Veiled symmetry of disordered Parity-Time lattices: protected PT-threshold and the fate of localization

Open, non-equilibrium systems with balanced gain and loss, known as parity-time ([Formula: see text])-symmetric systems, exhibit properties that are absent in closed, isolated systems. A key property is the [Formula: see text]-symmetry breaking transition, which occurs when the gain-loss strength, a measure of the openness of the system, exceeds the intrinsic energy-scale of the system. We analyze the fate of this transition in disordered lattices with non-Hermitian gain and loss potentials ±iγ at reflection-symmetric sites. Contrary to the popular belief, we show that the [Formula: see text]-symmetric phase is protected in the presence of a periodic disorder which leads to a positive [Formula: see text]-symmetry breaking threshold. We uncover a veiled symmetry of such disordered systems that is instrumental for the said protection, and show that this symmetry leads to new localization behavior across the [Formula: see text]-symmetry breaking transition. We elucidate the interplay between such localization and the [Formula: see text]-symmetry breaking phenomena in disordered [Formula: see text]-symmetric lattices, with Hermitian disorder or gain-loss disorder, and support our conclusions with a beampropagation- method analysis. Our theoretical predictions provide avenues for experimental realizations of -symmetric systems with engineered disorder