# Fermion parity gap and exponential ground state degeneracy of the one-dimensional Fermi gas with intrinsic attractive interaction

## TL;DR

This paper investigates a 1D Fermi gas with attractive interactions, spin-orbit coupling, and Zeeman field, revealing exponential ground state degeneracy and its sensitivity to local perturbations, with implications for ultracold atom systems.

## Contribution

It demonstrates the exponential ground state degeneracy in a number-conserving 1D Fermi system and analyzes its robustness against local impurities.

## Key findings

- Exponential decay of pair binding and excitation energy with system size.
- Ground state degeneracy is sensitive to local perturbations, vanishing as a power law.
- Comparison with Ising antiferromagnet highlights differences in order and degeneracy.

## Abstract

We examine the properties of a one-dimensional (1D) Fermi gas with attractive intrinsic (Hubbard) interactions in the presence of spin-orbit coupling and Zeeman field by numerically computing the pair binding energy, excitation gap, and susceptibility to local perturbations using the density matrix renormalization group. Such a system can, in principle, be realized in a system of ultracold atoms confined in a 1D optical lattice. We note that, in the presence of spatial interfaces introduced by a smooth parabolic potential, the pair binding and excitation energy of the system decays exponentially with the system size, pointing to the existence of an exponential ground state degeneracy, and is consistent with recent works. However, the susceptibility of the ground state degeneracy of this number-conserving system to local impurities indicates that the energy gap vanishes as a power law with the system size in the presence of local perturbations. We compare this system with the more familiar system of an Ising antiferromagnet in the presence of a transverse field realized with Rydberg atoms and argue that the exponential splitting in the clean number-conserving 1D Fermi system is similar to a phase with only conventional order.