Comparative Study of One-Dimensional Bose and Fermi Gases with Contact Interactions*

One-dimensional spinless Bose and Fermi gases with contact interactions have the close relationship via Girardeau’s Bose-Fermi mapping, leading to the correspondences in their energy spectra and thermodynamics. However, correlation functions are in general not identical between these systems. We derive in both systems the universal relations for correlation functions, which hold for any energy eigenstate and any statistical ensemble of the eigenstates[1]. These relations include the large-momentum tails of static structure factors and of momentum distributions as well as energy relations, which connect the sums of kinetic and interaction energies to the momentum distributions. The relations involve two- and three-body contacts, which measure local two- and three-body correlations, respectively. We clarify how the relations for bosons and fermions differ and are connected with each other. In particular, we find that the three-body contact makes no contribution to the bosonic energy relation, but it plays a crucial role in fermionic one.
[1] Y. Sekino, S. Tan, and Y. Nishida, Phys. Rev. A 97, 013621 (2018).