Universal Tan relations for quantum gases in one dimension

We investigate universal properties of one-dimensional multi-component systems comprised of fermions, bosons, or an arbitrary mixture, with contact interactions and subjected to an external potential. The masses and the coupling strengths between different types of particles are allowed to be different and we also take into account the presence of an arbitrary magnetic field. We show that the momentum distribution of these systems exhibits a universal $n_\sigma(k) \sim C_\sigma/k^4$ decay with $C_\sigma$ the contact of species $\sigma$ which can be computed from the derivatives of an appropriate thermodynamic potential with respect to the scattering lengths. In the case of integrable fermionic systems we argue that at fixed density and repulsive interactions the total contact reaches its maximum in the balanced system and monotonically decreases to zero as we increase the magnetic field. The converse effect is present in integrable bosonic systems: the contact is largest in the fully polarized state and reaches its minimum when all states are equally populated. We obtain short distance expansions for the Green's function and pair distribution function and show that the coefficients of these expansions can be expressed in terms of the density, kinetic energy and contact. In addition we derive universal thermodynamic identities relating the total energy of the system, pressure, trapping energy and contact. Our results are valid at zero and finite temperature, for homogeneous or trapped systems and for few-body or many-body states.