Tan's two-body contact in a planar Bose gas: experiment vs theory

We determine the two-body contact in a planar Bose gas confined by a transverse harmonic potential, using the nonperturbative functional renormalization group. We use the three-dimensional thermodynamic definition of the contact where the latter is related to the derivation of the pressure of the quasi-two-dimensional system with respect to the three-dimensional scattering length of the bosons. Without any free parameter, we find a remarkable agreement with the experimental data of Zou {\it et al.} [Nat. Comm. {\bf 12}, 760 (2021)] from low to high temperatures, including the vicinity of the Berezinskii-Kosterlitz-Thouless transition. We also show that the short-distance behavior of the pair distribution function and the high-momentum behavior of the momentum distribution are determined by two contacts: the three-dimensional contact for length scales smaller than the characteristic length $\ell_z=\sqrt{\hbar/m\omega_z}$ of the harmonic potential and, for length scales larger than $\ell_z$, an effective two-dimensional contact, related to the three-dimensional one by a geometric factor depending on $\ell_z$.