One-dimensional topological superconductors with charge conservation and the gapless Haldane phase

In this talk, I will address the question of the existence of gapless topological superconducting phases in one dimension. I will present a general approach to treat charge-conserving one-dimensional systems with pairing interactions, and argue that in presence of symmetries the notion of a topological phase can be well defined although the system is gapless. I will demonstrate our approach on systems with time-reversal symmetry and on systems with SO(3) symmetry, where a gapless analogue of the Haldane phase in spin-1 chains can be realized.