Systematic pathway to ${\mathcal{PT}}$ symmetry breaking in scattering systems

TL;DR

This paper introduces a non-local current as an invariant that acts as an order parameter for the transition from unbroken to broken T-symmetry in scattering systems, extending the phase diagram to higher dimensions.

Contribution

It derives a non-local current invariant that characterizes T-symmetry breaking, enabling analysis beyond eigenstates to higher-dimensional scattering states.

Findings

Identifies a non-local current as an order parameter for T-symmetry breaking.

Extends the phase diagram to include non-eigenstate scattering states.

Provides a framework for analyzing T-symmetry transitions in higher-dimensional systems.

Abstract

Recently [Phys. Rev. Lett. {\bf 106}, 093902 (2011)] it has been shown that $\mathcal{PT}$-symmetric scattering systems with balanced gain and loss, undergo a transition from $\mathcal{PT}$-symmetric scattering eigenstates, which are norm preserving, to symmetry broken pairs of eigenstates exhibiting net amplification and loss. In the present work we derive the existence of an invariant non-local current which can be directly associated with the observed transition playing the role of an "order parameter". The use of this current for the description of the $\mathcal{PT}$-symmetry breaking allows the extension of the known phase diagram to higher dimensions incorporating scattering states which are not eigenstates of the scattering matrix.