  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 $V<V_c$ (ballistic transport), $\delta \simeq 1/2$ at the critical point
$V=V_c$ (diffusive transport), and $\delta=0$ in the localized phase $V>V_c$
(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.
