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

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.

Automated summary

The study focuses on local quantum field theories for one-dimensional Bose and Fermi gases, connected by Girardeau's Bose-Fermi mapping. It is found that while bosons require only a two-body interaction term, fermions need a marginally relevant three-body interaction. The presence of this three-body interaction leads to the appearance of a three-body contact in the energy relation for fermions, impacting momentum distribution and sum rules. Additionally, universal large-energy and momentum asymptotics for dynamic structure factor and single-particle spectral density are derived using the operator product expansion, applicable to any 1D scattering length at any temperature. Discussions also include the behavior of Tonks-Girardeau gas with hardcore repulsion and the Bose-Fermi correspondence in the presence of three-body attractions.