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 ${\mathbb{Z}}_8$ 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, $\mathrm{tr}{r}_{\text{he}}$. With interactions, $\mathrm{tr}{r}_{\text{he}}$ 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 $\mathrm{tr}{r}_{\text{he}}=\pm 4$ 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.

Figure 1

(a) Schematic picture of 12 Majorana chains that interpolate between effective free-fermion classes $n=4$ and $n=-4$. The interactions terms coupling different Majorana end states are indicated by thick lines. The “colored” fermions constructed with Majorana fermions are indicated explicitly. (b) Energy spectrum as a function of the interpolation parameter $\theta$, showing that the spectrum remains gapped through the interpolation. The ground state is twofold degenerate. (c) Amplitudes $\alpha$, ${\alpha }^{\prime }$, $\beta$, ${\beta }^{\prime }$, $\eta$, and ${\eta }^{\prime }$ for the ground state wave functions, as a function of the interpolation parameter $\theta$; see Eq. (9).