  In this note we discuss the phase space of the O(2N) vector model in the
presence of a quadratic and a quartic interaction by writing the large-N
effective potential using large charge methods in dimensions 2<D<4 and 4<D<6.
Based on a simple discussion of the convexity properties of the grand
potential, we find very different behavior in the two regimes: while in 2<D<4,
the theory is well-behaved, the model in 4<D<6 leads to a complex CFT in the
UV, consistently with earlier results. We also find a new metastable massive
phase in the high-energy regime for the theory on the cylinder.
