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Two different gauge potential methods are engaged to calculate explicitly thespin Hall conductivity in graphene. The graphene Hamiltonian with spin-orbitinteraction is expressed in terms of kinematic momenta by introducing a gaugepotential. A formulation of the spin Hall conductivity is established byrequiring that the time evolution of this kinematic momentum vector vanishes.We then calculated the conductivity employing the Berry gauge fields. We showthat both of the gauge fields can be deduced from the pure gauge field arisingfrom the Foldy-Wouthuysen transformations.

Gauge Potential Formulations of the Spin Hall Effect in Graphene