Tan relations in one dimension

We derive exact relations that connect the universal C/k4C/k4-decay of the momentum distribution at large kk with both thermodynamic properties and correlation functions of two-component Fermi gases in one dimension with contact interactions. The relations are analogous to those obtained by Tan in the three-dimensional case and are derived from an operator product expansion of the one- and two-particle density matrix. They extend earlier results by Olshanii and Dunjko (2003) [24] for the bosonic Lieb–Liniger gas. As an application, we calculate the pair distribution function at short distances and the dimensionless contact in the limit of infinite repulsion. The ground state energy approaches a universal constant in this limit, a behavior that also holds in the three-dimensional case. In both one and three dimensions, a Stoner instability to a saturated ferromagnet for repulsive fermions with zero range interactions is ruled out at any finite coupling.

► We derive universal relations for the two-component, contact-interacting 1D Fermi gas.
► These relations connect the tail of the momentum distribution to thermodynamic properties.
► There is no saturated ferromagnetism at finite, repulsive couplings for the 3D model.