Theory of odd-frequency pairing in Kitaev chain.

We study odd-frequency pairing and resulting anomalous proximity effect in spin-triplet $p$-wave superconductor systems based on Kitaev chain. In semi-infinite Kitaev chain, we study the spectral and spatial profile of $s$-wave component of odd-frequency pair amplitude and its relevance to zero energy surface Andreev bound states (ZEABS) as a Majorana fermion. The spatial dependence of the odd-frequency pair amplitude at zero frequency and the probability density of Majorana fermion coincide in the topological regime. The magnitude of the odd-frequency pair amplitude which is prominent in topological regime is reduced drastically at topological critical point (TCP). We also show the presence of the anomalous proximity effect triggered by the odd-frequency pairing in normal metal/diffusive normal metal/Kitaev chain (N/DN/Kitaev) junction in topological regime, where local density of states (LDOS) of quasiparticle in DN has a zero energy peak (ZEP). The peak width of ZEP of LDOS is proportional to $\exp(-c_1L)$ or $\exp(-c_2W)$ where $L$ is the length of DN, $W$ is the strength of random potential and $c_1$ and $c_2$ are constants. The obtained zero voltage conductance is quantized in topological regime robust against impurity scattering. The peak width of conductance decreases exponentially with $L$ and $W$ similar to LDOS.