  Repulsive soft-core atomic systems may undergo clustering if their density is
high enough that core overlap is unavoidable. In one-dimensional quantum
systems, it has been shown that this instability triggers a transition from a
Luttinger liquid to various cluster Luttinger liquids. Here, we focus on the
Luttinger liquid regime and theoretically study the evolution of key
observables related to density fluctuations, that manifest a striking
dependence on density. We tune the interaction so that the low-density regime
corresponds to a Tonks-Girardeau gas, and show that as the density is increased
the system departs more and more from Tonks-Girardeau behavior, displaying a
much larger compressibility as well as rotonic excitations that finally drive
the clustering transition. We compare various theoretical approaches, which are
accurate in different regimes. Using quantum Monte Carlo methods and analytic
continuation as a benchmark, we investigate the regime of validity of the
mean-field Bogoliubov and the real-time multiconfiguration time-dependent
Hartree-Fock approaches. Part of the behavior that we describe should be
observable in ultracold Rydberg-dressed gases, provided that system losses are
prevented.
