Liquid State of One-Dimensional Bose Mixtures: A Quantum Monte Carlo Study

By using exact quantum Monte Carlo methods we calculate the ground-state properties of the liquid phase in one-dimensional Bose mixtures with contact interactions. We find that the liquid state can be formed if the ratio of coupling strengths between interspecies attractive and intraspecies repulsive interactions exceeds a critical value. As a function of this ratio we determine the density where the energy per particle has a minimum and the one where the compressibility diverges, thereby identifying the equilibrium density and the spinodal point in the phase diagram of the homogeneous liquid. Furthermore, in the stable liquid state, we calculate the chemical potential, the speed of sound, as well as structural and coherence properties, such as the pair correlation function, the static structure factor, and the one-body density matrix, thus providing a detailed description of the bulk region in self-bound droplets.

Figure 1

Energy per particle, in units of half of the binding energy, as a function of the density for different values of the ratio $|\tilde{g}|/g$ of coupling constants. Error bars are smaller than the symbol sizes. The dashed line is the result of the GGP approach at $|\tilde{g}|/g=0.9$ (cyan) and at $|\tilde{g}|/g=0.6$ (yellow).

Figure 2

Phase diagram of the homogeneous liquid phase: (blue) circles correspond to the equilibrium density of the liquid and (red) squares to the spinodal point where the compressibility diverges. Dashed lines refer to the predictions of the GGP theory.

Figure 3

Chemical potential ${\mu }_{\mathrm{eq}}$ in units of half of the dimer binding energy: (blue) circles and right vertical axis. Speed of sound $c_{\mathrm{eq}}$ in units of the Fermi velocity: (red) squares and left vertical axis. The dashed lines correspond to the results of the GGP theory.

Figure 4

Pair correlation function of parallel spins $g_{aa}$ (dashed lines) and antiparallel spins $g_{ab}$ (solid lines) in the liquid for different values of the ratio $|\tilde{g}|/g$ of coupling constants.

Figure 5

Density (full symbols) and magnetic (open symbols) static structure factor in the liquid as a function of $q/k_F$ for different values of the ratio $|\tilde{g}|/g$. Here $k_F=\hslash \pi n/2$ is the Fermi wave vector. Dashed lines correspond to the low-$q$ linear dependence fixed by the compressibility and by the magnetic susceptibility, respectively, for $S_D(q)$ and $S_M(q)$.

Figure 6

Spatial dependence of the OBDM for different values of the ratio $|\tilde{g}|/g$ of coupling constants. Dashed lines are power-law fits $1/x^{\alpha }$ to the long-range behavior. In the inset we report the values of the exponent $\alpha$ obtained from the fit.