Spontaneous Breaking of Classical PT Symmetry

TL;DR

This paper investigates classical trajectories of complex PT-symmetric Hamiltonians, revealing that at certain parameter values, new non-PT-symmetric trajectories emerge, causing unusual period variations.

Contribution

It uncovers the spontaneous breaking of PT symmetry in classical trajectories and identifies new types of trajectories at specific rational parameter values.

Findings

PT-symmetric orbits have unusual period dependence on epsilon

Rapid variation regions linked to non-PT-symmetric trajectories

New trajectories occur only at special rational epsilon values

Abstract

The classical trajectories of the family of complex PT-symmetric Hamiltonians $H=p^2+x^2(ix)^\epsilon$ ($\epsilon\geq0$) form closed orbits. All such complex orbits that have been studied in the past are PT symmetric (left-right symmetric). The periods of these orbits exhibit an unusual dependence on the parameter $\epsilon$. There are regions in $\epsilon$ of smooth behavior interspersed with regions of rapid variation. It is demonstrated that the onset of rapid variation is associated with strange new kinds of classical trajectories that have never been seen previously. These rare kinds of trajectories are not PT symmetric and occur only for special rational values of $\epsilon$.