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Majorana edge modes protected by emergent symmetry in a one dimensional fermi gas

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 $Z_2$ 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 $1<K<2$ the zero modes are unseperable from the gapless phonon continuum and therefore show up as an algebraic zero bias resonance in the response. For $K>2$, 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 $1/L^{K/2}$ and is therefore parametrically separated from the phonons.