Evolution of static and dynamical density correlations of one-dimensional soft-core bosons from the Tonks-Girardeau limit to a clustering fluid

Repulsive soft-core atomic systems may undergo clustering if their density is high enough that core overlap is unavoidable. In one-dimensional bosonic 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, which 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.

Figure 1

Energy per particle as a function of the density from QMC simulations, in units of the corresponding quantity for a Tonks-Girardeau gas $E_{\text{TG}}/N$, compared with the Tonks-Girardeau model at low density (solid line) and to the mean-field Bogoliubov result $E_B/N$ at high density (dashed line). Inset: PIGS results and Bogoliubov result in a wider range of densities (log-log scale).

Figure 2

Log-log plot of the Luttinger parameter $K_L$ as a function of the density $\rho$, extracted from the low-momentum behavior of the PIGS static structure factor, compared to the Tonks-Girardeau value $K_L=1$ and the Bogoliubov result (dashed line, see text).

Figure 3

Pair distribution function for increasing densities $\rho$ (symbols, PIGS). The distance between particles $r$ is in units of the typical clustering distance $b_c$, to highlight its role at high density. Lines are a guide to the eye. Datasets for $\rho =5.5,17.8,24.6$ are reproduced from Ref. [18].

Figure 4

Static structure factor for increasing densities $\rho$ (symbols, PIGS) compared to its Bogoliubov expression $S_B\left(q\right)$ (solid lines). Momentum is in units of $q_c$ to highlight its role at high density.

Figure 5

Feynman approximation of the dispersion relation ${\varepsilon }_{FA}\left(q\right)$, using the PIGS results for the static structure factor, for increasing densities, compared to the Bogoliubov expression ${\varepsilon }_B\left(q\right)$ (solid lines). Momenta are in units of $q_c$ to highlight its role with increasing density. Datasets for $\rho =5.5,17.8,24.6$ are reproduced from Ref. [18].

Figure 6

Panels (a)–(f): Dynamical structure factors for the densities $\rho =0.1,0.3,0.6,0.8,1.0,1.5,8.2$ from the analytic continuation of PIGS data in units of the corresponding $\hslash /E_F$. The structure factors have been represented by a color map in the $(q,\omega )$ domain, with higher values of $S(q,\omega )$ corresponding to darker shades of green. The figure also shows the curves corresponding to the ideal Fermi gas particle-hole boundaries ${\varepsilon }_{IFG}$ (blue dashed lines) and to the Feynman sum rule ${\varepsilon }_{FA}$ (red dotted line) and Bogoliubov ${\varepsilon }_B$ (purple dash-dotted line) approximations. In panels (c)–(f), an arrow indicates the momentum $q=q_c$ in units of $2k_F$, and ${\varepsilon }_B$ essentially coincides with ${\varepsilon }_{FA}$. Notice that the cutoff of $S(q,\omega )$ is customized for each panel to highlight the dominant features and that panel (f) is magnified to highlight the rotonic region. The dataset for $\rho =0.1$ is reproduced from Ref. [18].

Figure 7

Energy oscillations during the propagation of a 12-particle system at interaction $U=U_{\text{TG}}$ and density $\rho =2.05$, subject to the time-dependent perturbation (9) with amplitude $A={10}^{-4}{E}_c$. The simulations with ${\omega }_{\text{ext}}=1.750,1.850,1.950E_c/\hslash$ show that, as ${\omega }_{\text{ext}}$ approaches $\omega \left(k_{\min }\right)$, the oscillations grow in amplitude and decrease in frequency. If their frequencies ${\omega }_E$ are used to infer the excitation energy, the result for all three simulations is $\omega \left(k_{\min }\right)\simeq 2.062$. Indeed, a simulation with ${\omega }_{\text{ext}}=2.062$ (solid line, only partially shown) exhibits no oscillation during the time of the simulation but rather a steady energy growth.

Figure 8

Excitation energy of the low-lying excited states obtained by MCTDH simulations on a system of 12 particles compared to the Bogoliubov spectrum. Note that these results have been obtained for systems with different densities, which are reported on the upper $x$ axis.