  We study the effects of spin orbit interactions on the low energy electronic
structure of a single plane of graphene. We find that in an experimentally
accessible low temperature regime the symmetry allowed spin orbit potential
converts graphene from an ideal two dimensional semimetallic state to a quantum
spin Hall insulator. This novel electronic state of matter is gapped in the
bulk and supports the quantized transport of spin and charge in gapless edge
states that propagate at the sample boundaries. The edge states are non chiral,
but they are insensitive to disorder because their directionality is correlated
with spin. The spin and charge conductances in these edge states are calculated
and the effects of temperature, chemical potential, Rashba coupling, disorder
and symmetry breaking fields are discussed.
