Comparative study of one-dimensional Bose and Fermi gases with contact interactions from the viewpoint of universal relations for correlation functions

TL;DR

This paper derives universal relations for correlation functions in one-dimensional Bose and Fermi gases with contact interactions, highlighting their differences and computing the fermionic momentum distribution at unitarity.

Contribution

It provides exact universal relations for correlation functions in both Bose and Fermi gases, clarifies their differences, and calculates the fermionic momentum distribution at unitarity.

Findings

Universal relations hold for any eigenstate or ensemble.

Three-body contact is crucial for fermionic energy relations.

Exact fermionic momentum distribution computed at unitarity.

Abstract

One-dimensional spinless Bose and Fermi gases with contact interactions have the close interrelation 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 exact universal relations for correlation functions, which hold for any energy eigenstate and any statistical ensemble of the eigenstates with or without a trapping potential. These relations include the large-momentum behaviors 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 are the integrals of local pair and triad 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 the fermionic one. In addition, we compute the exact momentum distribution for any number of fermions in the unitary limit.