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.