A 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 a closed geometry, Fidkowski and Kitaev have shown that such systems have a ${\mathbb{Z}}_8$ topological classification. We show that in the weak coupling limit, these systems retain a unitary scattering matrix at zero temperature, with a topological index given by the trace of the Andreev reflection matrix, $tr{r}_{\mathrm{h}\mathrm{e}}$. With interactions, $tr{r}_{\mathrm{h}\mathrm{e}}$ generically takes on the finite set of values $0$, $\pm 1$, $\pm 2$, $\pm 3$, and $\pm 4$. We show that the two topologically equivalent phases with $tr{r}_{\mathrm{h}\mathrm{e}}=\pm 4$ 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.