We discuss a strong relationship between Majorana fermions and odd-frequency Cooper pairs which appear at a disordered normal (N) nanowire attached to a topologically nontrivial superconducting (S) one. The transport properties in superconducting nanowire junctions show universal behaviors irrespective of the degree of disorder: the quantized zero-bias differential conductance at 2e 2 /h in NS junction and the fractional current-phase (J -ϕ) relationship of the Josephson effect in SNS junction J ∝ sin(ϕ/2). Such behaviors are exactly the same as those found in the anomalous proximity effect of odd-parity spin-triplet superconductors. We show that the odd-frequency pairs exist wherever the Majorana fermions stay.

Motivated by a recent experiment [Keizer et al., Nature (London) 439, 825 (2006)], we study the Josephson effect in superconductor/diffusive half metal/superconductor junctions using the recursive Green function method. The spin-flip scattering at the junction interfaces opens the Josephson channel of the odd-frequency spin-triplet Cooper pairs. As a consequence, the local density of states in a half metal has a large peak at the Fermi energy. Therefore the odd-frequency pairs can be detected experimentally by using the scanning tunneling spectroscopy.

We present a theory of tunneling spectroscopy for normal metal/Larkin-Ovchinnikov state junctions in which the spatial periodic modulation in the pair potential amplitude is taken into account. The tunneling spectra show the characteristic line shapes reflecting the minigap structures under the periodic pair potentials depending on the boundary condition of the pair potentials at the junction interface. These features are qualitatively different from the tunneling spectra of the Fulde-Ferrell state. We propose an experimental setup which identifies the superconducting state of CeCoIn5.

We discuss the appearance of odd-frequency Cooper pairs in two-band superconductors by solving the Gor'kov equation analytically. We introduce the equal-time s-wave pair potentials as realized in MgB2 and iron pnictides. Although the order parameter symmetry is conventional, the band degree of freedom enriches the symmetry variety of pairing correlations. The hybridization and the asymmetry between the two conduction bands induce odd-frequency pairs as a subdominant pairing correlation in the uniform ground state. To study the magnetic response of odd-frequency Cooper pairs, we analyze the Meissner kernel represented by the Gor'kov Green function. In contrast to the even-frequency pairs linked to the pair potential, the induced odd-frequency Cooper pairs indicate a paramagnetic property. We also discuss the relation between the amplitude of the odd-frequency pairing correlation and the stability of superconducting states in terms of the self-consistent equation for the pair potential.