Exponentially FragilePTSymmetry in Lattices with Localized Eigenmodes

Abstract

We study the effect of localized modes in lattices of size $N$ with parity-time ($\mathcal{P}\mathcal{T}$) symmetry. Such modes are arranged in pairs of quasidegenerate levels with splitting $\delta \sim {exp}^{-N/\xi }$ where $\xi$ is their localization length. The level “evolution” with respect to the $\mathcal{P}\mathcal{T}$ breaking parameter $\gamma$ shows a cascade of bifurcations during which a pair of real levels becomes complex. The spontaneous $\mathcal{P}\mathcal{T}$ symmetry breaking occurs at ${\gamma }_{\mathcal{P}\mathcal{T}}\sim \min \{\delta \}$, thus resulting in an exponentially narrow exact $\mathcal{P}\mathcal{T}$ phase. As $N/\xi$ decreases, it becomes more robust with ${\gamma }_{\mathcal{P}\mathcal{T}}\sim 1/N^2$ and the distribution $\mathcal{P}({\gamma }_{\mathcal{P}\mathcal{T}})$ changes from log-normal to semi-Gaussian. Our theory can be tested in the frame of optical lattices.