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A semiclassical formulation of the spin Hall effect for physical systemssatisfying Dirac-like equation is introduced. We demonstrated that the maincontribution to the spin Hall conductivity is given by the spin Chern numberwhether the spin is conserved or not at the quantum level. We illustrated theformulation within the Kane-Mele model of graphene in the absence and in thepresence of the Rashba spin-orbit coupling term.

Relation Between the Spin Hall Conductivity and the Spin Chern Number for Dirac-like Systems