True spin and pseudo spin entanglement around Dirac Points in graphene with Rashba spin-orbit interaction
We analytically obtained the Schmidt decomposition of the entangled state between the pseudo spin and the true spin in graphene with Rashba spin-orbit coupling. The entangled state has the standard form of the Bell state, where the SU(2) spin symmetry is broken. These states can be explicitly expressed as the superposition of two nonorthogonal, but mirror symmetrical spin states entangled with the pseudo spin states. Because of the closely locking between the pseudo spin and the true spin, it is found that the orbit curve in the spin-polarization parameter space for the fixed equi-energy contour around Dirac points has the same shape as the δk→-contour. Due to the spin-orbit coupling that cause the topological transition in the local geometry of the dispersion relation, the new equi-energy contours around the new emergent Dirac Points can be obtained by squeezing the one around the original Dirac point. The spin texture in the momentum space around the Dirac points is analyzed under the Rashba spin-orbit interaction and it is found that the orientation of the spin polarization at each crystal momentum k→ is independent of the Rashba coupling strength.