Theta-vacuum and large N limit in CPN−1 σ models

The θ dependence of the vacuum energy density in CPN−1 models is re-analysed in the semiclassical approach, the 1/N expansion and arguments based on the nodal structure of vacuum wavefunctionals. The 1/N expansion is shown not to be in contradiction with instanton physics at finite (spacetime) volume V. The interplay of large volume V and large N parameter gives rise to two regimes with different θ dependence, one behaving as a dilute instanton gas and the other dominated by the traditional large N picture, where instantons reappear as resonances of the one-loop effective action, even in the absence of regular instantonic solutions. The realms of the two regimes are given in terms of the mass gap m by m2V≪N and m2V≫N, respectively. The small volume regime m2V≪N is relevant for physical effects associated to the physics of the boundary, like the leading rôle of edge states in the quantum Hall effect, which, however, do not play any rôle in the thermodynamic limit at large N. Depending on the order in which the limits N→∞ and V→∞ are taken, two different theories are obtained; this is the hallmark of a phase transition at 1/N=0.