Scaling properties of Tan's contact: Embedding pairs and correlation effect in the Tonks-Girardeau limit

We study Tan's contact of a one-dimensional quantum gas of $N$ repulsive identical bosons confined in a harmonic trap at finite temperature. This canonical ensemble framework corresponds to the experimental conditions, with the number of particles being fixed for each experimental sequence. We show that in the strongly interacting regime, the contact rescaled by the contact at the Tonks-Girardeau limit is a universal function of two parameters: the rescaled interaction strength and temperature. This means that all pair and correlation effects in Tan's contact are embedded in Tan's contact in the Tonks-Girardeau limit.

Figure 1

Canonical Tan's contact $C_2^c(g,T)$ as a function of $\tau =T/T_F$ [Eq. (5)] for different values of the interaction strength $z=a_{ho}/\left(\right|a_{1D}\left|\sqrt{N}\right)$. From bottom to top: $z=0.5$, 1, 2.5, 5, and 1000. The curve for $z=1000$ is indiscernible from the contact evaluated in the Tonks limit by means of Eq. (6).

Figure 2

Canonical contact (empty symbols) [Eq. (7)] and grand-canonical contact (full symbols) [46] as a function of $\tau$ for $N=2$ (violet squares), $N=3$ (green circles), $N=4$ (light-blue up-triangles), and $N=5$ (orange down-triangles) Tonks-Girardeau bosons. The grand-canonical case will be discussed in Sec. 4.

Figure 3

Canonical contact in the Tonks-Girardeau limit $C_N(\infty ,T)$, given by Eq. (7), as a function of $\tau$, scaled by the factor $s\left(N\right)=N^{5/2}-N^{3/4[1+exp(-2/\tau \left)\right]}$; see Eq. (23). Violet squares: $N=2$; green circles: $N=3$; light-blue up-triangles: $N=4$; and orange down-triangles: $N=5$. The blue dashed line corresponds to the high-temperature limit $h_2(\infty ,\tau \gg 1)$ [Eq. (20)]. The black cross and the black line correspond to $h_2(\infty ,0)={(2/\pi )}^{3/2}{a}_{ho}^{-3}{(2^{5/2}-2^{3/4})}^{-1}$ and $h_2(\infty ,\tau )$ [Eq. (24)], respectively.

Figure 4

$C_N^c(z,\tau )a_{ho}^3$ as a function of $\tau$, for the case $z=1$, rescaled by (a) $N^{5/2}-N^{3/4}$, (b) $N^{5/2}-N^{3/2}$, and (c) $s\left(N\right)$. (d) $f_N(z=1,\tau )$ as a function of $\tau$. The points (violet squares: $N=2$; green circles: $N=3$; light-blue up-triangles: $N=4$; and orange down-triangles: $N=5$) correspond to the QMC data. The continuous yellow line corresponds to the two-boson contact obtained by Eq. (5). Nonvisible QMC error bars are smaller than the symbol size.

Figure 5

$C_N^c(z,\tau )a_{ho}^3$ as a function of $\tau$, for the case $z=2.5$, rescaled by (a) $N^{5/2}-N^{3/4}$, (b) $N^{5/2}-N^{3/2}$, and (c) $s\left(N\right)$. (d) $f_N(z=2.5,\tau )$ as a function of $\tau$. The points (violet squares: $N=2$; green circles: $N=3$; light-blue up-triangles: $N=4$; and orange down-triangles: $N=5$) correspond to the QMC data. The continuous yellow line corresponds to the two-boson contact obtained by Eq. (5). Nonvisible QMC error bars are smaller than the symbol size.

Figure 6

Canonical (empty symbols) and grand-canonical contact (full symbols) as a function of $\tau$ for $N=2$ (violet squares), $N=3$ (green circles), $N=4$ (light-blue up-triangles), and $N=5$ (orange down-triangles) Tonks-Girardeau bosons. The canonical contact [Eq. (7)] is rescaled by a factor $N^{5/2}-N^{3/2}$, while the grand-canonical one (Eqs. (8) and (9) in [46]) is rescaled by $N^{5/2}$. The black continuous curve corresponds to $\sqrt{\tau }/{\pi }^{3/2}$ [Eq. (20)].

Figure 7

Canonical (empty symbols) and grand-canonical contact (full symbols) as a function of $\tau$ for $N=2$ (violet squares), $N=3$ (green circles), and $N=4$ (light-blue up-triangles) bosons. All points correspond to QMC data evaluated in the weakly interacting regime $z=0.5$. QMC error bars for the canonical data are smaller than the symbol size.

Figure 8

$C_N^{gc}(z=1,\tau )/C_N^{gc}(\infty ,\tau )$ as a function of $\tau$. The points (violet squares: $N=2$; green circles: $N=3$; light-blue up-triangles: $N=4$) correspond to the QMC data.