The Large-N Limit of PT-Symmetric O(N) Models

Abstract

We study a $PT$-symmetric quantum mechanical model with an O(N)-symmetric potential of the form $m^2{\vec{x}}^2/2-g({\vec{x}}^2{)}^2/N$ 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 $m^2/g^{2/3}$, given by $3\cdot 2^{1/3}$.

7 pages, 2 figures

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