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We investigate the change in the non-zero Chern number and out-of-plane spinpolarization of the edge currents in a honeycomb lattice with theHaldane-Rashba interaction. This interaction breaks the time-reversal symmetrydue to the Haldane phase caused by a current loop at site-I and -II atoms andalso accounts for the Rashba-type spin-orbit interaction. The Rashba spin-orbitinteraction increases the number of Dirac points and the band-touchingphenomenon can be generated by tuning the on-site potential in the non-zeroHaldane phase. By using the Pontryagin winding number and numerical Berrycurvature methods, we find that the Chern number pattern is $\{+2, -1, 0\}$ and$\{-2, +1, 0\}$ for the positive and negative Haldane phase, respectively. Anon-zero Chern number is called a Chern-insulating phase. We discovered thatchanges in both the Haldane phase and on-site potential leads to a change inthe 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 atomsat one outer edge and site-II atoms at the opposite outer edge, the spinpolarization of the edge states approximately obeys the properties of bulk spinpolarization regardless of the change in the Chern number. In addition, evenwhen the Chern number changes from $+2$ to $-1$ (or $-2$ to $+1$), by tuningthe strength of the on-site potential, the sign of the spin polarization of theedge states persists. This approximate bulk-edge correspondence of the spinpolarization in the Haldane-Rashba system would play an important role inspintronics, because it enables us to control the orientation of the spinpolarization in a single Chern-insulating phase.

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