  We consider a zero-temperature one-dimensional system of bosons interacting
via the soft-shoulder potential in the continuum, typical of dressed Rydberg
gases. We employ quantum Monte Carlo simulations, which allow for the exact
calculation of imaginary-time correlations, and a stochastic analytic
continuation method, to extract the dynamical structure factor. At finite
densities, in the weakly-interacting homogeneous regime, a rotonic spectrum
marks the tendency to clustering. With strong interactions, we indeed observe
cluster liquid phases emerging, characterized by the spectrum of a composite
harmonic chain. Luttinger theory has to be adapted by changing the reference
lattice density field. In both the liquid and cluster liquid phases, we find
convincing evidence of a secondary mode, which becomes gapless only at the
transition. In that region, we also measure the central charge and observe its
increase towards c = 3/2, as recently evaluated in a related extended
Bose-Hubbard model, and we note a fast reduction of the Luttinger parameter.
For 2-particle clusters, we then interpret such observations in terms of the
compresence of a Luttinger liquid and a critical transverse Ising model,
related to the instability of the reference lattice density field towards
coalescence of sites, typical of potentials which are flat at short distances.
Even in the absence of a true lattice, we are able to evaluate the spatial
correlation function of a suitable pseudo-spin operator, which manifests
ferromagnetic order in the cluster liquid phase, exponential decay in the
liquid phase, and algebraic order at criticality.
