Quantization of conductance minimum and index theorem

We discuss the minimum value of the zero-bias differential conductance $G_{\min }$ in a junction consisting of a normal metal and a nodal superconductor preserving time-reversal symmetry. Using the quasiclassical Green function method, we show that $G_{\min }$ is quantized at $(4e^2/h)N_{\mathrm{ZES}}$ in the limit of strong impurity scatterings in the normal metal at the zero temperature. The integer $N_{\mathrm{ZES}}$ represents the number of perfect transmission channels through the junction. An analysis of the chiral symmetry of the Hamiltonian indicates that $N_{\mathrm{ZES}}$ corresponds to the Atiyah-Singer index in mathematics.

Figure 1

(a) Schematic picture of a normal-metal/superconductor junction. (b) Numerical results on a two-dimensional tight-binding model. The index $|N_{\mathrm{ZES}}|$ is 9 and 5 for $p_x$- and $f$-wave symmetry, respectively.