  We derive the non-adiabatic ring polymer molecular dynamics (RPMD) approach
in the phase space of the SU(N) Lie Group. This method, which we refer to as
the spin mapping non-adiabatic RPMD (SM-NRPMD), is based on the spin-mapping
formalism for the electronic degrees of freedom (DOFs) and ring polymer
path-integral description for the nuclear DOFs. Using the Stratonovich-Weyl
transform for the electronic DOFs, and the Wigner transform for the nuclear
DOFs, we derived an exact expression of the Kubo-transformed time-correlation
function (TCF). We further derive the spin mapping non-adiabatic Matsubara
dynamics using the Matsubara approximation that removes the high frequency
nuclear normal modes in the TCF and derive the SM-NRPMD approach from the
non-adiabatic Matsubara dynamics by discarding the imaginary part of the
Liouvillian. The SM-NRPMD method has numerical advantages compared to the
original NRPMD method based on the MMST mapping formalism, due to a more
natural mapping using the SU(N) Lie Group that preserves the symmetry of the
original system. We numerically compute the Kubo-transformed position
auto-correlation function and electronic population correlation function for
three-state model systems. The numerical results demonstrate the accuracy of
the SM-NRPMD method, which outperforms the original MMST-based NRPMD. We
envision that the SM-NRPMD method will be a powerful approach to simulate
electronic non-adiabatic dynamics and nuclear quantum effects accurately.
