Universal scaling of density and momentum distributions in Lieb-Liniger gases

We present an exact numerical study of the scaling of density and momentum distribution functions of harmonically trapped one-dimensional bosons with repulsive contact interactions at zero and finite temperatures. We use path integral quantum Monte Carlo with worm updates in our calculations at finite interaction strengths, and the Bose-Fermi mapping in the Tonks-Girardeau regime. We discuss the homogeneous case and, within the local density approximation, use it to motivate the scaling in the presence of a harmonic trap. For the momentum distribution function, we pay special attention to the high momentum tails and their $k^{-4}$ asymptotic behavior.

Figure 1

Worm algorithm results for the ground-state kinetic and interaction energies (see text) of the Lieb-Liniger model (circles) compared with the exact analytical results (lines) [5]. The statistical errors (shown within the circles) can be seen to be much smaller than the circle sizes. The average number of particles in the system is $N_b\approx 20$.

Figure 2

(a) Ground-state one-particle correlations and (b) momentum distribution functions of bosons for $\gamma =8.33$. Results are reported for systems with the same density but different average number of particles. Dashed lines are the results obtained using the worm algorithm and solid lines the results obtained in numerical calculations via algebraic Bethe ansatz [24].

Figure 3

Density profiles (a) and momentum distribution functions (b) of the Tonks-Girardeau gas as calculated using the lattice approach (red solid lines) and in the continuum using the worm algorithm (dashed lines). The average number of particles is 5. Results are presented for different temperatures: $\tilde{T}\equiv k_{B}T/\hslash \omega =0.1$, 5, 10. Thin dotted lines indicate the $k^{-4}$ asymptotic behavior of the momentum tails. The statistical errors in the worm algorithm are of the order of the fluctuations seen in the results. They are not reported for clarity.

Figure 4

Tan's contact for the trapped systems in Fig. 3 compared with results in Ref. [45]. The contact from the worm algorithm simulation is obtained through the interaction energy using the expression $\mathcal{C}{a}_{\mathrm{HO}}^3=-a_{\mathrm{HO}}^3{N}_b\nu \left(\gamma \right)/\left(\pi a_{1\mathrm{D}}\right)$ [derived from Eq. (10)], where $\nu \left(\gamma \right)$ is the average interaction energy per particle [in units of ${\hslash }^2/\left(2m\right)$]. We calculated $\mathcal{C}{a}_{\mathrm{HO}}^3$ for two small values of $a_{\text{1D}}$ obtaining results that agree with the ones in Ref. [45]. The contact from the lattice calculation is obtained via a linear fit of the high momentum tails, and then extrapolated to the zero density limit via a quadratic fit to obtain the result for zero $a/a_{\mathrm{HO}}$ (see inset).

Figure 5

Scaled momentum distribution functions $m\left(k\right)/L$ in the Tonks-Girardeau limit as a function of the scaled momentum $k/\rho$. (a) Systems with different number of particles and densities, (b) same as (a) but in log-log scale, (c) systems with the same number of particles and different densities, (d) systems with different number of particles and the same density. Insets: (a) scaled average one-particle density matrix, (c) and (d) same as main panels but in log-log scale. Thin dotted lines indicate the asymptotic behavior discussed in the text. Here, and throughout this work, $\rho$ in the lattice is given in units of the inverse lattice spacing $a$.

Figure 6

Scaled momentum distribution functions $m\left(k\right)/L$ vs the scaled momentum $k/\rho$ for finite values of $\gamma$, plotted in [(a) and (c)] a linear scale and in [(b) and (d)] a log-log scale. Results are reported for [(a) and (b)] $\gamma =8.33$ and [(c) and (d)] $\gamma =1.0$. Insets: $g\left(r\right)/\rho$ vs $\rho r$ for the same systems as in the main panels. In all plots, thin dotted lines indicate the asymptotic behavior discussed in the text. A linear fit of the long distance behavior of the curves in the insets indicate that the Luttinger parameter is $K\approx 1.52$ for $\gamma =8.33$ and $K\approx 3.63$ for $\gamma =1.0$. Here, and throughout this work, $\rho$ in the continuum is reported in an arbitrary unit.

Figure 7

[(a) and (c)] Universal scaling of density and [(b) and (d)] momentum distribution functions in the Tonks-Girardeau limit. We report results from lattice calculations for [(a) and (b)] up to 25 bosons, and from the worm algorithm for [(c) and (d)] up to 100 bosons. In (a) and (c), we also report the LDA prediction for the density profiles. In (b) and (d), thin dotted line depict $k^{-4}$ behavior.

Figure 8

[(a) and (c)] Universal behavior of scaled density profiles and [(b) and (d)] momentum distribution functions for harmonically trapped systems with $\gamma \left(0\right)\approx 9.3$ in (a) and (b), and $\gamma \left(0\right)\approx 0.5$ in (c) and (d). In (b) and (d), thin dotted lines depict $k^{-4}$ behavior.

Figure 9

Finite temperature results for the scaled momentum distribution function of the homogeneous Tonks-Girardeau gas in systems with different average number of particles and densities. The scaled temperature in the plots is: (a) $\tilde{T}=2.51$ and (b) $\tilde{T}=15.06$. The black solid line in both panels was obtained using the lattice approach, while all other curves were obtained using the worm algorithm. Thin dotted lines depict $k^{-4}$ behavior.

Figure 10

Finite temperature results for the scaled momentum distribution function of the homogeneous Lieb-Liniger gas with different average number of particles and densities at finite $\gamma =8.33$. The scaled temperature in the plots is: (a) $\tilde{T}=2.51$ and (b) $\tilde{T}=15.06$. All results were obtained using the worm algorithm. Thin dotted lines depict $k^{-4}$ behavior.

Figure 11

Finite temperature results for [(a) and (c)] the scaled density profiles and [(b) and (d)] momentum distribution functions of harmonically trapped Tonks-Girardeau gases with [(a) and (b)] $\tilde{T}\left(0\right)\approx 2.63$ and [(c) and (d)] $\tilde{T}\left(0\right)\approx 37.48$. In (a) and (c), black solid lines show the LDA predictions. In (b) and (d), the black solid lines show results of the lattice calculations. All other curves were obtained using the worm algorithm. In (b) and (d), thin dotted lines depict $k^{-4}$ behavior.

Figure 12

Finite temperature results for [(a) and (c)] the scaled density profiles and [(b) and (d)] momentum distribution functions of harmonically trapped Lieb-Liniger gases with [(a) and (b)] $\gamma \left(0\right)\approx 9.3$, and $\tilde{T}\left(0\right)\approx 1.56$ and [(c) and (d)] $\tilde{T}\left(0\right)\approx 21.38$. All curves were obtained using the worm algorithm. In (b) and (d), thin dotted lines depict $k^{-4}$.