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Out-of-plane spin polarization of edge currents in Chern insulator with Rashba spin-orbit interaction

We investigate the change in the non-zero Chern number and out-of-plane spin polarization of the edge currents in a honeycomb lattice with the Haldane-Rashba interaction. This interaction breaks the time-reversal symmetry due to the Haldane phase caused by a current loop at site-I and -II atoms and also accounts for the Rashba-type spin-orbit interaction. The Rashba spin-orbit interaction increases the number of Dirac points and the band-touching phenomenon can be generated by tuning the on-site potential in the non-zero Haldane phase. By using the Pontryagin winding number and numerical Berry curvature methods, we find that the Chern number pattern is $\{+2, -1, 0\}$ and $\{-2, +1, 0\}$ for the positive and negative Haldane phase, respectively. A non-zero Chern number is called a Chern-insulating phase. We discovered that changes in both the Haldane phase and on-site potential leads to a change in the orientation of the bulk spin polarization of site-I and site-II atoms. Interestingly, in a ribbon with a zigzag edge, which naturally has site-I atoms at one outer edge and site-II atoms at the opposite outer edge, the spin polarization of the edge states approximately obeys the properties of bulk spin polarization regardless of the change in the Chern number. In addition, even when the Chern number changes from $+2$ to $-1$ (or $-2$ to $+1$), by tuning the strength of the on-site potential, the sign of the spin polarization of the edge states persists. This approximate bulk-edge correspondence of the spin polarization in the Haldane-Rashba system would play an important role in spintronics, because it enables us to control the orientation of the spin polarization in a single Chern-insulating phase.