PT-breaking threshold in spatially asymmetric Aubry-André and Harper models: Hidden symmetry and topological states

Aubry-Andr\'e-Harper lattice models, characterized by a reflection-asymmetric sinusoidally varying nearest-neighbor tunneling profile, are well known for their topological properties. We consider the fate of such models in the presence of balanced gain and loss potentials $\ifmmode\pm\else\textpm\fi{}i\ensuremath{\gamma}$ located at reflection-symmetric sites. We predict that these models have a finite $\mathcal{PT}$-breaking threshold only for specific locations of the gain-loss potential and uncover a hidden symmetry that is instrumental to the finite threshold strength. We also show that the topological edge states remain robust in the $\mathcal{PT}$-symmetry-broken phase. Our predictions substantially broaden the possible experimental realizations of a $\mathcal{PT}$-symmetric system.