We study the effect of $\mathcal{PT}$-symmetric imaginary potentials embedded in the two arms of an Aharonov-Bohm interferometer on the transmission phase by finding an exact solution for a concrete tight-binding system. It is observed that the spectral singularity always occurs at $k=\pm \pi /2$ for a wide range of fluxes and imaginary potentials. Critical behavior associated with the physics of the spectral singularity is also investigated. It is demonstrated that the quasispectral singularity corresponds to a transmission maximum and the transmission phase jumps abruptly by $\pi$ when the system is swept through this point. Moreover, we find that there exists a pulselike phase lapse when the imaginary potential approaches the boundary value of the spectral singularity.

• Received 16 September 2014

DOI:https://doi.org/10.1103/PhysRevA.91.012116