Universal Contact for a Tonks-Girardeau Gas at Finite Temperature

We determine the finite-temperature momentum distribution of a strongly interacting 1D Bose gas in the Tonks-Girardeau (impenetrable-boson) limit under harmonic confinement and explore its universal properties associated to the scale invariance of the model. We show that, at difference from the unitary Fermi gas in three dimensions, the weight of its large-momentum tails—given by Tan’s contact—increases with temperature and calculate the high-temperature universal second contact coefficient using a virial expansion.

Figure 1

Momentum distribution of a Tonks-Girardeau gas (in units $a_{\mathrm{HO}}$) as a function of wave vector (in units of $a_{\mathrm{HO}}^{-1}$) with $N=5$ particles under harmonic confinement at increasing temperature, from top to bottom in the main peak $k_{B}T/\hslash \omega =0.1$, 0.5, 0.7, 5, 10, 15. The last three curves at high temperature are obtained by the use of the classical limit of the fermionic one-body density matrix (10) into the series expansion (4) up to $j=1$. The inset shows the same curves in double-logarithmic scale.

Figure 2

Tan’s contact (in units of $a_{\mathrm{HO}}^{-3}$) as a function of temperature $k_{B}T$ (in units of $\hslash \omega$) for a TG gas under harmonic confinement. The full expression from (8, 9) (solid magenta line) is compared with the high-temperature limit (12) (dashed red line) and the data from the numerical calculation of the momentum distribution (blue crosses).