Mobility edges in $\mathcal{PT}$-symmetric cross-stitch flat band lattices

Abstract: We study the cross-stitch flat band lattice with a $\mathcal{PT}$-symmetric on-site potential and uncover mobility edges with exact solutions. Furthermore, we study the relationship between the $\mathcal{PT}$ symmetry broken point and the localization-delocalization transition point, and verify that mobility edges in this non-Hermitian model is available to signal the $\mathcal{PT}$ symmetry breaking.