Spin-orbit-phonon interaction as an origin of helical-symmetry breaking spin-triplet superconducting state

The excess increase of the local field distribution width $\Delta H\equiv\sigma/\gamma_{\mu}$ (with $\gamma_{\mu}=135.5\,$MHz/T being the gyromagnetic ratio of $\mu^{+}$), and/or the spontaneous magnetic field $H_{\rm spon}$, have been reported in the superconducting state of a series of compounds more than 10 by $\mu$SR (muon spin rotation or relaxation) experiment. Sizes of these quantities are of the order of 1G on the whole. We propose that the increase of the local field distribution $\Delta H$ in those compounds, including ions with strong spin-orbit interaction of 5d electrons, such as La in LaNiGa$_2$ or Re in Re$_6$Zr, is possible through the cooperation of the spin-orbit coupling between the quasiparticles and ionic vibrations (phonons) and the conventional quasiparticles-phonon coupling, inducing the spin-triplet p-wave pairing with helical symmetry breaking superconducting state with chiral super spin current around stopped $\mu^{+}$ site. The size of $\Delta H$ is estimated as of the order of 1G, in consistent with experimental observations.