Tan’s Contact for Trapped Lieb-Liniger Bosons at Finite Temperature

The universal Tan relations connect a variety of microscopic features of many-body quantum systems with two-body contact interactions to a single quantity, called the contact. The latter has become pivotal in the description of quantum gases. We provide a complete characterization of the Tan contact of the harmonically trapped Lieb-Liniger gas for arbitrary interactions and temperature. Combining thermal Bethe ansatz, local-density approximation, and exact quantum Monte Carlo calculations, we show that the contact is a universal function of only two scaling parameters, and determine the scaling function. We find that the temperature dependence of the contact, or equivalently the interaction dependence of the entropy, displays a maximum. The presence of this maximum provides an unequivocal signature of the crossover to the fermionized regime and it is accessible in current experiments.

Figure 1

Reduced Tan contact $a_{\mathrm{ho}}^{3}C/N^{5/2}$ for 1D Bose gases in a harmonic trap, versus the reduced temperature ${\xi }_T=-a_{1\mathrm{D}}/{\lambda }_T$ and the reduced interaction strength ${\xi }_{\gamma }=-a_{\mathrm{ho}}/a_{1\mathrm{D}}\sqrt{N}$. The results are found using thermal Bethe ansatz solutions combined with local-density approximation (see main text).

Figure 2

Reduced Tan contact $a_{\mathrm{ho}}^{3}C/N^{5/2}$ versus the scaling parameters, as found from LDA (solid lines) and QMC calculations (points). (a) Reduced contact versus the interaction ${\xi }_{\gamma }=-a_{\mathrm{ho}}/a_{1\mathrm{D}}\sqrt{N}$ at the fixed temperatures ${\xi }_T=-a_{1\mathrm{D}}/{\lambda }_T=0.0085$ (blue), 0.28 (green), and 18.8 (red). (b) Reduced contact versus the temperature via the quantity ${\xi }_T$ at the fixed interaction strengths ${\xi }_{\gamma }={10}^{-2}$ (blue), $1.58\times {10}^{-1}$ (green), and 15.0 (red). The black dashed, red dotted, and red dash-dotted lines correspond to Eqs. (8), (9), and (10), respectively. The QMC data are found from various sets of parameters, corresponding to the various symbols [33].

Figure 3

Behavior of the temperature at which the contact is maximum versus the interaction strength. Shown is the value of ${\xi }_T^{*}$ (solid black line with shaded gray error bars) as found from the data of Fig. 1, together with the asymptotic behaviors ${\xi }_T^{*}\simeq 0.49$ for the strongly interacting regime (dashed red line) and ${\xi }_T^{*}\propto {\xi }_T^{{\nu }_{\mathrm{fit}}}$, with ${\nu }_{\mathrm{fit}}\simeq 0.6$ for the weakly interacting regime (dotted blue line).

Figure 4

Log-log plots of momentum distributions found by QMC calculations in the strongly interacting regime. (a) Low temperature: ${\xi }_{\gamma }=4.47$ and ${\xi }_T=0.0085$. (b) Temperature at the maximum contact: ${\xi }_{\gamma }=1.26$ and ${\xi }_T=0.49$. The solid blue lines with shaded statistical error bars are the QMC results, the dashed red lines are algebraic fits to the large-$k$ tails, and the dotted green lines are the momentum distributions of the nondegenerate ideal gas. The insets show the same data in lin-lin scale.