  We develop an efficient real-time simulation method for the spin-charge
coupled system in the velocity gauge. This method enables us to compute the
real-time simulation for the two-dimensional system with the complex spin
texture. We focus on the effect of the collective excitation of the localized
spins on the electronic transport properties of the non-trivial topological
state in real space. To investigate this effect, we calculate the linear
optical conductivity by calculating the real-time evolution of the Kondo
lattice model on the triangular lattice, which hosts an all-in/all-out magnetic
structure. In the linear conductivity spectra, we observe multiple peaks below
the bandgap regime, attributed to the resonant contributions of collective
modes similar to the skyrmionic system, alongside broadband modifications
resulting from off-resonant spin dynamics. The result shows that the collective
excitation, similar to the skyrmionic system, influences the optical response
of the electron systems based on symmetry analysis. We elucidate the
interference between the contributions from the different spin excitations to
the optical conductivity in the multiple spin texture, pointing out the
mode-dependent electrical activity. We show the complex interplay between the
complex spin texture and the itinerant electrons in the two-dimensional
spin-charge coupled system.
