  We develop a framework to analyze one-dimensional topological superconductors
with charge conservation. In particular, we consider models with $N$ flavors of
fermions and $(\mathbb{Z}_2)^N$ symmetry, associated with the conservation of
the fermionic parity of each flavor. For a single flavor, we recover the result
that a distinct topological phase with exponentially localized zero modes does
not exist due to absence of a gap to single particles in the bulk. For $N>1$,
however, we show that the ends of the system can host low-energy,
exponentially-localized modes. The analysis can readily be generalized to
systems in other symmetry classes. To illustrate these ideas, we focus on
lattice models with $SO\left(N\right)$ symmetric interactions, and study the
phase transition between the trivial and the topological gapless phases using
bosonization and a weak-coupling renormalization group analysis. As a concrete
example, we study in detail the case of $N=3$. We show that in this case, the
topologically non-trivial superconducting phase corresponds to a gapless
analogue of the Haldane phase in spin-1 chains. In this phase, although the
bulk is gapless to single particle excitations, the ends host spin-$1/2$
degrees of freedom which are exponentially localized and protected by the spin
gap in the bulk. We obtain the full phase diagram of the model numerically,
using density matrix renormalization group calculations. Within this model, we
identify the self-dual line studied by Andrei and Destri [Nucl. Phys. B,
231(3), 445-480 (1984)], as a first-order transition line between the gapless
Haldane phase and a trivial gapless phase. This allows us to identify the
propagating spin-$1/2$ kinks in the Andrei-Destri model as the topological
end-modes present at the domain walls between the two phases.
