  We investigate a model of hard-core bosons with infinitely repulsive nearest-
and next-nearest-neighbor interactions in one dimension, introduced by Fendley,
Sengupta and Sachdev in Phys. Rev. B 69, 075106 (2004). Using a combination of
exact diagonalization, tensor network, and quantum Monte Carlo simulations, we
show how an intermediate incommensurate phase separates a crystalline and a
disordered phase. We base our analysis on a variety of diagnostics, including
entanglement measures, fidelity susceptibility, correlation functions, and
spectral properties. According to theoretical expectations, the
disordered-to-incommensurate-phase transition point is compatible with
Berezinskii-Kosterlitz-Thouless universal behaviour. The second transition is
instead non-relativistic, with dynamical critical exponent $z > 1$. For the
sake of comparison, we illustrate how some of the techniques applied here work
at the Potts critical point present in the phase diagram of the model for
finite next-nearest-neighbor repulsion. This latter application also allows to
quantitatively estimate which system sizes are needed to match the conformal
field theory spectra with experiments performing level spectroscopy.
