Exponentially fragile PT symmetry in lattices with localized eigenmodes.

We study the effect of localized modes in lattices of size N with parity-time (PT) symmetry. Such modes are arranged in pairs of quasidegenerate levels with splitting delta approximately exp(-N/xi) where xi is their localization length. The level "evolution" with respect to the PT breaking parameter gamma shows a cascade of bifurcations during which a pair of real levels becomes complex. The spontaneous PT symmetry breaking occurs at gammaPT approximately min{delta}, thus resulting in an exponentially narrow exact PT phase. As N/xi decreases, it becomes more robust with gammaPT approximately 1/N2 and the distribution P(gammaPT) changes from log-normal to semi-Gaussian. Our theory can be tested in the frame of optical lattices.