Scattering Matrix Formulation of the Topological Index of Interacting Fermions in One-Dimensional Superconductors

We construct a scattering matrix formulation for the topological classification of one-dimensional superconductors with effective time-reversal symmetry in the presence of interactions. For an isolated system, Fidkowski and Kitaev have shown that such systems have a Z(8) topological classification. We here show that these systems have a unitary scattering matrix at zero temperature when weakly coupled to a normal-metal lead, with a topological index given by the trace of the Andreev-reflection matrix, trr(hc). With interactions, trr(hc) generically takes on the finite set of values 0, +/- 1, +/- 2, +/- 3, and +/- 4. We show that the two topologically equivalent phases with trr(hc) = +/- 4 support emergent many-body end states, which we identify to be a topologically protected Kondo-like resonance. The path in phase space that connects these equivalent phases crosses a non-Fermi-liquid fixed point where a multiple-channel Kondo effect develops. Our results connect the topological index to transport properties, thereby highlighting the experimental signatures of interacting topological phases in one dimension.