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

We propose that the odd-frequency $s$ wave ($s^{\mathrm{o}\mathrm{d}\mathrm{d}}$ wave) superconducting gap function, which is usually unstable in the 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. The interference between bulk and edge gap functions triggers the $d+s^{\mathrm{o}\mathrm{d}\mathrm{d}}$ 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 on 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.