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

Abstract

We examine the topological properties of a one-dimensional (1D) chain of fermions with spin-orbit coupling, Zeeman field, and attractive Hubbard interaction by numerically computing the pair binding energy, excitation gap, and susceptibility to local perturbations by density matrix renormalization group. Such a system can in principle be realized in a 1D optical lattice. We find that, in the presence of spatial interfaces introduced by a smooth parabolic potential, the variation of the pair binding energy and the excitation gap with the system size indicate an exponentially vanishing fermion parity gap and topological ground state degeneracy in the thermodynamic limit, consistent with recent works. However, the susceptibility of the ground state degeneracy to local perturbations indicate that the vanishing of the fermion parity gap in this number conserving system scales as a power-law in system size. We compare the present 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 degeneracy of this conventional symmetry-breaking system is identical to the 1D chain of fermions considered. Therefore, the degeneracy properties of the 1D chain of fermions cannot be attributed to topology.