Generation of odd-frequency surface superconductivity with spontaneous spin current owing to zero-energy Andreev-bound-state.

We propose that the odd-frequency $s$ wave (= $s^{\rm odd}$ wave) superconducting gap function, which is usually unstable in bulk, naturally emerges at the edge of $d$ wave superconductors. This prediction is based on the surface spin fluctuation pairing mechanism owing to the zero-energy surface Andreev-bound state (SABS). The interference between bulk and edge gap functions triggers the $d+s^{\rm odd}$ state, and the generated spin current is a useful signal uncovering the "hidden" odd-frequency gap. In addition, the edge $s^{\rm odd}$ gap can be determined via the proximity effect into the diffusive normal metal. Furthermore, this study provides a decisive validation of the "Hermite odd-frequency gap function," which has been an open fundamental challenge to this field.