From one-dimensional charge conserving superconductors to the gapless Haldane phase

We discuss the existence of topological superconducting phases in one-dimensional, charge conserving systems. We develop a general approach to treat one-dimensional systems with multiple flavors and attractive pairing interactions. For a single flavor, i.e. a spinless system, we recover the result that a distinct topological phase does not exist due to absence of a gap to single particles in the bulk. For N flavors, we study the case of SO(N) symmetric interactions, and show that for N > 1, a distinct topological phase, with protected, exponentially localized end-modes, can exist. We discuss in detail the case of N = 3, and show that such a system can realize 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 introduce a lattice model that realizes this phase and analyze it both analytically, using Bosonization and RG, and numerically, using DMRG.