  We propose and investigate a simple one-dimensional model for a
single-channel quantum wire hosting electrons that interact repulsively and are
subject to a significant spin-orbit interaction. We show that an external
Zeeman magnetic field, applied at the right angle to the Rashba spin-orbit
axis, drives the wire into a correlated spin-density wave state with gapped
spin and gapless charge excitations. By computing the ground-state degeneracies
of the model with either (anti-)periodic or open boundary conditions, we
conclude that the correlated spin-density state realizes a gapless
symmetry-protected topological phase, as the ground state is unique in the ring
geometry while it is two-fold degenerate in the wire with open boundaries.
Microscopically the two-fold degeneracy is found to be protected by the
conservation of the magnetization parity. Open boundaries induce localized
zero-energy (midgap) states which are described, at the special Luther-Emery
point of the model, by Majorana fermions. We find that spin densities at the
open ends of the wire exhibit unusual long-ranged correlations despite the fact
that all correlations in the bulk of the wire decay in a power-law or
exponential fashion. Our study exposes the crucial importance of the
long-ranged string operator needed to implement the correct commutation
relations between spin densities at different points in the wire. Along the way
we rederive the low-energy theory of Galilean-invariant electron systems in
terms of current operators.
