Two-particle irreducible finite temperature effective potential of the O(N) linear sigma model in 1+1 dimensions at next-to-leading order of 1/N

We study the O(N) linear sigma model in 1+1 dimensions by using the two-particle irreducible formalism of Cornwall, Jackiw and Tomboulis in order to evaluate the effective potential at finite temperature. At next-to-leading order in a 1/N expansion, one has to include the sums over necklace and generalized 'sunset' diagrams. We find that--in contrast to the Hartree approximation--there is no spontaneous symmetry breaking in this approximation, as to be expected for the exact theory. The effective potential becomes convex throughout for all parameter sets which include N=4,10,100, couplings λ=0.1,0.5 and temperatures between 0.3 and 1 (in arbitrary units). The Green's functions obtained by solving the Schwinger-Dyson equations are enhanced in the infrared region. We also compare the effective potential as a function of the external field φ with those obtained in the one-particle irreducible and two-particle point-irreducible expansions