The large-N limit of PT-symmetric O(N) models

We study a PT-symmetric quantum-mechanical model with an O(N)-symmetric potential of the form using its equivalent Hermitian form. Although the corresponding classical model has finite-energy trajectories that escape to infinity, the spectrum of the quantum theory is proven to consist only of bound states for all N. We show that the model has two distinct phases in the large-N limit, with different scaling behaviors as N goes to infinity. The two phases are separated by a first-order phase transition at a critical value of the dimensionless parameter m²/g^2/3, given by 3 × 2 ^1/3.