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

TBDWe 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, $\mbox{tr}\, r_{\rm he}$. With interactions, $\mbox{tr}\, r_{\rm he}$ 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 $\mbox{tr}\, r_{\rm he} = \pm 4$  support  emergent {\it 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.