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

We develop a framework to analyze the topological properties of one-dimensional systems with charge conservation and tendency towards topological superconducting order. In particular, we consider models with N flavors of fermions and (Z2)N symmetry, associated with the conservation of the fermionic parity of each flavor. For N=1, and with no other symmetry other than charge conservation, we recover the result that there is no distinct topological phase with exponentially localized zero modes. For N>1, however, we show that the ends of the system can host low-energy, exponentially-localized modes. To illustrate these ideas, we focus on lattice models with SON 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. In this case, the topologically nontrivial superconducting phase corresponds to a gapless analog of the Haldane phase in spin-1 chains. In this phase, although the bulk hosts gapless modes, corresponding to composite fermionic excitations with an enlarged Fermi surface, 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 using density matrix renormalization group calculations. Within this model, we identify the self-dual line studied by Andrei and Destri [Nucl. Phys. B 231, 445 (1984)NUPBBO0550-321310.1016/0550-3213(84)90514-5] 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 at the domain walls between the two phases.