Phase transitions in a non-Hermitian Aubry-André-Harper model

The Aubry-Andr\'e-Harper model provides a paradigmatic example of aperiodic order in a one-dimensional lattice displaying a delocalization-localization
phase transition at a finite critical value $V_c$ of the quasiperiodic potential amplitude $V$.
In terms of dynamical behavior of the system, the phase transition is
discontinuous when one measures the quantum diffusion exponent $\delta$ of wave packet spreading, with $\delta=1$ in the delocalized phase $VV_c$ (dynamical localization). However, the phase transition turns out to be smooth when one measures, as a dynamical variable, the speed $v(V)$ of excitation transport in the lattice, which is a continuous function of potential amplitude $V$ and vanishes as the localized phase is approached. Here we consider a non-Hermitian extension of the Aubry-Andr\'e-Harper model, in which hopping along the lattice is asymmetric, and show that the dynamical localization-delocalization transition is discontinuous not only in the diffusion exponent $\delta$, but also in the speed $v$ of ballistic transport. This means that, even very close to the spectral phase transition point, rather counter-intuitively ballistic transport with a finite speed is allowed in the lattice. Also, we show that the ballistic velocity can increase as $V$ is increased above zero, i.e. surprisingly disorder in the lattice can result in an enhancement of transport.