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Theory of Tunneling Effect in 1D AIII-class Topological Insulator (Nanowire) Proximity Coupled with a Superconductor

We study the tunneling effect in an AIII-class insulator proximity coupled with a spin-singlet $s$-wave superconductor, in which three phases are characterized by the integer topological invariant $\mathcal{N}$. By solving the Bogoliubov-de Gennes equation explicitly, we analytically obtain a normal reflection coefficient $R_{σσ'}$ and an Andreev reflection coefficient $A_{σσ'}$, and derive a charge conductance formula,where $σ(σ')$ is the spin index of a reflected (injected) wave. The resulting conductance indicates a wide variety of line shapes: (i)gap structure without coherence peaks for $\mathcal{N}=0$, (ii)quantized zero-bias conductance peak (ZBCP) with height $2e^{2}/h$ for $\mathcal{N}=1$, and (iii)ZBCP spitting for $\mathcal{N}=2$. At zero bias voltage $eV=0$, $\sum_{σσ'} R_{σσ'} = \sum_{σσ'} A_{σσ'}$ is satisfied and the spin direction of an injected electron is rotated at approximately $90^\circ$ for the $\mathcal{N}=1$ state. Meanwhile, $A_{σσ'}=0$ is satisfied for the $\mathcal{N}=2$ state, and the spin rotation angle can become $180^\circ$.