Topological insulators beyond the Brillouin zone via Chern parity

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 analog 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 [Phys. Rev. Lett. 95, 226801 (2005)] 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.