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

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.