PT-symmetry breaking with divergent potentials: lattice and continuum cases

We investigate the parity- and time-reversal ($\mathcal{PT}$)-symmetry breaking in lattice models in the presence of long-ranged, non-hermitian, $\mathcal{PT}$-symmetric potentials that remain finite or become divergent in the continuum limit. By scaling analysis of the fragile $\mathcal{PT}$ threshold for an open finite lattice, we show that continuum loss-gain potentials $V_\alpha(x)\propto i |x|^\alpha \mathrm{sign}(x)$ have a positive $\mathcal{PT}$-breaking threshold for $\alpha>-2$, and a zero threshold for $\alpha\leq -2$. When $\alpha<0$ localized states with complex (conjugate) energies in the continuum energy-band occur at higher loss-gain strengths. We investigate the signatures of $\mathcal{PT}$-symmetry breaking in coupled waveguides, and show that the emergence of localized states dramatically shortens the relevant time-scale in the $\mathcal{PT}$-symmetry broken region.