Discrete solitons in self-defocusing systems with $\mathcal{PT}$-symmetric defects

We construct families of discrete solitons (DSs) in an array of self-defocusing waveguides with an embedded $\mathcal{PT}$ (parity-time)-symmetric dimer, which is represented by a pair of waveguides carrying mutually balanced gain and loss. Four types of states attached to the embedded defect are found, namely, staggered and unstaggered bright localized modes and gray or anti-gray DSs. Their existence and stability regions expand with the increase of the strength of the coupling between the dimer-forming sites. The existence of the gray and staggered bright DSs is qualitatively explained by dint of the continuum limit. All the gray and anti-gray DSs are stable (some of them are unstable if the dimer carries the nonlinear $\mathcal{PT}$ symmetry, represented by balanced nonlinear gain and loss; in that case, the instability does not lead to a blowup, but rather creates oscillatory dynamical states). The boundary between the gray and anti-gray DSs is predicted in an approximate analytical form.