f_{\pi }$ and the scalar condensate at low temperatures in both models, showing that they are independent of field redefinition.