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. 
P.A. Kalozoumis, G. Pappas, F.K. Diakonos and P. Schmelcher
physics.optics quant-ph