  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.
