  The topological insulator is an electronic phase stabilized by spin-orbit
coupling that supports propagating edge states and is not adiabatically
connected to the ordinary insulator. In several ways it is a spin-orbit-induced
analogue in time-reversal-invariant systems of the integer quantum Hall effect
(IQHE). This paper studies the topological insulator phase in disordered
two-dimensional systems, using a model graphene Hamiltonian introduced by Kane
and Mele as an example. The nonperturbative definition of a topological
insulator given here is distinct from previous efforts in that it involves
boundary phase twists that couple only to charge, does not refer to edge
states, and can be measured by pumping cycles of ordinary charge. In this
definition, the phase of a Slater determinant of electronic states is
determined by a Chern parity analogous to Chern number in the IQHE case.
Numerically we find, in agreement with recent network model studies, that the
direct transition between ordinary and topological insulators that occurs in
band structures is a consequence of the perfect crystalline lattice.
Generically these two phases are separated by a metallic phase, which is
allowed in two dimensions when spin-orbit coupling is present. The same
approach can be used to study three-dimensional topological insulators.
