Phase transition in theσmodel at finite temperature

We study the phase transition through which the spontaneously broken symmetry of the σ 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 σ model in 1 time and d space dimensions for 2<d≤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 $d=3$. 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 σ model with N fields [the O(N) model] is studied in the limit of large N. For $2<d<3$ this model undergoes a second-order transition whose critical exponents are computed to O(1/N). When $d=3$, however, the large-N approximation breaks down at high temperatures.