Phase transition through the splitting of self-dual spectral singularity in optical potentials

We consider optical media, which feature antilinear symmetries. We show that (i) spectral singularities of such media (if any) are always self-dual, i.e., correspond to coherent perfect absorber lasers; (ii) under the change of a system parameter, the self-dual spectral singularity may split into a pair of isolated complex conjugate eigenvalues, which corresponds to an unconventional and overlooked, in the most of the previous studies, scenario of the phase transition (known as parity-time [PT]-symmetry breaking in systems obeying PT symmetry); (iii) if the antilinear symmetry is local, i.e., does not involve any spatial reflection, then no spectral singularity is possible. Our findings are illustrated with several examples including a PT-symmetric bilayer and other complex potentials discussed in recent literature.