One-dimensional liquid $^4$He beyond Luttinger theory

We investigate zero temperature liquid $^4$He in strictly one dimension by means of state-of-the-art Quantum Monte Carlo and analytic continuation techniques. The system displays the unique feature of spanning all the possible values of the Tomonaga-Luttinger liquid parameter $K_L$ by only changing the density. We explore the behavior of the dynamical structure factor $S(q,\omega)$ beyond the limits of applicability of Tomonaga-Luttinger liquid theory in the whole range in $K_L$. We observe a crossover from a weakly interacting Bose gas regime at low density ($K_L\gg 1$) to a quasi-solid regime at high density ($K_L\ll 1$), which we interpret in terms of novel analytical expressions for the spectrum of hard-rods. During this transition the interplay between dimensionality and interaction makes $S(q,\omega)$ manifest a pseudo particle-hole continuum typical of a fermionic system, while the Bogoliubov mode evolves into a remnant of the roton mode. We provide also a perturbative estimation of the drag force experienced by a soft impurity moving along the system.