Scaling behavior of Tan's contact for trapped Lieb-Liniger bosons: From two to many

We show that the contact parameter of $N$ harmonically trapped interacting one-dimensional bosons at zero temperature can be analytically and accurately obtained by a simple rescaling of the exact two-boson solution, and that $N$-body effects can be almost factorized. The small deviations observed between our analytical results and density matrix renormalization group (DMRG) calculations are more pronounced when the interaction energy is maximal (i.e., at intermediate interaction strengths) but they remain bounded by the large-$N$ local-density approximation obtained from the Lieb-Liniger equation of state stemming from the Bethe ansatz. The rescaled two-body solution is so close to the exact ones, that is possible, within a simple expression interpolating the rescaled two-boson result to the local density, to obtain $N$-boson contact and ground-state energy functions in very good agreement with DMRG calculations. Our results suggest a change of paradigm in the study of interacting quantum systems, giving to the contact parameter a more fundamental role than energy.

Figure 1

Rescaled dimensionless contact $\mathcal{C}(N,z)$, Eq. (6) (left panel), and reduced contact parameter $f_N\left(z\right)$, Eq. (10) (right panel), as a function of the dimensionless scaling parameter $z=a_{\mathrm{ho}}/\left(\right|a_{1\mathrm{D}}|\sqrt{N})=a_{\mathrm{ho}}/\left(\right|a_{1\mathrm{D}}^{\prime }\left|\sqrt{2}\right)$. The different symbols correspond to DMRG calculations: $N=2$ (black squares), $N=3$ (brown circles), $N=4$ (purple triangles up), $N=5$ (light-blue triangles down), and $N=8$ (green diamonds). The black continuous line corresponds to Eq. (C1), and the orange dashed line corresponds to the LDA solution, Eq. (8) [14]. Top inset in the right panel: Reduced contact parameter $f_N\left(z\right)$ for $\mathrm{SU}\left(\kappa \right)$ fermions [11] (red points), $\kappa$ ranging from 2 to 6, superposed to all the data and curves of the main panel. Bottom inset in the right panel: Convergence rate $R_N\left(z\right)$ as a function of $N$, in a log-log scale, for $z=$ 0.14 (squares), 0.35 (stars), 0.70 (crosses), and $z\to \infty$ (plus).

Figure 2

Top panel: Difference ${\mathcal{D}}_N\left(z\right)=f_N\left(z\right)-f_2\left(z\right)$ for different values of $N$ as a function of the dimensionless scaling parameter $z$. Middle panel: Dimensionless interaction energy ${\mathcal{E}}_{\mathrm{int}}(N,z)$ for different values of $N$, Eq. (7), as a function of $z$. All curves display a clear maximum at intermediate dimensionless interaction strengths $z\simeq 0.5$. Bottom panel: Scaled difference ${\mathcal{D}}_N\left(z\right)/{\beta }_N$ with ${\beta }_N=1-2/N$ for different values of $N$ as a function of $z$. All curves collapse quite well onto the LDA prediction ${\mathcal{D}}_{\mathrm{LDA}}\left(z\right)$ (dashed orange curve) even if further corrections would be needed around the maximum. Symbols are the same as in Fig. 1.

Figure 3

Reduced contact parameter $f_N(\alpha /\sqrt{N})$, Eq. (10), for different $N$ and plotted as a function of $\alpha =a_{\mathrm{ho}}/\left|a_{1\mathrm{D}}\right|$ for better visibility. Solid lines: theoretical prediction, Eq. (15); symbols: DMRG results. Symbols are the same as in Fig. 1.

Figure 4

Rescaled ground-state energy ${\mathrm{\Delta }}_N\left(z\right)$ relative to its noninteracting value, Eq.(18), as a function of the dimensionless scaling parameter $z$ for different values of $N$. Solid lines: theoretical prediction, Eq. (17); symbols: DMRG results, same $N$ values and symbols as in Fig. 1.