Heating the O(N) nonlinear sigma model

The thermodynamics of the O(N) nonlinear sigma model in 1+1 dimensions is studied. We calculate the finite temperature effective potential in leading order in the 1/N expansion and show that at this order the effective potential can be made finite by temperature independent renormalization. We will show that this is not longer possible at next-to-leading order in 1/N. In that case one can only renormalize the minimum of the effective potential in a temperature independent way, which gives us finite physical quantities like the pressure.