Invisible defects in complex crystals

•We show the existence of invisible localized defects in complex crystals.•They turn out to be fully invisible to Bloch waves belonging to any lattice band.•An example of invisible defect is presented for a PT-symmetric Mathieu crystal.

We show that invisible localized defects, i.e. defects that cannot be detected by an outside observer, can be realized in a crystal with an engineered imaginary potential at the defect site. The invisible defects are synthesized by means of supersymmetric (Darboux) transformations of an ordinary crystal using band-edge wavefunctions to construct the superpotential. The complex crystal has an entire real-valued energy spectrum and Bragg scattering is not influenced by the defects. An example of complex crystal synthesis is presented for the Mathieu potential.