Embedded states in the continuum for $\mathcal{P}\mathcal{T}$-symmetric systems

We introduce the novel concept of a bound state in the continuum (BIC) for a binary lattice satisfying the $\mathcal{P}\mathcal{T}$ symmetry condition. We show how to build such state and the local potential necessary to sustain it. We find that an appropriate choice of the envelope function can bring the system from a $\mathcal{P}\mathcal{T}$-symmetric phase into a Hamiltonian one. For more general envelope functions, the BIC can still be created but the bounded state will force the system to undergo the $\mathcal{P}\mathcal{T}$ symmetry breaking transition.