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

M. D. Girardeau, Maxim Olshanii|arXiv (Cornell University)|Sep 17, 2003

Cold Atom Physics and Bose-Einstein Condensates2 references3 citations

TL;DR

This paper establishes a Fermi-Bose mapping for spin-polarized fermions in tight atom waveguides, showing that the strongly interacting 1D Fermi gas with zero-range p-wave interactions maps exactly to a 1D Bose gas with delta-function repulsion. The key result is the derivation of the exact N-particle ground state and equation of state for such a system, particularly near a confinement-induced resonance where it behaves as a 'fermionic Tonks-Girardeau gas' mapping to an ideal Bose gas.

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.

Motivation & Objective

• To derive the exact N-particle ground state of spin-polarized fermions with zero-range p-wave interactions in tight atom waveguides.
• To establish a Fermi-Bose duality for 1D fermions with p-wave interactions and 1D bosons with s-wave interactions under tight transverse confinement.
• To resolve inconsistencies in formulating contact conditions for odd-wave fermionic interactions using standard Hamiltonian formulations.
• To provide a rigorous variational reformulation that enables consistent perturbative and nonperturbative treatment of the system.
• To extend the Tonks-Girardeau mapping to the fermionic regime, identifying the 'fermionic TG gas' as dual to an ideal Bose gas.

Proposed method

• Use of a K-matrix derived by Granger and Blume to identify low-energy scattering behavior in 1D fermions confined in waveguides.
• Construction of a contact condition for odd-wave fermions using the scattering length $ a_{1D}^F $, derived from 3D p-wave scattering volume and transverse confinement parameters.
• Adoption of a two-slot functional formalism to replace standard bra-ket matrix elements, enabling consistent variational treatment of discontinuous wavefunctions.
• Application of the Fermi-Bose mapping $ \psi_B = A \psi_F $, originally used for hard-core bosons, to map the 1D Fermi gas with p-wave interactions to a 1D Bose gas with delta-function repulsion.
• Leveraging the Lieb-Liniger solution for the 1D Bose gas to obtain the exact ground state of the 1D Fermi gas via duality.
• Derivation of the equation of state for the ultracold spin-polarized fermionic gas in the 1D regime, particularly near a 1D confinement-induced resonance.

Experimental results

Research questions

• RQ1 Can the effective 1D dynamics of spin-polarized fermions with 3D p-wave interactions be mapped to a 1D Bose gas with s-wave interactions under tight transverse confinement?
• RQ2 How can consistent contact conditions be formulated for 1D fermions with odd-wave (p-wave) interactions, given the ill-defined nature of $ \delta' $-functions in discontinuous wavefunctions?
• RQ3 What is the exact N-particle ground state of a 1D spin-polarized Fermi gas with zero-range p-wave interactions in the absence of longitudinal trapping?
• RQ4 How does the Fermi-Bose duality extend beyond the Tonks-Girardeau limit, and what is the equation of state in the fermionic TG regime?
• RQ5 What is the role of confinement-induced resonances in tuning the effective 1D coupling constant and enabling mapping to an ideal Bose gas?

Key findings

• The effective 1D Hamiltonian for spin-polarized fermions with 3D p-wave interactions maps exactly to a 1D Bose gas with delta-function repulsion, generalizing the Tonks-Girardeau duality.
• Near a 1D confinement-induced resonance, the system enters a 'fermionic TG gas' regime where the ground state maps to that of an ideal Bose gas, with $ \gamma_B \gamma_F = 4 $.
• The low-energy scattering behavior is captured by a contact condition involving the 1D p-wave scattering length $ a_{1D}^F $, derived from the 3D p-wave scattering volume and transverse oscillator length.
• A critical resonance occurs at $ V_p^{\text{crit}}/a_\perp^3 = -0.4009\ldots $, where the 1D scattering length diverges, signaling a confinement-induced resonance.
• The variational formulation using two-slot functionals provides a consistent framework for perturbation theory, overcoming the ill-defined nature of $ \delta' $-interactions in discontinuous wavefunctions.
• For the untrapped fermionic TG gas, the ground state wavefunction exhibits discontinuities at particle collisions, while the corresponding bosonic state is smooth, illustrating the duality.