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.