Field-theoretical aspects of one-dimensional Bose and Fermi gases with contact interactions

TL;DR

This paper explores the quantum field theories of 1D Bose and Fermi gases with contact interactions, revealing the necessity of three-body interactions for fermions and deriving universal high-energy behaviors.

Contribution

It introduces the role of three-body interactions in fermionic systems and applies operator product expansion to derive universal asymptotics for dynamic properties.

Findings

Three-body contact appears in fermionic energy relations.

Universal large-energy and momentum asymptotics are derived.

Bose-Fermi correspondence extends to systems with three-body attractions.

Abstract

We investigate local quantum field theories for one-dimensional (1D) Bose and Fermi gases with contact interactions, which are closely connected with each other by Girardeau's Bose-Fermi mapping. While the Lagrangian for bosons includes only a two-body interaction, a marginally relevant three-body interaction term is found to be necessary for fermions. Because of this three-body coupling, the three-body contact characterizing a local triad correlation appears in the energy relation for fermions, which is one of the sum rules for a momentum distribution. In addition, we apply in both systems the operator product expansion to derive large-energy and momentum asymptotics of a dynamic structure factor and a single-particle spectral density. These behaviors are universal in the sense that they hold for any 1D scattering length at any temperature. The asymptotics for the Tonks-Girardeau gas, which is a Bose gas with a hardcore repulsion, as well as the Bose-Fermi correspondence in the presence of three-body attractions are also discussed.