Quantization of conductance minimum and index theorem

Abstract

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 ZES in the limit of strong impurity scatterings in the normal metal at the zero temperature. The integer N ZES represents the number of perfect transmission channels through the junction. An analysis of the chiral symmetry of the Hamiltonian indicates that N ZES corresponds to the Atiyah-Singer index in mathematics.

Chiral zero modes in superconducting nanowires with Dresselhaus spin-orbit coupling

“…Thus, it may also be applied to other setups, such as NS, SNS junctions, or the continuous limit of these systems as long as the existence of a chiral operator is intact. In particular, we expect the analysis would simpler in the continuous limit, and the results should be qualitatively similar to what we have obtained here [25]. Finally, we would like to mention that the two models we analyze here both belong to the BDI class, where the number of MZES may take on any integer values.…”

“…The two numbers in the parenthesis are the corresponding kx ky for the mode, and there are more than one pair of exact zero modes in some specific values of µ as predicted in Eq. (24) and (25). Again, the value of the exact zero mode energy is zero up to numerical inaccuracy, which is of the order 10 −16 .…”

“…When an in-plane Zeeman field is applied, such systems may achieve topologically non-trivial phase. As the system belongs to the BDI class, its number of MZES may take on arbitrary integer values [24,25]. Although some signature of MZES has been reported in experiments [26][27][28][29], it is still under debate whether the zero-bias anomaly really stems from Majorana fermions [30][31][32][33].…”

Current fluctuations in unconventional superconductor junctions with impurity scattering

“…Another manifestation of such an exotic odd-frequency Andreev resonance presented here is that the excess current maintains the ballistic maximum value in the presence of impurities, a feature unique to this pairing state. The special zero-energy state of p x -wave superconductors is connected to the emergence of a Majorana bound state [64]. Indeed, Majorana states are always accompanied by oddfrequency Cooper pairs [65,66].…”

Stability of flat zero-energy states at the dirty surface of a nodal superconductor

“…By paying attention to the chiral symmetry of a Bogoliubov-de Gennes (BdG) Hamiltonian [19,[33][34][35][36], we show that a mathematical index, N ZES , well characterizes the number of ZESs at a dirty surface. The index N ZES is an invariant defined in terms of the chirality of the surface ZESs and is closely related to the one-dimensional winding number W (k) [7].…”