0 Pathologies of the Large - N Limit for RP N − 1 , CP N − 1 , QP N − 1 and Mixed Isovector / Isotensor σ - Models Alan

We compute the phase diagram in the N → ∞ limit for lattice RPN−1, CP and QP σ-models with the quartic action, and more generally for mixed isovector/isotensor models. We show that the N = ∞ limit exhibits phase transitions that are forbidden for any finite N . We clarify the origin of these pathologies by examining the exact solution of the one-dimensional model: we find that there are complex zeros of the partition function that tend to the real axis as N → ∞. We conjecture the correct phase diagram for finite N as a function of the spatial dimension d. Along the way, we prove some new correlation inequalities for a class of N -component σ-models, and we obtain some new results concerning the complex zeros of confluent hypergeometric functions. PACS CODES: 11.15.Pg, 02.30.Gp, 05.50.+q, 11.10.Kk, 11.15.Ex, 64.60.Cn, 64.70.Md, 75.10.Hk.