  The quantum spin Hall (QSH) phase is a time reversal invariant electronic
state with a bulk electronic band gap that supports the transport of charge and
spin in gapless edge states. We show that this phase is associated with a novel
$Z_2$ topological invariant, which distinguishes it from an ordinary insulator.
The $Z_2$ classification, which is defined for time reversal invariant
Hamiltonians, is analogous to the Chern number classification of the quantum
Hall effect. We establish the $Z_2$ order of the QSH phase in the two band
model of graphene and propose a generalization of the formalism applicable to
multi band and interacting systems.
