Scalar O(N) Model at Finite Temperature -- 2PI Effective Potential in Different Approximations

We calculate the two-particle irreducible (2PI) effective potential of the O(N) linear sigma model in 1+1 dimensions. The approximations we use are the next-to-leading order of a 1/N expansion (for arbitrary N) and a kind of "resummed loop approximation" for N=1. We show that the effective potential of the 1/N expansion is convex for N=4 and N=10 whereas it is not for the "loop" expansion and the case N=1 of the 1/N expansion.