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

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. CONTENTS I. Introduction 1 II. Bose-Fermi correspondence 2 III. Bosons 3 A. Operator product expansion 4 B. OPE for G n(K ) 5 C. Dynamic structure factor 7 D. OPE for G φ (K) 7 E. Single-particle spectral density 9 IV. Fermions 10 A. Three-body problem 10 B. Contacts and the energy relation 11 C. Single-particle spectral density 12 V. Conclusion 13 Acknowledgments 14 A. Loop integrals 14 B. Fermions with a three-body attraction 15 References 15