One-Dimensional Liquid ${}^4\mathrm{He}$: Dynamical Properties beyond Luttinger-Liquid Theory

We compute the zero-temperature dynamical structure factor of one-dimensional liquid ${}^4\mathrm{He}$ by means of state-of-the-art quantum Monte Carlo and analytic continuation techniques. By increasing the density, the dynamical structure factor reveals a transition from a highly compressible critical liquid to a quasisolid regime. In the low-energy limit, the dynamical structure factor can be described by the quantum hydrodynamic Luttinger-liquid theory, with a Luttinger parameter spanning all possible values by increasing the density. At higher energies, our approach provides quantitative results beyond the Luttinger-liquid theory. In particular, as the density increases, the interplay between dimensionality and interaction makes the dynamical structure factor manifest a pseudo-particle-hole continuum typical of fermionic systems. At the low-energy boundary of such a region and moderate densities, we find consistency, within statistical uncertainties, with predictions of a power-law structure by the recently developed nonlinear Luttinger-liquid theory. In the quasisolid regime, we observe a novel behavior at intermediate momenta, which can be described by new analytical relations that we derive for the hard-rods model.

Figure 1

TLL parameter $K_L$, from the compressibility ${\kappa }_S^{-1}=\rho {\partial }_{\rho }[{\rho }^2{\partial }_{\rho }E(\rho )]$ (blue circles) and the low-$q$ behavior of $S(q)$ (orange triangles). Superimposed lines are described in the text. Inset: Equation of state $E(\rho )$.

Figure 2

Static structure factor $S(q)$ at $\rho =0.22$ and $0.30{\AA }^{-1}$ (red circles and green triangles, respectively). Inset: Scaling of $S(2k_F)$ with $N$ at the same densities (dashed lines, fit to a power law). Values of $K_L$ from $c$ and the scaling of $S(2k_F)$ are reported.

Figure 3

Color plot of $S(q,\omega )$ at several densities and corresponding $K_L$. Feynman approximation ${\omega }_F(q)$ (gray dash-dotted lines) and the free-particle dispersion $\hslash q^2/2m$ (green dotted lines) are drawn for comparison. Panels (a)–(d) show also the bounds ${\omega }^{\pm }(q)$ of the particle-hole band (blue dashed line), while panels (e) and (f) show the bounds ${\omega }_{*}^{\pm }(q)$ of the HR elementary excitations (violet solid line). Panel (f) shows the low-energy threshold ${\omega }_{\text{th}}(q)$ of HR with $K_L=0.125$ (double-dashed line) and momentum ${\mathcal{Q}}_1$ (red arrow). Values of $S(q,\omega )$ beyond the scale are plotted in black.

Figure 4

Analytical nonlinear TLL exponent Eq. (6) for HR with $K_L=0.125$ (solid line) and $\mathrm{PIGS}+\mathrm{GIFT}$ (circles) fitted exponents of ${}^4\mathrm{He}$ at density $\rho =0.3{\AA }^{-1}$.