  We study a 1D system with a power-law quasiparticle dispersion $\propto
|k|^\alpha\sign k$ in the presence of a short-range-correlated random potential
and demonstrate that for $\alpha<1/2$ it exhibits a disorder-driven quantum
phase transition with the critical properties similar to those of the
localisation transition near the edge of the band of a semiconductor in high
dimensions, studied in Refs. 1 and 2. Despite the absence of localisation in
the considered 1D system, the disorder-driven transition manifests itself, for
example, in a critical form of the disorder-averaged density of states. We
confirm the existence of the transition by numerical simulations and find the
critical exponents and the critical disorder strength as a function of
$\alpha$. The proposed system thus presents a convenient platform for numerical
studies of the recently predicted unconventional high-dimensional localisation
effects and has the potential for experimental realisations in chains of
ultracold atoms in optical traps.
