Tan relations in one dimension

TL;DR

This paper derives exact relations linking the momentum distribution decay, thermodynamics, and correlations in one-dimensional Fermi gases with contact interactions, extending Tan's relations to 1D systems.

Contribution

It introduces one-dimensional Tan relations for Fermi gases, connecting momentum decay to thermodynamic and correlation functions, and applies these to analyze short-distance pair distributions and the contact.

Findings

Universal $C/k^4$ decay in momentum distribution established.

Ground state energy approaches a universal constant at infinite repulsion.

No ferromagnetic transition occurs at finite coupling in 1D and 3D.

Abstract

We derive exact relations that connect the universal $C/k^4$-decay of the momentum distribution at large $k$ 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 [Phys. Rev. Lett. 91, 090401 (2003)] 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.