Majorana zero mode on the edge of an interacting quantum wire subject to spin-orbit interaction and transverse magnetic field

We revisit the problem of an interacting quantum wire subject to spin-orbit interaction and transverse magnetic field. Previous studies showed that the spin sector of the model is equivalent to that of a spin chain with uniform DM interaction (D) in a transverse magnetic field (h), the ground state of which contains two ordered Ising phases and a critical Tomonaga-Luttinger liquid phase, depending on the ratio D/h. Importantly, the charge sector of the wire is a gapless Tomonaga-Luttinger liquid. We show that a quantum wire with an open boundary supports a zero-energy bound state localized at the edge, provided that the spin sector of the problem is massive. We argue that as long as the charge sector is gapless, the wire is in a topological phase and that the bound state is a Majorana zero mode, and discuss physical implications of this finding.