Fermi-Bose mapping and N-particle ground state of spin-polarized fermions in tight atom waveguides

TL;DR

This paper explores the mapping between spin-polarized fermions and bosons in tight waveguides, providing a rigorous formulation of contact conditions, and derives the equation of state for ultracold fermionic gases, revealing a fermionic Tonks-Girardeau regime.

Contribution

It establishes a duality between 1D fermions and bosons with zero-range interactions, generalizing the Fermi-Bose mapping in tight waveguides and providing a rigorous variational reformulation.

Findings

Identifies a contact condition for 1D spinless fermions based on the K-matrix.

Shows the duality between fermionic and bosonic gases with zero-range interactions in tight waveguides.

Derives the equation of state for ultracold spin-polarized fermionic vapor in a 1D trap.

Abstract

A K-matrix for waveguide confined spin-polarized fermionic atoms recently computed by Granger and Blume is identified, in the low-energy domain, with a contact condition for one-dimensional (1D) spinless fermions. Difficulties in consistently formulating the contact conditions in terms of interaction potentials are discussed and a rigorous alternative variational reformulation is constructed. A duality between 1D fermions and bosons with zero-range interactions suggested by Cheon and Shigehara is shown to hold for the effective 1D dynamics of a spin-polarized Fermi gas with 3D p-wave interactions and that of a Bose gas with 3D s-wave interactions in a tight waveguide. This generalizes the mapping from impenetrable bosons (TG gas) to free fermions and is used to derive the equation of state of an ultracold spin-polarized fermionic vapor in a tight waveguide. Near a 1D confinement-induced resonance one has a "fermionic TG gas" which maps to an ideal Bose gas.