  In this letter we present, in a number conserving framework, a model of
interacting fermions in a two-wire geometry supporting non-local zero-energy
Majorana-like edge excitations. The model has an exactly solvable line, on
varying the density of fermions, described by a topologically non-trivial
ground state wave-function. Away from the exactly solvable line we study the
system by means of the numerical density matrix renormalization group. We
characterize its topological properties through the explicit calculation of a
degenerate entanglement spectrum and of the braiding operators which are
exponentially localized at the edges. Furthermore, we establish the presence of
a gap in its single particle spectrum while the Hamiltonian is gapless, and
compute the correlations between the edge modes as well as the superfluid
correlations. The topological phase covers a sizeable portion of the phase
diagram, the solvable line being one of its boundaries.
