Phase transition in theσmodel at finite temperature

We study the phase transition through which the spontaneously broken symmetry of the $\sigma$ model is restored at finite temperature. The methods of nonrelativistic many-body theory, in which the equations of motion are approximated in a self-consistent manner, are applied to the $\sigma$ model in 1 time and $d$ space dimensions for $2<d\le 3$. We consider several different approximations of this type and discuss difficulties associated with their renormalization. The Hartree approximation predicts a second-order transition for all $d$, but breaks down at high temperatures when $d=3\mathrm{}$. The "modified Hartree approximation," a variant of Hartree theory which incorporates more of the effects of thermal fluctuations, predicts a first-order transition for all $d$. This result is shown to be an artifact of the approximation. The $\sigma$ model with $N$ fields [the $\mathrm{O}\mathrm{}\left(N\right)\mathrm{}$ model] is studied in the limit of large $N$. For $2<d<\mathrm{}3\mathrm{}$ this model undergoes a second-order transition whose critical exponents are computed to $O\left(\frac{1}{N}\right)$. When $d=3\mathrm{}$, however, the large-$N$ approximation breaks down at high temperatures.