Dynamical correlations in one-dimensional 4He beyond Luttinger theory

We study a collection of $^4$He atoms confined to strictly one dimension at zero temperature. We use the exact Path Integral Ground State method to evaluate the equation of state and the radial distribution function, and we find that the system behaves as a Luttinger liquid with parameter $K_L = \hbar \pi \rho / m v$ that takes all the possible values $0<K_L <+\infty$, depending on the density $\rho$, being $m$ the $^4$He mass and $v$ the sound velocity\cite{uno}. Actually the system goes from $K_L \ll 1$ in the high density quasi--solid regime to $K_L \gg 1$ close to the low density spinodal decomposition. By inverting the imaginary--time intermediate scattering function with the Genetic Inversion via Falsification of Theories method\cite{due}, we also evaluate the dynamical structure factor $S(q,\omega)$ in the whole range in $K_L$, exploring the behavior of the dynamical correlations beyond the limits of applicability of Luttinger liquid theory. We find that the famous phonon--maxon--roton excitation spectrum of $^4$He is not present in 1D. On the contrary, $S(q,\omega)$ manifests a particle--hole continuum typical of a fermionic system, as expected from the Bose-Fermi mapping valid for 1D hard-core interactions. In qualitative agreement with recent non--linear Luttinger liquid theories, we find that the main weight of density fluctuations continuously shifts from the lower threshold branch in the quasi--solid regime, to the upper Bogoliubov branch in the compressible low--density regime. At an intermediate density near $\rho = 0.15$ \AA$^{-1}$, the system corresponds to $K_L = 1$ and $S(q,\omega)$ maps to a non interacting Fermi gas at very low energies $\hbar\omega$, while at higher energies display non--universal effects depending on the $^4$He interaction potential.