We show that a one dimensional ultra-cold Fermi gas with Rashba-like spin orbit coupling, a Zeeman field and intrinsic attractive interactions exhibits a novel topological superfluid state, which forms in spite of total number conservation and the absence of a single particle gap. Majorana zero modes are localized to the interface between a topological region in the middle of the trap and trivial regions at its wings. Unlike the realization of a topological superconductor in proximity coupled nano-wires, the Majorana modes do not carry a quantum number associated with the total fermion parity. Instead, the topological degeneracy is protected by an emergent
Z2 symmetry present only at low energies. We discuss the experimental implications of the novel zero modes, as manifest for example in the response to modulation of a local potential near the position of the Majorana bound states. For the range of interaction strength corresponding to Luttinger parameter
12, on the other hand the zero-mode can be detected as a sharp low frequency response at an energy which generically scales with system size as
1/LK/2 and is therefore parametrically separated from the phonons.
Based on the density-matrix renormalization group and the infinite time-evolving block decimation methods we study the interacting spin-orbit-coupled 1D Fermi gas in a transverse magnetic field. We find that the system with an attractive interaction can have a polarized insulator phase, a superconducting (SC) phase, a Luther-Emery (LE) phase, and a band insulator phase as we vary the chemical potential and the strength of the magnetic field. Spin-orbit coupling (SOC) enhances the triplet pairing order at zero momentum in both the SC and the LE phase, which leads to an algebraically decaying correlation with the same exponent as that of the singlet pairing one. In contrast to the Fulde-Ferrell-Larkin-Ovchinnikov phase found in the spin imbalanced system without SOC, pairings at finite momentum in these two phases have larger exponents hence do not dictate the long-range behavior. We also test for the presence of Majorana fermions in this system. Unlike results from the mean-field study, we do not find positive evidence of Majorana fermions.
We use the density matrix renormalization group method(DMRG) and the infinite time evolved block decimation method(iTEBD) to investigate the ground states of the spin-orbit coupled Fermi gas in a one dimensional optical lattice with a transverse magnetic field. We discover that the system with attractive interaction can have a polarized insulator(PI), a superconducting phase(SC), a Luther-Emery(LE) phase and a band insulator(BI) phase as we vary the chemical potential and the strength of magnetic field. We find that spin-orbit coupling induces a triplet pairing order at zero momentum with the same critical exponent as that of the singlet pairing one in both the SC and the LE phase. In contrast to the FFLO phase found in the spin imbalanced system without spin-orbit coupling, pairings at finite momentum in these two phases have a larger exponent hence do not dictate the long range behavior. We also find good agreements of the dominant correlations between numerical results and the prediction from the bosonization method. The presence of Majorana fermions is tested. However, unlike results from the mean field study, we do not find positive evidence of Majorana fermions in our system.