Origin of branch points in the spectrum of PT -symmetric periodic potentials

There exist multiple branch points in the energy spectrum for some PT -symmetric periodic potentials, where the real eigenvalues turn into complex ones. By studying the transmission amplitude for a localized complex potential, we elucidate the physical origin of the breakdown of the perturbation method and Born approximation. Most importantly, we derive an analytic criterion to determine why, when, and where the bifurcation will occur.