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

TL;DR

This paper investigates a PT-symmetric O(N) quantum model, revealing two large-N phases separated by a first-order transition, with all quantum states bound despite classical escape trajectories.

Contribution

It demonstrates the existence of two distinct large-N phases in a PT-symmetric O(N) model and characterizes the phase transition and spectral properties.

Findings

Quantum spectrum consists only of bound states for all N.

Two large-N phases with different scaling behaviors.

First-order phase transition at a specific critical parameter.

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 $3\cdot2^{1/3}$.