PT symmetry and large-N models

TL;DR

This paper explores PT-symmetric quantum models, demonstrating their large-N limits, spectral properties, and connections to Hermitian models, with implications for scalar field theories in higher dimensions.

Contribution

It introduces methods to analyze PT-symmetric matrix and vector models in the large-N limit, including explicit Hermitian forms and spectral properties.

Findings

Large-N limit exists for a wide class of PT-symmetric matrix models

Ground state energies show rapid convergence to the large-N limit

Explicit isospectral Hermitian matrix models are constructed

Abstract

Recently developed methods for PT-symmetric models can be applied to quantum-mechanical matrix and vector models. In matrix models, the calculation of all singlet wave functions can be reduced to the solution a one-dimensional PT-symmetric model. The large-N limit of a wide class of matrix models exists, and properties of the lowest-lying singlet state can be computed using WKB. For models with cubic and quartic interactions, the ground state energy appears to show rapid convergence to the large-N limit. For the special case of a quartic model, we find explicitly an isospectral Hermitian matrix model. The Hermitian form for a vector model with O(N) symmetry can also be found, and shows many unusual features. The effective potential obtained in the large-N limit of the Hermitian form is shown to be identical to the form obtained from the original PT-symmetric model using familiar constraint field methods. The analogous constraint field prescription in four dimensions suggests that PT-symmetric scalar field theories are asymptotically free.