Ground and excited states of spinor Fermi gases in tight waveguides and the Lieb-Liniger-Heisenberg model

TL;DR

This paper establishes an exact mapping between a 1D spin-1/2 Fermi gas with specific interactions and a Lieb-Liniger-Heisenberg model, providing analytical solutions for ground and excited states in different magnetic phases.

Contribution

It introduces an exact mapping of a complex spinor Fermi gas to a solvable Lieb-Liniger-Heisenberg model, enabling analytical solutions for ground and excited states.

Findings

Exact energy spectra for ground states in ferromagnetic and antiferromagnetic phases.

Identification of excitation branches including phonons and spin waves.

Quadratic and linear behaviors of spin waves in different magnetic phases.

Abstract

The ground and excited states of a one-dimensional (1D) spin-1/2 Fermi gas (SFG) with both attractive zero-range odd-wave interactions and repulsive zero-range even-wave interactions are mapped exactly to a 1D Lieb-Liniger-Heisenberg (LLH) model with delta-function repulsions depending on isotropic Heisenberg spin-spin interactions, such that the complete SFG and LLH energy spectra are identical. The ground state in the ferromagnetic phase is given exactly by the Lieb-Liniger (LL) Bethe ansatz, and that in the antiferromagnetic phase by a variational method combining Bethe ansatz solutions of the LL and 1D Heisenberg models. There are excitation branches corresponding to LL type I and II phonons and spin waves, the latter behaving quadratically for small wave number in the ferromagnetic phase and linearly in the antiferromagnetic phase.