  We investigate the nature of the phase transition between the period-three
charge-density wave and the disordered phase of a hard-boson model proposed in
the context of cold-atom experiments. Building on a density-matrix
renormalization group algorithm that takes full advantage of the hard-boson
constraints, we study systems with up to 9'000 sites and calculate the
correlation length and the wave-vector of the incommensurate short-range
correlations with unprecedented accuracy. We provide strong numerical evidence
that there is an intermediate floating phase far enough from the integrable
Potts point, while in its vicinity, our numerical data are consistent with a
unique transition in the Huse-Fisher chiral universality class.
