The Aubry-André-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 the dynamical behavior of the system, the phase transition is discontinuous when one measures the quantum diffusion exponent δ of wave-packet spreading, with δ = 1 in the delocalized phase V < V c (ballistic transport), δ 1/2 at the critical

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... nt V = V c (diffusive transport), and δ = 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-André-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 δ, but also in the speed v of ballistic transport. This means that even very close to the spectral phase transition point, rather counterintuitively, 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.