$\boldsymbol{d}$-vector precession induced pumping in topological $p$-wave superconductors

Time-reversal invariant $p$-wave superconductors (SCs) are characterized by their $\boldsymbol{d}$-vectors, whose orientations could be manipulated by a tiny magnetic field. We study in this paper the adiabatic pumping process induced by periodically rotating $\boldsymbol{d}$-vector in a topological $p$-wave SC, which is coupled to two normal leads. If $\boldsymbol{d}$-vector rotates nearly within a plane, the pumped spin $2S_z/\hbar$ over one cycle is nearly quantized at $2$ without net charge pumping. When the pumping lead is fully spin-polarized, both the pumped charge $Q/e$ and spin $2S_z/\hbar$ would peak nearly at $1$. When a mixing $s$-wave pairing component is taken into account, a topological phase transition can be driven by modulating the ratio between the pairing components. We found a sharp resonance phenomenon near the phase transition when the $p$-wave $\boldsymbol{d}$-vector is adiabatically rotating, which may help experimentally distinguish the topological SCs from trivial ones.