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We construct a scattering matrix formulation for the topologicalclassification of one-dimensional superconductors with effective time reversalsymmetry in the presence of interactions. For a closed geometry, Fidkowski andKitaev have shown that such systems have a $\mathbb{Z}_8$ topologicalclassification. We show that in the weak coupling limit, these systems retain aunitary scattering matrix at zero temperature, with a topological index givenby the trace of the Andreev reflection matrix, $\mbox{tr}\, r_{\rm he}$. Withinteractions, $\mbox{tr}\, r_{\rm he}$ generically takes on the finite set ofvalues $0$, $\pm 1$, $\pm 2$, $\pm 3$, and $\pm 4$. We show that the twotopologically equivalent phases with $\mbox{tr}\, r_{\rm he} = \pm 4$ supportemergent many-body end states, which we identify to be a topologicallyprotected Kondo-like resonance. The path in phase space that connects theseequivalent phases crosses a non-fermi liquid fixed point where a multiplechannel Kondo effect develops. Our results connect the topological index totransport properties, thereby highlighting the experimental signatures ofinteracting topological phases in one dimension.

A scattering matrix formulation of the topological index of interacting fermions in one-dimensional superconductors